KERE {KERE}

Fits the regularization paths for the kernel expectile regression.

Description

Fits a regularization path for the kernel expectile regression at a sequence of regularization parameters lambda.

Usage

KERE(x, y, kern, lambda = NULL, eps = 1e-08, maxit = 1e4,
omega = 0.5, gamma = 1e-06, option = c("fast", "normal"))

Arguments

x

matrix of predictors, of dimension N \times p; each row is an observation vector.

y

response variable.

kern

the built-in kernel classes in KERE. The kern parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments. KERE provides the most popular kernel functions which can be initialized by using the following functions:

• rbfdot Radial Basis kernel function,

• polydot Polynomial kernel function,

• vanilladot Linear kernel function,

• tanhdot Hyperbolic tangent kernel function,

• laplacedot Laplacian kernel function,

• besseldot Bessel kernel function,

• anovadot ANOVA RBF kernel function,

• splinedot the Spline kernel.

Objects can be created by calling the rbfdot, polydot, tanhdot, vanilladot, anovadot, besseldot, laplacedot, splinedot functions etc. (see example.)

lambda

a user supplied lambda sequence. It is better to supply a decreasing sequence of lambda values, if not, the program will sort user-defined lambda sequence in decreasing order automatically.

eps

convergence threshold for majorization minimization algorithm. Each majorization descent loop continues until the relative change in any coefficient ||alpha(new)-\alpha(old)||_2^2/||\alpha(old)||_2^2 is less than eps. Defaults value is 1e-8.

maxit

maximum number of loop iterations allowed at fixed lambda value. Default is 1e4. If models do not converge, consider increasing maxit.

omega

the parameter \omega in the expectile regression model. The value must be in (0,1). Default is 0.5.

gamma

a scalar number. If it is specified, the number will be added to each diagonal element of the kernel matrix as perturbation. The default is 1e-06.

option

users can choose which method to use to update the inverse matrix in the MM algorithm. "fast" uses a trick described in Yang, Zhang and Zou (2015) to update estimates for each lambda. "normal" uses a naive way for the computation.

Details

Note that the objective function in KERE is

Loss(y- \alpha_0 - K * \alpha )) + \lambda * \alpha^T * K * \alpha,

where the \alpha_0 is the intercept, \alpha is the solution vector, and K is the kernel matrix with K_{ij}=K(x_i,x_j). Users can specify the kernel function to use, options include Radial Basis kernel, Polynomial kernel, Linear kernel, Hyperbolic tangent kernel, Laplacian kernel, Bessel kernel, ANOVA RBF kernel, the Spline kernel. Users can also tweak the penalty by choosing different lambda.

For computing speed reason, if models are not converging or running slow, consider increasing eps before increasing maxit.

Value

An object with S3 class KERE.

call

the call that produced this object.

alpha

a nrow(x)*length(lambda) matrix of coefficients. Each column is a solution vector corresponding to a lambda value in the lambda sequence.

lambda

the actual sequence of lambda values used.

npass

total number of loop iterations corresponding to each lambda value.

jerr

error flag, for warnings and errors, 0 if no error.

Author(s)

Yi Yang, Teng Zhang and Hui Zou
Maintainer: Yi Yang <[e-mail]>

References

Y. Yang, T. Zhang, and H. Zou. "Flexible Expectile Regression in Reproducing Kernel Hilbert Space." ArXiv e-prints: stat.ME/1508.05987, August 2015.

Examples

# create data
N <- 200
X1 <- runif(N)
X2 <- 2*runif(N)
X3 <- 3*runif(N)
SNR <- 10 # signal-to-noise ratio
Y <- X1**1.5 + 2 * (X2**.5) + X1*X3
sigma <- sqrt(var(Y)/SNR)
Y <- Y + X2*rnorm(N,0,sigma)
X <- cbind(X1,X2,X3)

# set gaussian kernel 
kern <- rbfdot(sigma=0.1)

# define lambda sequence
lambda <- exp(seq(log(0.5),log(0.01),len=10))

# run KERE
m1 <- KERE(x=X, y=Y, kern=kern, lambda = lambda, omega = 0.5) 

# plot the solution paths
plot(m1)