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_complex
frame_distance_complex(x, resid, tau)matrix, covariate of quantile regression model
matrix, residuals of quantile regression models
singular or vectors, quantile
dataframe for residual-robust distance plot
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 \(\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_i\) and \(\bar{C}(X)=\frac{1}{n-1}\sum_{i=1}^{n}(x_i-\bar{x})^{T}(x_i- \bar{x})\) 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.