# Fits the regularization paths for the kernel expectile regression. Fits a regularization path for the kernel expectile regression at a sequence of regularization parameters lambda. ```r 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 $N*p$; 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`. ## Returns 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 ## 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 ```r # 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) ```

KERE function