krome: Fits the regularization paths for the kernel huber...

In emeryyi/rome: Robust Mean Estimation

Description

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

Usage

krome(x, y, kern, lambda = NULL, eps = 1e-08, maxit = 1e4,
delta = 2, gamma = 1e-06)

Arguments

x	matrix of predictors, of dimension N*p; each row is an observation vector.
y	response variable.
kern	the built-in kernel classes in krome. The kern parameter can be set to any function, of class kernel, which computes the inner product in feature space between two vector arguments. krome 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)-α(old)||_2^2/||α(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.
delta	the parameter delta in the huber regression loss. The value must be positive. Default is 2.
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.

Details

Note that the objective function in krome is

Loss(y- α_0 - K * α ) + λ * α^T * K * α,

where the α_0 is the intercept, α 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 krome.

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.

References

Y. Yang, T. Zhang, and H. Zou. (2017) "Flexible Expectile Regression in Reproducing Kernel Hilbert Space." Technometrics. Accepted.

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 krome
m1 <- krome(x=X, y=Y, kern=kern, lambda = lambda, delta = 2) 

# plot the solution paths
plot(m1)

emeryyi/rome documentation built on May 6, 2019, 9:53 a.m.