frame_distance: Residual-robust distance plot of quantile regression model
In quokar: Quantile Regression Outlier Diagnostics with K Left Out Analysis

Description Usage Arguments Details Value Author(s) See Also Examples

the standardized residuals from quantile regression against the robust MCD distance. This display is used to diagnose both vertical outlier and horizontal leverage points. Function frame_distance only work for linear quantile regression model. With non-linear model, use frame_distance_implement

1	frame_distance(object, tau)

`object`	model, quantile regression model
`tau`	singular or vectors, quantile

The generalized MCD algorithm based on the fast-MCD algorithm formulated by Rousseeuw and Van Driessen(1999), which is similar to the algorithm for least trimmed squares(LTS). The canonical Mahalanobis distance is defined as

MD(x_i)=[(x_i-\bar{x})^{T}\bar{C}(X)^{-1}(x_i-\bar{x})]^{1/2}

where \bar{x}=\frac{1}{n}∑_{i=1}^{n}x_i and \bar{C}(X)=\frac{1}{n-1}∑_{i=1}^{n}(x_i-\bar{x})^{T}(x_i- \bar{x}) are the empirical multivariate location and scatter, respectively. Here x_i=(x_{i1},...,x_{ip})^{T} exclueds the intercept. The relation between the Mahalanobis distance MD(x_i) and the hat matrix H=(h_{ij})=X(X^{T}X)^{-1}X^{T} is

h_{ii}=\frac{1}{n-1}MD^{2}_{i}+\frac{1}{n}

The canonical robust distance is defined as

RD(x_{i})=[(x_{i}-T(X))^{T}C(X)^{-1}(x_{i}-T(X))]^{1/2}

where T(X) and C(X) are the robust multivariate location and scatter, respectively, obtained by MCD. To achieve robustness, the MCD algorithm estimates the covariance of a multivariate data set mainly through as MCD h-point subset of data set. This subset has the smallest sample-covariance determinant among all the possible h-subsets. Accordingly, the breakdown value for the MCD algorithm equals \frac{(n-h)}{n}. This means the MCD estimates is reliable, even if up to \frac{100(n-h)}{n} set are contaminated.

dataframe for residual-robust distance plot

Wenjing Wang wenjingwangr@gmail.com

function frame_distance_complex

library(quantreg)
library(ggplot2)
library(ALDqr)
library(purrr)
library(robustbase)
library(tidyr)
library(gridExtra)
tau = c(0.1, 0.5, 0.9)
ais_female <- subset(ais, Sex == 1)
object <- rq(BMI ~ LBM + Ht, data = ais_female, tau = tau)
plot_distance <- frame_distance(object, tau = c(0.1, 0.5, 0.9))
distance <- plot_distance[[1]]
cutoff_v <- plot_distance[[2]]
cutoff_h <- plot_distance[[3]]
n <- nrow(object$model)
case <- rep(1:n, length(tau))
distance <- cbind(case, distance)
distance$residuals <- abs(distance$residuals)
distance1 <- subset(distance, tau_flag == "tau0.1")
p1 <- ggplot(distance1, aes(x = rd, y = residuals)) +
 geom_point() +
 geom_hline(yintercept = cutoff_h[1], colour = "red") +
 geom_vline(xintercept = cutoff_v, colour = "red") +
 geom_text(data = subset(distance1, residuals > cutoff_h[1]|rd > cutoff_v),
           aes(label = case), hjust = 0, vjust = 0) +
 xlab("Robust Distance") +
 ylab("|Residuals|")

distance2 <- subset(distance, tau_flag == "tau0.5")

p2 <- ggplot(distance1, aes(x = rd, y = residuals)) +
 geom_point() +
 geom_hline(yintercept = cutoff_h[2], colour = "red") +
 geom_vline(xintercept = cutoff_v, colour = "red") +
 geom_text(data = subset(distance1, residuals > cutoff_h[2]|rd > cutoff_v),
          aes(label = case), hjust = 0, vjust = 0) +
 xlab("Robust Distance") +
 ylab("|Residuals|")
distance3 <- subset(distance, tau_flag == "tau0.9")

p3 <- ggplot(distance1, aes(x = rd, y = residuals)) +
 geom_point() +
 geom_hline(yintercept = cutoff_h[3], colour = "red") +
 geom_vline(xintercept = cutoff_v, colour = "red") +
 geom_text(data = subset(distance1, residuals > cutoff_h[3]|rd > cutoff_v),
         aes(label = case), hjust = 0, vjust = 0) +
xlab("Robust Distance") +
 ylab("|Residuals|")
grid.arrange(p1, p2, p3, ncol = 3)