Capítulo 3 Regresión Lineal Múltiple

Como se mencionó en el Capítulo 2, los modelos de regresión lineal pueden utilizarse para predecir futuros valores de una variable respuesta continua a partir de valores específicos de las variables controlables del proceso.

En la práctica, pueden existir múltiples variables controlables en un proceso de producción o de servicios. Por ejemplo, en un proceso de pintura electrostática, puede ser de interés determinar el espesor de la capa de pintura (variable respuesta $y$, en micrones) con la que se recubre una lámina de área determinada, a partir de valores conocidos de la presión de aire (variable $x_1$ en psi) y la velocidad de la banda transportadora (variable $x_2$ en m/s) en la que se desplaza dicha lámina. En este caso, el interés es:

Determinar la magnitud de la influencia de las variables $x_1$ y $x_2$ sobre el espesor de capa esperado;
construir una función $f(x_1,x_2)$ que permita predecir el espesor de capa esperado; y
construir intervalos de confianza y predicción para dicho valor.

Si la variable respuesta $y$ es continua y aproximadamente simétrica, podemos desarrollar 1, 2 y 3 a partir de la estimación de un modelo de regresión lineal. Puesto que el número de variables controlables es $k>1$, una posibilidad es utilizar el modelo de Regresión Lineal Múltiple (RLM).

El modelo de RLM es una extensión del modelo de RLS cuando se tiene más de una variable controlable.

3.1 Formulación básica del modelo de RLM

Matemáticamente, el modelo de RLM puede expresarse como:

\[\begin{align} \label{mod1} y_i &= \beta_0 + \beta_1X_{1i} + \beta_2X_{2i} + \cdots + \beta_1X_{ki} + \epsilon_i,\\ \epsilon_i &\sim N(0, \sigma^2), \\ \sigma^2 &= \text{constante}. \end{align}\]

Este modelo es equivalente a

\[\begin{align} \label{mod2} \mathbf{y} &= \mathbf{X}\mathbf{\beta} + \mathbf{\epsilon} \end{align}\]

donde $\mathbf{y} = (y_1,y_2,\ldots,y_n)$ es el vector respuesta, $\mathbf{X} = (\mathbf{1}, \mathbf{x}_1, \mathbf{x}_2,\ldots,\mathbf{x}_k)_{n\times p}$ es la matriz de diseño y $\mathbf{\epsilon} = (\epsilon_1,\epsilon_2,\ldots,\epsilon_n)$ es el error aleatorio.

3.1.1 Estimación

Similar a RLS, la estimación del modelo de RLM se realiza utilizando el método de mínimos cuadrados ordinarios (MCO).

A partir de una muestra aleatoria de tamaño $n$ del proceso de producción, los datos se registran en una estructura rectangular similar a:

Figura 3.1: Estructura de datos en RLM.

De esta forma, se tienen $n$ unidades experimentales para cada una de estas se determina el valor de la variable respuesta $y_i$ para condiciones fijas $\mathbf{X}_i$. Por ejemplo, para la quinta unidad experimental, se obtuvo el valor $y_6$ cuando las variables controlables tomaron los valores fijos $(x_{1,6}, x_{2,6}, \ldots, x_{k,6})$.

Al igual que en RLS, la estimación del modelo de RLM realiza utilizando minimos cuadrados La idea fundamental consiste en minimizar

\[\begin{eqnarray*}\label{L} L &=&\sum_{i=1}^n\epsilon_i^2 = \sum_{i=1}^n(Y_i-\beta_0-\beta_1X_{1,i} -\beta_2X_{2,i} - \ldots -\beta_kX_{k,i} )^2. \end{eqnarray*}\]

Los estimadores de mínimos cuadrados deben satisfacer las siguientes dos condiciones fundamentales:

\[\begin{eqnarray*} \frac{\partial L}{\partial \beta_0} | _{\hat{\beta}_0,\hat{\beta}_1,\ldots \hat{\beta}_k} &=& -2\sum_{i=1}^n \left(y_i-\hat{\beta}_0-\sum_{j=1}^k{\hat{\beta}_jx_{ij}}\right) = 0 \\ \frac{\partial L}{\partial \beta_j} | _{\hat{\beta}_0,\hat{\beta}_1,\ldots \hat{\beta}_k} &=& -2\sum_{i=1}^n \left(y_i-\hat{\beta}_0-\sum_{j=1}^k{\hat{\beta}_jx_{ij}}\right)x_{ij} = 0 \end{eqnarray*}\]

La solución al sistema de ecuaciones de condiciones fundamentales da origen al sistema de ecuaciones normales de mínimos cuadrados dado por

Figura 3.2: Ecuaciones normales de mínimos cuadrados.

Las solución de estas ecuaciones permite determinar $\hat{\mathbf{\beta}}$. Es fácil llegar a que el vector de coeficientes estimado para el modelo de RLM puede obtenerse como

\[ \hat{\mathbf{\beta}} = (\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime \mathbf{y} \]

Finalmente, el modelo estimado es \[\hat{y}_i = \hat{\beta}_0+\sum_{j=1}^k\hat{\beta}_jx_{ij},\]

que, matricialmente, puede representarse como

\[\hat{\mathbf{y}} = \mathbf{X}\hat{\mathbf{\beta}}\]

A partir del modelo ajustado, un valor específico $y_i$ puede calculase como:

\[\hat{y}_i = \hat{\beta}_0 + \hat{\beta}_1x_1 + \hat{\beta}_2x_2+\cdots+\hat{\beta}_kx_k\]

Como ilustración, consideremos los siguientes datos provenientes de un proceso de pintura electrostática:

## lectura de datos
url <- 'https://www.dropbox.com/s/st5xk1prkxg1pj4/datalab2.txt?dl=1'
datos <- read.table(url, header = TRUE)
head(datos)

##          y        x1 x2
## 1 91.88308 18.722091  6
## 2 94.37544 19.056131  3
## 3 74.63026  9.292093  3
## 4 93.66348 17.456714  4
## 5 76.84981 14.626183  9
## 6 76.05198 12.786439  9

En este caso, la variable respuesta $y$ representa el espesor de la capa de pintura en micrones, $x_1$ es la presión de aire en psi y $x_2$ es la velocidad de la banda transportadora.

Una forma de comenzar a establecer si existe relación lineal entre $y$ y las variables $x_1$ y $x_2$ es a través de una red, como se muestra a continuación:

## matriz de correlación
require(IsingSampler)
require(qgraph)
corMat <- cor(datos)
par(mfrow = c(1, 1), mar = c(0.1, 0.1, 0.1, 0.1))
qgraph(corMat, graph = "cor", layout = "spring", sampleSize = nrow(datos), legend.cex = 1,
    alpha = 0.05)

Figura 3.3: Red de correlación para $x_1$, $x_2$ y $y$. Correlaciones positivas se muestran en verde, y las negativas en rojo.

O utilizando una matriz de dispersión:

## pairs plot
panel.cor <- function(x, y, digits = 2, prefix = "", cex.cor, ...) {
    usr <- par("usr")
    on.exit(par(usr))
    par(usr = c(0, 1, 0, 1))
    r <- cor(x, y)
    txt <- format(c(r, 0.123456789), digits = digits)[1]
    txt <- paste0(prefix, txt)
    text(0.5, 0.5, txt, cex = 1.5)
}
pairs(datos, lower.panel = panel.smooth, upper.panel = panel.cor, las = 1)

Figura 3.4: Matriz de dispersión para $x_1$, $x_2$ y $y$.

Este último gráfico indica que $x_1$ e $y$ están linealmente relacionados y la correlación es $\hat\rho = 0.89$, mientras la correlación de $y$ y $x_2$ es $\hat\rho = -0.32$.

Para otras posibilidades relacionadas con correlogramas, se recomienda revisar las alternativas en R Graph Gallery.

Para estimar el modelo de RLM, utilizamos la función lm como se muestra a continuación:

## ajuste del modelo de RLM
fit <- lm(y ~ x1 + x2, data = datos)
summary(fit)

## 
## Call:
## lm(formula = y ~ x1 + x2, data = datos)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -5.0571 -1.6610 -0.1362  1.5409  8.7008 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 66.28483    1.10350  60.068  < 2e-16 ***
## x1           1.48275    0.06144  24.132  < 2e-16 ***
## x2          -1.05521    0.13660  -7.725 1.03e-11 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.767 on 97 degrees of freedom
## Multiple R-squared:  0.8716, Adjusted R-squared:  0.8689 
## F-statistic: 329.1 on 2 and 97 DF,  p-value: < 2.2e-16

A partir de estos resultados podemos realizar las pruebas de significancia global y marginales tal y como se mostró en el Capítulo 2.

3.2 Propiedades de los estimadores de ${\mathbf{\beta}}$

Cuando estimamos $\hat{\mathbf{\beta}}$, los cálculos están basados en los resultados obtenidos al tomar una muestra aleatoria de tamaño $n$. Como consecuencia, el valor de los estimadores $\mathbf{\hat{\beta}} = (\hat{\beta}_0, \hat{\beta}_1, \ldots, \hat{\beta}_k)$ cambian si cambiamos la muestra.

Desde el punto de vista formal, los estimadores $\mathbf{\hat{\beta}}$ cumplen con las siguientes propiedades:

Los estimadores $\mathbf{\hat\beta}$ son insesgados. Esta propiedad implica que, al aumentar $n$, el valor de los estimadores de $\mathbf{\beta}$ se aproximan a los verdaderos valores de los parámetros. Matemáticamente se tiene que:

\[\begin{eqnarray} E[\mathbf{\hat\beta}] &=& E[(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime\mathbf{y}]\\\nonumber &=& E[(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime(\mathbf{X\beta} + \mathbf{\epsilon})]\\\nonumber &=& E[(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime\mathbf{X\beta} + (\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime\mathbf{\epsilon}]\\\nonumber &=& E[(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime\mathbf{X\beta}] + E[(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime\mathbf{\epsilon}]\\\nonumber &=& E[\mathbf\beta] + \mathbf{0} = \mathbf{\beta}\nonumber \end{eqnarray}\]

La varianza de $\hat{\beta}_j$ y la covarianza entre $\hat\beta_{i}$ y $\hat\beta_{j}$ están dadas por:

\[\begin{eqnarray*} V(\hat\beta_{j}) &=& \sigma^2(\mathbf{X}^\prime\mathbf{X})^{-1}_{jj} \hspace{0.5cm} 0,1,2,\ldots,p;\\ \text{cov}(\hat\beta_{i}, \hat\beta_{j}) &=& \sigma^2(\mathbf{X}^\prime\mathbf{X})^{-1}_{ij} \hspace{0.5cm} i\neq j. \end{eqnarray*}\]

Ahora, a partir de $E[\hat{\beta}_j]$ y $V(\hat\beta_{j})$, es posible hacer inferencia para el parámetro $\beta_j$, $j=1,2,\ldots,k.$ Sin embargo, observe que $V(\hat\beta_{j})$ depende de $\sigma^2$, la varianza del modelo de RLM, que se estima a través del MSE.

Los coeficientes del modelo de RLM ajustado para los datos provenientes del proceso de pintura electrostática se obtienen haciendo:

## coeficientes estimados
coefficients(fit)

## (Intercept)          x1          x2 
##   66.284831    1.482753   -1.055206

A partir de estos coeficientes, el modelo ajustado será

\[\hat{y} = 66.285 + 1.483x_1 - 1.055x_2\] Note que esta es la ecuación de un plano en el espacio $(x_1, x_2, y)$.

Los coeficientes estimados del modelo de RLM se interpretan en términos de una derivada parcial.

En particular, si incrementamos $x_1$ en una unidad y mantenemos constante $x_2$, esperaríamos que el espesor de la capa de pintura aumentara, en promedio, $\hat{\beta}_1 = 1.483$ micrones. Por otro lado, si incrementamos $x_2$ en una unidad y mantenemos constante $x_1$, se espera que el espesor de la capa de pintura disminuya, en promedio, $\hat{\beta}_2 = -1.055$ micrones.

3.3 Estimación de $\sigma^2$

Similar a como se observó en RLS, en RLM, también se cumple la misma relación que en RLS en cuanto que

\[SST = SSR + SSE \]

Adicionalmente, $\mathbf{\hat{\beta}}$, se tiene que

\[\begin{eqnarray*}\label{L2} L &=&\sum_{i=1}^n\hat{\epsilon}_i^2 = \sum_{i=1}^n(Y_i-\hat{Y}_i^2)^2 \\ &=&\sum_{i=1}^n(Y_i-\hat{\beta}_0-\hat{\beta}_1X_{1,i} -\hat{\beta}_2X_{2,i} - \cdots -\hat{\beta}_kX_{k,i})^2 \\ &=& SSE \end{eqnarray*}\]

Por lo tanto,

\[\hat{\sigma}^2 = \frac{SSE}{n-p} = MSE\]

donde $p = k+1$ es el número de coeficientes del modelo ajustado.

Este resulado indica que la varianza de los errores, también conocida como la varianza del modelo, puede estimarse utilizando el MSE. El MSE se obtiene de la tabla ANOVA que tiene la siguiente forma:

Figura 3.5: Tabla ANOVA en RLM.

En el caso del modelo de RLM para el proceso de pintura electrostática, podemos obtener el valor de $\hat{\sigma}^2$ haciendo:

## estimación de sigma^2
summary(fit)$sigma^2

## [1] 7.65454

Por lo tanto, el modelo estimado será:

\[ \begin{eqnarray} \hat{y} &=& 66.285 + 1.483x_1 - 1.055x_2 \\\nonumber \epsilon &\sim& N(0, \sigma^2) \\\nonumber \hat{\sigma}^2 &=& 7.655.\nonumber \end{eqnarray} \]

Recordemos que, adicional al MSE, a partir de la tabla ANOVA es posible calcular el porcentaje de variabilidad de la respuesta explicado por el modelo de RLM, también conocido como coeficiente de determinación o, simplemente, como $R^2$:

\[R^2 = SSR/SST = 1 - SSE/SST\]

Puesto que $R^2$ incrementa a medida que el número de variables aumenta, se recomienda utilizar el $R^2$ ajustado en RLM:

\[R^2_{\text{ajustado}} = 1 - \frac{SSE/(n-p)}{SST/(n-1)}\]

La interpretación de esta medida de desempeño es similar a la interpretación de $R^2$ discutida en el Capítulo 2.

El ejemplo de la pintura electrostática, $R^2_{\text{adj}}=0.8689$, por lo que podemos afirmar que incluir las variables $x_1$ y $x_2$ en un modelo de regresión permite explica cerca del 87% de la variabilidad de la respuesta.

3.4 Inferencia para $\mathbf{\beta}$

Uno de los propósitos de la inferencia estadística es determinar el valor de los verdaderos parámetros de una población a partir de los resultados obtenidos en una muestra. En este caso, los parámetros poblacionales son $\mathbf{{\beta}} = (\beta_0, \beta_1, \ldots, \beta_k)$, además de $\sigma^2$.

Con los valores muestrales, podemos construir pruebas de hipótesis de dos tipos para los parámetros del modelo de RLM: la prueba de significancia total y las pruebas de significancia marginal.

3.4.1 Prueba de significancia global

Esta prueba se utiliza para determinar la significancia total del modelo, es decir, para detemrinar si incluir las variables controlables en el modelo de regresión es mejor que no incluirlas para explicar la respuesta $Y$. La idea fundamental es determinar si, en la población,

\[\begin{eqnarray} H_0&:& \beta_1=\beta_2=\cdots\beta_k=0 \\\nonumber H_1&:& \text{Al menos un $\beta_j \neq 0$}\nonumber \end{eqnarray}\]

Este procedimiento de prueba de hipótesis se realiza a través de la tabla de ANOVA utilizando el estadístico $F$ dado por

\[F_0 = \frac{SSR/k}{SSE/(n-p)} = \frac{MSR}{MSE} \sim F_{k, n-p}\]

Rechazamos $H_0: \beta_1=\beta_2=\cdots\beta_k=0$ si $F_0 > F_{\alpha,k,n-p}$, donde $\alpha\in(0,1)$ es un nivel de significancia predeterminado. Cuando esto ocurre, concluimos que al menos un $\beta_j$ es estad'isticamente significativo al $100(1-\alpha)\%$.

Observe que en la prueba de significancia global del modelo, $F_{\text{calc}} = 329.1$ y $p < 2.2\times 10^{-16}$. Esto indica que tener este modelo de RLM para explicar el espesor de la capa de pintura es mejor que no tenerlo.

3.4.2 Prueba de significancia marginal

Esta prueba se realiza si rechazamos la prueba de significancia global. Lo que intentamos hacer es determinar si, a nivel poblacional, los coeficientes asociados a cada $x_j$ son o diferentes de cero. Esto es equivalente a probar:

\[\begin{eqnarray*} H_0&:& \beta_j=0 \\\nonumber H_1&:& \beta_j \neq 0\nonumber \end{eqnarray*}\]

Para $j$ fijo, el estadístico de prueba es

\[t_j = \frac{\hat{\beta}_j - 0}{\text{s.e.}(\hat\beta_j)} = \frac{\hat{\beta}_j}{\sqrt{\hat{\sigma}^2(\mathbf{X}^\prime\mathbf{X})^{-1}_{jj}}}\sim t_{n-p}\]

Por lo tanto, rechazamos $H_0$ con un nivel de significancia de $100\times(1-\alpha)\%$ si $|t_j| > t_{\alpha/2, n-p}$.

Para el problema de la pintura electrostática, tendríamos:

## ajuste del modelo de RLM
summary(fit)$coefficients

##              Estimate Std. Error   t value     Pr(>|t|)
## (Intercept) 66.284831 1.10349995 60.067815 1.529216e-78
## x1           1.482753 0.06144364 24.131927 8.750958e-43
## x2          -1.055206 0.13659758 -7.724922 1.026117e-11

Según estos resultados, todas las pruebas marginales del tipo $H_0: \beta_j = 0$ vs. $H_1: \beta_j \neq 0$, el valor $p$ es $< 0.05$. Por lo tanto, la magnitud de la influencia de cada variable controlable sobre $E[y|x_1, x_2]$ es estadísticamente diferente de cero a nivel poblacional. En otras palabras, controlando las variables $x_1$ y $x_2$ en el proceso, permitiría modificar satisfactoriamente el espesor de la capa de pintura.

3.4.3 Intervalos de confianza para $\beta_j$

Otra forma de realizar inferencia para $\mathbf{\beta}$ es a través de la construcción de intervalos de confianza del $100\times(1-\alpha)100\%$. Es fácil mostrar que, para $j$ fijo,

\[\begin{equation}\label{eq:icbeta} \beta_j\in \left( \hat{\beta}_j - t_{\alpha/2, n-p} \sqrt{\hat{\sigma}^2(\mathbf{X}^\prime\mathbf{X})^{-1}_{jj}}, \hat{\beta}_j + t_{\alpha/2, n-p} \sqrt{\hat{\sigma}^2(\mathbf{X}^\prime\mathbf{X})^{-1}_{jj}} \right) \end{equation}\]

Otra alternativa para construir intervalos de confianza es vía bootstrap o likelihood profiling. Finalmente concluimos, con un nivel de confianza del $100\times(1-\alpha)\%$, que $\beta_j$ está en el intervalo anterior.

Uan forma de construir los intervalos de confianza para los coeficientes del modelo es utilizando la función confint.default:

## intervalos de confianza del 95% para los coeficientes
confint.default(fit)

##                 2.5 %     97.5 %
## (Intercept) 64.122011 68.4476513
## x1           1.362326  1.6031807
## x2          -1.322932 -0.7874793

Por lo tanto, con una confianza del 95%,

\[\begin{align} \beta_0 &\in(64.12, 68.45) \\ \beta_1 &\in(1.36, 1.60) \\ \beta_2 &\in(-1.323, -0.787). \end{align}\]

3.5 Inferencia para la respuesta

El modelo de RLM ajustado puede utilizarse para predecir $E[\mathbf{Y}|\mathbf{x}_0]$ sólo si se validan todos los supuestos. Para más detalles ver Análisis de Residuales.

3.5.1 Intervalos de confianza para $E[\mathbf{Y}|\mathbf{x}_0]$

A partir del modelo ajustado y para valores fijos de las variables controlables, digamos $\mathbf{x}_0$, se tiene que \[\begin{eqnarray} \hat{\mu}_{\mathbf{Y} | \mathbf{x}_0} &=& \hat{E[\mathbf{Y} | \mathbf{x}_0]} = \mathbf{x}_0^\prime\hat{\mathbf{\beta}} \\\nonumber V[\hat{\mu}_{\mathbf{Y} | \mathbf{x}_0}] &=& \hat{\sigma}^2\mathbf{x}_0^\prime(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{x}_0 \end{eqnarray}\]

Finalmente, el intervalo de confianza del $100\times(1-\alpha)\%$ puede calcularse como

\[\begin{eqnarray} \hat{\mu}_{\mathbf{Y} | \mathbf{x}_0} \pm t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2\mathbf{x}_0^\prime(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{x}_0} \end{eqnarray}\]

Supongamos que queremos determinar el espesor de la capa de pintura para las condiciones $\mathbf{x}_0 = (10, 8)$, es decir, cuando $x_1 = 10$ y $x_2 = 8$.

En R procedemos de la siguiente manera:

## cálculo de E[y|x_1 = 10, x_2 = 8]
predict(fit, newdata = data.frame(x1 = 10, x2 = 8))

##        1 
## 72.67072

Si trabajamos bajo las condiciones $\mathbf{x}_0$, se espera que, en promedio, el espesor de la capa de pintura sea 72.67 micrones.

Para otros argumentos y opciones, se sugiere al lector escribir ?predict.lm en la consola de R.

Ahora, si es de interés calcular un intervalo de confianza del 95%, agregamos el argumento interval = 'confidence' a la instrucción anterior:

## cálculo de E[y|x_1 = 10, x_2 = 8]
predict(fit, newdata = data.frame(x1 = 10, x2 = 8), interval = 'confidence')

##        fit      lwr      upr
## 1 72.67072 71.64637 73.69507

Por lo tanto, se espera que, si continuamos trabajando bajo las condiciones $x_1 = 10$ y $x_2 = 8$, el espesor de capa promedio sea 72.67 micrones. A nivel poblacional, dicho promedio se encontrará en el intervalo $(71.646, 73.695)$ con una confianza del 95%.

3.5.2 Intervalos de predicción para $E[\mathbf{Y}|\mathbf{x}_0]$

Sea $\hat{y}_0 = \hat{\mu}_{\mathbf{Y} | \mathbf{x}_0}$, donde $\mathbf{x}_0$ es el vector de covariables futuro. Un intervalo de predicción del $100\times(1-\alpha)\%$ para $Y_0$ está dado por:

\[\begin{eqnarray} \hat{y}_0 \pm t_{\alpha/2,n-p}\sqrt{\hat{\sigma}^2(1 + \mathbf{x}_0^\prime(\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{x}_0)} \end{eqnarray}\]

Otra posibilidad para construir dicho intervalo es vía bootstrap. Observe que lo único que cambia en este intervalo en relación con el intervalo de confianza es la varianza de $Y_0$ $-$ existe más incertidumbre. Observe que en el intervalo de predicción estamos interesados en $Y_0 | \mathbf{x}_0$ y no $\hat{\mu}_{\mathbf{Y} | \mathbf{x}_0}$.

Para calcular un intervalo de predicción del 95%, agregamos el argumento interval = 'prediction' a la instrucción anterior:

## intervalo de predicción para E[y|x_1 = 10, x_2 = 8]
predict(fit, newdata = data.frame(x1 = 10, x2 = 8), interval = 'prediction')

##        fit      lwr      upr
## 1 72.67072 67.08489 78.25655

Este resultado indica que, bajo las condiciones $x_1 = 10$ y $x_2 = 8$, el valor del espesor de la capa de pintura para la próxima unidad experimental será 72.67 micrones. A nivel poblacional, dicho valor se encontrará en el intervalo $(67.085, 78.257)$ con una confianza del 95%.

3.6 Análisis de Residuales

El análisis de residuales en RLM es fundamental para:

Validar los supuestos del error;
identicar observaciones outlier; e
identificar observaciones influenciales.

3.6.1 Validación de supuestos

La validación de los supestos del error en el modelo de RLM se realiza de manera similar a como se mostró para el modelo de RLS. Para más detalles, ver Análisis de Residuales.

3.6.2 Identificación de outliers

Los outliers son también conocidos como observaciones atípicas en los datos. A partir del modelo ajustado, podemos calcular:

Residuales crudos \[\hat{\epsilon}_i = y_i - \hat{y}_i\]
Residuales estandarizados \[d_i = \frac{\hat{\epsilon}_i}{\sqrt{\hat{\sigma}^2}} = \frac{\hat{\epsilon}_i}{\sqrt{\text{MSE}}} \]
Residuales estudentizados \[r_i = \frac{\hat{\epsilon}_i}{\sqrt{\hat{\sigma}^2(1-h_{ii})}} = \frac{\hat{\epsilon}_i}{\sqrt{\text{MSE}(1-h_{ii})}}\]

donde $h_{ii} = \mathbf{X}(\mathbf{X^\prime\mathbf{X}})^{-1}\mathbf{X}^\prime_{ii}$ y $\mathbf{X}(\mathbf{X^\prime\mathbf{X}})^{-1}\mathbf{X}$ denominada la matriz hat.

La mejor manera de identificar outliers es a partir del cálculo de los residuales estudentizados. Decimos que la $i$-ésima observación es un outlier si $r_i\notin (-3, 3)$.

En R las funciones clave son:

## residuales crudos
r_crudo <- residuals(fit)

## residuales estudentizados
r_est <- rstudent(fit)

## residuales estandarizados
r_normal <- rstandard(fit)

donde fit es el objeto R que contiene en modelo de RLM estimado. Para más detalles, se sugiere consultar la ayuda de cada función: ?residuals, ?rstudent y ?rstandard.

En nuestro caso,

## residuales estudentizados
r <- rstudent(fit)
which(r < -3 | r > 3)

## 18 
## 18

la observación 18 podría considerarse un outlier. En la práctica, el siguiente paso es evaluar la trazabilidad de esa observación y determinar si existen o no causas asignables para que esta sea un outlier. En caso de que exista una causa asignable, dicha observación debería removerse de la base de datos y, con los datos reducidos, estimar nuevamente el modelo de RLM.

Otra forma de detectar outliers es a través de la prueba de Bonferroni. Esta prueba está implementada en la función outlierTest del paquete car. En nuestro ejemplo, procedemos de la siguiente manera:

## prueba Bonferroni para outliers
require(car)
outlierTest(fit, n.max = 5)

## No Studentized residuals with Bonferroni p < 0.05
## Largest |rstudent|:
##    rstudent unadjusted p-value Bonferroni p
## 18 3.360964           0.001116       0.1116

En la parte superior vemos que la observación 18 tiene el mayor valor del residual estudentizado. Sin embargo, el valor $p$ es superior a 0.05, por lo que no tenemos evidencia suficiente para concluir que dicha observación representa un outlier.

Gráficamente es posible identificar gráficamente cuáles son las observaciones influenciales utilizando la función influenceIndexPlot del paquete car:

## gráfico de variables influenciales
influenceIndexPlot(fit, vars = "Bonf", las = 1)

Figura 3.6: Gráfico de observaciones influenciales. El eje $y$ corresponde al valor $p$ de la prueba de Bonferroni.

Para más detalles, se recomienda escribir ?viainfluenceIndexPlot en la consola del R.

3.6.3 Identificación de observaciones influenciales

En ciertas ocasiones encontramos observaciones que lucen algo anormales y es importante determinar si estas son influenciales o no.

A diferencia de los outliers, las observaciones influenciales controlan el modelo y por ello es importante determinar si el modelo ajustado es consistente cuando estas se remueven.

La identificación de este tipo de observaciones se realiza utilizando, principalmente, la Distancia de Cook.

Para la $i$-ésima observación, esta distancia se calcula como

\[D_i = \frac{r_i^2}{p}\frac{h_{ii}}{(1-h_{ii})} = \frac{\hat{\epsilon}_i^2 \,h_{ii}}{p\,\hat{\sigma}^2(1-h_{ii})^2}\]

donde $p$ es el número de variables controlables incluídas en el modelo.

Usualmente, cuando $D_i>\frac{4}{n-p-2}$ decimos que la $i$-ésima observacion es influencial. Este criterio es el utilizado por la función cooks.distance() del R.

3.7 Análisis de Multicolinealidad

Cuando se utiliza el modelo de RLM, se asume que las variables controlables $X_1, X_2,\ldots,X_k$ son independientes. Desde el punto de vista práctico, esto tiene consideraciones importantes puesto que permite evaluar la magnitud del efecto de sobre $\widehat{E[Y]}$ cuando modificamos, en una unidad, digamos $x_j$, mientras se mantienen el resto de ellas constantes. Este efecto corresponde, sin duda, a $\hat{\beta}_j$. Sin embargo, cuando estas variables controlables no son independientes, este efecto no puede calcularse de la misma forma.

Buscamos modelos en los que las covariables estén altamente correlacionadas con la respuesta, pero mínimamente entre ellas.

Desde el punto de vista teórico, la existencia de no independencia en las variables controlables tiene consecuencias importantes sobre los estimadores de los parámetros del modelo dados por $\mathbf{\beta} = (\beta_0, \beta_1, \beta_2, \ldots, \beta_k)$. Cuando se usa el método de mínimos cuadrados, los estimadores de $\mathbf{\beta}$ están dados por:

\[ \hat{\mathbf{\beta}}_{\text{OLS}} = (\mathbf{X}^\prime\mathbf{X})^{-1}\mathbf{X}^\prime \mathbf{y} \]

La existencia de multicolinealidad es sinónimo de que no existe independencia en las variables controlables del modelo. Si esto es cierto, las columnas de la matriz de diseño $\mathbf{X}$ no son independientes, es decir, que la columna $x_j$ puede expresarse como una combinación lineal de las demás. Matemáticamente, esto es equivalente a escribir $x_j \sim x_{-j}$ para algún $j$. Por ejemplo, para $j=1$ tendríamos

\[ x_1 \sim x_2 + x_3 + \cdots + x_k. \]

Esta expresión indica que la variable independiente/controlable $x_1$ puede escribirse como una combinación lineal de las demás variables controlables. O, en otras palabras, que la información contenida en $x_1$ puede explicarse por las demás variables controlables medidas en el proceso durante la etapa de muestreo.

Una manera de interpretar la multicolinealidad es como sinónimo de redundancia. Esta redundancia se refiere a que existen variables controlables en el proceso de producción que contienen la misma información que las demás. Por lo tanto, basta con medir sólo aquellas que realmente determinan dicho proceso.

En la expresión de $\hat{\mathbf{\beta}}_{\text{OLS}}$, el término $(\mathbf{X}^\prime\mathbf{X})^{-1}$ se refiere a la inversa de $\mathbf{X}^\prime\mathbf{X}$. Si existe multicolinealidad,

\[ \text{det}(\mathbf{X}^\prime \mathbf{X}) \approx 0 \quad \Rightarrow \quad \frac{1}{|\mathbf{X}^\prime \mathbf{X}|} \rightarrow\infty \]

Por lo tanto,

\[ \hat{\mathbf{\beta}}_{\text{OLS}} \rightarrow \infty. \]

3.7.1 Cómo detectarla?

Existen varios indicadores para sospechar que existe multicolinealidad:

Una alta correlación en las variables independientes. Esto es posible determinarlo gráficamente a través de una matriz de dispersión (ver por ejemplo ?pairs en la consola del R) o utilizando una prueba de independencia completa.
Que se rechace la prueba de significancia global pero no todas las pruebas de significancia marginal.
Que ocurran ambios considerables en $\hat{\mathbf{\beta}}$ cuando se agrega o elimina una variable predictora.

Para probar que efectivamente existe, podemos usar tres aproximaciones:

El número de condición de la matriz $\mathbf{X}^\prime \mathbf{X}$. También conocido como I-ll condicion number o ICN, este número mide qué tan “enferma” se encuentra la matriz que debe ser invertida para poder calcular $\hat{\mathbf{\beta}}_{\text{OLS}}$. El ICN se calcula como

\[\text{ICN}(\mathbf{X}^\prime\mathbf{X}) = \sqrt{\frac{\lambda_\text{máx}}{\lambda_\text{min}}} \]

con $\lambda_\text{máx}$ y $\lambda_\text{min}$ los valores propios máximos y mínimos de $\mathbf{X}^\prime \mathbf{X}$, obtenidos a partir de la descomposición espectral de dicha matriz.

Decimos que existe multicolinealidad cuando $\text{ICN}(\mathbf{X}^\prime \mathbf{X}) > 30$. El inconveniente con el ICN es que no nos da información acerca de cuál de las variables independentes es la más multicolineal (o redundante) en el sistema.

En R, la función clave para calcular el ICN es kappa. Para más detalles, se recomienda escribir ?kappa en la consola. En el caso del ejemplo del espesor de pintura, tendríamos los siguientes resultados:

## ICN para el modelo ajustado
kappa(fit)

## [1] 48.62071

Puesto que $\text{ICN}(\mathbf{X}^\prime \mathbf{X}) > 30$, podemos concluir que existe evidencia para sospechar que, efectivamente, existe multicolinealidad.

El factor de inflación de varianza (VIF). A través de este indicador podemos detectar cuál de las variables independientes es la más colineal de las $k$ medidas. Para la $j$-ésima variable independiente,

\[\text{VIF}_j = \frac{1}{1-R_j^2}\]

donde $R_j^2$ es el $R^2_\text{adjusted}$ del modelo $x_j\sim x_{-j}$.

Decimos que la variable $x_j$ es responsable por la multicolineal en el sistema si $\text{VIF}_j > 5$.

En R, el VIF puede calcularse a través de la función vif del paquete car. Para más detalles, se recomienda escribir ?vif en la consola. En nuestro ejemplo tendríamos los siguientes resultados:

## cálculo del VIF
car:::vif(fit)

##       x1       x2 
## 1.001645 1.001645

Puesto que en ninguna de las dos variables controlables el $\text{VIF}>5$, concluimos que no existe multicolinealidad.

Pruebas complementarias. En algunos casos, el ICN indica que existe multicolinealidad, pero para ninguno de los predictores el $\text{VIF}>5$. Cuando esto ocurre, lo mejor es utilizar pruebas complementaria, más robustas, que permitan decidir si efectivamente existe multicolinealidad en el modelo de RLM.

Las pruebas complementarias pueden realizarse en R con la función mctest del paquete mctest. En nuestro ejemplo tendríamos los siguientes resultados:

## pruebas complementarias de multicolinealidad
require(mctest)
mctest(fit)

## 
## Call:
## omcdiag(mod = mod, Inter = TRUE, detr = detr, red = red, conf = conf, 
##     theil = theil, cn = cn)
## 
## 
## Overall Multicollinearity Diagnostics
## 
##                        MC Results detection
## Determinant |X'X|:         0.9984         0
## Farrar Chi-Square:         0.1602         0
## Red Indicator:             0.0405         0
## Sum of Lambda Inverse:     2.0033         0
## Theil's Method:           -0.8683         0
## Condition Number:          8.2787         0
## 
## 1 --> COLLINEARITY is detected by the test 
## 0 --> COLLINEARITY is not detected by the test

De acuerdo con estos resultados, podemos concluir que no existe multicolinealidad.

3.8 Selección de Modelos

3.8.1 Método de Todas las Regresiones Posibles

El Método de Todas las Regresiones Posibles permite, a partir de un conjunto de variables independientes $X_1, X_2, \ldots, X_k$ que potencialmente podrían explicar una respuesta continua $Y$, ajustar hasta $2^{k}-1$ modelos de regresión y seleccionar el mejor de estos utilizando algún criterio.

En la práctica, algunos de los criterios más utilizados incluyen $R^2$, $R^2_\text{adj}$, $\sqrt{MSE}$, AIC, BIC, PRESS y

\[\begin{equation} C_p = p+\frac{\text{SSE}_p}{\text{MSE}_\text{todos}} - (n-2p) = \begin{cases} = p & \text{ para el modelo completo } \\ \approx p & \text{ el sesgo es pequeño $\rightarrow$ ideal!} \\ > p & \text{ sesgo es alto }\\ < p & \text{ no hay sesgo }\\ \end{cases} \end{equation}\]

también conocido como el estadístico de Mallows. En la expresión anterior, $\text{MSE}_\text{todos}$ es $\hat{\sigma}^2$ usando todas las covariables, y $\text{SSE}_p$ es el SSE del modelo con sólo $p^\prime < p$ de ellas.

Como ejemplo, consideraremos los siguientes datos que corresponden al peso del producto terminado en gramos (variable $y$) cuando se controlan los parámetros $x_1, x_2, \ldots, x_{10}$ de una inyectora:

## datos inyectora
d <- read.table("https://www.dropbox.com/s/a9gzu54luabtubo/inyectora.txt?dl=1", 
                header = TRUE)

## primeras 3 filas
head(d, 3)

##          y         x1          x2         x3         x4         x5         x6
## 1 22.49879 -0.4248450  0.19997792 -0.5225479  0.5691505  0.9721086 -0.2927878
## 2 17.60173  0.5766103 -0.33435292  0.9247179 -0.9811402 -0.7258651 -0.2671171
## 3 22.50649 -0.1820462 -0.02277393  0.2027315  0.5581318  0.8106192 -0.4257997
##           x7         x8          x9        x10
## 1 -0.5255406  0.6898671 -0.05863633 0.84739843
## 2  0.3729807 -0.4797351 -0.26830905 0.08519674
## 3 -0.5483632 -0.9537110 -0.75745589 0.70472920

En total se tienen 100 unidades experimentales. Dado que en los datos sólo existen la variable respuesta y las covariables, podemos utilizar la siguiente sintaxis para ajustar el modelo de RLM:

## modelo de RLM ajustado
fit_inyectora <- lm(y ~ ., data = d)
summary(fit_inyectora)

## 
## Call:
## lm(formula = y ~ ., data = d)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -2.09900 -0.61503 -0.04698  0.49843  2.35473 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) 19.97501    0.09496 210.353  < 2e-16 ***
## x1           3.04897    0.17692  17.234  < 2e-16 ***
## x2           0.85418    0.18412   4.639 1.20e-05 ***
## x3           4.98623    0.17088  29.179  < 2e-16 ***
## x4           4.97033    0.17161  28.963  < 2e-16 ***
## x5           2.99030    0.17537  17.052  < 2e-16 ***
## x6           4.03875    0.16979  23.786  < 2e-16 ***
## x7           2.08153    0.16505  12.612  < 2e-16 ***
## x8           0.89152    0.16997   5.245 1.05e-06 ***
## x9           2.90108    0.18030  16.090  < 2e-16 ***
## x10          2.89888    0.17135  16.918  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.9418 on 89 degrees of freedom
## Multiple R-squared:  0.9778, Adjusted R-squared:  0.9753 
## F-statistic: 391.2 on 10 and 89 DF,  p-value: < 2.2e-16

Como el número de variables independientes es $k=10$, debemos estimar $2^k-1= 1023$ modelos diferentes al utilizar el Método de Todas las Regresiones Posibles. Para encontrar el mejor modelo, usamos la función ols_step_all_possible del paquete olsrr. Para más detalles, se recomienda escribir ?ols_step_all_possible en la consola del R.

## método de todas las regresiones posibles
require(olsrr)
k <- ols_step_all_possible(fit_inyectora)
head(k, 10)

## $result
##      mindex  n                     predictors     rsquare         adjr
## 4         1  1                             x4 0.381431483  0.375119559
## 10        2  1                            x10 0.221939980  0.214000593
## 5         3  1                             x5 0.150758161  0.142092428
## 3         4  1                             x3 0.122800863  0.113849851
## 9         5  1                             x9 0.065647890  0.056113685
## 6         6  1                             x6 0.061473423  0.051896621
## 7         7  1                             x7 0.049953135  0.040258779
## 8         8  1                             x8 0.037568650  0.027747921
## 1         9  1                             x1 0.005957422 -0.004185870
## 2        10  1                             x2 0.005662949 -0.004483347
## 40       11  2                         x4 x10 0.521877556  0.512019361
## 28       12  2                          x3 x4 0.479958467  0.469235961
## 36       13  2                          x4 x6 0.463712088  0.452654605
## 35       14  2                          x4 x5 0.460691166  0.449571396
## 37       15  2                          x4 x7 0.443771776  0.432303152
## 38       16  2                          x4 x8 0.410519226  0.398364983
## 39       17  2                          x4 x9 0.398723458  0.386326004
## 13       18  2                          x1 x4 0.397481773  0.385058717
## 21       19  2                          x2 x4 0.390969000  0.378411660
## 45       20  2                         x5 x10 0.360661309  0.347479068
## 34       21  2                         x3 x10 0.339693118  0.326078543
## 29       22  2                          x3 x5 0.299562577  0.285120568
## 49       23  2                         x6 x10 0.272555214  0.257556352
## 55       24  2                         x9 x10 0.265598346  0.250456044
## 54       25  2                         x8 x10 0.255804930  0.240460702
## 52       26  2                         x7 x10 0.247829404  0.232320732
## 33       27  2                          x3 x9 0.227021235  0.211083529
## 19       28  2                         x1 x10 0.226266172  0.210312897
## 27       29  2                         x2 x10 0.222154952  0.206116910
## 30       30  2                          x3 x6 0.220932150  0.204868895
## 44       31  2                          x5 x9 0.211028430  0.194760975
## 42       32  2                          x5 x7 0.196199207  0.179625995
## 41       33  2                          x5 x6 0.190688488  0.174001653
## 31       34  2                          x3 x7 0.171408947  0.154324596
## 14       35  2                          x1 x5 0.168292013  0.151143394
## 32       36  2                          x3 x8 0.166994041  0.149818660
## 43       37  2                          x5 x8 0.163402726  0.146153297
## 22       38  2                          x2 x5 0.154742731  0.137314746
## 48       39  2                          x6 x9 0.146734025  0.129140912
## 20       40  2                          x2 x3 0.135450160  0.117624390
## 12       41  2                          x1 x3 0.133072501  0.115197707
## 46       42  2                          x6 x7 0.120437501  0.102302192
## 51       43  2                          x7 x9 0.106580735  0.088159719
## 47       44  2                          x6 x8 0.104784791  0.086326745
## 53       45  2                          x8 x9 0.090696676  0.071948154
## 50       46  2                          x7 x8 0.085916349  0.067069263
## 18       47  2                          x1 x9 0.083581943  0.064686726
## 15       48  2                          x1 x6 0.071713905  0.052573986
## 26       49  2                          x2 x9 0.068376014  0.049167272
## 23       50  2                          x2 x6 0.067114560  0.047879809
## 16       51  2                          x1 x7 0.057543941  0.038111857
## 24       52  2                          x2 x7 0.052909882  0.033382250
## 25       53  2                          x2 x8 0.041934796  0.022180874
## 17       54  2                          x1 x8 0.041924853  0.022170726
## 11       55  2                          x1 x2 0.012731126 -0.007624933
## 125      56  3                      x3 x4 x10 0.619057967  0.607153528
## 145      57  3                      x4 x5 x10 0.600178034  0.587683598
## 121      58  3                       x3 x4 x6 0.599101154  0.586573065
## 149      59  3                      x4 x6 x10 0.591930193  0.579178012
## 120      60  3                       x3 x4 x5 0.578830536  0.565668990
## 152      61  3                      x4 x7 x10 0.560847296  0.547123774
## 154      62  3                      x4 x8 x10 0.549099996  0.535009371
## 122      63  3                       x3 x4 x7 0.540559455  0.526201938
## 146      64  3                       x4 x6 x7 0.538177791  0.523745847
## 76       65  3                      x1 x4 x10 0.534544419  0.519998933
## 155      66  3                      x4 x9 x10 0.532915933  0.518319556
## 104      67  3                      x2 x4 x10 0.524072273  0.509199531
## 141      68  3                       x4 x5 x6 0.522895504  0.507985988
## 142      69  3                       x4 x5 x7 0.518279654  0.503225893
## 124      70  3                       x3 x4 x9 0.518231935  0.503176682
## 123      71  3                       x3 x4 x8 0.514557721  0.499387650
## 130      72  3                      x3 x5 x10 0.503017883  0.487487191
## 64       73  3                       x1 x3 x4 0.501574150  0.485998343
## 147      74  3                       x4 x6 x8 0.498433927  0.482759987
## 92       75  3                       x2 x3 x4 0.496996513  0.481277654
## 148      76  3                       x4 x6 x9 0.492038069  0.476164259
## 72       77  3                       x1 x4 x6 0.488139440  0.472143798
## 71       78  3                       x1 x4 x5 0.487520048  0.471505049
## 144      79  3                       x4 x5 x9 0.478714761  0.462424598
## 143      80  3                       x4 x5 x8 0.473485387  0.457031805
## 100      81  3                       x2 x4 x6 0.473343194  0.456885169
## 150      82  3                       x4 x7 x8 0.471150620  0.454624077
## 99       83  3                       x2 x4 x5 0.468240425  0.451622938
## 73       84  3                       x1 x4 x7 0.463020642  0.446240037
## 151      85  3                       x4 x7 x9 0.455420365  0.438402251
## 101      86  3                       x2 x4 x7 0.449357124  0.432149535
## 74       87  3                       x1 x4 x8 0.424072190  0.406074446
## 134      88  3                      x3 x6 x10 0.423254387  0.405231087
## 75       89  3                       x1 x4 x9 0.423014250  0.404983445
## 153      90  3                       x4 x8 x9 0.422311204  0.404258429
## 102      91  3                       x2 x4 x8 0.418508525  0.400336917
## 140      92  3                      x3 x9 x10 0.415061487  0.396782158
## 57       93  3                       x1 x2 x4 0.409488296  0.391034805
## 103      94  3                       x2 x4 x9 0.405929402  0.387364696
## 165      95  3                      x5 x9 x10 0.400613914  0.381883099
## 129      96  3                       x3 x5 x9 0.400360755  0.381622028
## 159      97  3                      x5 x6 x10 0.392785005  0.373809537
## 162      98  3                      x5 x7 x10 0.383947359  0.364695714
## 139      99  3                      x3 x8 x10 0.379770406  0.360388231
## 81      100  3                      x1 x5 x10 0.374833233  0.355296772
## 164     101  3                      x5 x8 x10 0.371920374  0.352292886
## 126     102  3                       x3 x5 x6 0.370628336  0.350960471
## 133     103  3                       x3 x6 x9 0.366363426  0.346562283
## 137     104  3                      x3 x7 x10 0.364858048  0.345009862
## 109     105  3                      x2 x5 x10 0.360681948  0.340703259
## 70      106  3                      x1 x3 x10 0.347713340  0.327329382
## 127     107  3                       x3 x5 x7 0.343209435  0.322684729
## 98      108  3                      x2 x3 x10 0.342376482  0.321825747
## 171     109  3                      x6 x9 x10 0.331484156  0.310593036
## 65      110  3                       x1 x3 x5 0.326589540  0.305545463
## 128     111  3                       x3 x5 x8 0.314769082  0.293355616
## 170     112  3                      x6 x8 x10 0.311468665  0.289952061
## 93      113  3                       x2 x3 x5 0.310185864  0.288629172
## 168     114  3                      x6 x7 x10 0.305003924  0.283285297
## 175     115  3                      x8 x9 x10 0.289797204  0.267603366
## 174     116  3                      x7 x9 x10 0.286945450  0.264662496
## 131     117  3                       x3 x6 x7 0.280755403  0.258279009
## 173     118  3                      x7 x8 x10 0.280750716  0.258274176
## 85      119  3                      x1 x6 x10 0.280273987  0.257782549
## 91      120  3                      x1 x9 x10 0.278363642  0.255812506
## 132     121  3                       x3 x6 x8 0.274376326  0.251700586
## 113     122  3                      x2 x6 x10 0.272810349  0.250085673
## 158     123  3                       x5 x6 x9 0.266770539  0.243857118
## 119     124  3                      x2 x9 x10 0.265601410  0.242651454
## 136     125  3                       x3 x7 x9 0.264140936  0.241145341
## 69      126  3                       x1 x3 x9 0.259017559  0.235861858
## 90      127  3                      x1 x8 x10 0.258855303  0.235694531
## 118     128  3                      x2 x8 x10 0.255844736  0.232589884
## 138     129  3                       x3 x8 x9 0.254844586  0.231558480
## 88      130  3                      x1 x7 x10 0.253240274  0.229904033
## 161     131  3                       x5 x7 x9 0.248287204  0.224796179
## 116     132  3                      x2 x7 x10 0.247833663  0.224328465
## 80      133  3                       x1 x5 x9 0.246218036  0.222662350
## 156     134  3                       x5 x6 x7 0.243466657  0.219824990
## 66      135  3                       x1 x3 x6 0.239483931  0.215717803
## 97      136  3                       x2 x3 x9 0.234917494  0.211008666
## 94      137  3                       x2 x3 x6 0.234874359  0.210964182
## 63      138  3                      x1 x2 x10 0.226696563  0.202530831
## 163     139  3                       x5 x8 x9 0.217144272  0.192680030
## 78      140  3                       x1 x5 x7 0.216144217  0.191648724
## 135     141  3                       x3 x7 x8 0.213846588  0.189279294
## 77      142  3                       x1 x5 x6 0.212764586  0.188163479
## 108     143  3                       x2 x5 x9 0.212727583  0.188125320
## 160     144  3                       x5 x7 x8 0.208211556  0.183468167
## 157     145  3                       x5 x6 x8 0.207662064  0.182901504
## 106     146  3                       x2 x5 x7 0.198076934  0.173016839
## 167     147  3                       x6 x7 x9 0.195644408  0.170508296
## 105     148  3                       x2 x5 x6 0.194760838  0.169597115
## 67      149  3                       x1 x3 x7 0.183745402  0.158237446
## 95      150  3                       x2 x3 x7 0.179825629  0.154195179
## 79      151  3                       x1 x5 x8 0.178446530  0.152772984
## 96      152  3                       x2 x3 x8 0.177703405  0.152006637
## 84      153  3                       x1 x6 x9 0.176135218  0.150389444
## 169     154  3                       x6 x8 x9 0.175326276  0.149555222
## 68      155  3                       x1 x3 x8 0.175068716  0.149289614
## 58      156  3                       x1 x2 x5 0.173838990  0.148021458
## 107     157  3                       x2 x5 x8 0.166819842  0.140782962
## 166     158  3                       x6 x7 x8 0.162283261  0.136104613
## 112     159  3                       x2 x6 x9 0.149092854  0.122502006
## 56      160  3                       x1 x2 x3 0.148120017  0.121498767
## 82      161  3                       x1 x6 x7 0.133406429  0.106325380
## 172     162  3                       x7 x8 x9 0.131213542  0.104063966
## 87      163  3                       x1 x7 x9 0.126012650  0.098700545
## 110     164  3                       x2 x6 x7 0.123175041  0.095774261
## 83      165  3                       x1 x6 x8 0.112928246  0.085207253
## 111     166  3                       x2 x6 x8 0.109039295  0.081196773
## 115     167  3                       x2 x7 x9 0.107783343  0.079901573
## 89      168  3                       x1 x8 x9 0.105072988  0.077106518
## 117     169  3                       x2 x8 x9 0.092878577  0.064531032
## 86      170  3                       x1 x7 x8 0.091697922  0.063313482
## 114     171  3                       x2 x7 x8 0.088002197  0.059502266
## 62      172  3                       x1 x2 x9 0.087397378  0.058878546
## 59      173  3                       x1 x2 x6 0.078819974  0.050033098
## 60      174  3                       x1 x2 x7 0.061372135  0.032040014
## 61      175  3                       x1 x2 x8 0.047156896  0.017380549
## 324     176  4                   x3 x4 x6 x10 0.723095467  0.711436329
## 320     177  4                   x3 x4 x5 x10 0.716724226  0.704796825
## 321     178  4                    x3 x4 x6 x7 0.673981038  0.660253924
## 316     179  4                    x3 x4 x5 x6 0.673446388  0.659696762
## 323     180  4                    x3 x4 x6 x9 0.664056408  0.649911415
## 327     181  4                   x3 x4 x7 x10 0.656750191  0.642297568
## 354     182  4                   x4 x5 x6 x10 0.651824499  0.637164478
## 329     183  4                   x3 x4 x8 x10 0.651590585  0.636920715
## 330     184  4                   x3 x4 x9 x10 0.647481152  0.632638254
## 322     185  4                    x3 x4 x6 x8 0.642483377  0.627430045
## 363     186  4                   x4 x6 x7 x10 0.640999576  0.625883769
## 209     187  4                   x1 x3 x4 x10 0.636693512  0.621396397
## 357     188  4                   x4 x5 x7 x10 0.635451268  0.620101847
## 205     189  4                    x1 x3 x4 x6 0.634176348  0.618773247
## 317     190  4                    x3 x4 x5 x7 0.634015368  0.618605489
## 265     191  4                   x2 x3 x4 x10 0.625425300  0.609653734
## 365     192  4                   x4 x6 x8 x10 0.624274990  0.608454990
## 229     193  4                   x1 x4 x5 x10 0.622497931  0.606603107
## 319     194  4                    x3 x4 x5 x9 0.620881235  0.604918340
## 261     195  4                    x2 x3 x4 x6 0.617945217  0.601858700
## 204     196  4                    x1 x3 x4 x5 0.615317518  0.599120361
## 360     197  4                   x4 x5 x9 x10 0.611819893  0.595475467
## 359     198  4                   x4 x5 x8 x10 0.611800092  0.595454832
## 233     199  4                   x1 x4 x6 x10 0.611686421  0.595336375
## 366     200  4                   x4 x6 x9 x10 0.611609629  0.595256350
## 285     201  4                   x2 x4 x5 x10 0.601486740  0.584707235
## 289     202  4                   x2 x4 x6 x10 0.594381358  0.577302679
## 318     203  4                    x3 x4 x5 x8 0.593909444  0.576810895
## 260     204  4                    x2 x3 x4 x5 0.593617338  0.576506489
## 351     205  4                    x4 x5 x6 x7 0.591442102  0.574239664
## 368     206  4                   x4 x7 x8 x10 0.586883865  0.569489501
## 340     207  4                   x3 x5 x9 x10 0.576402914  0.558567247
## 236     208  4                   x1 x4 x7 x10 0.576058853  0.558208699
## 325     209  4                    x3 x4 x7 x8 0.573269699  0.555302108
## 361     210  4                    x4 x6 x7 x8 0.571264270  0.553212240
## 326     211  4                    x3 x4 x7 x9 0.570065430  0.551962921
## 369     212  4                   x4 x7 x9 x10 0.568623255  0.550460024
## 230     213  4                    x1 x4 x6 x7 0.567700229  0.549498133
## 206     214  4                    x1 x3 x4 x7 0.565768085  0.547484636
## 334     215  4                   x3 x5 x6 x10 0.562917357  0.544513877
## 292     216  4                   x2 x4 x7 x10 0.561882750  0.543435708
## 238     217  4                   x1 x4 x8 x10 0.559641575  0.541100167
## 362     218  4                    x4 x6 x7 x9 0.559182393  0.540621652
## 225     219  4                    x1 x4 x5 x6 0.557261992  0.538620391
## 370     220  4                   x4 x8 x9 x10 0.555953617  0.537256927
## 208     221  4                    x1 x3 x4 x9 0.555719249  0.537012691
## 262     222  4                    x2 x3 x4 x7 0.552155247  0.533298626
## 239     223  4                   x1 x4 x9 x10 0.551557639  0.532675855
## 294     224  4                   x2 x4 x8 x10 0.550624204  0.531703118
## 353     225  4                    x4 x5 x6 x9 0.550575343  0.531652199
## 226     226  4                    x1 x4 x5 x7 0.548700467  0.529698381
## 328     227  4                    x3 x4 x8 x9 0.544235113  0.525045013
## 286     228  4                    x2 x4 x6 x7 0.543507363  0.524286620
## 352     229  4                    x4 x5 x6 x8 0.541224059  0.521907178
## 188     230  4                   x1 x2 x4 x10 0.537908217  0.518451721
## 295     231  4                   x2 x4 x9 x10 0.534312223  0.514704316
## 207     232  4                    x1 x3 x4 x8 0.533150225  0.513493392
## 346     233  4                   x3 x6 x9 x10 0.532569825  0.512888554
## 264     234  4                    x2 x3 x4 x9 0.531401801  0.511671351
## 281     235  4                    x2 x4 x5 x6 0.530760840  0.511003401
## 232     236  4                    x1 x4 x6 x9 0.530745705  0.510987629
## 356     237  4                    x4 x5 x7 x9 0.530721110  0.510961999
## 355     238  4                    x4 x5 x7 x8 0.530360965  0.510586690
## 263     239  4                    x2 x3 x4 x8 0.529534108  0.509725018
## 214     240  4                   x1 x3 x5 x10 0.525675915  0.505704374
## 337     241  4                   x3 x5 x7 x10 0.525323675  0.505337303
## 176     242  4                    x1 x2 x3 x4 0.522773080  0.502679315
## 282     243  4                    x2 x4 x5 x7 0.522518876  0.502414408
## 231     244  4                    x1 x4 x6 x8 0.519751337  0.499530341
## 364     245  4                    x4 x6 x8 x9 0.519296434  0.499056284
## 339     246  4                   x3 x5 x8 x10 0.516668710  0.496317919
## 228     247  4                    x1 x4 x5 x9 0.516654037  0.496302628
## 333     248  4                    x3 x5 x6 x9 0.506640243  0.485867200
## 287     249  4                    x2 x4 x6 x8 0.506370532  0.485586133
## 270     250  4                   x2 x3 x5 x10 0.505003979  0.484162041
## 184     251  4                    x1 x2 x4 x6 0.500874383  0.479858567
## 288     252  4                    x2 x4 x6 x9 0.498690578  0.477582813
## 183     253  4                    x1 x2 x4 x5 0.497804323  0.476659242
## 227     254  4                    x1 x4 x5 x8 0.497231068  0.476061850
## 234     255  4                    x1 x4 x7 x8 0.487698023  0.466127413
## 358     256  4                    x4 x5 x8 x9 0.487647282  0.466074536
## 284     257  4                    x2 x4 x5 x9 0.484149931  0.462429928
## 235     258  4                    x1 x4 x7 x9 0.481951858  0.460139304
## 283     259  4                    x2 x4 x5 x8 0.480244464  0.458360021
## 367     260  4                    x4 x7 x8 x9 0.478496409  0.456538363
## 290     261  4                    x2 x4 x7 x8 0.475633661  0.453555079
## 345     262  4                   x3 x6 x8 x10 0.471622210  0.449374724
## 185     263  4                    x1 x2 x4 x7 0.470601321  0.448310850
## 291     264  4                    x2 x4 x7 x9 0.459652792  0.436901331
## 213     265  4                    x1 x3 x5 x9 0.459205082  0.436434770
## 343     266  4                   x3 x6 x7 x10 0.456822596  0.433951969
## 376     267  4                   x5 x6 x9 x10 0.444799508  0.421422645
## 237     268  4                    x1 x4 x8 x9 0.442563283  0.419092263
## 350     269  4                   x3 x8 x9 x10 0.441911906  0.418413460
## 224     270  4                   x1 x3 x9 x10 0.439675083  0.416082455
## 218     271  4                   x1 x3 x6 x10 0.438177916  0.414522249
## 349     272  4                   x3 x7 x9 x10 0.434222501  0.410400290
## 186     273  4                    x1 x2 x4 x8 0.434222368  0.410400151
## 336     274  4                    x3 x5 x7 x9 0.433351819  0.409492948
## 187     275  4                    x1 x2 x4 x9 0.432311495  0.408408821
## 293     276  4                    x2 x4 x8 x9 0.428627300  0.404569502
## 249     277  4                   x1 x5 x9 x10 0.427912752  0.403824868
## 274     278  4                   x2 x3 x6 x10 0.426741653  0.402604459
## 331     279  4                    x3 x5 x6 x7 0.423789595  0.399528104
## 217     280  4                    x1 x3 x6 x9 0.423392348  0.399114131
## 373     281  4                   x5 x6 x7 x10 0.421242842  0.396874120
## 379     282  4                   x5 x7 x9 x10 0.419806541  0.395377343
## 280     283  4                   x2 x3 x9 x10 0.416300719  0.391723907
## 342     284  4                    x3 x6 x7 x9 0.412938223  0.388219832
## 243     285  4                   x1 x5 x6 x10 0.410693021  0.385880096
## 375     286  4                   x5 x6 x8 x10 0.407733428  0.382795888
## 210     287  4                    x1 x3 x5 x6 0.406788759  0.381811444
## 380     288  4                   x5 x8 x9 x10 0.406776783  0.381798963
## 269     289  4                    x2 x3 x5 x9 0.406746863  0.381767783
## 338     290  4                    x3 x5 x8 x9 0.406702022  0.381721054
## 348     291  4                   x3 x7 x8 x10 0.403903539  0.378804740
## 305     292  4                   x2 x5 x9 x10 0.400736308  0.375504152
## 246     293  4                   x1 x5 x7 x10 0.399840778  0.374570916
## 344     294  4                    x3 x6 x8 x9 0.399824448  0.374553899
## 378     295  4                   x5 x7 x8 x10 0.394834230  0.369353566
## 299     296  4                   x2 x5 x6 x10 0.392823677  0.367258358
## 332     297  4                    x3 x5 x6 x8 0.392727577  0.367158212
## 223     298  4                   x1 x3 x8 x10 0.385964884  0.360110773
## 302     299  4                   x2 x5 x7 x10 0.384001350  0.358064564
## 248     300  4                   x1 x5 x8 x10 0.383986017  0.358048586
## 266     301  4                    x2 x3 x5 x6 0.382416952  0.356413455
## 279     302  4                   x2 x3 x8 x10 0.381678955  0.355644384
## 193     303  4                   x1 x2 x5 x10 0.375061599  0.348748403
## 273     304  4                    x2 x3 x6 x9 0.374624304  0.348292696
## 221     305  4                   x1 x3 x7 x10 0.374306546  0.347961558
## 211     306  4                    x1 x3 x5 x7 0.373103127  0.346707469
## 304     307  4                   x2 x5 x8 x10 0.371920433  0.345474978
## 277     308  4                   x2 x3 x7 x10 0.366403545  0.339725799
## 384     309  4                   x6 x8 x9 x10 0.358857477  0.331862002
## 383     310  4                   x6 x7 x9 x10 0.358848639  0.331852793
## 335     311  4                    x3 x5 x7 x8 0.357720016  0.330676648
## 255     312  4                   x1 x6 x9 x10 0.353305449  0.326076205
## 182     313  4                   x1 x2 x3 x10 0.351434573  0.324126555
## 267     314  4                    x2 x3 x5 x7 0.350185972  0.322825381
## 382     315  4                   x6 x7 x8 x10 0.343109189  0.315450629
## 177     316  4                    x1 x2 x3 x5 0.340715905  0.312956575
## 212     317  4                    x1 x3 x5 x8 0.338481440  0.310628027
## 341     318  4                    x3 x6 x7 x8 0.332584392  0.304482682
## 311     319  4                   x2 x6 x9 x10 0.331493689  0.303346055
## 268     320  4                    x2 x3 x5 x8 0.324431288  0.295986289
## 254     321  4                   x1 x6 x8 x10 0.317488630  0.288751309
## 252     322  4                   x1 x6 x7 x10 0.314637516  0.285780148
## 242     323  4                    x1 x5 x6 x9 0.313034985  0.284110142
## 310     324  4                   x2 x6 x8 x10 0.311518658  0.282529970
## 385     325  4                   x7 x8 x9 x10 0.310881783  0.281866280
## 372     326  4                    x5 x6 x7 x9 0.310733424  0.281711673
## 308     327  4                   x2 x6 x7 x10 0.305007556  0.275744717
## 215     328  4                    x1 x3 x6 x7 0.303008398  0.273661383
## 258     329  4                   x1 x7 x9 x10 0.300845659  0.271407582
## 259     330  4                   x1 x8 x9 x10 0.299628808  0.270139495
## 220     331  4                    x1 x3 x7 x9 0.297857709  0.268293824
## 347     332  4                    x3 x7 x8 x9 0.291514090  0.261683105
## 216     333  4                    x1 x3 x6 x8 0.290030642  0.260137195
## 315     334  4                   x2 x8 x9 x10 0.289848107  0.259946975
## 271     335  4                    x2 x3 x6 x7 0.289837806  0.259936240
## 314     336  4                   x2 x7 x9 x10 0.287100620  0.257083804
## 272     337  4                    x2 x3 x6 x8 0.286134622  0.256077133
## 245     338  4                    x1 x5 x7 x9 0.285236912  0.255141624
## 257     339  4                   x1 x7 x8 x10 0.284723015  0.254606090
## 222     340  4                    x1 x3 x8 x9 0.281914509  0.251679330
## 197     341  4                   x1 x2 x6 x10 0.280858375  0.250578728
## 313     342  4                   x2 x7 x8 x10 0.280786473  0.250503799
## 203     343  4                   x1 x2 x9 x10 0.278411297  0.248028615
## 374     344  4                    x5 x6 x8 x9 0.275713701  0.245217435
## 181     345  4                    x1 x2 x3 x9 0.269759956  0.239013007
## 276     346  4                    x2 x3 x7 x9 0.269298234  0.238531844
## 240     347  4                    x1 x5 x6 x7 0.268947673  0.238166523
## 298     348  4                    x2 x5 x6 x9 0.268297041  0.237488495
## 278     349  4                    x2 x3 x8 x9 0.261792331  0.230709903
## 371     350  4                    x5 x6 x7 x8 0.260050115  0.228894331
## 202     351  4                   x1 x2 x8 x10 0.258988834  0.227788364
## 178     352  4                    x1 x2 x3 x6 0.256911238  0.225623290
## 377     353  4                    x5 x7 x8 x9 0.254363792  0.222968583
## 200     354  4                   x1 x2 x7 x10 0.253313472  0.221874039
## 247     355  4                    x1 x5 x8 x9 0.249262489  0.217652489
## 192     356  4                    x1 x2 x5 x9 0.249120191  0.217504199
## 301     357  4                    x2 x5 x7 x9 0.248890649  0.217264993
## 296     358  4                    x2 x5 x6 x7 0.245269744  0.213491628
## 251     359  4                    x1 x6 x7 x9 0.227831622  0.195319269
## 241     360  4                    x1 x5 x6 x8 0.226702986  0.194143112
## 244     361  4                    x1 x5 x7 x8 0.225556715  0.192948577
## 381     362  4                    x6 x7 x8 x9 0.223929481  0.191252828
## 219     363  4                    x1 x3 x7 x8 0.223775994  0.191092878
## 275     364  4                    x2 x3 x7 x8 0.220778372  0.187969040
## 190     365  4                    x1 x2 x5 x7 0.219153906  0.186276176
## 303     366  4                    x2 x5 x8 x9 0.218659555  0.185761010
## 189     367  4                    x1 x2 x5 x6 0.218634274  0.185734664
## 297     368  4                    x2 x5 x6 x8 0.211074650  0.177856740
## 300     369  4                    x2 x5 x7 x8 0.209723481  0.176448680
## 253     370  4                    x1 x6 x8 x9 0.200049178  0.166367039
## 307     371  4                    x2 x6 x7 x9 0.196492579  0.162660688
## 179     372  4                    x1 x2 x3 x7 0.194217005  0.160289300
## 180     373  4                    x1 x2 x3 x8 0.187795360  0.153597270
## 191     374  4                    x1 x2 x5 x8 0.183280413  0.148892220
## 196     375  4                    x1 x2 x6 x9 0.179801043  0.145266350
## 309     376  4                    x2 x6 x8 x9 0.177139265  0.142492498
## 250     377  4                    x1 x6 x7 x8 0.172885566  0.138059695
## 306     378  4                    x2 x6 x7 x8 0.164117140  0.128922073
## 256     379  4                    x1 x7 x8 x9 0.146962701  0.111045341
## 194     380  4                    x1 x2 x6 x7 0.137251887  0.100925651
## 312     381  4                    x2 x7 x8 x9 0.132069802  0.095525372
## 199     382  4                    x1 x2 x7 x9 0.127971933  0.091254962
## 195     383  4                    x1 x2 x6 x8 0.118357742  0.081235962
## 201     384  4                    x1 x2 x8 x9 0.108174811  0.070624277
## 198     385  4                    x1 x2 x7 x8 0.094448667  0.056320190
## 585     386  5                x3 x4 x5 x6 x10 0.797981238  0.787235560
## 597     387  5                x3 x4 x6 x9 x10 0.774040361  0.762021231
## 594     388  5                x3 x4 x6 x7 x10 0.772900650  0.760820897
## 596     389  5                x3 x4 x6 x8 x10 0.763721574  0.751153572
## 450     390  5                x1 x3 x4 x6 x10 0.752447126  0.739279419
## 588     391  5                x3 x4 x5 x7 x10 0.750226131  0.736940287
## 591     392  5                x3 x4 x5 x9 x10 0.748442678  0.735061969
## 446     393  5                x1 x3 x4 x5 x10 0.747877348  0.734466569
## 582     394  5                 x3 x4 x5 x6 x7 0.741674160  0.727933424
## 584     395  5                 x3 x4 x5 x6 x9 0.739700125  0.725854387
## 520     396  5                x2 x3 x4 x6 x10 0.731014819  0.716707097
## 590     397  5                x3 x4 x5 x8 x10 0.730519862  0.716185812
## 449     398  5                 x1 x3 x4 x6 x9 0.729218754  0.714815496
## 593     399  5                 x3 x4 x6 x7 x9 0.727491362  0.712996221
## 442     400  5                 x1 x3 x4 x5 x6 0.722788363  0.708043063
## 516     401  5                x2 x3 x4 x5 x10 0.721770745  0.706971316
## 592     402  5                 x3 x4 x6 x7 x8 0.715539331  0.700408444
## 447     403  5                 x1 x3 x4 x6 x7 0.715192649  0.700043322
## 583     404  5                 x3 x4 x5 x6 x8 0.696555640  0.680414983
## 619     405  5                x4 x5 x6 x7 x10 0.695945550  0.679772441
## 595     406  5                 x3 x4 x6 x8 x9 0.695625454  0.679435318
## 512     407  5                 x2 x3 x4 x5 x6 0.690036731  0.673549323
## 599     408  5                x3 x4 x7 x8 x10 0.687971478  0.671374216
## 517     409  5                 x2 x3 x4 x6 x7 0.686561244  0.669888969
## 480     410  5                x1 x4 x5 x6 x10 0.680619981  0.663631682
## 600     411  5                x3 x4 x7 x9 x10 0.679719704  0.662683518
## 445     412  5                 x1 x3 x4 x5 x9 0.679617491  0.662575868
## 519     413  5                 x2 x3 x4 x6 x9 0.677775254  0.660635640
## 456     414  5                x1 x3 x4 x9 x10 0.677708059  0.660564870
## 453     415  5                x1 x3 x4 x7 x10 0.677292891  0.660127620
## 443     416  5                 x1 x3 x4 x5 x7 0.674494313  0.657180181
## 448     417  5                 x1 x3 x4 x6 x8 0.673711476  0.656355704
## 601     418  5                x3 x4 x8 x9 x10 0.672866818  0.655466117
## 628     419  5                x4 x6 x7 x8 x10 0.672261498  0.654828599
## 622     420  5                x4 x5 x6 x9 x10 0.670938996  0.653435751
## 621     421  5                x4 x5 x6 x8 x10 0.668301835  0.650658316
## 587     422  5                 x3 x4 x5 x7 x9 0.667185131  0.649482212
## 455     423  5                x1 x3 x4 x8 x10 0.666604122  0.648870299
## 607     424  5                x3 x5 x6 x9 x10 0.665108572  0.647295198
## 489     425  5                x1 x4 x6 x7 x10 0.664995700  0.647176322
## 523     426  5                x2 x3 x4 x7 x10 0.660998078  0.642966061
## 483     427  5                x1 x4 x5 x7 x10 0.660713560  0.642666409
## 518     428  5                 x2 x3 x4 x6 x8 0.658961600  0.640821260
## 387     429  5                 x1 x2 x3 x4 x6 0.658798318  0.640649292
## 525     430  5                x2 x3 x4 x8 x10 0.656795179  0.638539603
## 629     431  5                x4 x6 x7 x9 x10 0.656282704  0.637999869
## 550     432  5                x2 x4 x5 x6 x10 0.653403429  0.634967441
## 526     433  5                x2 x3 x4 x9 x10 0.652052892  0.633545067
## 586     434  5                 x3 x4 x5 x7 x8 0.648312293  0.629605500
## 624     435  5                x4 x5 x7 x8 x10 0.646616980  0.627820011
## 391     436  5                x1 x2 x3 x4 x10 0.645554909  0.626701447
## 513     437  5                 x2 x3 x4 x5 x7 0.644015042  0.625079672
## 625     438  5                x4 x5 x7 x9 x10 0.643877143  0.624934438
## 486     439  5                x1 x4 x5 x9 x10 0.642507334  0.623491767
## 492     440  5                x1 x4 x6 x9 x10 0.642348895  0.623324900
## 559     441  5                x2 x4 x6 x7 x10 0.642099816  0.623062572
## 491     442  5                x1 x4 x6 x8 x10 0.641367816  0.622291636
## 630     443  5                x4 x6 x8 x9 x10 0.638013273  0.618758659
## 553     444  5                x2 x4 x5 x7 x10 0.635954172  0.616590033
## 386     445  5                 x1 x2 x3 x4 x5 0.635013805  0.615599646
## 515     446  5                 x2 x3 x4 x5 x9 0.631872270  0.612291008
## 485     447  5                x1 x4 x5 x8 x10 0.631448160  0.611844338
## 477     448  5                 x1 x4 x5 x6 x7 0.631036513  0.611410796
## 589     449  5                 x3 x4 x5 x8 x9 0.629784732  0.610092431
## 444     450  5                 x1 x3 x4 x5 x8 0.626576923  0.606713993
## 561     451  5                x2 x4 x6 x8 x10 0.625962831  0.606067237
## 411     452  5                x1 x2 x4 x5 x10 0.625020797  0.605075095
## 466     453  5                x1 x3 x5 x9 x10 0.624395201  0.604416222
## 626     454  5                x4 x5 x8 x9 x10 0.620487613  0.600300784
## 415     455  5                x1 x2 x4 x6 x10 0.615734276  0.595294610
## 562     456  5                x2 x4 x6 x9 x10 0.612968864  0.592382101
## 555     457  5                x2 x4 x5 x8 x10 0.612814377  0.592219398
## 556     458  5                x2 x4 x5 x9 x10 0.612513708  0.591902735
## 618     459  5                 x4 x5 x6 x7 x9 0.612188040  0.591559744
## 452     460  5                 x1 x3 x4 x7 x9 0.609921281  0.589172413
## 617     461  5                 x4 x5 x6 x7 x8 0.609342102  0.588562427
## 514     462  5                 x2 x3 x4 x5 x8 0.607564776  0.586690562
## 488     463  5                 x1 x4 x6 x7 x9 0.601948410  0.580775453
## 479     464  5                 x1 x4 x5 x6 x9 0.601745162  0.580561394
## 494     465  5                x1 x4 x7 x8 x10 0.599773543  0.578484902
## 487     466  5                 x1 x4 x6 x7 x8 0.597389841  0.575974407
## 547     467  5                 x2 x4 x5 x6 x7 0.595663679  0.574156428
## 598     468  5                 x3 x4 x7 x8 x9 0.595505676  0.573990021
## 451     469  5                 x1 x3 x4 x7 x8 0.595245369  0.573715868
## 460     470  5                x1 x3 x5 x6 x10 0.593407449  0.571780186
## 610     471  5                x3 x5 x7 x9 x10 0.593156561  0.571515953
## 604     472  5                x3 x5 x6 x7 x10 0.592174800  0.570481970
## 631     473  5                x4 x7 x8 x9 x10 0.591294626  0.569554979
## 459     474  5                 x1 x3 x5 x6 x9 0.590941919  0.569183510
## 495     475  5                x1 x4 x7 x9 x10 0.589254557  0.567406395
## 564     476  5                x2 x4 x7 x8 x10 0.587495366  0.565553631
## 627     477  5                 x4 x6 x7 x8 x9 0.586051937  0.564033423
## 521     478  5                 x2 x3 x4 x7 x8 0.583280881  0.561114970
## 611     479  5                x3 x5 x8 x9 x10 0.582782191  0.560589755
## 606     480  5                x3 x5 x6 x8 x10 0.582611074  0.560409536
## 388     481  5                 x1 x2 x3 x4 x7 0.580927287  0.558636185
## 522     482  5                 x2 x3 x4 x7 x9 0.579141319  0.556755219
## 472     483  5                x1 x3 x6 x9 x10 0.578144562  0.555705443
## 418     484  5                x1 x2 x4 x7 x10 0.577981090  0.555533275
## 536     485  5                x2 x3 x5 x9 x10 0.577199015  0.554709601
## 454     486  5                 x1 x3 x4 x8 x9 0.576558062  0.554034555
## 412     487  5                 x1 x2 x4 x6 x7 0.575476290  0.552895241
## 557     488  5                 x2 x4 x6 x7 x8 0.575408564  0.552823913
## 390     489  5                 x1 x2 x3 x4 x9 0.572910527  0.550193002
## 478     490  5                 x1 x4 x5 x6 x8 0.571686715  0.548904094
## 496     491  5                x1 x4 x8 x9 x10 0.571105242  0.548291691
## 482     492  5                 x1 x4 x5 x7 x9 0.570911868  0.548088031
## 565     493  5                x2 x4 x7 x9 x10 0.569233593  0.546320486
## 407     494  5                 x1 x2 x4 x5 x6 0.568343442  0.545382987
## 530     495  5                x2 x3 x5 x6 x10 0.565544060  0.542434702
## 615     496  5                x3 x6 x8 x9 x10 0.564570384  0.541409234
## 620     497  5                 x4 x5 x6 x8 x9 0.563498583  0.540280423
## 558     498  5                 x2 x4 x6 x7 x9 0.562747431  0.539489316
## 420     499  5                x1 x2 x4 x8 x10 0.562095970  0.538803202
## 614     500  5                x3 x6 x7 x9 x10 0.559188974  0.535741579
## 481     501  5                 x1 x4 x5 x7 x8 0.557582841  0.534050013
## 566     502  5                x2 x4 x8 x9 x10 0.556980074  0.533415184
## 524     503  5                 x2 x3 x4 x8 x9 0.556196384  0.532589808
## 549     504  5                 x2 x4 x5 x6 x9 0.555802020  0.532174468
## 408     505  5                 x1 x2 x4 x5 x7 0.555072188  0.531405815
## 421     506  5                x1 x2 x4 x9 x10 0.553906475  0.530178096
## 490     507  5                 x1 x4 x6 x8 x9 0.552750220  0.528960338
## 389     508  5                 x1 x2 x3 x4 x8 0.551862710  0.528025621
## 463     509  5                x1 x3 x5 x7 x10 0.550081903  0.526150089
## 548     510  5                 x2 x4 x5 x6 x8 0.548136702  0.524101420
## 603     511  5                 x3 x5 x6 x7 x9 0.547896419  0.523848356
## 414     512  5                 x1 x2 x4 x6 x9 0.539940978  0.515469753
## 623     513  5                 x4 x5 x7 x8 x9 0.539687994  0.515203313
## 609     514  5                x3 x5 x7 x8 x10 0.538564735  0.514020307
## 465     515  5                x1 x3 x5 x8 x10 0.536460188  0.511803815
## 542     516  5                x2 x3 x6 x9 x10 0.534186392  0.509409072
## 551     517  5                 x2 x4 x5 x7 x8 0.534041142  0.509256096
## 552     518  5                 x2 x4 x5 x7 x9 0.533745924  0.508945175
## 413     519  5                 x1 x2 x4 x6 x8 0.530421055  0.505443451
## 396     520  5                x1 x2 x3 x5 x10 0.529225011  0.504183788
## 533     521  5                x2 x3 x5 x7 x10 0.526402838  0.501211499
## 560     522  5                 x2 x4 x6 x8 x9 0.525021815  0.499757018
## 410     523  5                 x1 x2 x4 x5 x9 0.524245851  0.498939779
## 484     524  5                 x1 x4 x5 x8 x9 0.521663159  0.496219710
## 535     525  5                x2 x3 x5 x8 x10 0.518283258  0.492660027
## 605     526  5                 x3 x5 x6 x8 x9 0.517210661  0.491530377
## 529     527  5                 x2 x3 x5 x6 x9 0.513453416  0.487573279
## 409     528  5                 x1 x2 x4 x5 x8 0.506559105  0.480312249
## 613     529  5                x3 x6 x7 x8 x10 0.504308488  0.477941918
## 493     530  5                 x1 x4 x7 x8 x9 0.500843766  0.474292902
## 462     531  5                 x1 x3 x5 x7 x9 0.494057915  0.467146102
## 416     532  5                 x1 x2 x4 x7 x8 0.493911043  0.466991418
## 554     533  5                 x2 x4 x5 x8 x9 0.492707766  0.465724136
## 417     534  5                 x1 x2 x4 x7 x9 0.487831829  0.460588841
## 471     535  5                x1 x3 x6 x8 x10 0.484113925  0.456673176
## 563     536  5                 x2 x4 x7 x8 x9 0.482067716  0.454518126
## 502     537  5                x1 x5 x6 x9 x10 0.481094930  0.453493597
## 469     538  5                x1 x3 x6 x7 x10 0.474473912  0.446520397
## 541     539  5                x2 x3 x6 x8 x10 0.474170690  0.446201046
## 468     540  5                 x1 x3 x6 x7 x9 0.473611169  0.445611763
## 395     541  5                 x1 x2 x3 x5 x9 0.469056415  0.440814735
## 634     542  5                x5 x6 x7 x9 x10 0.468875311  0.440623998
## 457     543  5                 x1 x3 x5 x6 x7 0.464347780  0.435855640
## 476     544  5                x1 x3 x8 x9 x10 0.462290127  0.433688537
## 464     545  5                 x1 x3 x5 x8 x9 0.461660842  0.433025780
## 616     546  5                x3 x7 x8 x9 x10 0.460791722  0.432110431
## 475     547  5                x1 x3 x7 x9 x10 0.460234410  0.431523474
## 539     548  5                x2 x3 x6 x7 x10 0.458842938  0.430057988
## 635     549  5                x5 x6 x8 x9 x10 0.453469447  0.424398673
## 419     550  5                 x1 x2 x4 x8 x9 0.450763736  0.421549041
## 470     551  5                 x1 x3 x6 x8 x9 0.450065585  0.420813754
## 505     552  5                x1 x5 x7 x9 x10 0.448553389  0.419221122
## 612     553  5                 x3 x6 x7 x8 x9 0.446048504  0.416582999
## 602     554  5                 x3 x5 x6 x7 x8 0.445447174  0.415949684
## 572     555  5                x2 x5 x6 x9 x10 0.444930880  0.415405927
## 400     556  5                x1 x2 x3 x6 x10 0.443360542  0.413752060
## 546     557  5                x2 x3 x8 x9 x10 0.442803491  0.413165379
## 406     558  5                x1 x2 x3 x9 x10 0.442094464  0.412418638
## 499     559  5                x1 x5 x6 x7 x10 0.441653470  0.411954186
## 608     560  5                 x3 x5 x7 x8 x9 0.439652967  0.409847274
## 532     561  5                 x2 x3 x5 x7 x9 0.437437438  0.407513897
## 633     562  5                x5 x6 x7 x8 x10 0.436022276  0.406023461
## 399     563  5                 x1 x2 x3 x6 x9 0.435710127  0.405694708
## 545     564  5                x2 x3 x7 x9 x10 0.434839991  0.404778289
## 527     565  5                 x2 x3 x5 x6 x7 0.431406386  0.401162045
## 506     566  5                x1 x5 x8 x9 x10 0.431307645  0.401058052
## 431     567  5                x1 x2 x5 x9 x10 0.427913980  0.397483872
## 636     568  5                x5 x7 x8 x9 x10 0.425938961  0.395403799
## 458     569  5                 x1 x3 x5 x6 x8 0.424640534  0.394036307
## 501     570  5                x1 x5 x6 x8 x10 0.423089673  0.392402953
## 392     571  5                 x1 x2 x3 x5 x6 0.422985648  0.392293395
## 569     572  5                x2 x5 x6 x7 x10 0.421289565  0.390507095
## 575     573  5                x2 x5 x7 x9 x10 0.420254791  0.389417280
## 538     574  5                 x2 x3 x6 x7 x9 0.418092565  0.387140042
## 534     575  5                 x2 x3 x5 x8 x9 0.412724896  0.381486858
## 474     576  5                x1 x3 x7 x8 x10 0.411358364  0.380047638
## 425     577  5                x1 x2 x5 x6 x10 0.411025893  0.379697483
## 504     578  5                x1 x5 x7 x8 x10 0.408523010  0.377061468
## 571     579  5                x2 x5 x6 x8 x10 0.407735239  0.376231794
## 540     580  5                 x2 x3 x6 x8 x9 0.407029398  0.375488408
## 576     581  5                x2 x5 x8 x9 x10 0.406956680  0.375411822
## 544     582  5                x2 x3 x7 x8 x10 0.404897449  0.373243058
## 528     583  5                 x2 x3 x5 x6 x8 0.403349143  0.371612395
## 428     584  5                x1 x2 x5 x7 x10 0.399852303  0.367929553
## 574     585  5                x2 x5 x7 x8 x10 0.394966574  0.362783945
## 405     586  5                x1 x2 x3 x8 x10 0.388665971  0.356148203
## 637     587  5                x6 x7 x8 x9 x10 0.386057063  0.353400524
## 461     588  5                 x1 x3 x5 x7 x8 0.384194864  0.351439272
## 430     589  5                x1 x2 x5 x8 x10 0.384095021  0.351334118
## 393     590  5                 x1 x2 x3 x5 x7 0.382985238  0.350165303
## 509     591  5                x1 x6 x7 x9 x10 0.382880917  0.350055434
## 510     592  5                x1 x6 x8 x9 x10 0.376751447  0.343599929
## 403     593  5                x1 x2 x3 x7 x10 0.376694829  0.343540298
## 531     594  5                 x2 x3 x5 x7 x8 0.363958316  0.330126312
## 498     595  5                 x1 x5 x6 x7 x9 0.359939526  0.325893756
## 579     596  5                x2 x6 x7 x9 x10 0.359083026  0.324991698
## 580     597  5                x2 x6 x8 x9 x10 0.358935835  0.324836678
## 437     598  5                x1 x2 x6 x9 x10 0.353373025  0.318977973
## 467     599  5                 x1 x3 x6 x7 x8 0.351653240  0.317166710
## 394     600  5                 x1 x2 x3 x5 x8 0.351420118  0.316921188
## 508     601  5                x1 x6 x7 x8 x10 0.350829471  0.316299124
## 578     602  5                x2 x6 x7 x8 x10 0.343155010  0.308216446
## 537     603  5                 x2 x3 x6 x7 x8 0.339989738  0.304882809
## 511     604  5                x1 x7 x8 x9 x10 0.321725430  0.285646996
## 473     605  5                 x1 x3 x7 x8 x9 0.320206061  0.284046809
## 632     606  5                 x5 x6 x7 x8 x9 0.319811720  0.283631493
## 500     607  5                 x1 x5 x6 x8 x9 0.317875317  0.281592089
## 436     608  5                x1 x2 x6 x8 x10 0.317698578  0.281405950
## 424     609  5                 x1 x2 x5 x6 x9 0.315889359  0.279500495
## 397     610  5                 x1 x2 x3 x6 x7 0.315041113  0.278607129
## 434     611  5                x1 x2 x6 x7 x10 0.314749936  0.278300465
## 581     612  5                x2 x7 x8 x9 x10 0.311197431  0.274558997
## 568     613  5                 x2 x5 x6 x7 x9 0.311165356  0.274525215
## 402     614  5                 x1 x2 x3 x7 x9 0.305352831  0.268403514
## 398     615  5                 x1 x2 x3 x6 x8 0.304804366  0.267825874
## 440     616  5                x1 x2 x7 x9 x10 0.300859465  0.263671139
## 441     617  5                x1 x2 x8 x9 x10 0.299629443  0.262375690
## 543     618  5                 x2 x3 x7 x8 x9 0.295926247  0.258475516
## 404     619  5                 x1 x2 x3 x8 x9 0.291429244  0.253739310
## 503     620  5                 x1 x5 x7 x8 x9 0.288188299  0.250325974
## 427     621  5                 x1 x2 x5 x7 x9 0.286616191  0.248670244
## 439     622  5                x1 x2 x7 x8 x10 0.284723087  0.246676443
## 497     623  5                 x1 x5 x6 x7 x8 0.282298141  0.244122510
## 570     624  5                 x2 x5 x6 x8 x9 0.277026062  0.238570001
## 422     625  5                 x1 x2 x5 x6 x7 0.272023727  0.233301585
## 567     626  5                 x2 x5 x6 x7 x8 0.261431320  0.222145751
## 573     627  5                 x2 x5 x7 x8 x9 0.254859800  0.215224683
## 429     628  5                 x1 x2 x5 x8 x9 0.251945397  0.212155258
## 507     629  5                 x1 x6 x7 x8 x9 0.251238775  0.211411050
## 401     630  5                 x1 x2 x3 x7 x8 0.232432382  0.191604317
## 423     631  5                 x1 x2 x5 x6 x8 0.231726024  0.190860387
## 433     632  5                 x1 x2 x6 x7 x9 0.229519823  0.188536835
## 426     633  5                 x1 x2 x5 x7 x8 0.228076770  0.187017024
## 577     634  5                 x2 x6 x7 x8 x9 0.224467385  0.183215651
## 435     635  5                 x1 x2 x6 x8 x9 0.202977628  0.160582821
## 432     636  5                 x1 x2 x6 x7 x8 0.175575579  0.131723217
## 438     637  5                 x1 x2 x7 x8 x9 0.148432666  0.103136531
## 825     638  6             x3 x4 x5 x6 x9 x10 0.850054497  0.840380593
## 822     639  6             x3 x4 x5 x6 x7 x10 0.842239784  0.832061705
## 711     640  6             x1 x3 x4 x5 x6 x10 0.840488029  0.830196934
## 723     641  6             x1 x3 x4 x6 x9 x10 0.828405784  0.817335189
## 710     642  6              x1 x3 x4 x5 x6 x9 0.825409556  0.814145657
## 824     643  6             x3 x4 x5 x6 x8 x10 0.818970531  0.807291211
## 832     644  6             x3 x4 x6 x7 x9 x10 0.816576813  0.804743059
## 831     645  6             x3 x4 x6 x7 x8 x10 0.812325281  0.800217234
## 720     646  6             x1 x3 x4 x6 x7 x10 0.807490073  0.795070077
## 833     647  6             x3 x4 x6 x8 x9 x10 0.804605119  0.791998998
## 767     648  6             x2 x3 x4 x5 x6 x10 0.804425255  0.791807529
## 717     649  6             x1 x3 x4 x5 x9 x10 0.797829040  0.784785752
## 719     650  6              x1 x3 x4 x6 x7 x9 0.797481101  0.784415366
## 708     651  6              x1 x3 x4 x5 x6 x7 0.797260985  0.784181049
## 821     652  6              x3 x4 x5 x6 x7 x9 0.796831443  0.783723794
## 722     653  6             x1 x3 x4 x6 x8 x10 0.789719995  0.776153543
## 714     654  6             x1 x3 x4 x5 x7 x10 0.784695733  0.770805136
## 779     655  6             x2 x3 x4 x6 x9 x10 0.779436991  0.765207120
## 776     656  6             x2 x3 x4 x6 x7 x10 0.778178734  0.763867685
## 828     657  6             x3 x4 x5 x7 x9 x10 0.776443066  0.762020037
## 778     658  6             x2 x3 x4 x6 x8 x10 0.770247196  0.755424434
## 820     659  6              x3 x4 x5 x6 x7 x8 0.764297430  0.749090812
## 646     660  6             x1 x2 x3 x4 x6 x10 0.764145837  0.748929439
## 827     661  6             x3 x4 x5 x7 x8 x10 0.763520093  0.748263325
## 830     662  6              x3 x4 x6 x7 x8 x9 0.758606375  0.743032593
## 716     663  6             x1 x3 x4 x5 x8 x10 0.758308947  0.742715976
## 829     664  6             x3 x4 x5 x8 x9 x10 0.757087835  0.741416082
## 642     665  6             x1 x2 x3 x4 x5 x10 0.755865894  0.740115306
## 823     666  6              x3 x4 x5 x6 x8 x9 0.753946869  0.738072474
## 721     667  6              x1 x3 x4 x6 x8 x9 0.753735071  0.737847012
## 770     668  6             x2 x3 x4 x5 x7 x10 0.753522081  0.737620280
## 764     669  6              x2 x3 x4 x5 x6 x7 0.752731698  0.736778905
## 718     670  6              x1 x3 x4 x6 x7 x8 0.752592994  0.736631251
## 773     671  6             x2 x3 x4 x5 x9 x10 0.751803972  0.735791325
## 766     672  6              x2 x3 x4 x5 x6 x9 0.751438088  0.735401835
## 645     673  6              x1 x2 x3 x4 x6 x9 0.748566766  0.732345267
## 638     674  6              x1 x2 x3 x4 x5 x6 0.745649990  0.729240312
## 709     675  6              x1 x3 x4 x5 x6 x8 0.740855227  0.724136209
## 775     676  6              x2 x3 x4 x6 x7 x9 0.736626611  0.719634780
## 772     677  6             x2 x3 x4 x5 x8 x10 0.734962337  0.717863133
## 733     678  6             x1 x3 x5 x6 x9 x10 0.734912974  0.717810585
## 643     679  6              x1 x2 x3 x4 x6 x7 0.732681205  0.715434832
## 745     680  6             x1 x4 x5 x6 x7 x10 0.729163875  0.711690576
## 713     681  6              x1 x3 x4 x5 x7 x9 0.728125384  0.710585086
## 774     682  6              x2 x3 x4 x6 x7 x8 0.726290893  0.708632241
## 748     683  6             x1 x4 x5 x6 x9 x10 0.712889665  0.694366417
## 842     684  6             x4 x5 x6 x7 x8 x10 0.712252648  0.693688303
## 726     685  6             x1 x3 x4 x7 x9 x10 0.712161699  0.693591486
## 765     686  6              x2 x3 x4 x5 x6 x8 0.711735405  0.693137689
## 843     687  6             x4 x5 x6 x7 x9 x10 0.710960696  0.692312999
## 777     688  6              x2 x3 x4 x6 x8 x9 0.707995569  0.689156574
## 725     689  6             x1 x3 x4 x7 x8 x10 0.705687592  0.686699695
## 834     690  6             x3 x4 x7 x8 x9 x10 0.704704623  0.685653308
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## 801     692  6             x2 x4 x5 x6 x7 x10 0.696550363  0.676972968
## 644     693  6              x1 x2 x3 x4 x6 x8 0.695430102  0.675780431
## 641     694  6              x1 x2 x3 x4 x5 x9 0.695057985  0.675384306
## 747     695  6             x1 x4 x5 x6 x8 x10 0.693734041  0.673974947
## 754     696  6             x1 x4 x6 x7 x8 x10 0.693319446  0.673533604
## 781     697  6             x2 x3 x4 x7 x8 x10 0.691323716  0.671409117
## 755     698  6             x1 x4 x6 x7 x9 x10 0.690755484  0.670804225
## 639     699  6              x1 x2 x3 x4 x5 x7 0.688623052  0.668534217
## 837     700  6             x3 x5 x6 x7 x9 x10 0.688189204  0.668072378
## 712     701  6              x1 x3 x4 x5 x7 x8 0.684862619  0.664531175
## 652     702  6             x1 x2 x3 x4 x9 x10 0.684713134  0.664372046
## 715     703  6              x1 x3 x4 x5 x8 x9 0.683766095  0.663363908
## 676     704  6             x1 x2 x4 x5 x6 x10 0.683755160  0.663352267
## 649     705  6             x1 x2 x3 x4 x7 x10 0.683690590  0.663283532
## 844     706  6             x4 x5 x6 x8 x9 x10 0.683175143  0.662734829
## 782     707  6             x2 x3 x4 x7 x9 x10 0.682762976  0.662296072
## 846     708  6             x4 x6 x7 x8 x9 x10 0.682446012  0.661958658
## 783     709  6             x2 x3 x4 x8 x9 x10 0.676751327  0.655896574
## 751     710  6             x1 x4 x5 x7 x9 x10 0.676681648  0.655822399
## 826     711  6              x3 x4 x5 x7 x8 x9 0.676121590  0.655226209
## 838     712  6             x3 x5 x6 x8 x9 x10 0.675330711  0.654384305
## 769     713  6              x2 x3 x4 x5 x7 x9 0.674690632  0.653702931
## 651     714  6             x1 x2 x3 x4 x8 x10 0.673953946  0.652918717
## 810     715  6             x2 x4 x6 x7 x8 x10 0.672887777  0.651783763
## 804     716  6             x2 x4 x5 x6 x9 x10 0.671682876  0.650501126
## 803     717  6             x2 x4 x5 x6 x8 x10 0.669508346  0.648186304
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## 685     719  6             x1 x2 x4 x6 x7 x10 0.667293793  0.645828876
## 744     720  6              x1 x4 x5 x6 x7 x9 0.667209038  0.645738654
## 789     721  6             x2 x3 x5 x6 x9 x10 0.666211173  0.644676410
## 756     722  6             x1 x4 x6 x8 x9 x10 0.664153389  0.642485866
## 679     723  6             x1 x2 x4 x5 x7 x10 0.662065357  0.640263122
## 768     724  6              x2 x3 x4 x5 x7 x8 0.657431351  0.635330148
## 811     725  6             x2 x4 x6 x7 x9 x10 0.656794460  0.634652168
## 845     726  6             x4 x5 x7 x8 x9 x10 0.652589869  0.630176312
## 752     727  6             x1 x4 x5 x8 x9 x10 0.647665502  0.624934245
## 806     728  6             x2 x4 x5 x7 x8 x10 0.646950013  0.624172594
## 688     729  6             x1 x2 x4 x6 x9 x10 0.644967140  0.622061794
## 640     730  6              x1 x2 x3 x4 x5 x8 0.644904115  0.621994703
## 743     731  6              x1 x4 x5 x6 x7 x8 0.644788293  0.621871408
## 687     732  6             x1 x2 x4 x6 x8 x10 0.644347118  0.621401771
## 807     733  6             x2 x4 x5 x7 x9 x10 0.644089035  0.621127037
## 682     734  6             x1 x2 x4 x5 x9 x10 0.644084164  0.621121852
## 736     735  6             x1 x3 x5 x7 x9 x10 0.642798670  0.619753423
## 771     736  6              x2 x3 x4 x5 x8 x9 0.640241071  0.617030817
## 812     737  6             x2 x4 x6 x8 x9 x10 0.638983366  0.615691971
## 673     738  6              x1 x2 x4 x5 x6 x7 0.637725351  0.614352793
## 729     739  6              x1 x3 x5 x6 x7 x9 0.635844727  0.612350839
## 681     740  6             x1 x2 x4 x5 x8 x10 0.633529894  0.609886661
## 724     741  6              x1 x3 x4 x7 x8 x9 0.630086374  0.606220978
## 737     742  6             x1 x3 x5 x8 x9 x10 0.627174754  0.603121513
## 662     743  6             x1 x2 x3 x5 x9 x10 0.626632408  0.602544176
## 730     744  6             x1 x3 x5 x6 x7 x10 0.626025068  0.601897653
## 841     745  6              x4 x5 x6 x7 x8 x9 0.625449995  0.601285478
## 753     746  6              x1 x4 x6 x7 x8 x9 0.623327619  0.599026175
## 648     747  6              x1 x2 x3 x4 x7 x9 0.622385988  0.598023793
## 808     748  6             x2 x4 x5 x8 x9 x10 0.621054043  0.596605917
## 800     749  6              x2 x4 x5 x6 x7 x9 0.614864790  0.590017357
## 799     750  6              x2 x4 x5 x6 x7 x8 0.612885380  0.587910244
## 836     751  6             x3 x5 x6 x7 x8 x10 0.611683444  0.586630763
## 675     752  6              x1 x2 x4 x5 x6 x9 0.609453298  0.584256736
## 732     753  6             x1 x3 x5 x6 x8 x10 0.609442787  0.584245547
## 746     754  6              x1 x4 x5 x6 x8 x9 0.609424749  0.584226345
## 647     755  6              x1 x2 x3 x4 x7 x8 0.608451460  0.583190263
## 757     756  6             x1 x4 x7 x8 x9 x10 0.608270966  0.582998125
## 740     757  6             x1 x3 x6 x7 x9 x10 0.607897485  0.582600549
## 684     758  6              x1 x2 x4 x6 x7 x9 0.607435376  0.582108626
## 741     759  6             x1 x3 x6 x8 x9 x10 0.604229472  0.578695889
## 683     760  6              x1 x2 x4 x6 x7 x8 0.603653910  0.578083194
## 780     761  6              x2 x3 x4 x7 x8 x9 0.603607258  0.578033533
## 655     762  6              x1 x2 x3 x5 x6 x9 0.602223592  0.576560598
## 690     763  6             x1 x2 x4 x7 x8 x10 0.601056394  0.575318097
## 839     764  6             x3 x5 x7 x8 x9 x10 0.599505423  0.573667063
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## 786     767  6             x2 x3 x5 x6 x7 x10 0.593632306  0.567415036
## 792     768  6             x2 x3 x5 x7 x9 x10 0.593501296  0.567275573
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## 813     770  6             x2 x4 x7 x8 x9 x10 0.591673481  0.565329834
## 840     771  6             x3 x6 x7 x8 x9 x10 0.591004855  0.564618072
## 691     772  6             x1 x2 x4 x7 x9 x10 0.590550570  0.564134478
## 809     773  6              x2 x4 x6 x7 x8 x9 0.588955164  0.562436143
## 788     774  6             x2 x3 x5 x6 x8 x10 0.584756047  0.557966114
## 793     775  6             x2 x3 x5 x8 x9 x10 0.583450743  0.556576598
## 668     776  6             x1 x2 x3 x6 x9 x10 0.581740033  0.554755519
## 674     777  6              x1 x2 x4 x5 x6 x8 0.581581949  0.554587236
## 749     778  6              x1 x4 x5 x7 x8 x9 0.575845346  0.548480530
## 678     779  6              x1 x2 x4 x5 x7 x9 0.575587593  0.548206147
## 692     780  6             x1 x2 x4 x8 x9 x10 0.572911634  0.545357546
## 802     781  6              x2 x4 x5 x6 x8 x9 0.568277340  0.540424265
## 797     782  6             x2 x3 x6 x8 x9 x10 0.565759301  0.537743772
## 677     783  6              x1 x2 x4 x5 x7 x8 0.563253560  0.535076370
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## 735     785  6             x1 x3 x5 x7 x8 x10 0.560357862  0.531993853
## 796     786  6             x2 x3 x6 x7 x9 x10 0.559980571  0.531592221
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## 661     791  6             x1 x2 x3 x5 x8 x10 0.539459429  0.509747134
## 791     792  6             x2 x3 x5 x7 x8 x10 0.539383760  0.509666584
## 680     793  6              x1 x2 x4 x5 x8 x9 0.528821261  0.498422633
## 787     794  6              x2 x3 x5 x6 x8 x9 0.523547698  0.492808840
## 739     795  6             x1 x3 x6 x7 x8 x10 0.519295050  0.488281828
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## 689     797  6              x1 x2 x4 x7 x8 x9 0.505882953  0.474004434
## 795     798  6             x2 x3 x6 x7 x8 x10 0.505648516  0.473754872
## 658     799  6              x1 x2 x3 x5 x7 x9 0.500967366  0.468771712
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## 761     803  6             x1 x5 x6 x8 x9 x10 0.486141414  0.452989247
## 742     804  6             x1 x3 x7 x8 x9 x10 0.482461465  0.449071882
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## 816     812  6             x2 x5 x6 x7 x9 x10 0.469396892  0.435164434
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## 671     814  6             x1 x2 x3 x7 x9 x10 0.461742994  0.427016735
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## 759     817  6             x1 x5 x6 x7 x8 x10 0.453716530  0.418472436
## 817     818  6             x2 x5 x6 x8 x9 x10 0.453672976  0.418426071
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## 790     823  6              x2 x3 x5 x7 x8 x9 0.443449240  0.407542740
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## 654     825  6              x1 x2 x3 x5 x6 x8 0.439351141  0.403180246
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## 758     837  6              x1 x5 x6 x7 x8 x9 0.364769237  0.323786607
## 663     838  6              x1 x2 x3 x6 x7 x8 0.361592636  0.320405064
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## 669     841  6              x1 x2 x3 x7 x8 x9 0.326707171  0.283268924
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## 963     853  7          x3 x4 x5 x6 x8 x9 x10 0.863574752  0.853194570
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## 851     858  7          x1 x2 x3 x4 x5 x6 x10 0.850997737  0.839660608
## 965     859  7          x3 x4 x6 x7 x8 x9 x10 0.846861360  0.835209507
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## 904     871  7           x1 x3 x4 x5 x6 x7 x8 0.814577854  0.800469648
## 848     872  7           x1 x2 x3 x4 x5 x6 x7 0.813576406  0.799392003
## 960     873  7           x3 x4 x5 x6 x7 x8 x9 0.811404551  0.797054898
## 859     874  7           x1 x2 x3 x4 x6 x7 x9 0.811284652  0.796925875
## 945     875  7          x2 x3 x4 x6 x8 x9 x10 0.809198461  0.794680953
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## 857     877  7          x1 x2 x3 x4 x5 x9 x10 0.803854222  0.788930086
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## 862     879  7          x1 x2 x3 x4 x6 x8 x10 0.799562557  0.784311882
## 911     880  7          x1 x3 x4 x5 x7 x8 x10 0.794502581  0.778866908
## 854     881  7          x1 x2 x3 x4 x5 x7 x10 0.790467825  0.774525160
## 964     882  7          x3 x4 x5 x7 x8 x9 x10 0.785130576  0.768781815
## 940     883  7          x2 x3 x4 x5 x7 x9 x10 0.778609329  0.761764387
## 932     884  7           x2 x3 x4 x5 x6 x7 x8 0.774229804  0.757051637
## 861     885  7           x1 x2 x3 x4 x6 x8 x9 0.771366843  0.753970842
## 858     886  7           x1 x2 x3 x4 x6 x7 x8 0.767776848  0.750107695
## 942     887  7           x2 x3 x4 x6 x7 x8 x9 0.766662873  0.748908961
## 939     888  7          x2 x3 x4 x5 x7 x8 x10 0.766348620  0.748570797
## 856     889  7          x1 x2 x3 x4 x5 x8 x10 0.765474779  0.747630469
## 935     890  7           x2 x3 x4 x5 x6 x8 x9 0.765001906  0.747121617
## 849     891  7           x1 x2 x3 x4 x5 x6 x8 0.761941669  0.743828535
## 921     892  7          x1 x3 x5 x6 x7 x9 x10 0.761290780  0.743128122
## 941     893  7          x2 x3 x4 x5 x8 x9 x10 0.760161387  0.741912797
## 927     894  7          x1 x4 x5 x6 x7 x9 x10 0.756574903  0.738053428
## 926     895  7          x1 x4 x5 x6 x7 x8 x10 0.741872660  0.722232537
## 922     896  7          x1 x3 x5 x6 x8 x9 x10 0.739897908  0.720107532
## 853     897  7           x1 x2 x3 x4 x5 x7 x9 0.739356313  0.719524728
## 873     898  7          x1 x2 x3 x5 x6 x9 x10 0.738174872  0.718253395
## 918     899  7          x1 x3 x4 x7 x8 x9 x10 0.732426814  0.712067985
## 910     900  7           x1 x3 x4 x5 x7 x8 x9 0.732222968  0.711848629
## 885     901  7          x1 x2 x4 x5 x6 x7 x10 0.730865957  0.710388367
## 967     902  7          x4 x5 x6 x7 x8 x9 x10 0.723523348  0.702487082
## 928     903  7          x1 x4 x5 x6 x8 x9 x10 0.720484641  0.699217168
## 866     904  7          x1 x2 x3 x4 x7 x9 x10 0.717254756  0.695741531
## 888     905  7          x1 x2 x4 x5 x6 x9 x10 0.714746980  0.693042946
## 954     906  7          x2 x4 x5 x6 x7 x8 x10 0.712637695  0.690773172
## 930     907  7          x1 x4 x6 x7 x8 x9 x10 0.712059983  0.690151503
## 955     908  7          x2 x4 x5 x6 x7 x9 x10 0.711160094  0.689183144
## 865     909  7          x1 x2 x3 x4 x7 x8 x10 0.710874903  0.688876255
## 946     910  7          x2 x3 x4 x7 x8 x9 x10 0.707202308  0.684924223
## 867     911  7          x1 x2 x3 x4 x8 x9 x10 0.704553348  0.682073711
## 855     912  7           x1 x2 x3 x4 x5 x8 x9 0.698624718  0.675693990
## 966     913  7          x3 x5 x6 x7 x8 x9 x10 0.698527531  0.675589408
## 852     914  7           x1 x2 x3 x4 x5 x7 x8 0.697907874  0.674922603
## 887     915  7          x1 x2 x4 x5 x6 x8 x10 0.696295809  0.673187881
## 894     916  7          x1 x2 x4 x6 x7 x8 x10 0.694861128  0.671644040
## 895     917  7          x1 x2 x4 x6 x7 x9 x10 0.692135610  0.668711145
## 949     918  7          x2 x3 x5 x6 x7 x9 x10 0.688674788  0.664987001
## 956     919  7          x2 x4 x5 x6 x8 x9 x10 0.683764490  0.659703093
## 938     920  7           x2 x3 x4 x5 x7 x8 x9 0.683181709  0.659075970
## 958     921  7          x2 x4 x6 x7 x8 x9 x10 0.682733083  0.658593208
## 929     922  7          x1 x4 x5 x7 x8 x9 x10 0.681767211  0.657553846
## 891     923  7          x1 x2 x4 x5 x7 x9 x10 0.677464714  0.652923986
## 950     924  7          x2 x3 x5 x6 x8 x9 x10 0.676250562  0.651617452
## 925     925  7           x1 x4 x5 x6 x7 x8 x9 0.674981464  0.650251793
## 884     926  7           x1 x2 x4 x5 x6 x7 x9 0.671719023  0.646741122
## 890     927  7          x1 x2 x4 x5 x7 x8 x10 0.670131938  0.645033281
## 896     928  7          x1 x2 x4 x6 x8 x9 x10 0.666169610  0.640769471
## 957     929  7          x2 x4 x5 x7 x8 x9 x10 0.652733654  0.626311215
## 883     930  7           x1 x2 x4 x5 x6 x7 x8 0.650592533  0.624007183
## 892     931  7          x1 x2 x4 x5 x8 x9 x10 0.649039112  0.622335566
## 923     932  7          x1 x3 x5 x7 x8 x9 x10 0.645506388  0.618534048
## 876     933  7          x1 x2 x3 x5 x7 x9 x10 0.644206483  0.617135237
## 869     934  7           x1 x2 x3 x5 x6 x7 x9 0.643606866  0.616489997
## 920     935  7          x1 x3 x5 x6 x7 x8 x10 0.641687925  0.614425050
## 864     936  7           x1 x2 x3 x4 x7 x8 x9 0.641308248  0.614016484
## 919     937  7           x1 x3 x5 x6 x7 x8 x9 0.640627356  0.613283786
## 924     938  7          x1 x3 x6 x7 x8 x9 x10 0.633605041  0.605727164
## 877     939  7          x1 x2 x3 x5 x8 x9 x10 0.629215512  0.601003649
## 870     940  7          x1 x2 x3 x5 x6 x7 x10 0.629198251  0.600985075
## 893     941  7           x1 x2 x4 x6 x7 x8 x9 0.627949064  0.599640840
## 953     942  7           x2 x4 x5 x6 x7 x8 x9 0.627797617  0.599477871
## 886     943  7           x1 x2 x4 x5 x6 x8 x9 0.616592335  0.587420012
## 872     944  7          x1 x2 x3 x5 x6 x8 x10 0.613503448  0.584096101
## 948     945  7          x2 x3 x5 x6 x7 x8 x10 0.612791121  0.583329576
## 880     946  7          x1 x2 x3 x6 x7 x9 x10 0.610187409  0.580527755
## 897     947  7          x1 x2 x4 x7 x8 x9 x10 0.609181960  0.579445804
## 881     948  7          x1 x2 x3 x6 x8 x9 x10 0.607094818  0.577199858
## 871     949  7           x1 x2 x3 x5 x6 x8 x9 0.606470484  0.576528021
## 951     950  7          x2 x3 x5 x7 x8 x9 x10 0.599768241  0.569315825
## 952     951  7          x2 x3 x6 x7 x8 x9 x10 0.591508680  0.560427819
## 889     952  7           x1 x2 x4 x5 x7 x8 x9 0.580183602  0.548241050
## 947     953  7           x2 x3 x5 x6 x7 x8 x9 0.562405438  0.529110200
## 875     954  7          x1 x2 x3 x5 x7 x8 x10 0.562238157  0.528930191
## 879     955  7          x1 x2 x3 x6 x7 x8 x10 0.521716082  0.485324914
## 931     956  7          x1 x5 x6 x7 x8 x9 x10 0.512651665  0.475570814
## 900     957  7          x1 x2 x5 x6 x7 x9 x10 0.507702667  0.470245261
## 878     958  7           x1 x2 x3 x6 x7 x8 x9 0.507038585  0.469530651
## 874     959  7           x1 x2 x3 x5 x7 x8 x9 0.503043119  0.465231182
## 868     960  7           x1 x2 x3 x5 x6 x7 x8 0.491600752  0.452918201
## 901     961  7          x1 x2 x5 x6 x8 x9 x10 0.486141439  0.447043505
## 882     962  7          x1 x2 x3 x7 x8 x9 x10 0.483540492  0.444244659
## 959     963  7          x2 x5 x6 x7 x8 x9 x10 0.478318476  0.438625317
## 899     964  7          x1 x2 x5 x6 x7 x8 x10 0.453716582  0.412151539
## 902     965  7          x1 x2 x5 x7 x8 x9 x10 0.451988422  0.410291889
## 903     966  7          x1 x2 x6 x7 x8 x9 x10 0.406073923  0.360883895
## 898     967  7           x1 x2 x5 x6 x7 x8 x9 0.365814494  0.317561249
## 998     968  8       x1 x3 x4 x5 x6 x7 x9 x10 0.965030418  0.961956169
## 973     969  8       x1 x2 x3 x4 x5 x6 x9 x10 0.931287405  0.925246737
## 999     970  8       x1 x3 x4 x5 x6 x8 x9 x10 0.930550543  0.924445097
## 997     971  8       x1 x3 x4 x5 x6 x7 x8 x10 0.906272895  0.898033149
## 1012    972  8       x3 x4 x5 x6 x7 x8 x9 x10 0.901185703  0.892498732
## 969     973  8        x1 x2 x3 x4 x5 x6 x7 x9 0.899494918  0.890659306
## 1001    974  8       x1 x3 x4 x6 x7 x8 x9 x10 0.899355851  0.890508014
## 970     975  8       x1 x2 x3 x4 x5 x6 x7 x10 0.897859330  0.888879931
## 996     976  8        x1 x3 x4 x5 x6 x7 x8 x9 0.894353242  0.885065615
## 1006    977  8       x2 x3 x4 x5 x6 x7 x9 x10 0.890000589  0.880330312
## 980     978  8       x1 x2 x3 x4 x6 x7 x9 x10 0.882357074  0.872014839
## 1007    979  8       x2 x3 x4 x5 x6 x8 x9 x10 0.867352280  0.855690942
## 1005    980  8       x2 x3 x4 x5 x6 x7 x8 x10 0.866659531  0.854937292
## 972     981  8       x1 x2 x3 x4 x5 x6 x8 x10 0.866460250  0.854720492
## 981     982  8       x1 x2 x3 x4 x6 x8 x9 x10 0.860666506  0.848417408
## 979     983  8       x1 x2 x3 x4 x6 x7 x8 x10 0.850298987  0.837138458
## 1009    984  8       x2 x3 x4 x6 x7 x8 x9 x10 0.849747268  0.836538237
## 971     985  8        x1 x2 x3 x4 x5 x6 x8 x9 0.849469895  0.836236479
## 978     986  8        x1 x2 x3 x4 x6 x7 x8 x9 0.833726042  0.819108551
## 976     987  8       x1 x2 x3 x4 x5 x7 x9 x10 0.832428136  0.817696544
## 1000    988  8       x1 x3 x4 x5 x7 x8 x9 x10 0.832297706  0.817554647
## 968     989  8        x1 x2 x3 x4 x5 x6 x7 x8 0.829447192  0.814453539
## 1004    990  8        x2 x3 x4 x5 x6 x7 x8 x9 0.818611046  0.802664764
## 977     991  8       x1 x2 x3 x4 x5 x8 x9 x10 0.807772899  0.790873813
## 975     992  8       x1 x2 x3 x4 x5 x7 x8 x10 0.799611585  0.781995021
## 1008    993  8       x2 x3 x4 x5 x7 x8 x9 x10 0.787064887  0.768345317
## 1002    994  8       x1 x3 x5 x6 x7 x8 x9 x10 0.766256455  0.745707572
## 1003    995  8       x1 x4 x5 x6 x7 x8 x9 x10 0.764254033  0.743529112
## 985     996  8       x1 x2 x3 x5 x6 x7 x9 x10 0.763382788  0.742581274
## 991     997  8       x1 x2 x4 x5 x6 x7 x9 x10 0.757475891  0.736155090
## 990     998  8       x1 x2 x4 x5 x6 x7 x8 x10 0.743168251  0.720589636
## 974     999  8        x1 x2 x3 x4 x5 x7 x8 x9 0.742959052  0.720362045
## 986    1000  8       x1 x2 x3 x5 x6 x8 x9 x10 0.742853531  0.720247248
## 982    1001  8       x1 x2 x3 x4 x7 x8 x9 x10 0.736722111  0.713576802
## 1011   1002  8       x2 x4 x5 x6 x7 x8 x9 x10 0.723644738  0.699349770
## 992    1003  8       x1 x2 x4 x5 x6 x8 x9 x10 0.722079702  0.697647148
## 994    1004  8       x1 x2 x4 x6 x7 x8 x9 x10 0.713021156  0.687792247
## 1010   1005  8       x2 x3 x5 x6 x7 x8 x9 x10 0.698892434  0.672421439
## 993    1006  8       x1 x2 x4 x5 x7 x8 x9 x10 0.682410036  0.654490039
## 989    1007  8        x1 x2 x4 x5 x6 x7 x8 x9 0.679072875  0.650859502
## 983    1008  8        x1 x2 x3 x5 x6 x7 x8 x9 0.647934675  0.616983877
## 987    1009  8       x1 x2 x3 x5 x7 x8 x9 x10 0.646761639  0.615707717
## 984    1010  8       x1 x2 x3 x5 x6 x7 x8 x10 0.644283095  0.613011279
## 988    1011  8       x1 x2 x3 x6 x7 x8 x9 x10 0.635328776  0.603269767
## 995    1012  8       x1 x2 x5 x6 x7 x8 x9 x10 0.512786784  0.469954853
## 1021   1013  9    x1 x3 x4 x5 x6 x7 x8 x9 x10 0.972378125  0.969615937
## 1015   1014  9    x1 x2 x3 x4 x5 x6 x7 x9 x10 0.970881812  0.967969993
## 1016   1015  9    x1 x2 x3 x4 x5 x6 x8 x9 x10 0.938006896  0.931807585
## 1014   1016  9    x1 x2 x3 x4 x5 x6 x7 x8 x10 0.913055412  0.904360953
## 1013   1017  9     x1 x2 x3 x4 x5 x6 x7 x8 x9 0.906226185  0.896848803
## 1018   1018  9    x1 x2 x3 x4 x6 x7 x8 x9 x10 0.905090586  0.895599645
## 1022   1019  9    x2 x3 x4 x5 x6 x7 x8 x9 x10 0.903530255  0.893883280
## 1017   1020  9    x1 x2 x3 x4 x5 x7 x8 x9 x10 0.836353127  0.819988439
## 1019   1021  9    x1 x2 x3 x5 x6 x7 x8 x9 x10 0.768104154  0.744914569
## 1020   1022  9    x1 x2 x4 x5 x6 x7 x8 x9 x10 0.764972963  0.741470259
## 1023   1023 10 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 0.977757114  0.975257913
##           rmse       predrsq         cp      aic       sbic      sbc       msep
## 4    4.6854216  0.3563272191 2379.06542 598.6789 309.211140 606.4944 2240.12921
## 10   5.2548621  0.1898676142 3017.23548 621.6185 332.085655 629.4340 2817.72339
## 5    5.4899762  0.1135647217 3302.05382 630.3725 340.818481 638.1880 3075.50643
## 3    5.5796103  0.0863676362 3413.91878 633.6115 344.050090 641.4270 3176.75303
## 9    5.7585091  0.0285222602 3642.60379 639.9234 350.348273 647.7389 3383.73098
## 6    5.7713586  0.0200230436 3659.30700 640.3692 350.793116 648.1847 3398.84870
## 7    5.8066719  0.0125528922 3705.40289 641.5892 352.010581 649.4047 3440.56912
## 8    5.8443962  0.0009190039 3754.95667 642.8844 353.303050 650.6999 3485.41920
## 1    5.9396009 -0.0408208639 3881.44202 646.1161 356.528259 653.9316 3599.89840
## 2    5.9404806 -0.0304214677 3882.62029 646.1457 356.557819 653.9612 3600.96483
## 40   4.1193087  0.4908138068 1819.10146 574.9248 283.654317 585.3455 1749.54408
## 28   4.2960941  0.4477989315 1986.83144 583.3290 292.017211 593.7496 1902.93427
## 36   4.3626842  0.4281993472 2051.83774 586.4052 295.079176 596.8259 1962.38297
## 35   4.3749545  0.4242993042 2063.92529 586.9669 295.638336 597.3876 1973.43712
## 37   4.4430508  0.4092383001 2131.62450 590.0560 298.713528 600.4766 2035.34850
## 38   4.5739308  0.3754474140 2264.67724 595.8623 304.494967 606.2830 2157.02612
## 39   4.6194673  0.3592822121 2311.87540 597.8436 306.468075 608.2643 2200.18915
## 13   4.6242347  0.3584735696 2316.84373 598.0499 306.673528 608.4706 2204.73272
## 21   4.6491598  0.3573550589 2342.90315 599.1250 307.744290 609.5457 2228.56424
## 45   4.7634352  0.3204348339 2464.17269 603.9815 312.581667 614.4022 2339.46604
## 34   4.8409175  0.2998174031 2548.07228 607.2086 315.796480 617.6292 2416.19278
## 29   4.9858527  0.2560600895 2708.64579 613.1086 321.675147 623.5293 2563.03833
## 49   5.0810654  0.2243045998 2816.70979 616.8919 325.445393 627.3126 2661.86358
## 55   5.1053038  0.2189585074 2844.54616 617.8437 326.393982 628.2644 2687.32013
## 54   5.1392313  0.2110460809 2883.73235 619.1684 327.714274 629.5891 2723.15616
## 52   5.1666965  0.2016617860 2915.64466 620.2344 328.776757 630.6551 2752.34018
## 33   5.2376751  0.1800300562 2998.90396 622.9632 331.496772 633.3839 2828.48136
## 19   5.2402326  0.1720289408 3001.92518 623.0609 331.594095 633.4816 2831.24429
## 27   5.2541361  0.1761189965 3018.37532 623.5908 332.122352 634.0115 2846.28805
## 30   5.2582643  0.1686437673 3023.26809 623.7479 332.278934 634.1686 2850.76253
## 44   5.2915810  0.1591361195 3062.89564 625.0111 333.538173 635.4318 2887.00218
## 42   5.3410788  0.1434108920 3122.23150 626.8732 335.394521 637.2939 2941.26523
## 41   5.3593563  0.1381360991 3144.28143 627.5565 336.075673 637.9772 2961.43003
## 31   5.4228163  0.1194335059 3221.42425 629.9108 338.422859 640.3314 3031.97767
## 14   5.4330063  0.1111120948 3233.89598 630.2862 338.797209 640.7069 3043.38314
## 32   5.4372440  0.1161816246 3239.08953 630.4422 338.952686 640.8629 3048.13268
## 43   5.4489521  0.1104034548 3253.45938 630.8724 339.381614 641.2931 3061.27401
## 22   5.4770818  0.1017421807 3288.11044 631.9022 340.408414 642.3229 3092.96263
## 48   5.5029680  0.0908819204 3320.15551 632.8452 341.348698 643.2659 3122.26806
## 20   5.5392350  0.0849217229 3365.30541 634.1590 342.658684 644.5797 3163.55794
## 12   5.5468467  0.0756157440 3374.81909 634.4336 342.932540 644.8543 3172.25827
## 46   5.5871216  0.0651114933 3425.37525 635.8806 344.375373 646.3012 3218.49222
## 51   5.6309598  0.0526060168 3480.82004 637.4437 345.934151 647.8644 3269.19685
## 47   5.6366166  0.0482497153 3488.00612 637.6445 346.134414 648.0652 3275.76856
## 53   5.6807956  0.0386262501 3544.37660 639.2060 347.691594 649.6266 3327.31974
## 50   5.6957083  0.0325025774 3563.50402 639.7303 348.214509 650.1510 3344.81190
## 18   5.7029766  0.0241967508 3572.84463 639.9854 348.468877 650.4060 3353.35395
## 15   5.7397859  0.0046127904 3620.33197 641.2721 349.752160 651.6928 3396.78144
## 26   5.7500961  0.0175498955 3633.68780 641.6310 350.110136 652.0517 3408.99544
## 23   5.7539877  0.0088878170 3638.73523 641.7663 350.245090 652.1870 3413.61135
## 16   5.7834279 -0.0046552578 3677.02995 642.7870 351.263087 653.2077 3448.63213
## 24   5.7976290 -0.0005455932 3695.57212 643.2775 351.752299 653.6982 3465.58906
## 25   5.8311244 -0.0103303022 3739.48650 644.4297 352.901467 654.8504 3505.74906
## 17   5.8311546 -0.0195779756 3739.52629 644.4307 352.902502 654.8514 3505.78544
## 11   5.9193292 -0.0497268293 3856.33854 647.4323 355.896488 657.8530 3612.61103
## 125  3.6769203  0.5854435056 1432.25549 554.2028 261.193768 567.2287 1408.61491
## 145  3.7669347  0.5631357432 1507.79937 559.0400 265.994600 572.0659 1478.42751
## 121  3.7720043  0.5635291498 1512.10827 559.3090 266.261605 572.3349 1482.40951
## 149  3.8055900  0.5553671006 1540.80128 561.0819 268.021641 574.1078 1508.92568
## 120  3.8661901  0.5430997525 1593.21667 564.2416 271.158945 577.2675 1557.36447
## 152  3.9478673  0.5217108209 1665.17264 568.4228 275.311542 581.4487 1623.86136
## 154  4.0003214  0.5115179954 1712.17687 571.0627 277.933923 584.0885 1667.29952
## 122  4.0380290  0.5015489971 1746.34996 572.9390 279.798180 585.9649 1698.88000
## 146  4.0484817  0.4985987516 1755.87966 573.4561 280.311919 586.4819 1707.68672
## 76   4.0643761  0.4911489095 1770.41780 574.2398 281.090609 587.2656 1721.12189
## 155  4.0714799  0.4913969653 1776.93382 574.5890 281.437662 587.6149 1727.14357
## 104  4.1098434  0.4841832049 1812.31978 576.4647 283.301617 589.4905 1759.84490
## 141  4.1149212  0.4800948541 1817.02836 576.7116 283.547042 589.7375 1764.19625
## 142  4.1347787  0.4734976975 1835.49767 577.6745 284.503945 590.7003 1781.26435
## 124  4.1349834  0.4752895878 1835.68860 577.6844 284.513790 590.7102 1781.44080
## 123  4.1507213  0.4747493836 1850.39016 578.4441 285.268918 591.4700 1795.02699
## 130  4.1997666  0.4637328314 1896.56428 580.7935 287.604182 593.8193 1837.69802
## 64   4.2058623  0.4596296023 1902.34105 581.0836 287.892542 594.1094 1843.03653
## 147  4.2190906  0.4547797264 1914.90596 581.7116 288.516889 594.7375 1854.64818
## 92   4.2251319  0.4584982619 1920.65746 581.9978 288.801384 595.0237 1859.96333
## 148  4.2459058  0.4453020439 1940.49758 582.9787 289.776600 596.0046 1878.29824
## 72   4.2621684  0.4417608792 1956.09708 583.7433 290.536751 596.7692 1892.71426
## 71   4.2647464  0.4414394193 1958.57544 583.8643 290.656989 596.8901 1895.00459
## 144  4.3012282  0.4307863556 1993.80785 585.5678 292.350842 598.5937 1927.56403
## 143  4.3227487  0.4263113499 2014.73204 586.5660 293.343389 599.5919 1946.90076
## 100  4.3233323  0.4284937778 2015.30099 586.5930 293.370241 599.6189 1947.42655
## 150  4.3323224  0.4279596258 2024.07409 587.0085 293.783372 600.0343 1955.53406
## 99   4.3442262  0.4235605249 2035.71860 587.5572 294.329097 600.5831 1966.29514
## 73   4.3654957  0.4174703908 2056.60441 588.5341 295.300520 601.5599 1985.59640
## 151  4.3962813  0.4076999258 2087.01524 589.9395 296.698298 602.9654 2013.70006
## 101  4.4206872  0.4068747094 2111.27596 591.0467 297.799554 604.0726 2036.12019
## 74   4.5210446  0.3758872942 2212.44804 595.5363 302.265583 608.5622 2129.61666
## 134  4.5242534  0.3734063127 2215.72030 595.6782 302.406752 608.7041 2132.64066
## 75   4.5251952  0.3716761506 2216.68116 595.7198 302.448166 608.7457 2133.52862
## 153  4.5279513  0.3739089040 2219.49424 595.8416 302.569317 608.8675 2136.12829
## 102  4.5428296  0.3754139376 2234.70982 596.4977 303.222071 609.5236 2150.18951
## 140  4.5562745  0.3652887516 2248.50238 597.0888 303.810118 610.1146 2162.93567
## 57   4.5779287  0.3618923274 2270.80228 598.0370 304.753619 611.0629 2183.54374
## 103  4.5917031  0.3583279198 2285.04241 598.6379 305.351490 611.6638 2196.70351
## 165  4.6121997  0.3490645383 2306.31116 599.5287 306.237854 612.5545 2216.35867
## 129  4.6131736  0.3501085897 2307.32412 599.5709 306.279873 612.5968 2217.29478
## 159  4.6422232  0.3399677405 2337.63681 600.8264 307.529190 613.8522 2245.30774
## 162  4.6758835  0.3309155303 2372.99870 602.2713 308.967150 615.2972 2277.98684
## 139  4.6917084  0.3288420163 2389.71186 602.9471 309.639647 615.9729 2293.43202
## 81   4.7103449  0.3183355993 2409.46686 603.7399 310.428751 616.7658 2311.68827
## 164  4.7213057  0.3199289908 2421.12203 604.2048 310.891407 617.2306 2322.45919
## 126  4.7261594  0.3178418127 2426.29183 604.4103 311.095941 617.4361 2327.23678
## 133  4.7421457  0.3100520134 2443.35693 605.0856 311.768136 618.1115 2343.00720
## 137  4.7477755  0.3124906522 2449.38037 605.3229 312.004324 618.3488 2348.57366
## 109  4.7633583  0.3075096437 2466.09011 605.9783 312.656635 619.0041 2364.01569
## 70   4.8114283  0.2903796817 2517.98113 607.9865 314.655635 621.0124 2411.96990
## 127  4.8280107  0.2863853201 2536.00252 608.6746 315.340629 621.7005 2428.62405
## 98   4.8310712  0.2895132942 2539.33539 608.8014 315.466799 621.8272 2431.70407
## 171  4.8709158  0.2705365761 2582.91863 610.4441 317.102211 623.4700 2471.98079
## 65   4.8887148  0.2700745024 2602.50336 611.1736 317.828485 624.1994 2490.07968
## 128  4.9314342  0.2585970387 2649.80031 612.9137 319.560981 625.9395 2533.78836
## 170  4.9432961  0.2512691280 2663.00620 613.3942 320.039403 626.4200 2545.99236
## 93   4.9478989  0.2534419388 2668.13905 613.5803 320.224739 626.6062 2550.73579
## 168  4.9664486  0.2447369516 2688.87344 614.3287 320.969937 627.3546 2569.89713
## 175  5.0204883  0.2305442389 2749.71977 616.4931 323.125243 629.5190 2626.12724
## 174  5.0305579  0.2263725536 2761.13043 616.8939 323.524311 629.9197 2636.67221
## 131  5.0523459  0.2175104961 2785.89854 617.7582 324.385090 630.7841 2659.56124
## 173  5.0523624  0.2223969271 2785.91730 617.7589 324.385739 630.7847 2659.57857
## 85   5.0540365  0.2108790334 2787.82482 617.8252 324.451725 630.8510 2661.34138
## 91   5.0607394  0.2109793716 2795.46865 618.0902 324.715710 631.1161 2668.40529
## 132  5.0747014  0.2102928005 2811.42301 618.6412 325.264470 631.6671 2683.14924
## 113  5.0801743  0.2099513938 2817.68892 618.8568 325.479169 631.8827 2688.93978
## 158  5.1012279  0.2016681025 2841.85589 619.6840 326.302949 632.7098 2711.27327
## 119  5.1052932  0.2067789795 2846.53390 619.8433 326.461627 632.8691 2715.59638
## 136  5.1103670  0.2028913900 2852.37766 620.0420 326.659495 633.0678 2720.99679
## 69   5.1281265  0.1976527795 2872.87773 620.7358 327.350541 633.7616 2739.94158
## 90   5.1286879  0.1938300483 2873.52696 620.7577 327.372348 633.7835 2740.54155
## 118  5.1390939  0.1973987305 2885.57308 621.1631 327.776112 634.1889 2751.67377
## 138  5.1425462  0.1958504967 2889.57496 621.2974 327.909887 634.3232 2755.37204
## 88   5.1480791  0.1852913812 2895.99426 621.5124 328.124100 634.5383 2761.30433
## 161  5.1651239  0.1803504472 2915.81288 622.1735 328.782568 635.1994 2779.61937
## 116  5.1666818  0.1877596169 2917.62762 622.2338 328.842646 635.2597 2781.29644
## 80   5.1722278  0.1791248541 2924.09219 622.4484 329.056367 635.4743 2787.27056
## 156  5.1816587  0.1776067919 2935.10123 622.8128 329.419282 635.8386 2797.44438
## 66   5.1952801  0.1692752300 2951.03723 623.3378 329.942292 636.3637 2812.17136
## 97   5.2108540  0.1755251790 2969.30882 623.9365 330.538607 636.9623 2829.05674
## 94   5.2110009  0.1693224266 2969.48142 623.9421 330.544224 636.9679 2829.21624
## 63   5.2387749  0.1586330487 3002.20306 625.0052 331.603269 638.0311 2859.45540
## 163  5.2710318  0.1507039307 3040.42445 626.2329 332.826278 639.2588 2894.77704
## 78   5.2743975  0.1463123999 3044.42595 626.3606 332.953458 639.3864 2898.47496
## 135  5.2821219  0.1480555767 3053.61940 626.6533 333.245042 639.6791 2906.97094
## 77   5.2857556  0.1407129973 3057.94879 626.7908 333.382061 639.8167 2910.97187
## 108  5.2858799  0.1469740279 3058.09685 626.7955 333.386743 639.8214 2911.10870
## 160  5.3010189  0.1392800343 3076.16674 627.3675 333.956584 640.3934 2927.80768
## 157  5.3028580  0.1399145579 3078.36541 627.4369 334.025699 640.4627 2929.83954
## 106  5.3348366  0.1298088896 3116.71819 628.6394 335.223691 641.6652 2965.28262
## 167  5.3429218  0.1266863056 3126.45141 628.9422 335.525449 641.9681 2974.27740
## 105  5.3458555  0.1255086844 3129.98682 629.0520 335.634832 642.0779 2977.54459
## 67   5.3822962  0.1126943528 3174.06266 630.4107 336.988556 643.4366 3018.27653
## 95   5.3952040  0.1132873473 3189.74676 630.8898 337.465883 643.9156 3032.77074
## 79   5.3997380  0.1047494899 3195.26492 631.0578 337.633281 644.0836 3037.87025
## 96   5.4021796  0.1121897236 3198.23837 631.1482 337.723367 644.1741 3040.61811
## 84   5.4073283  0.1019845464 3204.51312 631.3387 337.913207 644.3646 3046.41682
## 169  5.4099824  0.1059240668 3207.74993 631.4369 338.010994 644.4627 3049.40806
## 68   5.4108271  0.1053057997 3208.78050 631.4681 338.042108 644.4940 3050.36044
## 58   5.4148586  0.1013118773 3213.70098 631.6171 338.190532 644.6429 3054.90762
## 107  5.4378125  0.0979030646 3241.78655 632.4631 339.033526 645.4889 3080.86242
## 166  5.4525966  0.0923498867 3259.93868 633.0061 339.574609 646.0319 3097.63740
## 112  5.4953563  0.0787330087 3312.71718 634.5684 341.131412 647.5942 3146.41176
## 56   5.4984968  0.0768129412 3316.60977 634.6827 341.245278 647.7085 3150.00904
## 82   5.5457783  0.0548861963 3375.48294 636.3951 342.951802 649.4210 3204.41569
## 172  5.5527906  0.0615701073 3384.25730 636.6478 343.203664 649.6737 3212.52436
## 87   5.5693863  0.0517780931 3405.06752 637.2447 343.798484 650.2705 3231.75577
## 110  5.5784202  0.0508979395 3416.42159 637.5688 344.121532 650.5947 3242.24844
## 83   5.6109209  0.0326156846 3457.42188 638.7307 345.279468 651.7565 3280.13816
## 111  5.6232067  0.0362751439 3472.98266 639.1681 345.715451 652.1940 3294.51839
## 115  5.6271687  0.0393540862 3478.00807 639.3090 345.855848 652.3348 3299.16255
## 89   5.6357093  0.0315626013 3488.85296 639.6123 346.158157 652.6382 3309.18467
## 117  5.6739758  0.0275841354 3537.64620 640.9657 347.507109 653.9916 3354.27612
## 86   5.6776671  0.0149612677 3542.37034 641.0958 347.636752 654.1216 3358.64184
## 114  5.6892061  0.0187094334 3557.15796 641.5018 348.041484 654.5277 3372.30758
## 62   5.6910923  0.0149798679 3559.57801 641.5681 348.107564 654.5940 3374.54403
## 59   5.7177745 -0.0050134427 3593.89860 642.5036 349.040026 655.5295 3406.26082
## 60   5.7716700 -0.0166849078 3663.71228 644.3800 350.910389 657.4059 3470.77795
## 61   5.8152109 -0.0299410684 3720.59142 645.8831 352.408781 658.9090 3523.34184
## 324  3.1348719  0.6918178229 1017.97238 524.3054 229.764942 539.9364 1034.80663
## 320  3.1707317  0.6848861452 1043.46550 526.5802 232.012056 542.2112 1058.61629
## 321  3.4015448  0.6378023571 1214.49293 540.6336 245.907400 556.2647 1218.34980
## 316  3.4043329  0.6391114583 1216.63222 540.7975 246.069541 556.4285 1220.34782
## 323  3.4529314  0.6243688831 1254.20415 543.6324 248.875151 559.2634 1255.43866
## 327  3.4902772  0.6177699949 1283.43837 545.7839 251.004970 561.4150 1282.74238
## 354  3.5152310  0.6115738790 1303.14744 547.2088 252.415665 562.8398 1301.14994
## 329  3.5164117  0.6135789275 1304.08339 547.2759 252.482164 562.9069 1302.02409
## 330  3.5370886  0.6060740826 1320.52638 548.4485 253.643278 564.0795 1317.38126
## 322  3.5620736  0.6023393059 1340.52388 549.8563 255.037460 565.4873 1336.05820
## 363  3.5694578  0.6005826096 1346.46098 550.2704 255.447666 565.9015 1341.60324
## 209  3.5908011  0.5946879821 1363.69074 551.4628 256.628666 567.0938 1357.69523
## 357  3.5969348  0.5916792595 1368.66131 551.8041 256.966793 567.4351 1362.33756
## 205  3.6032190  0.5932791713 1373.76262 552.1532 257.312629 567.7843 1367.10200
## 317  3.6040117  0.5927579529 1374.40674 552.1972 257.356211 567.8282 1367.70359
## 265  3.6460615  0.5845524381 1408.77801 554.5172 259.654700 570.1482 1399.80512
## 365  3.6516557  0.5825691072 1413.38072 554.8238 259.958522 570.4549 1404.10389
## 229  3.6602811  0.5773214955 1420.49123 555.2957 260.426074 570.9267 1410.74485
## 319  3.6681105  0.5789247566 1426.96008 555.7230 260.849544 571.3541 1416.78653
## 261  3.6822866  0.5770595185 1438.70791 556.4945 261.614033 572.1255 1427.75858
## 204  3.6949280  0.5752209816 1449.22207 557.1799 262.293320 572.8109 1437.57843
## 360  3.7116875  0.5668347010 1463.21704 558.0850 263.190392 573.7161 1450.64924
## 359  3.7117822  0.5670518269 1463.29627 558.0901 263.195448 573.7212 1450.72323
## 233  3.7123256  0.5667064146 1463.75110 558.1194 263.224466 573.7504 1451.14803
## 366  3.7126927  0.5675236085 1464.05837 558.1392 263.244065 573.7702 1451.43500
## 285  3.7607647  0.5564266163 1504.56287 560.7122 265.794579 576.3432 1489.26476
## 289  3.7941433  0.5486709749 1532.99349 562.4794 267.546737 578.1105 1515.81795
## 318  3.7963497  0.5502881919 1534.88175 562.5957 267.662028 578.2267 1517.58151
## 260  3.7977149  0.5524869131 1536.05055 562.6676 267.733325 578.2986 1518.67313
## 351  3.8078653  0.5449086886 1544.75428 563.2015 268.262662 578.8325 1526.80210
## 368  3.8290483  0.5427483458 1562.99305 564.3110 269.362889 579.9420 1543.83647
## 340  3.8773165  0.5324117363 1604.93027 566.8164 271.847689 582.4474 1583.00433
## 236  3.8788908  0.5265783611 1606.30695 566.8976 271.928220 582.5286 1584.29010
## 325  3.8916297  0.5282537687 1617.46713 567.5533 272.578672 583.1843 1594.71331
## 361  3.9007634  0.5250651449 1625.49142 568.0222 273.043753 583.6532 1602.20771
## 326  3.9062133  0.5216744164 1630.28831 568.3014 273.320747 583.9324 1606.68784
## 369  3.9127593  0.5196138854 1636.05885 568.6363 273.652950 584.2673 1612.07732
## 230  3.9169432  0.5208847414 1639.75214 568.8500 273.864991 584.4810 1615.52671
## 206  3.9256867  0.5202951236 1647.48319 569.2960 274.307398 584.9270 1622.74723
## 334  3.9385517  0.5176944866 1658.88975 569.9503 274.956583 585.5813 1633.40055
## 292  3.9432104  0.5144733529 1663.02950 570.1867 275.191151 585.8178 1637.26693
## 238  3.9532832  0.5110642636 1671.99707 570.6970 275.697395 586.3280 1645.64232
## 362  3.9553438  0.5098576922 1673.83438 570.8012 275.800801 586.4322 1647.35831
## 225  3.9639501  0.5071626762 1681.51844 571.2359 276.232113 586.8669 1654.53495
## 370  3.9698029  0.5081978419 1686.75362 571.5310 276.524905 587.1620 1659.42441
## 208  3.9708503  0.5065031472 1687.69139 571.5838 276.577263 587.2148 1660.30026
## 262  3.9867455  0.5074496980 1701.95195 572.3828 277.370091 588.0138 1673.61912
## 239  3.9894046  0.4986166437 1704.34315 572.5161 277.502418 588.1471 1675.85241
## 294  3.9935544  0.5046457072 1708.07809 572.7240 277.708758 588.3551 1679.34071
## 353  3.9937715  0.4988406874 1708.27359 572.7349 277.719547 588.3659 1679.52330
## 226  4.0020933  0.4977261986 1715.77550 573.1512 278.132669 588.7822 1686.52981
## 328  4.0218438  0.4953249438 1733.64262 574.1358 279.109773 589.7668 1703.21707
## 286  4.0250535  0.4953533879 1736.55456 574.2953 279.268117 589.9264 1705.93671
## 352  4.0351073  0.4888638673 1745.69069 574.7943 279.763299 590.4253 1714.46953
## 188  4.0496631  0.4856799303 1758.95830 575.5144 280.478071 591.1455 1726.86101
## 295  4.0653898  0.4846889590 1773.34688 576.2896 281.247501 591.9207 1740.29943
## 207  4.0704587  0.4846372031 1777.99636 576.5388 281.494871 592.1699 1744.64187
## 346  4.0729881  0.4786242085 1780.31870 576.6631 281.618201 592.2941 1746.81086
## 264  4.0780738  0.4822401171 1784.99229 576.9127 281.865934 592.5437 1751.17582
## 281  4.0808619  0.4789185883 1787.55695 577.0493 282.001618 592.6804 1753.57113
## 232  4.0809277  0.4773804901 1787.61751 577.0526 282.004820 592.6836 1753.62769
## 356  4.0810346  0.4750210900 1787.71592 577.0578 282.010023 592.6888 1753.71960
## 355  4.0826003  0.4752236896 1789.15697 577.1345 282.086175 592.7656 1755.06548
## 263  4.0861927  0.4833387037 1792.46545 577.3104 282.260792 592.9415 1758.15549
## 214  4.1029135  0.4766025534 1807.90316 578.1272 283.071570 593.7582 1772.57376
## 337  4.1044366  0.4756768616 1809.31257 578.2014 283.145264 593.8324 1773.89010
## 176  4.1154491  0.4748620890 1819.51821 578.7373 283.677279 594.3683 1783.42180
## 282  4.1165451  0.4703000396 1820.53535 578.7906 283.730147 594.4216 1784.37178
## 231  4.1284578  0.4663491827 1831.60905 579.3685 284.303926 594.9995 1794.71421
## 364  4.1304127  0.4659803542 1833.42924 579.4632 284.397924 595.0942 1796.41421
## 339  4.1416865  0.4679606857 1843.94350 580.0083 284.939178 595.6393 1806.23415
## 228  4.1417494  0.4618445102 1844.00221 580.0114 284.942193 595.6424 1806.28898
## 333  4.1844331  0.4529833408 1884.07020 582.0620 286.978311 597.6930 1843.71105
## 287  4.1855767  0.4543441126 1885.14939 582.1166 287.032582 597.7476 1844.71898
## 270  4.1913664  0.4544338097 1890.61735 582.3931 287.307109 598.0241 1849.82586
## 184  4.2088137  0.4456686952 1907.14102 583.2239 288.132153 598.8549 1865.25837
## 288  4.2180110  0.4432978838 1915.87903 583.6604 288.565714 599.2915 1873.41936
## 183  4.2217378  0.4434345810 1919.42518 583.8371 288.741132 599.4681 1876.73135
## 227  4.2241467  0.4396332584 1921.71894 583.9512 288.854433 599.5822 1878.87363
## 234  4.2640058  0.4342737453 1959.86331 585.8295 290.720022 601.4605 1914.49912
## 358  4.2642170  0.4296385383 1960.06634 585.8394 290.729860 601.4704 1914.68874
## 284  4.2787461  0.4283686578 1974.06022 586.5197 291.405581 602.1507 1927.75852
## 235  4.2878524  0.4255148021 1982.85533 586.9449 291.827944 602.5759 1935.97283
## 283  4.2949126  0.4249625721 1989.68708 587.2739 292.154796 602.9050 1942.35345
## 367  4.3021289  0.4224358569 1996.68154 587.6097 292.488329 603.2407 1948.88603
## 290  4.3139209  0.4250151768 2008.13619 588.1571 293.032156 603.7882 1959.58426
## 345  4.3303904  0.4125236680 2024.18713 588.9192 293.789257 604.5503 1974.57526
## 185  4.3345718  0.4171364252 2028.27199 589.1123 293.981023 604.7433 1978.39038
## 291  4.3791642  0.4036837769 2072.08012 591.1593 296.014822 606.7903 2019.30560
## 213  4.3809780  0.4038260369 2073.87153 591.2421 296.097114 606.8731 2020.97872
## 343  4.3906176  0.3976668878 2083.40452 591.6817 296.533899 607.3127 2029.88219
## 376  4.4389443  0.3824482234 2131.51225 593.8710 298.709463 609.5020 2074.81309
## 237  4.4478748  0.3828162747 2140.46002 594.2730 299.108936 609.9040 2083.16999
## 350  4.4504728  0.3818462402 2143.06636 594.3898 299.224997 610.0208 2085.60421
## 224  4.4593826  0.3762735923 2152.01651 594.7898 299.622526 610.4208 2093.96334
## 218  4.4653363  0.3744931013 2158.00709 595.0566 299.887723 610.6876 2099.55833
## 349  4.4810275  0.3732065510 2173.83382 595.7582 300.584991 611.3892 2114.33992
## 186  4.4810280  0.3777463947 2173.83435 595.7582 300.585014 611.3892 2114.34042
## 336  4.4844741  0.3704709385 2177.31766 595.9120 300.737826 611.5430 2117.59370
## 187  4.4885888  0.3730812452 2181.48028 596.0954 300.920133 611.7264 2121.48145
## 293  4.5031303  0.3725888648 2196.22178 596.7423 301.563093 612.3733 2135.24948
## 249  4.5059452  0.3620645985 2199.08088 596.8672 301.687318 612.4983 2137.91978
## 274  4.5105548  0.3648184323 2203.76678 597.0717 301.890581 612.7028 2142.29624
## 331  4.5221537  0.3615233784 2215.57879 597.5854 302.401133 613.2164 2153.32823
## 217  4.5237122  0.3603130508 2217.16828 597.6543 302.469637 613.2853 2154.81276
## 373  4.5321363  0.3572102960 2225.76905 598.0264 302.839503 613.6574 2162.84558
## 379  4.5377565  0.3550688842 2231.51610 598.2743 303.085888 613.9053 2168.21311
## 280  4.5514455  0.3554504571 2245.54387 598.8767 303.684742 614.5077 2181.31455
## 342  4.5645364  0.3482779638 2258.99816 599.4511 304.255764 615.0821 2193.88037
## 243  4.5732565  0.3423741094 2267.98183 599.8328 304.635239 615.4638 2202.27081
## 375  4.5847260  0.3434937802 2279.82399 600.3338 305.133267 615.9648 2213.33096
## 210  4.5883808  0.3440866659 2283.60388 600.4931 305.291711 616.1242 2216.86124
## 380  4.5884272  0.3436632200 2283.65180 600.4952 305.293718 616.1262 2216.90599
## 269  4.5885429  0.3450505273 2283.77152 600.5002 305.298732 616.1312 2217.01781
## 338  4.5887163  0.3459068867 2283.95094 600.5078 305.306246 616.1388 2217.18538
## 348  4.5995256  0.3410657500 2295.14845 600.9783 305.774085 616.6094 2227.64346
## 305  4.6117287  0.3377876997 2307.82143 601.5083 306.300942 617.1393 2239.47957
## 246  4.6151733  0.3325203623 2311.40469 601.6576 306.449407 617.2886 2242.82620
## 344  4.6152361  0.3343578691 2311.47003 601.6603 306.452113 617.2913 2242.88723
## 378  4.6343833  0.3301497010 2331.43729 602.4883 307.275385 618.1194 2261.53593
## 299  4.6420753  0.3263519153 2339.48207 602.8200 307.605174 618.4510 2269.04948
## 332  4.6424427  0.3293293190 2339.86660 602.8358 307.620910 618.4669 2269.40861
## 223  4.6682207  0.3198573528 2366.92602 603.9433 308.722101 619.5743 2294.68114
## 302  4.6756786  0.3184907828 2374.78267 604.2626 309.039571 619.8936 2302.01897
## 248  4.6757368  0.3160305761 2374.84402 604.2651 309.042046 619.8961 2302.07627
## 266  4.6816879  0.3171851960 2381.12229 604.5194 309.295008 620.1505 2307.93995
## 279  4.6844843  0.3173927782 2384.07522 604.6389 309.413766 620.2699 2310.69788
## 193  4.7094846  0.3061170272 2410.55311 605.7034 310.472371 621.3344 2335.42729
## 273  4.7111320  0.3072814204 2412.30285 605.7733 310.541933 621.4044 2337.06148
## 221  4.7123287  0.3052753315 2413.57429 605.8241 310.592450 621.4552 2338.24896
## 211  4.7168582  0.3060504329 2418.38950 606.0163 310.783538 621.6473 2342.74620
## 304  4.7213055  0.3071505874 2423.12179 606.2048 310.970980 621.8358 2347.16599
## 277  4.7419955  0.3008534766 2445.19640 607.0793 311.840723 622.7103 2367.78288
## 384  4.7701503  0.2861688926 2475.39033 608.2633 313.018249 623.8943 2395.98293
## 383  4.7701832  0.2865236619 2475.42569 608.2646 313.019620 623.8957 2396.01596
## 335  4.7743798  0.2883983252 2479.94163 608.4405 313.194548 624.0715 2400.23368
## 255  4.7907595  0.2753553248 2497.60554 609.1255 313.875846 624.7565 2416.73115
## 182  4.7976843  0.2813340714 2505.09144 609.4144 314.163184 625.0454 2423.72271
## 267  4.8023003  0.2803658384 2510.08744 609.6067 314.354491 625.2377 2428.38880
## 382  4.8283791  0.2716344885 2538.40362 610.6899 315.431913 626.3209 2454.83510
## 177  4.8371669  0.2717450216 2547.97982 611.0535 315.793674 626.6846 2463.77893
## 212  4.8453571  0.2691336708 2556.92054 611.3919 316.130250 627.0229 2472.12924
## 341  4.8669059  0.2585910491 2580.51628 612.2794 317.013119 627.9104 2494.16682
## 311  4.8708811  0.2578192482 2584.88049 612.4427 317.175561 628.0737 2498.24283
## 268  4.8965426  0.2543180347 2613.13913 613.4936 318.221051 629.1246 2524.63539
## 254  4.9216386  0.2383884048 2640.91864 614.5160 319.238269 630.1470 2550.58046
## 252  4.9319076  0.2345620495 2652.32675 614.9329 319.653028 630.5639 2561.23523
## 242  4.9376702  0.2363053356 2658.73892 615.1664 319.885399 630.7974 2567.22397
## 310  4.9431166  0.2373480213 2664.80616 615.3869 320.104774 631.0179 2572.89056
## 385  4.9454024  0.2375318525 2667.35448 615.4794 320.196770 631.1104 2575.27059
## 372  4.9459347  0.2322589957 2667.94811 615.5009 320.218189 631.1319 2575.82502
## 308  4.9664356  0.2304254220 2690.85890 616.3282 321.041337 631.9592 2597.22288
## 215  4.9735735  0.2246260122 2698.85809 616.6154 321.327147 632.2464 2604.69384
## 258  4.9812840  0.2206588191 2707.51181 616.9252 321.635426 632.5563 2612.77611
## 259  4.9856169  0.2209766056 2712.38078 617.0991 321.808460 632.7302 2617.32355
## 220  4.9919168  0.2247788516 2719.46744 617.3517 322.059773 632.9827 2623.94223
## 347  5.0144162  0.2177091223 2744.85003 618.2511 322.954766 633.8821 2647.64867
## 216  5.0196631  0.2106032870 2750.78572 618.4603 323.162909 634.0913 2653.19239
## 315  5.0203084  0.2186449409 2751.51609 618.4860 323.188490 634.1170 2653.87453
## 271  5.0203448  0.2130666682 2751.55731 618.4874 323.189934 634.1185 2653.91303
## 314  5.0300105  0.2142597599 2762.50956 618.8721 323.572751 634.5031 2664.14203
## 272  5.0334172  0.2087939815 2766.37478 619.0075 323.707505 634.6386 2667.75202
## 245  5.0365811  0.2051978125 2769.96677 619.1332 323.832570 634.7642 2671.10681
## 257  5.0383913  0.2063249380 2772.02302 619.2051 323.904093 634.8361 2673.02727
## 222  5.0482732  0.2096625804 2783.26064 619.5970 324.294076 635.2280 2683.52280
## 197  5.0519842  0.1970778817 2787.48652 619.7439 324.440336 635.3749 2687.46963
## 313  5.0522368  0.2087196385 2787.77422 619.7539 324.450286 635.3849 2687.73833
## 203  5.0605723  0.1994682982 2797.27796 620.0836 324.778402 635.7146 2696.61449
## 374  5.0700227  0.1967339879 2808.07180 620.4568 325.149759 636.0878 2706.69554
## 181  5.0908183  0.1973867901 2831.89440 621.2754 325.964514 636.9064 2728.94500
## 276  5.0924274  0.1947370557 2833.74188 621.3386 326.027423 636.9697 2730.67048
## 240  5.0936489  0.1869969654 2835.14457 621.3866 326.075160 637.0176 2731.98054
## 298  5.0959150  0.1887223872 2837.74793 621.4756 326.163698 637.1066 2734.41199
## 278  5.1185158  0.1904284397 2863.77509 622.3606 327.044580 637.9916 2758.72043
## 371  5.1245522  0.1787668865 2870.74618 622.5963 327.279203 638.2274 2765.23118
## 202  5.1282259  0.1804195070 2874.99266 622.7397 327.421855 638.3707 2769.19724
## 178  5.1354099  0.1738883807 2883.30571 623.0196 327.700529 638.6507 2776.96133
## 377  5.1442050  0.1705875308 2893.49875 623.3619 328.041168 638.9929 2786.48126
## 200  5.1478268  0.1717904089 2897.70138 623.5026 328.181277 639.1337 2790.40637
## 247  5.1617721  0.1666896428 2913.91049 624.0437 328.719834 639.6747 2805.54510
## 192  5.1622613  0.1691908403 2914.47986 624.0627 328.738700 639.6937 2806.07687
## 301  5.1630503  0.1670204181 2915.39832 624.0932 328.769123 639.7242 2806.93468
## 296  5.1754802  0.1630120965 2929.88658 624.5741 329.247826 640.2052 2820.46619
## 251  5.2349288  0.1435132680 2999.66138 626.8584 331.521677 642.4894 2885.63336
## 241  5.2387532  0.1390551229 3004.17736 627.0044 331.667079 642.6354 2889.85113
## 244  5.2426345  0.1389604586 3008.76391 627.1525 331.814538 642.7836 2894.13481
## 381  5.2481394  0.1406118504 3015.27493 627.3624 332.023495 642.9935 2900.21587
## 219  5.2486584  0.1408864652 3015.88907 627.3822 332.043182 643.0132 2900.78945
## 275  5.2587832  0.1397828495 3027.88340 627.7676 332.426901 643.3987 2911.99172
## 190  5.2642620  0.1343481903 3034.38334 627.9759 332.634231 643.6069 2918.06244
## 303  5.2659281  0.1384322564 3036.36138 628.0392 332.697240 643.6702 2919.90985
## 189  5.2660133  0.1303131713 3036.46253 628.0424 332.700461 643.6734 2920.00433
## 297  5.2914260  0.1268793294 3066.71070 629.0053 333.659047 644.6363 2948.25504
## 300  5.2959553  0.1252586037 3072.11710 629.1764 333.829417 644.8074 2953.30443
## 253  5.3282724  0.1132064345 3110.82669 630.3931 335.040845 646.0241 2989.45780
## 307  5.3401040  0.1120789685 3125.05764 630.8367 335.482543 646.4678 3002.74899
## 179  5.3476604  0.1088807072 3134.16284 631.1195 335.764129 646.7506 3011.25293
## 180  5.3689271  0.1032930813 3159.85764 631.9133 336.554512 647.5443 3035.25095
## 191  5.3838290  0.0939670595 3177.92320 632.4677 337.106500 648.0987 3052.12354
## 196  5.3952848  0.0920777240 3191.84514 632.8928 337.529811 648.5238 3065.12613
## 309  5.4040324  0.0941965470 3202.49565 633.2168 337.852446 648.8478 3075.07333
## 250  5.4179822  0.0813413419 3219.51589 633.7324 338.365887 649.3634 3090.96963
## 306  5.4466251  0.0779024005 3254.60081 634.7869 339.416030 650.4180 3123.73769
## 256  5.5022305  0.0576244470 3323.24051 636.8184 341.439163 652.4494 3187.84472
## 194  5.5334601  0.0419254801 3362.09619 637.9504 342.566534 653.5814 3224.13454
## 312  5.5500535  0.0485904662 3382.83117 638.5492 343.162985 654.1802 3243.50026
## 199  5.5631402  0.0399648607 3399.22788 639.0203 343.632138 654.6513 3258.81421
## 195  5.5937232  0.0217498233 3437.69695 640.1167 344.724274 655.7477 3294.74294
## 201  5.6259341  0.0219632256 3478.44170 641.2651 345.868156 656.8961 3332.79708
## 198  5.6690633  0.0019455480 3533.36383 642.7925 347.389639 658.4235 3384.09240
## 585  2.6776304  0.7714353789  720.33350 494.7741 198.935214 513.0103  763.07240
## 597  2.8318498  0.7414535554  816.12763 505.9737 209.931219 524.2099  853.50273
## 594  2.8389825  0.7411442672  820.68794 506.4768 210.425702 524.7130  857.80769
## 596  2.8957883  0.7304652119  857.41598 510.4392 214.321473 528.6754  892.47922
## 450  2.9640721  0.7186761816  902.52820 515.1005 218.907719 533.3367  935.06546
## 588  2.9773389  0.7148836450  911.41502 515.9937 219.786896 534.2299  943.45468
## 591  2.9879495  0.7132329498  918.55112 516.7052 220.487319 534.9414  950.19121
## 446  2.9913051  0.7132164948  920.81316 516.9297 220.708323 535.1659  952.32659
## 582  3.0278803  0.7074076156  945.63384 519.3603 223.101775 537.5965  975.75749
## 584  3.0394272  0.7051938013  953.53251 520.1215 223.851579 538.3577  983.21388
## 520  3.0897187  0.6938499261  988.28484 523.4037 227.085343 541.6399 1016.02033
## 590  3.0925601  0.6933181878  990.26531 523.5876 227.266516 541.8238 1017.88990
## 449  3.1000169  0.6904362491  995.47140 524.0692 227.741211 542.3054 1022.80449
## 593  3.1098891  0.6895889047 1002.38318 524.7051 228.367968 542.9413 1029.32926
## 442  3.1366098  0.6881449147 1021.20120 526.4162 230.054733 544.6524 1047.09359
## 516  3.1423617  0.6841054847 1025.27297 526.7826 230.415994 545.0188 1050.93737
## 592  3.1773561  0.6769546466 1050.20660 528.9976 232.600167 547.2338 1074.47489
## 447  3.1792916  0.6774501220 1051.59377 529.1194 232.720292 547.3556 1075.78439
## 583  3.2816655  0.6570199209 1126.16564 535.4579 238.974422 553.6941 1146.18076
## 619  3.2849628  0.6525804386 1128.60679 535.6588 239.172685 553.8950 1148.48521
## 595  3.2866915  0.6537925532 1129.88758 535.7640 239.276551 554.0002 1149.69429
## 512  3.3167282  0.6513519949 1152.24962 537.5835 241.072826 555.8197 1170.80421
## 599  3.3277594  0.6463499094 1160.51328 538.2476 241.728538 556.4837 1178.60516
## 517  3.3352709  0.6460485680 1166.15602 538.6985 242.173826 556.9347 1183.93194
## 480  3.3667327  0.6354488866 1189.92867 540.5763 244.028332 558.8125 1206.37349
## 600  3.3714745  0.6339554016 1193.53093 540.8577 244.306367 559.0939 1209.77405
## 445  3.3720125  0.6383514488 1193.93991 540.8897 244.337885 559.1258 1210.16013
## 519  3.3816933  0.6333526843 1201.31122 541.4630 244.904253 559.6992 1217.11870
## 456  3.3820459  0.6303235675 1201.58008 541.4839 244.924851 559.7201 1217.37251
## 453  3.3842235  0.6324054452 1203.24128 541.6126 245.052020 559.8488 1218.94069
## 443  3.3988661  0.6323779717 1214.43918 542.4761 245.905052 560.7123 1229.51158
## 448  3.4029508  0.6298870076 1217.57153 542.7163 246.142371 560.9525 1232.46854
## 601  3.4073525  0.6277870062 1220.95124 542.9748 246.397801 561.2110 1235.65901
## 628  3.4105035  0.6285207986 1223.37330 543.1597 246.580454 561.3959 1237.94545
## 622  3.4173777  0.6252021798 1228.66499 543.5624 246.978355 561.7986 1242.94085
## 621  3.4310441  0.6214923124 1239.21701 544.3606 247.767102 562.5968 1252.90203
## 587  3.4368148  0.6205287845 1243.68526 544.6967 248.099230 562.9329 1257.12008
## 455  3.4398134  0.6218241298 1246.01004 544.8712 248.271598 563.1073 1259.31469
## 607  3.4475199  0.6205218420 1251.99415 545.3187 248.713920 563.5549 1264.96373
## 489  3.4481009  0.6192933256 1252.44578 545.3524 248.747223 563.5886 1265.39008
## 523  3.4686130  0.6153117810 1268.44138 546.5387 249.919649 564.7749 1280.49004
## 483  3.4700683  0.6116871959 1269.57982 546.6226 250.002571 564.8588 1281.56473
## 518  3.4790159  0.6141964348 1276.58990 547.1376 250.511666 565.3738 1288.18229
## 387  3.4798486  0.6140930105 1277.24324 547.1855 250.558981 565.4217 1288.79905
## 525  3.4900485  0.6115141112 1285.25836 547.7708 251.137633 566.0070 1296.36538
## 629  3.4926532  0.6092604134 1287.30891 547.9200 251.285136 566.1562 1298.30111
## 550  3.5072514  0.6046735465 1298.82970 548.7542 252.109845 566.9904 1309.17681
## 526  3.5140779  0.6042447659 1304.23357 549.1431 252.494347 567.3793 1314.27811
## 586  3.5329164  0.6001267034 1319.20075 550.2124 253.551652 568.4486 1328.40723
## 624  3.5414214  0.5958944524 1325.98417 550.6933 254.027187 568.9295 1334.81083
## 391  3.5467392  0.5966364438 1330.23382 550.9934 254.323949 569.2296 1338.82252
## 513  3.5544351  0.5983319023 1336.39525 551.4269 254.752658 569.6631 1344.63896
## 625  3.5551235  0.5923647735 1336.94703 551.4657 254.790961 569.7019 1345.15984
## 486  3.5619543  0.5908368870 1342.42802 551.8496 255.170641 570.0858 1350.33393
## 492  3.5627435  0.5928833810 1343.06198 551.8939 255.214464 570.1301 1350.93239
## 559  3.5639839  0.5934258057 1344.05861 551.9635 255.283318 570.1997 1351.87322
## 491  3.5676267  0.5929695407 1346.98755 552.1678 255.485395 570.4040 1354.63816
## 630  3.5842731  0.5895241064 1360.41001 553.0988 256.406265 571.3350 1367.30906
## 553  3.5944529  0.5849406270 1368.64905 553.6661 256.967346 571.9023 1375.08676
## 386  3.5990923  0.5900886906 1372.41172 553.9240 257.222542 572.1602 1378.63875
## 515  3.6145483  0.5848213392 1384.98188 554.7811 258.070396 573.0173 1390.50507
## 485  3.6166299  0.5787745024 1386.67886 554.8962 258.184308 573.1324 1392.10703
## 477  3.6186491  0.5809856878 1388.32598 555.0079 258.294749 573.2441 1393.66192
## 589  3.6247823  0.5813202964 1393.33470 555.3466 258.629841 573.5827 1398.39019
## 444  3.6404523  0.5782616156 1406.17004 556.2093 259.483458 574.4455 1410.50684
## 561  3.6434444  0.5760242858 1408.62719 556.3736 259.646044 574.6098 1412.82641
## 411  3.6480297  0.5719280373 1412.39653 556.6251 259.894942 574.8613 1416.38469
## 466  3.6510715  0.5760791340 1414.89972 556.7918 260.059891 575.0280 1418.74772
## 626  3.6700142  0.5680416220 1430.53507 557.8268 261.084082 576.0630 1433.50760
## 415  3.6929259  0.5622538078 1449.55450 559.0715 262.315968 577.3077 1451.46207
## 562  3.7061904  0.5606666682 1460.61969 559.7886 263.025738 578.0248 1461.90768
## 555  3.7069300  0.5603366165 1461.23783 559.8285 263.065240 578.0647 1462.49121
## 556  3.7083690  0.5602387818 1462.44089 559.9061 263.142076 578.1423 1463.62691
## 618  3.7099271  0.5575861967 1463.74398 559.9901 263.225235 578.2263 1464.85704
## 452  3.7207535  0.5590964086 1472.81392 560.5729 263.802137 578.8091 1473.41912
## 617  3.7235147  0.5541104004 1475.13137 560.7213 263.949009 578.9575 1475.60681
## 514  3.7319753  0.5582131372 1482.24295 561.1752 264.398373 579.4114 1482.32019
## 488  3.7585857  0.5500312231 1504.71560 562.5963 265.805236 580.8324 1503.53453
## 479  3.7595451  0.5478637772 1505.52885 562.6473 265.855779 580.8835 1504.30224
## 494  3.7688397  0.5464672102 1513.41785 563.1411 266.344753 581.3773 1511.74951
## 487  3.7800464  0.5449185011 1522.95571 563.7350 266.932752 581.9712 1520.75332
## 547  3.7881411  0.5409502568 1529.86256 564.1628 267.356404 582.3990 1527.27343
## 598  3.7888812  0.5419451751 1530.49478 564.2019 267.395093 582.4381 1527.87025
## 451  3.7901001  0.5449062587 1531.53634 564.2662 267.458800 582.5024 1528.85349
## 460  3.7986955  0.5420959082 1538.89037 564.7192 267.907457 582.9554 1535.79575
## 610  3.7998673  0.5395812379 1539.89424 564.7809 267.968546 583.0171 1536.74341
## 604  3.8044493  0.5394875321 1543.82254 565.0220 268.207238 583.2581 1540.45175
## 631  3.8085525  0.5371169658 1547.34437 565.2375 268.420747 583.4737 1543.77638
## 459  3.8101955  0.5400247277 1548.75565 565.3238 268.506178 583.5600 1545.10864
## 495  3.8180459  0.5313291041 1555.50725 565.7355 268.913874 583.9716 1551.48220
## 564  3.8262134  0.5354368070 1562.54627 566.1628 269.337163 584.3990 1558.12707
## 627  3.8329019  0.5308198799 1568.32183 566.5121 269.683142 584.7483 1563.57925
## 521  3.8457096  0.5325632836 1579.40960 567.1793 270.344005 585.4155 1574.04618
## 611  3.8480100  0.5301841480 1581.40500 567.2989 270.462474 585.5351 1575.92985
## 606  3.8487990  0.5295574508 1582.08968 567.3399 270.503093 585.5761 1576.57620
## 388  3.8565544  0.5302217721 1588.82699 567.7425 270.901899 585.9787 1582.93626
## 522  3.8647634  0.5245111302 1595.97314 568.1678 271.323176 586.4040 1589.68228
## 472  3.8693374  0.5191520866 1599.96145 568.4044 271.557523 586.6406 1593.44726
## 418  3.8700870  0.5203580884 1600.61554 568.4431 271.595905 586.6793 1594.06474
## 536  3.8736713  0.5237021590 1603.74484 568.6282 271.779324 586.8644 1597.01882
## 454  3.8766064  0.5221152581 1606.30947 568.7797 271.929395 587.0159 1599.43985
## 412  3.8815550  0.5207067516 1610.63795 569.0349 272.182170 587.2711 1603.52596
## 557  3.8818646  0.5217547080 1610.90894 569.0508 272.197974 587.2870 1603.78177
## 390  3.8932671  0.5187011953 1620.90428 569.6374 272.779161 587.8736 1613.21745
## 478  3.8988412  0.5117017715 1625.80110 569.9236 273.062663 588.1598 1617.84007
## 496  3.9014868  0.5131419123 1628.12773 570.0592 273.197082 588.2954 1620.03643
## 482  3.9023662  0.5120329790 1628.90148 570.1043 273.241744 588.3405 1620.76685
## 565  3.9099903  0.5122943059 1635.61672 570.4947 273.628530 588.7309 1627.10609
## 407  3.9140281  0.5097377162 1639.17846 570.7011 273.833073 588.9373 1630.46840
## 530  3.9266993  0.5096967939 1650.37957 571.3475 274.473617 589.5837 1641.04232
## 615  3.9310969  0.5024236300 1654.27553 571.5714 274.695452 589.8076 1644.72013
## 620  3.9359321  0.5034128247 1658.56410 571.8173 274.939075 590.0534 1648.76857
## 558  3.9393172  0.5045947354 1661.56967 571.9892 275.109460 590.2254 1651.60585
## 420  3.9422507  0.5049816197 1664.17635 572.1381 275.256998 590.3743 1654.06657
## 614  3.9553143  0.4984485500 1675.80805 572.7997 275.912711 591.0359 1665.04698
## 481  3.9625135  0.4956703321 1682.23464 573.1634 276.273161 591.3996 1671.11372
## 566  3.9652119  0.5014364257 1684.64648 573.2996 276.408100 591.5358 1673.39052
## 524  3.9687175  0.5012058701 1687.78224 573.4763 276.583269 591.7125 1676.35070
## 549  3.9704804  0.4958119224 1689.36020 573.5651 276.671301 591.8013 1677.84030
## 408  3.9737409  0.4967660114 1692.28046 573.7293 276.834014 591.9655 1680.59705
## 421  3.9789431  0.4932256460 1696.94480 573.9910 277.093356 592.2271 1685.00022
## 490  3.9840964  0.4934886170 1701.57130 574.2498 277.349931 592.4860 1689.36767
## 389  3.9880474  0.4969877270 1705.12248 574.4481 277.546426 592.6842 1692.72000
## 463  3.9959634  0.4927335919 1712.24798 574.8446 277.939536 593.0808 1699.44653
## 548  4.0045923  0.4874486253 1720.03128 575.2761 278.367177 593.5123 1706.79401
## 603  4.0056569  0.4866372626 1720.99272 575.3292 278.419875 593.5654 1707.70162
## 414  4.0407460  0.4788231215 1752.82466 577.0736 280.149137 595.3098 1737.75119
## 623  4.0418568  0.4736653644 1753.83692 577.1285 280.203640 595.3647 1738.70677
## 609  4.0467853  0.4799402061 1758.33139 577.3723 280.445278 595.6085 1742.94958
## 465  4.0560032  0.4786817735 1766.75227 577.8273 280.896447 596.0635 1750.89895
## 542  4.0659390  0.4706295807 1775.85036 578.3166 281.381623 596.5528 1759.48762
## 551  4.0665729  0.4717602788 1776.43155 578.3478 281.412536 596.5840 1760.03626
## 552  4.0678609  0.4705144281 1777.61280 578.4112 281.475337 596.6474 1761.15137
## 413  4.0823391  0.4689443312 1790.91653 579.1217 282.179917 597.3579 1773.71018
## 396  4.0875348  0.4695884988 1795.70224 579.3761 282.432164 597.6123 1778.22792
## 533  4.0997683  0.4657192895 1806.99454 579.9738 283.024854 598.2100 1788.88793
## 560  4.1057415  0.4641748612 1812.52039 580.2650 283.313610 598.5012 1794.10438
## 410  4.1090939  0.4619375346 1815.62525 580.4282 283.475490 598.6644 1797.03538
## 484  4.1202321  0.4558728530 1825.95931 580.9696 284.012405 599.2058 1806.79081
## 535  4.1347632  0.4580022970 1839.48324 581.6737 284.710726 599.9099 1819.55749
## 605  4.1393639  0.4552376396 1843.77501 581.8961 284.931320 600.1323 1823.60895
## 529  4.1554397  0.4497899712 1858.80879 582.6714 285.700230 600.9075 1837.80094
## 409  4.1847772  0.4406519801 1886.39486 584.0784 287.095937 602.3146 1863.84238
## 613  4.1943099  0.4370397862 1895.40020 584.5335 287.547374 602.7697 1872.34349
## 493  4.2089428  0.4364419115 1909.26352 585.2300 288.238381 603.4662 1885.43056
## 462  4.2374557  0.4293735732 1936.41561 586.5803 289.578077 604.8165 1911.06232
## 416  4.2380707  0.4328946300 1937.00328 586.6093 289.606875 604.8455 1911.61709
## 554  4.2431060  0.4270401310 1941.81793 586.8468 289.842502 605.0830 1916.16215
## 417  4.2634489  0.4234158738 1961.32792 587.8034 290.791669 606.0396 1934.57971
## 471  4.2788954  0.4141251538 1976.20429 588.5267 291.509403 606.7629 1948.62311
## 563  4.2873729  0.4185155894 1984.39175 588.9225 291.902233 607.1587 1956.35212
## 502  4.2913973  0.4079320348 1988.28413 589.1102 292.088448 607.3464 1960.02656
## 469  4.3186888  0.4034451480 2014.77667 590.3781 293.346743 608.6143 1985.03570
## 541  4.3199345  0.4032539609 2015.98995 590.4358 293.403992 608.6719 1986.18105
## 468  4.3222323  0.4057254105 2018.22875 590.5421 293.509544 608.7783 1988.29449
## 395  4.3408918  0.4043034295 2036.45359 591.4037 294.364665 609.6399 2005.49887
## 634  4.3416321  0.3952607603 2037.17824 591.4378 294.398515 609.6740 2006.18294
## 457  4.3600977  0.3952019959 2055.29416 592.2866 295.241056 610.5228 2023.28449
## 476  4.3684642  0.3901181759 2063.52740 592.6700 295.621636 610.9062 2031.05673
## 464  4.3710196  0.3956878595 2066.04535 592.7870 295.737739 611.0232 2033.43369
## 616  4.3745466  0.3893358771 2069.52294 592.9483 295.897870 611.1845 2036.71656
## 475  4.3768067  0.3874103451 2071.75290 593.0516 296.000418 611.2878 2038.82166
## 539  4.3824446  0.3877966559 2077.32057 593.3090 296.255995 611.5452 2044.07757
## 635  4.4041489  0.3799734298 2098.82141 594.2971 297.236879 612.5333 2064.37451
## 419  4.4150373  0.3834462156 2109.64771 594.7910 297.727165 613.0272 2074.59462
## 470  4.4178424  0.3791234962 2112.44121 594.9180 297.853283 613.1542 2077.23170
## 505  4.4239123  0.3720281772 2118.49193 595.1926 298.125912 613.4288 2082.94362
## 612  4.4339485  0.3718254534 2128.51467 595.6458 298.575881 613.8820 2092.40516
## 602  4.4363544  0.3732157084 2130.92076 595.7543 298.683601 613.9905 2094.67652
## 572  4.4384191  0.3706347489 2132.98660 595.8474 298.775995 614.0835 2096.62669
## 400  4.4446930  0.3681725287 2139.26996 596.1299 299.056495 614.3661 2102.55823
## 546  4.4469164  0.3714312277 2141.49888 596.2299 299.155809 614.4661 2104.66234
## 406  4.4497448  0.3688089256 2144.33589 596.3571 299.282075 614.5932 2107.34050
## 499  4.4515031  0.3648128063 2146.10043 596.4361 299.360529 614.6723 2109.00624
## 608  4.4594706  0.3654982693 2154.10500 596.7937 299.715651 615.0299 2116.56261
## 532  4.4682780  0.3630027121 2162.96995 597.1883 300.107477 615.4245 2124.93119
## 633  4.4738946  0.3608504008 2168.63241 597.4396 300.356954 615.6758 2130.27658
## 399  4.4751325  0.3640088576 2169.88141 597.4949 300.411898 615.7311 2131.45564
## 545  4.4785815  0.3623972256 2173.36306 597.6490 300.564901 615.8852 2134.74235
## 527  4.4921656  0.3570181260 2187.10188 598.2547 301.166383 616.4909 2147.71189
## 506  4.4925557  0.3539375958 2187.49697 598.2721 301.183626 616.5082 2148.08485
## 431  4.5059404  0.3517766123 2201.07597 598.8670 301.774476 617.1032 2160.90353
## 636  4.5137116  0.3490993346 2208.97857 599.2117 302.116734 617.4479 2168.36364
## 458  4.5188133  0.3517023955 2214.17394 599.4376 302.341107 617.6738 2173.26810
## 501  4.5248994  0.3440238775 2220.37937 599.7068 302.608442 617.9430 2179.12607
## 392  4.5253073  0.3489433109 2220.79560 599.7248 302.626349 617.9610 2179.51900
## 569  4.5319533  0.3441667943 2227.58211 600.0183 302.917850 618.2545 2185.92550
## 575  4.5360032  0.3444830799 2231.72252 600.1970 303.095277 618.4332 2189.83408
## 538  4.5444541  0.3413886061 2240.37419 600.5692 303.465009 618.8054 2198.00132
## 534  4.5653656  0.3401791077 2261.85174 601.4874 304.376994 619.7236 2218.27627
## 474  4.5706741  0.3339977545 2267.31962 601.7198 304.607850 619.9560 2223.43799
## 425  4.5719647  0.3296447370 2268.64992 601.7763 304.663935 620.0125 2224.69380
## 504  4.5816689  0.3297126165 2278.66466 602.2004 305.085149 620.4366 2234.14778
## 571  4.5847189  0.3302628306 2281.81675 602.3335 305.217358 620.5697 2237.12338
## 540  4.5874501  0.3313201463 2284.64102 602.4526 305.335668 620.6888 2239.78951
## 576  4.5877314  0.3325300377 2284.93198 602.4648 305.347849 620.7010 2240.06418
## 544  4.5956895  0.3283349448 2293.17154 602.8115 305.692170 621.0477 2247.84238
## 528  4.6016640  0.3273860368 2299.36674 603.0713 305.950281 621.3075 2253.69070
## 428  4.6151290  0.3206502356 2313.35858 603.6557 306.530784 621.8919 2266.89908
## 574  4.6338765  0.3179262002 2332.90774 604.4665 307.336252 622.7027 2285.35363
## 405  4.6579418  0.3092394417 2358.11822 605.5024 308.365490 623.7386 2309.15249
## 637  4.6678702  0.3018876112 2368.55718 605.9283 308.788587 624.1645 2319.00695
## 461  4.6749441  0.3046514036 2376.00836 606.2311 309.089496 624.4673 2326.04091
## 430  4.6753231  0.3037859119 2376.40787 606.2474 309.105604 624.4836 2326.41805
## 393  4.6795334  0.3037440437 2380.84842 606.4274 309.284474 624.6636 2330.60996
## 509  4.6799289  0.2958737150 2381.26583 606.4443 309.301271 624.6805 2331.00400
## 510  4.7031130  0.2884904649 2405.79155 607.4326 310.283302 625.6688 2354.15645
## 403  4.7033266  0.2948435196 2406.01810 607.4417 310.292328 625.6779 2354.37031
## 531  4.7511371  0.2810079230 2456.98044 609.4645 312.302395 627.7007 2402.47912
## 498  4.7661233  0.2735326441 2473.06075 610.0944 312.928343 628.3305 2417.65904
## 579  4.7693112  0.2740009962 2476.48784 610.2281 313.061242 628.4643 2420.89424
## 580  4.7698588  0.2741605534 2477.07679 610.2510 313.084063 628.4872 2421.45021
## 437  4.7905092  0.2637509203 2499.33515 611.1150 313.942752 629.3512 2442.46226
## 467  4.7968755  0.2648163863 2506.21649 611.3807 314.206737 629.6169 2448.95829
## 394  4.7977378  0.2686397159 2507.14928 611.4166 314.242467 629.6528 2449.83884
## 508  4.7999219  0.2614957660 2509.51262 611.5076 314.332938 629.7438 2452.06986
## 578  4.8282107  0.2580017516 2540.22029 612.6829 315.501072 630.9191 2481.05810
## 537  4.8398301  0.2525685961 2552.88542 613.1636 315.978913 631.3998 2493.01408
## 511  4.9063388  0.2297229015 2625.96602 615.8933 318.692430 634.1295 2562.00267
## 473  4.9118310  0.2354896067 2632.04544 616.1171 318.914879 634.3533 2567.74168
## 632  4.9132554  0.2261055771 2633.62331 616.1751 318.972533 634.4112 2569.23120
## 500  4.9202441  0.2266045521 2641.37140 616.4593 319.255161 634.6955 2576.54545
## 436  4.9208815  0.2248358875 2642.07858 616.4852 319.280918 634.7214 2577.21303
## 424  4.9274014  0.2259599141 2649.31777 616.7501 319.544195 634.9862 2584.04688
## 397  4.9304553  0.2235726529 2652.71184 616.8740 319.667394 635.1102 2587.25091
## 434  4.9315031  0.2207703617 2653.87692 616.9165 319.709649 635.1527 2588.35075
## 581  4.9442697  0.2258179011 2668.09148 617.4336 320.223752 635.6698 2601.76940
## 568  4.9443848  0.2178605339 2668.21983 617.4382 320.228382 635.6744 2601.89055
## 402  4.9652018  0.2201734684 2691.47736 618.2785 321.063848 636.5147 2623.84583
## 398  4.9671616  0.2122022419 2693.67193 618.3574 321.142323 636.5936 2625.91751
## 440  4.9812348  0.2091263758 2709.45657 618.9233 321.704954 637.1595 2640.81833
## 441  4.9856147  0.2095078414 2714.37824 619.0990 321.879738 637.3352 2645.46441
## 543  4.9987780  0.2089780991 2729.19576 619.6264 322.404116 637.8626 2659.45225
## 404  5.0147165  0.2079040720 2747.18952 620.2631 323.037225 638.4993 2676.43849
## 503  5.0261718  0.1913593411 2760.15745 620.7194 323.491027 638.9556 2688.68030
## 427  5.0317192  0.1937236067 2766.44789 620.9400 323.710416 639.1762 2694.61853
## 439  5.0383911  0.1929096070 2774.02273 621.2051 323.973964 639.4413 2701.76923
## 497  5.0469245  0.1845373110 2783.72562 621.5435 324.310540 639.7797 2710.92882
## 570  5.0654274  0.1840256489 2804.82068 622.2754 325.038405 640.5116 2730.84270
## 422  5.0829213  0.1741942542 2824.83642 622.9649 325.724166 641.2011 2749.73770
## 567  5.1197672  0.1641047297 2867.21960 624.4095 327.160919 642.6457 2789.74771
## 573  5.1424937  0.1573058634 2893.51408 625.2953 328.042014 643.5315 2814.56989
## 429  5.1525406  0.1564678084 2905.17542 625.6857 328.430299 643.9219 2825.57827
## 507  5.1549736  0.1530159833 2908.00282 625.7801 328.524215 644.0163 2828.24735
## 401  5.2193100  0.1343299537 2983.25245 628.2607 330.991882 646.4969 2899.28352
## 423  5.2217110  0.1279104046 2986.07878 628.3527 331.083390 646.5889 2901.95160
## 433  5.2292031  0.1307785146 2994.90641 628.6395 331.368664 646.8757 2910.28494
## 426  5.2340977  0.1262841742 3000.68047 628.8266 331.554819 647.0628 2915.73570
## 577  5.2463203  0.1269256540 3015.12262 629.2931 332.018921 647.5293 2929.36919
## 435  5.3185106  0.1033367485 3101.10914 632.0264 334.738400 650.2626 3010.54106
## 432  5.4091646  0.0677817257 3210.75240 635.4066 338.102126 653.6428 3114.04505
## 438  5.4974877  0.0457870025 3319.35878 638.6459 341.326091 656.8821 3216.57022
## 825  2.3068614  0.8259739339  513.97384 466.9653 170.122147 487.8066  572.53575
## 822  2.3662113  0.8165089492  545.24269 472.0457 175.059337 492.8871  602.37461
## 711  2.3793121  0.8165483408  552.25195 473.1500 176.133389 493.9913  609.06332
## 723  2.4677780  0.7989612460  600.59639 480.4513 183.242835 501.2927  655.19686
## 710  2.4892299  0.7989004920  612.58513 482.1824 184.930330 503.0237  666.63734
## 824  2.5347166  0.7894875300  638.34947 485.8041 188.463237 506.6454  691.22341
## 832  2.5514196  0.7855790324  647.92741 487.1177 189.745411 507.9591  700.36333
## 831  2.5808197  0.7809289794  664.93898 489.4091 191.982942 510.2505  716.59692
## 720  2.6138541  0.7766145990  684.28599 491.9529 194.468244 512.7942  735.05916
## 833  2.6333668  0.7713987190  695.82950 493.4403 195.922211 514.2817  746.07476
## 767  2.6345786  0.7731910076  696.54919 493.5324 196.012164 514.3737  746.76153
## 717  2.6786388  0.7637667202  722.94248 496.8495 199.256316 517.6908  771.94781
## 719  2.6809428  0.7649790052  724.33469 497.0214 199.424552 517.8628  773.27634
## 708  2.6823994  0.7659153614  725.21543 497.1300 199.530836 517.9714  774.11681
## 821  2.6852394  0.7641450788  726.93415 497.3417 199.737919 518.1830  775.75692
## 722  2.7318306  0.7562663308  755.38904 500.7821 203.105453 521.6235  802.91051
## 714  2.7642740  0.7495797736  775.49251 503.1433 205.418056 523.9847  822.09461
## 779  2.7978286  0.7422521981  796.53421 505.5564 207.782628 526.3978  842.17402
## 776  2.8057977  0.7415880025  801.56885 506.1253 208.340203 526.9666  846.97842
## 828  2.8167535  0.7383262436  808.51374 506.9047 209.104268 527.7461  853.60571
## 778  2.8555198  0.7318907779  833.30515 509.6385 211.785116 530.4798  877.26335
## 820  2.8922573  0.7265357049  857.11182 512.1952 214.293546 533.0365  899.98129
## 646  2.8931873  0.7260169815  857.71839 512.2595 214.356643 533.1008  900.56012
## 827  2.8970227  0.7239956838  860.22216 512.5244 214.616673 533.3658  902.94939
## 830  2.9269659  0.7195980711  879.88332 514.5810 216.635416 535.4223  921.71140
## 716  2.9287686  0.7188292911  881.07342 514.7041 216.756312 535.5455  922.84707
## 829  2.9361579  0.7172143076  885.95943 515.2081 217.251135 536.0494  927.50963
## 642  2.9435336  0.7159118572  890.84875 515.7099 217.743854 536.5512  932.17536
## 823  2.9550798  0.7163365690  898.52731 516.4928 218.512789 537.3342  939.50276
## 721  2.9563514  0.7138135329  899.37477 516.5789 218.597293 537.4202  940.31146
## 770  2.9576296  0.7131195900  900.22700 516.6653 218.682201 537.5067  941.12472
## 764  2.9623679  0.7152038637  903.38955 516.9855 218.996656 537.8269  944.14264
## 718  2.9631987  0.7139597757  903.94454 517.0416 219.051738 537.8829  944.67225
## 773  2.9679200  0.7117038548  907.10164 517.3600 219.364496 538.2013  947.68497
## 766  2.9701068  0.7133886496  908.56564 517.5073 219.509196 538.3486  949.08202
## 645  2.9872125  0.7078400580  920.05460 518.6558 220.637540 539.4972  960.04557
## 638  3.0044892  0.7083488986  931.72544 519.8092 221.770856 540.6506  971.18267
## 709  3.0326759  0.7013402313  950.91063 521.6768 223.606397 542.5181  989.49048
## 775  3.0573187  0.6939685372  967.83050 523.2954 225.197705 544.1367 1005.63656
## 772  3.0669632  0.6918366187  974.48973 523.9253 225.817124 544.7666 1011.99125
## 733  3.0672488  0.6949525731  974.68724 523.9439 225.835438 544.7853 1012.17973
## 643  3.0801333  0.6921685461  983.61717 524.7823 226.659942 545.6236 1020.70128
## 745  3.1003310  0.6844156300  997.69099 526.0895 227.945738 546.9308 1034.13147
## 713  3.1062693  0.6861216170 1001.84628 526.4722 228.322229 547.3136 1038.09673
## 774  3.1167315  0.6842420545 1009.18659 527.1447 228.983854 547.9860 1045.10136
## 748  3.1921198  0.6659376342 1062.80865 531.9247 233.688729 552.7661 1096.27116
## 842  3.1956591  0.6635250971 1065.35753 532.1464 233.906954 552.9877 1098.70348
## 726  3.1961640  0.6637124594 1065.72144 532.1780 233.938071 553.0193 1099.05075
## 765  3.1985300  0.6683094761 1067.42717 532.3260 234.083797 553.1673 1100.67847
## 843  3.2028251  0.6625478700 1070.52699 532.5944 234.348084 553.4357 1103.63653
## 777  3.2192114  0.6624479681 1082.39129 533.6150 235.353211 554.4564 1114.95825
## 725  3.2319085  0.6596503003 1091.62616 534.4023 236.128643 555.2436 1123.77078
## 834  3.2373011  0.6564004991 1095.55929 534.7357 236.457083 555.5771 1127.52405
## 727  3.2710632  0.6488266266 1120.33299 536.8107 238.501397 557.6521 1151.16479
## 801  3.2816940  0.6456565536 1128.18676 537.4597 239.140857 558.3010 1158.65939
## 644  3.2877460  0.6480844441 1132.66924 537.8282 239.504000 558.6695 1162.93688
## 641  3.2897539  0.6502692379 1134.15818 537.9503 239.624334 558.7916 1164.35773
## 747  3.2968876  0.6426372572 1139.45565 538.3835 240.051295 559.2248 1169.41293
## 754  3.2991183  0.6456539160 1141.11456 538.5188 240.184625 559.3601 1170.99598
## 781  3.3098354  0.6428604595 1149.10004 539.1674 240.823956 560.0088 1178.61626
## 755  3.3128805  0.6421409449 1151.37369 539.3513 241.005244 560.1927 1180.78593
## 639  3.3242831  0.6427802287 1159.90615 540.0385 241.682655 560.8799 1188.92818
## 837  3.3265982  0.6380292443 1161.64209 540.1777 241.819916 561.0191 1190.58474
## 712  3.3442962  0.6357577759 1174.95269 541.2390 242.866166 562.0803 1203.28662
## 652  3.3450893  0.6320869608 1175.55082 541.2864 242.912924 562.1277 1203.85739
## 715  3.3501094  0.6356628948 1179.34019 541.5863 243.208650 562.4277 1207.47346
## 676  3.3501673  0.6307957035 1179.38394 541.5898 243.212059 562.4311 1207.51522
## 649  3.3505093  0.6325601624 1179.64230 541.6102 243.232190 562.4515 1207.76176
## 844  3.3532381  0.6315419632 1181.70475 541.7730 243.392741 562.6144 1209.72989
## 782  3.3554186  0.6307485159 1183.35395 541.9030 243.520936 562.7444 1211.30366
## 846  3.3570944  0.6322607708 1184.62221 542.0029 243.619410 562.8442 1212.51392
## 783  3.3870620  0.6252134219 1207.40823 543.7803 245.372292 564.6216 1234.25790
## 751  3.3874270  0.6217910200 1207.68704 543.8018 245.393551 564.6432 1234.52395
## 826  3.3903596  0.6231655431 1209.92799 543.9749 245.564258 564.8163 1236.66242
## 838  3.3944966  0.6239353257 1213.09252 544.2188 245.804826 565.0602 1239.68223
## 769  3.3978410  0.6233911959 1215.65365 544.4158 245.999100 565.2571 1242.12623
## 651  3.4016862  0.6219647176 1218.60134 544.6420 246.222229 565.4833 1244.93912
## 810  3.4072434  0.6218125739 1222.86738 544.9684 246.544275 565.8098 1249.01006
## 804  3.4135128  0.6184268161 1227.68852 545.3361 246.906982 566.1775 1253.61073
## 803  3.4247985  0.6149277117 1236.38943 545.9962 247.558261 566.8376 1261.91372
## 750  3.4270030  0.6131731722 1238.09238 546.1249 247.685235 566.9663 1263.53879
## 685  3.4362537  0.6140953108 1245.25047 546.6641 248.217196 567.5054 1270.36953
## 744  3.4366914  0.6145804706 1245.58960 546.6895 248.242328 567.5309 1270.69315
## 789  3.4418399  0.6133523478 1249.58234 546.9889 248.537753 567.8303 1274.50329
## 756  3.4524330  0.6116487503 1257.81610 547.6035 249.144232 568.4449 1282.36051
## 679  3.4631486  0.6059921341 1266.17090 548.2233 249.755890 569.0647 1290.33322
## 768  3.4868124  0.6046427114 1284.71286 549.5853 251.100132 570.4267 1308.02721
## 811  3.4900521  0.6018934241 1287.26123 549.7710 251.283478 570.6124 1310.45905
## 845  3.5113653  0.5938988153 1304.08498 550.9887 252.485521 571.8301 1326.51341
## 752  3.5361636  0.5886227057 1323.78874 552.3962 253.875209 573.2376 1345.31608
## 806  3.5397523  0.5892574511 1326.65162 552.5991 254.075527 573.4404 1348.04803
## 688  3.5496787  0.5884168859 1334.58565 553.1591 254.628592 574.0005 1355.61921
## 640  3.5499938  0.5911431983 1334.83783 553.1769 254.646121 574.0182 1355.85986
## 743  3.5505727  0.5857788906 1335.30126 553.2095 254.678326 574.0508 1356.30210
## 687  3.5527769  0.5882096629 1337.06653 553.3336 254.800903 574.1750 1357.98664
## 807  3.5540657  0.5857814959 1338.09919 553.4062 254.872540 574.2475 1358.97207
## 682  3.5540900  0.5856853069 1338.11868 553.4075 254.873892 574.2489 1358.99067
## 736  3.5605026  0.5878587319 1343.26230 553.7680 255.229942 574.6094 1363.89906
## 771  3.5732266  0.5864488755 1353.49597 554.4815 255.934587 575.3229 1373.66473
## 812  3.5794671  0.5831539001 1358.52840 554.8305 256.279285 575.6719 1378.46701
## 673  3.5856983  0.5801731394 1363.56207 555.1784 256.622883 576.0197 1383.27048
## 729  3.5949932  0.5810026766 1371.08697 555.6961 257.134341 576.5375 1390.45125
## 681  3.6064013  0.5730723771 1380.34926 556.3298 257.760317 577.1711 1399.28996
## 724  3.6233054  0.5747417825 1394.12775 557.2650 258.684320 578.1064 1412.43833
## 737  3.6375371  0.5709869349 1405.77795 558.0491 259.458991 578.8904 1423.55574
## 662  3.6401819  0.5707478803 1407.94803 558.1944 259.602628 579.0358 1425.62658
## 730  3.6431413  0.5700062016 1410.37817 558.3570 259.763234 579.1983 1427.94558
## 841  3.6459413  0.5625561817 1412.67919 558.5106 259.915069 579.3520 1430.14138
## 753  3.6562565  0.5658646229 1421.17142 559.0757 260.473450 579.9170 1438.24523
## 648  3.6608238  0.5663782539 1424.93914 559.3253 260.720190 580.1667 1441.84065
## 808  3.6672744  0.5614992622 1430.26863 559.6775 261.068170 580.5188 1446.92640
## 800  3.6971016  0.5520938449 1455.03356 561.2975 262.669457 582.1389 1470.55878
## 799  3.7065901  0.5503644236 1462.95373 561.8102 263.176206 582.6515 1478.11675
## 836  3.7123398  0.5521071906 1467.76301 562.1202 263.482665 582.9615 1482.70609
## 675  3.7229848  0.5482665148 1476.68645 562.6928 264.048813 583.5342 1491.22144
## 732  3.7230349  0.5516463647 1476.72851 562.6955 264.051474 583.5369 1491.26157
## 746  3.7231208  0.5466097301 1476.80068 562.7002 264.056040 583.5415 1491.33045
## 647  3.7277568  0.5525241342 1480.69508 562.9490 264.302101 583.7904 1495.04675
## 757  3.7286159  0.5459803909 1481.41729 562.9951 264.347666 583.8365 1495.73593
## 740  3.7303930  0.5453153074 1482.91169 563.0904 264.441884 583.9318 1497.16199
## 684  3.7325905  0.5476499736 1484.76072 563.2082 264.558337 584.0496 1498.92646
## 741  3.7478008  0.5388656602 1497.58844 564.0215 265.362525 584.8629 1511.16753
## 683  3.7505250  0.5441332106 1499.89142 564.1669 265.506220 585.0082 1513.36520
## 780  3.7507457  0.5441987368 1500.07808 564.1786 265.517858 585.0200 1513.54333
## 655  3.7572862  0.5444395744 1505.61452 564.5271 265.862423 585.3685 1518.82657
## 690  3.7627947  0.5397770505 1510.28481 564.8201 266.152161 585.6615 1523.28327
## 839  3.7701019  0.5371392671 1516.49068 565.2081 266.535870 586.0495 1529.20533
## 656  3.7762447  0.5374330489 1521.71697 565.5337 266.857875 586.3751 1534.19262
## 731  3.7878083  0.5369337832 1531.57831 566.1452 267.462645 586.9866 1543.60298
## 786  3.7976449  0.5308179254 1539.99065 566.6639 267.975677 587.5053 1551.63061
## 792  3.7982571  0.5305852754 1540.51487 566.6962 268.007559 587.5375 1552.13085
## 650  3.8041493  0.5326221713 1545.56522 567.0062 268.314201 587.8475 1556.95023
## 813  3.8067869  0.5299504532 1547.82846 567.1448 268.451314 587.9862 1559.10998
## 840  3.8099024  0.5221908928 1550.50382 567.3084 268.613151 588.1498 1561.66298
## 691  3.8120177  0.5251162627 1552.32154 567.4194 268.722959 588.2608 1563.39758
## 809  3.8194372  0.5263937047 1558.70521 567.8083 269.107642 588.6497 1569.48930
## 788  3.8388967  0.5212761293 1575.50705 568.8247 270.113112 589.6661 1585.52276
## 793  3.8449257  0.5211184969 1580.72993 569.1386 270.423616 589.9799 1590.50679
## 668  3.8528129  0.5155018929 1587.57496 569.5484 270.829103 590.3898 1597.03878
## 674  3.8535409  0.5136170457 1588.20750 569.5862 270.866490 590.4276 1597.64240
## 749  3.8798675  0.5057174579 1611.16125 570.9479 272.213849 591.7893 1619.54642
## 678  3.8810461  0.5095774170 1612.19259 571.0087 272.273964 591.8500 1620.53060
## 692  3.8932621  0.5073109411 1622.89985 571.6372 272.895939 592.4785 1630.74819
## 802  3.9143278  0.5007092788 1641.44296 572.7164 273.964024 593.5578 1648.44328
## 797  3.9257264  0.4944629039 1651.51834 573.2980 274.539619 594.1394 1658.05789
## 677  3.9370367  0.4940905720 1661.54450 573.8734 275.109132 594.7147 1667.62554
## 686  3.9481330  0.4946550302 1671.40914 574.4363 275.666322 595.2776 1677.03905
## 735  3.9500667  0.4946501098 1673.13100 574.5342 275.763261 595.3756 1678.68216
## 796  3.9517612  0.4894835971 1674.64065 574.6200 275.848176 595.4613 1680.12277
## 835  3.9579574  0.4884982833 1680.16612 574.9333 276.158360 595.7747 1685.39555
## 659  3.9857349  0.4847637988 1705.04358 576.3321 277.543109 597.1734 1709.13531
## 785  3.9870726  0.4804904281 1706.24601 576.3992 277.609557 597.2405 1710.28275
## 805  4.0297911  0.4692878807 1744.85686 578.5306 279.720087 599.3720 1747.12785
## 661  4.0428602  0.4706254060 1756.75147 579.1782 280.361390 600.0195 1758.47849
## 791  4.0431923  0.4693588216 1757.05424 579.1946 280.377661 600.0360 1758.76741
## 680  4.0892872  0.4555227891 1799.31775 581.4618 282.623255 602.3032 1799.09812
## 787  4.1121077  0.4518678343 1820.41875 582.5749 283.725822 603.4162 1819.23412
## 739  4.1304186  0.4427679067 1837.43478 583.4635 284.606172 604.3048 1835.47198
## 760  4.1802565  0.4266289530 1884.13138 585.8622 286.983013 606.7036 1880.03304
## 689  4.1876434  0.4340476337 1891.10033 586.2153 287.332935 607.0567 1886.68328
## 795  4.1886367  0.4266670759 1892.03838 586.2628 287.379942 607.1041 1887.57843
## 658  4.2084216  0.4267232573 1910.76897 587.2052 288.313974 608.0466 1905.45243
## 738  4.2135583  0.4232143866 1915.64636 587.4492 288.555768 608.2906 1910.10678
## 734  4.2274689  0.4205330098 1928.88456 588.1084 289.209113 608.9498 1922.73956
## 667  4.2626406  0.4065351846 1962.55095 589.7655 290.851671 610.6068 1954.86632
## 761  4.2704789  0.4021950828 1970.09174 590.1329 291.215911 610.9743 1962.06225
## 742  4.2857429  0.4003427304 1984.81625 590.8465 291.923342 611.6879 1976.11337
## 664  4.2875248  0.4043546334 1986.53854 590.9296 292.005762 611.7710 1977.75690
## 728  4.2895587  0.4029015728 1988.50539 591.0245 292.099803 611.8658 1979.63380
## 698  4.2913688  0.3974835773 1990.25651 591.1089 292.183455 611.9502 1981.30484
## 665  4.3046717  0.3956216073 2003.14888 591.7279 292.797185 612.5692 1993.60761
## 847  4.3055859  0.3923914529 2004.03640 591.7704 292.839297 612.6117 1994.45455
## 653  4.3140403  0.3954983737 2012.25241 592.1627 293.228293 613.0040 2002.29482
## 660  4.3323630  0.3952786008 2030.11371 593.0103 294.068773 613.8517 2019.33928
## 816  4.3394997  0.3841483184 2037.09125 593.3395 294.395198 614.1809 2025.99773
## 672  4.3609226  0.3815448379 2058.10523 594.3244 295.371891 615.1658 2046.05069
## 671  4.3706861  0.3786085517 2067.71664 594.7717 295.815453 615.6131 2055.22255
## 798  4.3729896  0.3780334397 2069.98738 594.8771 295.919960 615.7184 2057.38945
## 666  4.3737765  0.3818003013 2070.76337 594.9131 295.955648 615.7544 2058.12995
## 759  4.4031532  0.3661835068 2099.83276 596.2519 297.283485 617.0933 2085.86993
## 817  4.4033288  0.3685553623 2100.00703 596.2599 297.291392 617.1012 2086.03623
## 784  4.4094658  0.3677970489 2106.10463 596.5384 297.567675 617.3798 2091.85498
## 762  4.4106270  0.3629957567 2107.25939 596.5911 297.619911 617.4324 2092.95693
## 794  4.4165572  0.3652057639 2113.16113 596.8598 297.886455 617.7012 2098.58877
## 701  4.4235839  0.3622491857 2120.16438 597.1777 298.201828 618.0191 2105.27175
## 790  4.4443388  0.3575788980 2140.91505 598.1139 299.130483 618.9553 2125.07344
## 695  4.4513860  0.3524699600 2147.98289 598.4308 299.444830 619.2722 2131.81805
## 654  4.4606715  0.3546551863 2157.31269 598.8476 299.858274 619.6889 2140.72119
## 815  4.4733547  0.3483153200 2170.08778 599.4154 300.421634 620.2568 2152.91204
## 702  4.4925500  0.3436082166 2189.49126 600.2718 301.271270 621.1132 2171.42816
## 818  4.5115366  0.3386908152 2208.76544 601.1153 302.108157 621.9566 2189.82089
## 697  4.5242428  0.3314857155 2221.70946 601.6778 302.666282 622.5191 2202.17295
## 670  4.5643894  0.3220262164 2262.84690 603.4447 304.419659 624.2860 2241.42909
## 700  4.5816644  0.3178537672 2280.66006 604.2002 305.169463 625.0415 2258.42761
## 763  4.5915489  0.3079750507 2290.88278 604.6312 305.597239 625.4726 2268.18282
## 657  4.6408501  0.3004842272 2342.19978 606.7672 307.717411 627.6086 2317.15299
## 819  4.6661866  0.2903199976 2368.78541 607.8561 308.798375 628.6975 2342.52280
## 705  4.6798704  0.2843302915 2383.20401 608.4418 309.379781 629.2831 2356.28201
## 706  4.7031004  0.2772820835 2407.77821 609.4321 310.362970 630.2735 2379.73237
## 758  4.7481074  0.2623959956 2455.73573 611.3369 312.254300 632.1783 2425.49669
## 663  4.7599645  0.2614003577 2468.44619 611.8357 312.749631 632.6771 2437.62588
## 694  4.7615639  0.2613623958 2470.16309 611.9029 312.816352 632.7443 2439.26426
## 704  4.7999164  0.2481693529 2511.50675 613.5074 314.409710 634.3488 2478.71718
## 669  4.8882878  0.2297571138 2608.03268 617.1561 318.033807 637.9975 2570.82878
## 707  4.9059985  0.2183432461 2627.58955 617.8794 318.752342 638.7208 2589.49127
## 696  4.9109321  0.2162809864 2633.04998 618.0805 318.952043 638.9218 2594.70198
## 814  4.9121020  0.2121752079 2634.34558 618.1281 318.999369 638.9695 2595.93834
## 699  5.0218212  0.1797288419 2757.22890 622.5462 323.389142 643.3876 2713.20194
## 693  5.0381597  0.1713424540 2775.75990 623.1959 324.034718 644.0372 2730.88547
## 703  5.1508078  0.1408293070 2905.16261 627.6184 328.430250 648.4598 2854.37031
## 909  1.6500438  0.9095391346  222.95895 401.9481 106.463640 425.3946  296.13997
## 906  1.9744481  0.8702038388  355.52215 437.8455 140.504647 461.2920  424.03089
## 962  1.9995463  0.8656409599  366.76714 440.3718 142.922511 463.8183  434.87955
## 905  2.0032299  0.8661505544  368.42948 440.7399 143.275032 464.1864  436.48329
## 916  2.1007969  0.8521520376  413.57376 450.2511 152.402074 473.6976  480.03643
## 963  2.2004026  0.8378802530  461.87553 459.5158 161.324788 482.9623  526.63577
## 961  2.2047217  0.8362727057  464.02061 459.9080 161.703163 483.3545  528.70525
## 908  2.2523672  0.8314107386  487.96269 464.1841 165.831942 487.6306  551.80347
## 937  2.2746569  0.8268193588  499.33916 466.1536 167.735610 489.6001  562.77897
## 917  2.2868560  0.8243068614  505.61286 467.2233 168.770112 490.6698  568.83155
## 851  2.2995942  0.8240738625  512.19968 468.3342 169.844849 491.7808  575.18621
## 965  2.3312946  0.8167214234  528.75048 471.0725 172.495454 494.5190  591.15366
## 934  2.3341623  0.8171577689  530.25887 471.3183 172.733566 494.7649  592.60890
## 915  2.3585375  0.8146139993  543.15504 473.3961 174.746504 496.8426  605.05053
## 850  2.3609410  0.8160276485  544.43389 473.5998 174.943926 497.0463  606.28430
## 863  2.4007603  0.8063761006  565.81079 476.9448 178.187547 500.3913  626.90776
## 907  2.4363903  0.8033470267  585.24181 479.8912 181.047317 503.3378  645.65392
## 912  2.4703332  0.7948572051  604.01897 482.6583 183.735258 506.1049  663.76927
## 936  2.4947432  0.7910769059  617.68312 484.6249 185.646829 508.0714  676.95181
## 914  2.5181287  0.7884886283  630.89979 486.4909 187.461670 509.9374  689.70266
## 944  2.5267970  0.7849598746  635.83014 487.1782 188.130332 510.6247  694.45923
## 943  2.5518522  0.7810420055  650.17633 489.1516 190.050929 512.5981  708.29978
## 860  2.5551282  0.7818944015  652.06256 489.4082 190.300727 512.8547  710.11952
## 904  2.5652847  0.7802662892  657.92580 490.2016 191.073248 513.6481  715.77611
## 848  2.5722028  0.7808353742  661.93287 490.7402 191.597794 514.1868  719.64195
## 960  2.5871426  0.7763185527  670.62307 491.8985 192.726017 515.3450  728.02586
## 859  2.5879649  0.7766484302  671.10282 491.9621 192.787932 515.4086  728.48870
## 945  2.6022302  0.7722113676  679.45025 493.0615 193.859139 516.5080  736.54193
## 933  2.6333246  0.7686443323  697.80441 495.4371 196.174893 518.8837  754.24918
## 857  2.6384220  0.7664607027  700.83405 495.8239 196.552039 519.2704  757.17203
## 913  2.6500015  0.7639383299  707.73815 496.6997 197.406225 520.1463  763.83280
## 862  2.6671301  0.7621899040  718.00620 497.9883 198.663256 521.4348  773.73894
## 911  2.7005857  0.7556865707  738.25257 500.4814 201.096493 523.9280  793.27172
## 854  2.7269686  0.7509109917  754.39676 502.4258 202.995167 525.8724  808.84689
## 964  2.7614811  0.7429569514  775.75259 504.9411 205.452629 528.3877  829.45001
## 940  2.8030731  0.7360941131  801.84592 507.9310 208.375515 531.3775  854.62367
## 932  2.8306624  0.7338900886  819.36962 509.8899 210.291575 533.3364  871.52973
## 861  2.8485535  0.7301078262  830.82513 511.1500 211.524576 534.5965  882.58148
## 858  2.8708303  0.7268453952  845.18970 512.7080 213.049510 536.1545  896.43977
## 942  2.8777078  0.7243404853  849.64702 513.1865 213.518008 536.6331  900.73999
## 939  2.8796450  0.7216423186  850.90444 513.3211 213.649776 536.7676  901.95309
## 856  2.8850248  0.7203452406  854.40092 513.6944 214.015271 537.1409  905.32633
## 935  2.8879319  0.7244488946  856.29302 513.8958 214.212501 537.3424  907.15174
## 849  2.9066749  0.7199343843  868.53788 515.1897 215.479569 538.6362  918.96503
## 921  2.9106459  0.7195075512  871.14227 515.4627 215.747007 538.9092  921.47762
## 941  2.9175232  0.7152314943  875.66128 515.9347 216.209361 539.3812  925.83736
## 927  2.9392562  0.7110654207  890.01181 517.4190 217.663595 540.8656  939.68209
## 926  3.0267167  0.6921207196  948.83959 523.2834 223.413348 546.7299  996.43645
## 922  3.0382723  0.6944534402  956.74112 524.0455 224.161049 547.4920 1004.05949
## 853  3.0414339  0.6942058442  958.90820 524.2535 224.365140 547.7001 1006.15018
## 873  3.0483191  0.6930643192  963.63548 524.7058 224.808906 548.1523 1010.71084
## 918  3.0815986  0.6824283040  986.63506 526.8774 226.940299 550.3239 1032.89979
## 910  3.0827722  0.6836963710  987.45070 526.9536 227.015057 550.4001 1033.68668
## 885  3.0905736  0.6793873859  992.88048 527.4590 227.511310 550.9056 1038.92509
## 967  3.1324489  0.6698385843 1022.26031 530.1507 230.154572 553.5973 1067.26941
## 928  3.1496160  0.6680518499 1034.41903 531.2438 231.228358 554.6903 1078.99958
## 866  3.1677611  0.6636219639 1047.34270 532.3927 232.357199 555.8393 1091.46775
## 888  3.1817782  0.6615290514 1057.37701 533.2757 233.224959 556.7223 1101.14839
## 954  3.1935202  0.6571322635 1065.81685 534.0125 233.949045 557.4590 1109.29077
## 930  3.1967287  0.6610022706 1068.12844 534.2133 234.146453 557.6598 1111.52088
## 955  3.2017201  0.6557553008 1071.72915 534.5253 234.453178 557.9719 1114.99468
## 865  3.2033004  0.6581273531 1072.87027 534.6240 234.550189 558.0706 1116.09558
## 946  3.2235811  0.6526450215 1087.56535 535.8863 235.791121 559.3328 1130.27273
## 867  3.2381302  0.6492185406 1098.16458 536.7869 236.676714 560.2335 1140.49838
## 855  3.2704581  0.6464947138 1121.88669 538.7737 238.630789 562.2203 1163.38438
## 966  3.2709854  0.6416499982 1122.27556 538.8060 238.662506 562.2525 1163.75955
## 852  3.2743453  0.6445989049 1124.75498 539.0113 238.864494 562.4578 1166.15158
## 887  3.2830702  0.6379777038 1131.20530 539.5435 239.388071 562.9900 1172.37456
## 894  3.2908155  0.6404169729 1136.94586 540.0148 239.851742 563.4613 1177.91279
## 895  3.3054797  0.6368489868 1147.85142 540.9040 240.726716 564.3506 1188.43397
## 949  3.3240069  0.6304986724 1161.69914 542.0219 241.826826 565.4684 1201.79361
## 956  3.3501179  0.6252224073 1181.34661 543.5868 243.367210 567.0333 1220.74860
## 938  3.3532034  0.6256204474 1183.67848 543.7709 243.548466 567.2175 1222.99828
## 958  3.3555767  0.6257272564 1185.47356 543.9124 243.687773 567.3590 1224.73009
## 929  3.3606806  0.6194356871 1189.33828 544.2164 243.987040 567.6629 1228.45860
## 891  3.3833224  0.6163836163 1206.55377 545.5593 245.309353 569.0059 1245.06732
## 950  3.3896845  0.6168189255 1211.41194 545.9351 245.679364 569.3816 1249.75425
## 925  3.3963218  0.6133739522 1216.48995 546.3263 246.064663 569.7728 1254.65329
## 884  3.4133249  0.6120023066 1229.54389 547.3251 247.048380 570.7716 1267.24713
## 890  3.4215659  0.6072606679 1235.89426 547.8074 247.523454 571.2539 1273.37368
## 896  3.4420542  0.6072091515 1251.74864 549.0014 248.699768 572.4479 1288.66926
## 957  3.5106386  0.5874971751 1305.50965 552.9473 252.588588 576.3938 1340.53543
## 883  3.5214446  0.5847725611 1314.07688 553.5620 253.194566 577.0085 1348.80069
## 892  3.5292639  0.5832383127 1320.29255 554.0056 253.631926 577.4521 1354.79729
## 923  3.5469819  0.5823167950 1334.42796 555.0071 254.619488 578.4537 1368.43449
## 876  3.5534793  0.5817623284 1339.62925 555.3731 254.980432 578.8197 1373.45245
## 869  3.5564723  0.5815958119 1342.02848 555.5415 255.146489 578.9881 1375.76713
## 920  3.5660341  0.5799215441 1349.70670 556.0785 255.676073 579.5251 1383.17472
## 864  3.5679229  0.5808458950 1351.22589 556.1844 255.780523 579.6310 1384.64037
## 919  3.5713078  0.5774634653 1353.95033 556.3741 255.967566 579.8206 1387.26878
## 924  3.6060315  0.5641215276 1382.04858 558.3093 257.876478 581.7558 1414.37669
## 877  3.6275679  0.5650071854 1399.61231 559.5002 259.051459 582.9467 1431.32137
## 870  3.6276523  0.5642758619 1399.68137 559.5048 259.056052 582.9514 1431.38800
## 893  3.6337578  0.5638607601 1404.67972 559.8412 259.387909 583.2877 1436.21018
## 953  3.6344973  0.5577979740 1405.28570 559.8819 259.428068 583.3284 1436.79480
## 886  3.6888005  0.5470209653 1450.12117 562.8480 262.355447 586.2945 1480.05001
## 872  3.7036299  0.5462241684 1462.48067 563.6504 263.147576 587.0969 1491.97389
## 948  3.7070413  0.5432576664 1465.33089 563.8345 263.329361 587.2811 1494.72365
## 880  3.7194841  0.5401122955 1475.74906 564.5047 263.991030 587.9512 1504.77464
## 897  3.7242778  0.5395275273 1479.77215 564.7623 264.245373 588.2088 1508.65592
## 881  3.7342092  0.5345409072 1488.12339 565.2949 264.771289 588.7415 1516.71281
## 871  3.7371749  0.5408476724 1490.62152 565.4537 264.928072 588.9002 1519.12290
## 951  3.7688647  0.5278417717 1517.43906 567.1425 266.595840 590.5890 1544.99524
## 952  3.8075550  0.5134985683 1550.48788 569.1852 268.613590 592.6317 1576.87923
## 889  3.8599748  0.5031065631 1595.80268 571.9198 271.315660 595.3664 1620.59688
## 947  3.9408575  0.4824910757 1666.93808 576.0674 275.415388 599.5139 1689.22506
## 875  3.9416107  0.4856613210 1667.60741 576.1056 275.453177 599.5521 1689.87081
## 879  4.1200042  0.4338231822 1829.74756 584.9585 284.210446 608.4051 1846.29621
## 931  4.1588620  0.4199857685 1866.01682 586.8360 286.068670 610.2825 1881.28713
## 900  4.1799252  0.4166965102 1885.81915 587.8464 287.068839 611.2929 1900.39151
## 878  4.1827435  0.4213921907 1888.47632 587.9812 287.202288 611.4277 1902.95504
## 874  4.1996599  0.4171863756 1904.46330 588.7884 288.001455 612.2349 1918.37854
## 868  4.2477331  0.4016848411 1950.24740 591.0648 290.255422 614.5113 1962.54895
## 901  4.2704788  0.3919255660 1972.09164 592.1329 291.313178 615.5794 1983.62328
## 882  4.2812729  0.3905218311 1982.49876 592.6378 291.813211 616.0843 1993.66359
## 959  4.3028628  0.3817723641 2003.39350 593.6438 292.809657 617.0904 2013.82188
## 899  4.4031530  0.3541324526 2101.83255 598.2519 297.374974 621.6984 2108.79138
## 902  4.4101122  0.3531857800 2108.74740 598.5677 297.687964 622.0143 2115.46251
## 903  4.5911446  0.2970574558 2292.46414 606.6136 305.663916 630.0601 2292.70402
## 898  4.7441993  0.2504384236 2453.55336 613.1722 312.169590 636.6188 2448.11554
## 998  1.1140380  0.9577212031   57.92307 325.3860  36.299641 351.4377  136.49129
## 973  1.5616113  0.9174228892  192.93829 392.9313  96.880850 418.9830  268.19510
## 999  1.5699622  0.9163261301  195.88668 393.9980  97.868208 420.0497  271.07117
## 997  1.8238427  0.8863001528  293.02833 423.9768 125.913150 450.0285  365.83031
## 1012 1.8726847  0.8789998649  313.38361 429.2623 130.910895 455.3140  385.68635
## 969  1.8886383  0.8785770900  320.14891 430.9589 132.518157 457.0106  392.28572
## 1001 1.8899444  0.8777536408  320.70535 431.0972 132.649211 457.1489  392.82852
## 970  1.9039438  0.8760863221  326.69335 432.5732 134.048753 458.6249  398.66966
## 996  1.9363455  0.8714824611  340.72219 435.9482 137.252931 461.9999  412.35442
## 1006 1.9758317  0.8656390650  358.13837 439.9856 141.093072 466.0373  429.34345
## 980  2.0433261  0.8573842227  388.72221 446.7035 147.499191 472.7552  459.17718
## 1007 2.1697249  0.8388841564  448.76058 458.7078 158.994073 484.7595  517.74304
## 1005 2.1753831  0.8369479681  451.53246 459.2287 159.494162 485.2804  520.44694
## 972  2.1770081  0.8381748901  452.32984 459.3780 159.637560 485.4297  521.22476
## 981  2.2237325  0.8309390956  475.51223 463.6251 163.719170 489.6768  543.83857
## 979  2.3049799  0.8187329350  516.99557 470.8021 170.631464 496.8538  584.30449
## 1009 2.3092235  0.8164715476  519.20315 471.1700 170.986256 497.2217  586.45793
## 971  2.3113539  0.8199404002  520.31300 471.3544 171.164151 497.4061  587.54055
## 978  2.4292202  0.7997758876  583.30855 481.3018 180.775615 507.3535  648.99106
## 976  2.4386828  0.7962985593  588.50184 482.0793 181.528252 508.1310  654.05698
## 1000 2.4396317  0.7953355742  589.02372 482.1571 181.603574 508.2088  654.56606
## 968  2.4602781  0.7941133288  600.42943 483.8426 183.235713 509.8943  665.69203
## 1004 2.5372321  0.7812082220  643.78787 490.0025 189.207970 516.0542  707.98706
## 977  2.6119333  0.7657782676  687.15433 495.8058 194.844611 521.8575  750.28990
## 975  2.6668039  0.7559443900  719.81002 499.9638 198.888823 526.0155  782.14467
## 1008 2.7490232  0.7404158187  770.01287 506.0368 204.803703 532.0885  831.11623
## 1002 2.8802129  0.7187801129  853.27321 515.3605 213.902229 541.4122  912.33452
## 1003 2.8925236  0.7133840043  861.28547 516.2136 214.735663 542.2653  920.15026
## 985  2.8978636  0.7166435555  864.77156 516.5825 215.096128 542.6342  923.55086
## 991  2.9338117  0.7063229640  888.40670 519.0482 217.506349 545.0999  946.60632
## 990  3.0191113  0.6872582594  945.65557 524.7802 223.114361 550.8319 1002.45109
## 974  3.0203407  0.6910259263  946.49263 524.8616 223.194070 550.9133 1003.26762
## 986  3.0209605  0.6922345615  946.91485 524.9027 223.234252 550.9544 1003.67948
## 982  3.0567644  0.6811713274  971.44838 527.2591 225.541751 553.3108 1027.61130
## 1011 3.1317612  0.6637193129 1023.77460 532.1068 230.292305 558.1585 1078.65415
## 992  3.1406165  0.6637694545 1030.03674 532.6715 230.846001 558.7232 1084.76271
## 994  3.1913888  0.6560712970 1066.28252 535.8789 233.991971 561.9306 1120.11951
## 1010 3.2690052  0.6343085489 1122.81548 540.6848 238.709368 566.7365 1175.26594
## 993  3.3572846  0.6139058766 1188.76616 546.0142 243.945351 572.0659 1239.59910
## 989  3.3748773  0.6111793985 1202.11907 547.0595 244.972900 573.1112 1252.62452
## 983  3.5348126  0.5778359201 1326.71171 556.3198 254.083691 582.3715 1374.16137
## 987  3.5406965  0.5756158001 1331.40535 556.6524 254.411205 582.7041 1378.73990
## 984  3.5530967  0.5735148034 1341.32270 557.3516 255.099712 583.4033 1388.41401
## 988  3.5975391  0.5584876551 1377.15143 559.8377 257.548336 585.8894 1423.36400
## 995  4.1582855  0.4102939701 1867.47617 588.8083 286.143814 614.8600 1901.66294
## 1021 0.9901052  0.9653333027   30.52284 303.7989  18.088867 332.4558  109.02351
## 1015 1.0165692  0.9640209204   36.51001 309.0744  22.412433 337.7313  114.92945
## 1016 1.4832913  0.9238501179  168.05173 384.6404  88.203055 413.2973  244.68670
## 1014 1.7566128  0.8916005021  267.88958 418.4652 119.410973 447.1221  343.17018
## 1013 1.8242971  0.8838131240  295.21523 426.0267 126.493221 454.6835  370.12513
## 1018 1.8353099  0.8825713805  299.75908 427.2304 127.623791 455.8873  374.60733
## 1022 1.8503349  0.8793972764  306.00240 428.8610 129.156676 457.5179  380.76596
## 1017 2.4099533  0.7960608864  574.79685 481.7092 179.540087 510.3661  645.91400
## 1019 2.8688065  0.7157046677  847.88006 516.5669 213.342529 545.2238  915.29261
## 1020 2.8881097  0.7089557061  860.40883 517.9081 214.649907 546.5650  927.65141
## 1023 0.8884855  0.9715035501   11.00000 284.1403   3.041157 315.4024   88.79029
##             fpe        apc        hsp
## 4    22.8492237 0.64381621 0.23094020
## 10   28.7406601 0.80981757 0.29048575
## 5    31.3700362 0.88390477 0.31706121
## 3    32.4027473 0.91300318 0.32749896
## 9    34.5139137 0.97248893 0.34883681
## 6    34.6681138 0.97683378 0.35039533
## 7    35.0936603 0.98882429 0.35469638
## 8    35.5511292 1.00171426 0.35932008
## 1    36.7188123 1.03461574 0.37112202
## 2    36.7296898 1.03492224 0.37123196
## 40   18.0183145 0.50769703 0.18222405
## 28   19.5980591 0.55220905 0.19820044
## 36   20.2103131 0.56946036 0.20439232
## 35   20.3241582 0.57266814 0.20554367
## 37   20.9617750 0.59063409 0.21199206
## 38   22.2149161 0.62594350 0.22466541
## 39   22.6594463 0.63846890 0.22916107
## 13   22.7062399 0.63978740 0.22963430
## 21   22.9516774 0.64670302 0.23211648
## 45   24.0938399 0.67888541 0.24366747
## 34   24.8840380 0.70115061 0.25165896
## 29   26.3963801 0.74376345 0.26695368
## 49   27.4141679 0.77244137 0.27724684
## 55   27.6763414 0.77982856 0.27989827
## 54   28.0454118 0.79022775 0.28363078
## 52   28.3459740 0.79869661 0.28667045
## 33   29.1301416 0.82079188 0.29460095
## 19   29.1585966 0.82159365 0.29488872
## 27   29.3135302 0.82595917 0.29645561
## 30   29.3596122 0.82725761 0.29692164
## 44   29.7328395 0.83777394 0.30069619
## 42   30.2916871 0.85352043 0.30634797
## 41   30.4993617 0.85937202 0.30844824
## 31   31.2259221 0.87984411 0.31579614
## 14   31.3433855 0.88315384 0.31698408
## 32   31.3923003 0.88453210 0.31747877
## 43   31.5276411 0.88834556 0.31884750
## 22   31.8539979 0.89754122 0.32214804
## 48   32.1558105 0.90604531 0.32520035
## 20   32.5810493 0.91802715 0.32950090
## 12   32.6706528 0.92055188 0.33040709
## 46   33.1468099 0.93396843 0.33522259
## 51   33.6690099 0.94868231 0.34050374
## 47   33.7366911 0.95058935 0.34118822
## 53   34.2676096 0.96554889 0.34655754
## 50   34.4477590 0.97062491 0.34837944
## 18   34.5357324 0.97310371 0.34926914
## 15   34.9829861 0.98570585 0.35379234
## 26   35.1087764 0.98925021 0.35506449
## 23   35.1563150 0.99058969 0.35554526
## 16   35.5169892 1.00075231 0.35919285
## 24   35.6916264 1.00567301 0.36095900
## 25   36.1052287 1.01732697 0.36514188
## 17   36.1056034 1.01733753 0.36514567
## 11   37.2057855 1.04833705 0.37627210
## 125  14.6463883 0.41268720 0.14824280
## 145  15.3722804 0.43314046 0.15558988
## 121  15.4136841 0.43430708 0.15600895
## 149  15.6893919 0.44207562 0.15879951
## 120  16.1930451 0.45626692 0.16389722
## 152  16.8844613 0.47574876 0.17089536
## 154  17.3361193 0.48847500 0.17546679
## 122  17.6644845 0.49772726 0.17879033
## 146  17.7560542 0.50030739 0.17971715
## 76   17.8957494 0.50424355 0.18113107
## 155  17.9583611 0.50600774 0.18176479
## 104  18.2983806 0.51558837 0.18520628
## 141  18.3436248 0.51686320 0.18566422
## 142  18.5210941 0.52186371 0.18746047
## 124  18.5229288 0.52191540 0.18747904
## 123  18.6641943 0.52589580 0.18890885
## 130  19.1078758 0.53839729 0.19339955
## 64   19.1633842 0.53996134 0.19396138
## 147  19.2841189 0.54336325 0.19518339
## 92   19.3393843 0.54492044 0.19574276
## 148  19.5300257 0.55029209 0.19767232
## 72   19.6799194 0.55451561 0.19918947
## 71   19.7037337 0.55518661 0.19943050
## 144  20.0422777 0.56472568 0.20285706
## 143  20.2433357 0.57039083 0.20489206
## 100  20.2488027 0.57054487 0.20494740
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## 651  13.3134104 0.37512826 0.13524391
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## 654  22.8929265 0.64504761 0.23255716
## 815  23.0232958 0.64872099 0.23388151
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## 670  23.9698993 0.67539318 0.24349756
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## 657  24.7796926 0.69821050 0.25172382
## 819  25.0509980 0.70585499 0.25447987
## 705  25.1981394 0.71000095 0.25597460
## 706  25.4489182 0.71706708 0.25852213
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## 694  26.0855537 0.73500538 0.26498937
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## 699  29.0150502 0.81754898 0.29474858
## 693  29.2041583 0.82287743 0.29666963
## 703  30.5247083 0.86008620 0.31008440
## 909   3.1961481 0.09005697 0.03252084
## 906   4.5764357 0.12894895 0.04656528
## 962   4.6935219 0.13224806 0.04775663
## 905   4.7108307 0.13273576 0.04793275
## 916   5.1808864 0.14598039 0.05271557
## 963   5.6838188 0.16015138 0.05783291
## 961   5.7061540 0.16078071 0.05806018
## 908   5.9554461 0.16780495 0.06059672
## 937   6.0739013 0.17114263 0.06180201
## 917   6.1392250 0.17298323 0.06246668
## 851   6.2078089 0.17491570 0.06316452
## 965   6.3801407 0.17977145 0.06491800
## 934   6.3958466 0.18021399 0.06507780
## 915   6.5301253 0.18399752 0.06644409
## 850   6.5434411 0.18437272 0.06657958
## 863   6.7660237 0.19064437 0.06884436
## 907   6.9683453 0.19634513 0.07090298
## 912   7.1638588 0.20185405 0.07289234
## 936   7.3061339 0.20586290 0.07433999
## 914   7.4437498 0.20974047 0.07574023
## 944   7.4950861 0.21118695 0.07626258
## 943   7.6444629 0.21539590 0.07778249
## 860   7.6641028 0.21594929 0.07798232
## 904   7.7251527 0.21766948 0.07860351
## 848   7.7668755 0.21884509 0.07902804
## 960   7.8573605 0.22139466 0.07994872
## 859   7.8623558 0.22153541 0.07999955
## 945   7.9492717 0.22398442 0.08088392
## 933   8.1403807 0.22936924 0.08282846
## 857   8.1719262 0.23025809 0.08314943
## 913   8.2438138 0.23228364 0.08388089
## 862   8.3507277 0.23529613 0.08496874
## 911   8.5615390 0.24123610 0.08711375
## 854   8.7296371 0.24597255 0.08882415
## 964   8.9520003 0.25223802 0.09108669
## 940   9.2236919 0.25989340 0.09385116
## 932   9.4061539 0.26503458 0.09570771
## 861   9.5254320 0.26839544 0.09692137
## 858   9.6750003 0.27260979 0.09844323
## 942   9.7214113 0.27391750 0.09891546
## 939   9.7345039 0.27428640 0.09904868
## 856   9.7709103 0.27531222 0.09941911
## 935   9.7906113 0.27586733 0.09961957
## 849   9.9181085 0.27945978 0.10091685
## 921   9.9452262 0.28022387 0.10119278
## 941   9.9922795 0.28154968 0.10167155
## 927  10.1417015 0.28575990 0.10319192
## 926  10.7542338 0.30301905 0.10942444
## 922  10.8365070 0.30533724 0.11026157
## 853  10.8590712 0.30597302 0.11049116
## 873  10.9082930 0.30735993 0.11099199
## 918  11.1477715 0.31410765 0.11342869
## 910  11.1562643 0.31434695 0.11351510
## 885  11.2128008 0.31593996 0.11409036
## 967  11.5187123 0.32455955 0.11720301
## 928  11.6453125 0.32812673 0.11849117
## 866  11.7798776 0.33191833 0.11986037
## 888  11.8843579 0.33486224 0.12092346
## 954  11.9722360 0.33733836 0.12181762
## 930  11.9963049 0.33801654 0.12206252
## 955  12.0337966 0.33907293 0.12244400
## 865  12.0456783 0.33940772 0.12256490
## 946  12.1986879 0.34371903 0.12412177
## 867  12.3090502 0.34682868 0.12524471
## 855  12.5560518 0.35378837 0.12775796
## 966  12.5601009 0.35390246 0.12779915
## 852  12.5859174 0.35462989 0.12806184
## 887  12.6530800 0.35652231 0.12874522
## 894  12.7128525 0.35820650 0.12935340
## 895  12.8264044 0.36140602 0.13050880
## 949  12.9705909 0.36546873 0.13197589
## 956  13.1751663 0.37123299 0.13405745
## 938  13.1994464 0.37191712 0.13430450
## 958  13.2181373 0.37244377 0.13449468
## 929  13.2583780 0.37357762 0.13490413
## 891  13.4376308 0.37862838 0.13672803
## 950  13.4882154 0.38005369 0.13724273
## 925  13.5410892 0.38154350 0.13778072
## 884  13.6770107 0.38537332 0.13916372
## 890  13.7431327 0.38723642 0.13983652
## 896  13.9082132 0.39188785 0.14151621
## 957  14.4679889 0.40766049 0.14721193
## 883  14.5571934 0.41017398 0.14811959
## 892  14.6219128 0.41199756 0.14877811
## 923  14.7690950 0.41614467 0.15027569
## 876  14.8232523 0.41767065 0.15082674
## 869  14.8482339 0.41837455 0.15108093
## 920  14.9281818 0.42062722 0.15189440
## 864  14.9440001 0.42107293 0.15205535
## 919  14.9723677 0.42187223 0.15234399
## 924  15.2649350 0.43011582 0.15532087
## 877  15.4478138 0.43526875 0.15718166
## 870  15.4485330 0.43528901 0.15718898
## 893  15.5005773 0.43675545 0.15771853
## 953  15.5068869 0.43693323 0.15778273
## 886  15.9737271 0.45008726 0.16253284
## 872  16.1024179 0.45371334 0.16384227
## 948  16.1320953 0.45454955 0.16414423
## 880  16.2405724 0.45760609 0.16524799
## 897  16.2824620 0.45878640 0.16567422
## 881  16.3694176 0.46123652 0.16655899
## 871  16.3954289 0.46196943 0.16682366
## 951  16.6746612 0.46983728 0.16966485
## 952  17.0187752 0.47953329 0.17316621
## 889  17.4906065 0.49282795 0.17796710
## 947  18.2312894 0.51369796 0.18550356
## 875  18.2382588 0.51389434 0.18557447
## 879  19.9265103 0.56146373 0.20275245
## 931  20.3041567 0.57210457 0.20659500
## 900  20.5103444 0.57791426 0.20869296
## 878  20.5380118 0.57869384 0.20897448
## 874  20.7044729 0.58338416 0.21066822
## 868  21.1811907 0.59681651 0.21551883
## 901  21.4086394 0.60322527 0.21783312
## 882  21.5170014 0.60627855 0.21893571
## 959  21.7345637 0.61240875 0.22114941
## 899  22.7595404 0.64128923 0.23157856
## 902  22.8315399 0.64331794 0.23231115
## 903  24.7444534 0.69721757 0.25177506
## 898  26.4217624 0.74447864 0.26884170
## 998   1.4865691 0.04188664 0.01515361
## 973   2.9209962 0.08230410 0.02977570
## 999   2.9523205 0.08318671 0.03009501
## 997   3.9843717 0.11226653 0.04061541
## 1012  4.2006300 0.11835998 0.04281988
## 969   4.2725059 0.12038521 0.04355256
## 1001  4.2784177 0.12055178 0.04361282
## 970   4.3420353 0.12234432 0.04426132
## 996   4.4910803 0.12654392 0.04578064
## 1006  4.6761131 0.13175754 0.04766680
## 980   5.0010416 0.14091296 0.05097902
## 1007  5.6389006 0.15888573 0.05748115
## 1005  5.6683496 0.15971551 0.05778134
## 972   5.6768211 0.15995421 0.05786770
## 981   5.9231151 0.16689397 0.06037834
## 979   6.3638420 0.17931220 0.06487097
## 1009  6.3872957 0.17997305 0.06511005
## 971   6.3990870 0.18030529 0.06523024
## 978   7.0683636 0.19916331 0.07205264
## 976   7.1235380 0.20071795 0.07261507
## 1000  7.1290827 0.20087418 0.07267159
## 968   7.2502590 0.20428853 0.07390682
## 1004  7.7109073 0.21726809 0.07860252
## 977   8.1716407 0.23025004 0.08329909
## 975   8.5185810 0.24002568 0.08683569
## 1008  9.0519455 0.25505415 0.09227264
## 1002  9.9365191 0.27997853 0.10128970
## 1003 10.0216428 0.28237704 0.10215742
## 985  10.0586798 0.28342062 0.10253496
## 991  10.3097840 0.29049591 0.10509464
## 990  10.9180067 0.30763363 0.11129467
## 974  10.9268998 0.30788421 0.11138532
## 986  10.9313856 0.30801061 0.11143105
## 982  11.1920344 0.31535483 0.11408802
## 1011 11.7479580 0.33101894 0.11975492
## 992  11.8144882 0.33289354 0.12043311
## 994  12.1995702 0.34374389 0.12435851
## 1010 12.8001870 0.36066730 0.13048101
## 993  13.5008595 0.38040996 0.13762344
## 989  13.6427234 0.38440722 0.13906956
## 983  14.9664190 0.42170462 0.15256289
## 987  15.0162852 0.42310969 0.15307121
## 984  15.1216490 0.42607849 0.15414525
## 988  15.5023002 0.43680399 0.15802549
## 995  20.7116028 0.58358506 0.21112745
## 1021  1.1981547 0.03376007 0.01223856
## 1015  1.2630603 0.03558890 0.01290153
## 1016  2.6890762 0.07576935 0.02746758
## 1014  3.7713972 0.10626561 0.03852295
## 1013  4.0676287 0.11461244 0.04154881
## 1018  4.1168876 0.11600039 0.04205197
## 1022  4.1845701 0.11790747 0.04274331
## 1017  7.0985138 0.20001285 0.07250780
## 1019 10.0589509 0.28342826 0.10274720
## 1020 10.1947726 0.28725527 0.10413455
## 1023  0.9845406 0.02774113 0.01007924

En el objeto k se encuentran todos los resultados. La función ols_step_all_possible permite estimar 11 indicadores diferentes que facilitan la elección del mejor modelo. Por faciidad, sólo se muestran 3 de ellos.

Por ejemplo, si queremos seleccionar el mejor modelo utilizando el $R^2_\text{adj}$, basta con escribir

## mejor modelo basado en el R^2_{adj}
results <- k$result
results[which.max(results$adjr), ]

##      mindex  n                     predictors   rsquare      adjr      rmse
## 1023   1023 10 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 0.9777571 0.9752579 0.8884855
##        predrsq cp      aic     sbic      sbc     msep       fpe        apc
## 1023 0.9715036 11 284.1403 3.041157 315.4024 88.79029 0.9845406 0.02774113
##             hsp
## 1023 0.01007924

Si usamos el AIC, el mejor modelo será:

## mejor modelo basado en el Cp
results[which.min(with(results, (cp-n)^2)), ]

##      mindex  n                     predictors   rsquare      adjr      rmse
## 1023   1023 10 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 0.9777571 0.9752579 0.8884855
##        predrsq cp      aic     sbic      sbc     msep       fpe        apc
## 1023 0.9715036 11 284.1403 3.041157 315.4024 88.79029 0.9845406 0.02774113
##             hsp
## 1023 0.01007924

Otra posibilidad es utilizar dos criterios selección a la vez:

## selección usando el R^2_{adj} y el AIC
require(ggplot2)
ggplot(results, aes(x = adjr, y = aic)) + geom_point() + theme_minimal() + xlab(expression(R^2 *
    " ajustado")) + ylab("AIC")

Figura 3.7: AIC vs. $R^2$ ajustado en todas las regresiones posibles.

De acuerdo con la definición de cada criterio, el mejor modelo debe tener un $R^2_{\text{adj}}\rightarrow1$ y $\text{AIC}\rightarrow 0$. De la gráfica, es posible observar que hay cuatro modelos que sobresalen:

## selección de modelos competitivos
subset(results, aic < 350 & adjr > 0.8, select = n:aic)

##       n                     predictors   rsquare      adjr      rmse   predrsq
## 998   8       x1 x3 x4 x5 x6 x7 x9 x10 0.9650304 0.9619562 1.1140380 0.9577212
## 1021  9    x1 x3 x4 x5 x6 x7 x8 x9 x10 0.9723781 0.9696159 0.9901052 0.9653333
## 1015  9    x1 x2 x3 x4 x5 x6 x7 x9 x10 0.9708818 0.9679700 1.0165692 0.9640209
## 1023 10 x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 0.9777571 0.9752579 0.8884855 0.9715036
##            cp      aic
## 998  57.92307 325.3860
## 1021 30.52284 303.7989
## 1015 36.51001 309.0744
## 1023 11.00000 284.1403

Por ejemplo, el modelo 968 no incluye x2 ni x8 y tiene un $R^2_{\text{adj}}=0.9619$, mientras el modelo 1023 tiene el $R^2_{\text{adj}}$ más alto e incluye 10 variables predictoras. A pesar de que este modelo es levemente mejor el 968, la ganancia en $R^2_{\text{adj}}$ es ínfima y podría no justificar la medición en proceso de dos factores controlables más. En términos del AIC, se tiene que los modelos 1021 y 1015 son similares.

3.8.2 Selección secuencial

Need to add some introductory remarks

Método stepwise

To be completed

Método forward

To be completed

Método backward

To be completed

Modelos de Regresión: Una aproximación práctica con R

Capítulo 3 Regresión Lineal Múltiple

3.1 Formulación básica del modelo de RLM

3.1.1 Estimación

3.2 Propiedades de los estimadores de \({\mathbf{\beta}}\)

3.3 Estimación de \(\sigma^2\)

3.4 Inferencia para \(\mathbf{\beta}\)

3.4.1 Prueba de significancia global

3.4.2 Prueba de significancia marginal

3.4.3 Intervalos de confianza para \(\beta_j\)

3.5 Inferencia para la respuesta

3.5.1 Intervalos de confianza para \(E[\mathbf{Y}|\mathbf{x}_0]\)

3.5.2 Intervalos de predicción para \(E[\mathbf{Y}|\mathbf{x}_0]\)

3.6 Análisis de Residuales

3.6.1 Validación de supuestos

3.6.2 Identificación de outliers

3.6.3 Identificación de observaciones influenciales

3.7 Análisis de Multicolinealidad

3.7.1 Cómo detectarla?

3.8 Selección de Modelos

3.8.1 Método de Todas las Regresiones Posibles

3.8.2 Selección secuencial

Método stepwise

Método forward

Método backward