KERE (version 1.0.0)

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

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

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

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.

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.

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)