Getting started with fastkqr

This package provides tools for fitting kernel quantile regression.

The strengths and improvements that this package offers relative to other quantile regression packages are as follows:

  • Compiled Fortran code significantly speeds up the kernel quantile regression estimation process.

  • Solve non-crossing kernel quantile regression.

For this getting-started vignette, first, we will use a real data set named as GAGurine in the package MASS, which collects the concentration of chemical GAGs in the urine of 314 children aged 0 to 17 years. We used the concentration of GAG as the response variable.

library(fastkqr)
library(MASS)
data(GAGurine)
x <- as.matrix(GAGurine$Age)
y <- GAGurine$GAG

Then the kernel quantile regression model is formulated as the sum of check loss and an \(\ell_2\) penalty:

\[ \min_{\alpha\in\mathbb{R}^{n},b\in\mathbb{R}}\frac{1}{n} \sum_{i=1}^{n}\rho_{\tau}(y_{i}-b-\mathbf{K}_{i}^{\top}\alpha) +\frac{\lambda}{2} \alpha^{\top}\mathbf{K}\alpha \qquad (*). \]

kqr()

Given an input matrix x, a quantile level tau, and a response vector y, a kernel quantile regression model is estimated for a sequence of penalty parameter values. The other main arguments the users might supply are:

  • lambda: a user-supplied lambda sequence.
  • is_exact: exact or approximated solutions.
lambda <- 10^(seq(1, -4, length.out=10))
fit <- kqr(x, y, lambda=lambda, tau=0.1, is_exact=TRUE)

cv.kqr()

This function performs k-fold cross-validation (cv). It takes the same arguments as kqr.

cv.fit <- cv.kqr(x, y, lambda=lambda, tau=0.1)

Methods

A number of S3 methods are provided for nckqr object.

  • coef() and predict() return a matrix of coefficients and predictions \(\hat{y}\) given a matrix x at each lambda respectively. The optional s argument may provide a specific value of \(\lambda\) (not necessarily part of the original sequence).
coef <- coef(fit, s = c(0.02, 0.03))
predict(fit, x, tail(x), s = fit$lambda[2:3])
#>            s1       s2
#> [1,] 4.699991 4.700007
#> [2,] 4.700649 4.702397
#> [3,] 4.700883 4.703242
#> [4,] 4.701311 4.704785
#> [5,] 4.701704 4.706201
#> [6,] 4.704191 4.715169

nckqr()

Given an input matrix x, a sequence of quantile levels tau, and a response vector y, a non-crossing kernel quantile regression model is estimated for two sequences of penalty parameter values. It takes the same arguments x, y,is_exact, which are specified above. The other main arguments the users might supply are:

  • lambda2: a user-supplied lambda1 sequence for the L2 penalty.

  • lambda1: a user-supplied lambda2 sequence for the smooth ReLU penalty.

l2 <- 1e-4
tau <- c(0.1, 0.3, 0.5)
l1_list <- 10^seq(-8, 2, length.out=10)
fit1 <- nckqr(x ,y, lambda1 = l1_list, lambda2 = l2,  tau = tau)

cv.nckqr()

This function performs k-fold cross-validation (cv) for selecting the tuning parameter ‘lambda2’ of non-crossing kernel quantile regression. It takes the same arguments as nckqr.

l2_list <- 10^(seq(1, -4, length.out=10))
cv.fit1 <- cv.nckqr(x, y, lambda1=10, lambda2=l2_list, tau=tau)

Methods

A number of S3 methods are provided for nckqr object.

  • coef() and predict() return an array of coefficients and predictions \(\hat{y}\) given a matrix X and lambda2 at each lambda1 respectively. The optional s1 argument may provide a specific value of \(\lambda_1\) (not necessarily part of the original sequence).
coef <- coef(fit1, s2=1e-4, s1 = l1_list[2:3])
predict(fit1, x, tail(x), s1=l1_list[1:3], s2=l2)
#> , , 1
#> 
#>          [,1]     [,2]     [,3]
#> [1,] 2.142394 2.273287 2.338837
#> [2,] 2.013088 2.032177 2.073685
#> [3,] 1.972057 2.065825 2.231912
#> [4,] 1.932055 2.207069 2.647345
#> [5,] 1.931853 2.402301 3.121885
#> [6,] 2.650067 4.616806 7.272911
#> 
#> , , 2
#> 
#>          [,1]     [,2]     [,3]
#> [1,] 2.142393 2.273286 2.338839
#> [2,] 2.013086 2.032177 2.073687
#> [3,] 1.972055 2.065825 2.231914
#> [4,] 1.932053 2.207070 2.647346
#> [5,] 1.931851 2.402302 3.121886
#> [6,] 2.650068 4.616805 7.272911
#> 
#> , , 3
#> 
#>          [,1]     [,2]     [,3]
#> [1,] 2.142350 2.273265 2.338919
#> [2,] 2.012995 2.032160 2.073763
#> [3,] 1.971969 2.065819 2.231972
#> [4,] 1.931984 2.207082 2.647372
#> [5,] 1.931804 2.402326 3.121885
#> [6,] 2.650180 4.616830 7.272831