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,] 4.700012 4.699970
#> [2,] 4.700631 4.702216
#> [3,] 4.700860 4.703043
#> [4,] 4.701284 4.704572
#> [5,] 4.701676 4.705986
#> [6,] 4.704176 4.715000


## 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,] 2.155597 2.436787 2.299304
#> [2,] 1.841000 1.895155 1.938847
#> [3,] 1.804191 1.940970 2.175125
#> [4,] 1.821835 2.184524 2.795772
#> [5,] 1.912900 2.533088 3.500514
#> [6,] 3.568363 6.173995 9.190227
#>
#> , , 2
#>
#>          [,1]     [,2]     [,3]
#> [1,] 2.155595 2.436764 2.299327
#> [2,] 1.840997 1.895141 1.938859
#> [3,] 1.804188 1.940961 2.175132
#> [4,] 1.821833 2.184523 2.795771
#> [5,] 1.912899 2.533092 3.500508
#> [6,] 3.568363 6.173998 9.190227
#>
#> , , 3
#>
#>          [,1]     [,2]     [,3]
#> [1,] 2.155563 2.436068 2.299875
#> [2,] 1.840905 1.894621 1.939124
#> [3,] 1.804106 1.940613 2.175253
#> [4,] 1.821779 2.184459 2.795682
#> [5,] 1.912875 2.533239 3.500290
#> [6,] 3.568487 6.174461 9.190205