# Kernel Logistic Regression The function performs a kernel logistic regression for binary outputs. ```r KLR(X, y, xnew, lambda = 0.01, kernel = c("matern", "exponential")[1], nu = 1.5, power = 1.95, rho = 0.1) ``` ## Arguments - `X`: a design matrix with dimension `n` by `d`. - `y`: a response vector with length `n`. The values in the vector are 0 or 1. - `xnew`: a testing matrix with dimension `n_new` by `d` in which each row corresponds to a predictive location. - `lambda`: a positive value specifing the tuning parameter for KLR. The default is 0.01. - `kernel`: "matern" or "exponential" which specifies the matern kernel or power exponential kernel. The default is "matern". - `nu`: a positive value specifying the order of matern kernel if `kernel` == "matern". The default is 1.5 if matern kernel is chosen. - `power`: a positive value (between 1.0 and 2.0) specifying the power of power exponential kernel if `kernel` == "exponential". The default is 1.95 if power exponential kernel is chosen. - `rho`: a positive value specifying the scale parameter of matern and power exponential kernels. The default is 0.1. ## Returns Predictive probabilities at given locations `xnew`. ## Details This function performs a kernel logistic regression, where the kernel can be assigned to Matern kernel or power exponential kernel by the argument `kernel`. The arguments `power` and `rho` are the tuning parameters in the power exponential kernel function, and `nu` and `rho` are the tuning parameters in the Matern kernel function. The power exponential kernel has the form $$ K_{ij}=\exp(-\frac{\sum_{k}{|x_{ik}-x_{jk}|^{power}}}{rho}), $$ and the Matern kernel has the form $$ K_{ij}=\prod_{k}\frac{1}{\Gamma(nu)2^{nu-1}}(2\sqrt{nu}\frac{|x_{ik}-x_{jk}|}{rho})^{nu} \kappa(2\sqrt{nu}\frac{|x_{ik}-x_{jk}|}{rho}). $$ The argument `lambda` is the tuning parameter for the function smoothness. ## Examples ```r library(calibrateBinary) set.seed(1) np <- 10 xp <- seq(0,1,length.out = np) eta_fun <- function(x) exp(exp(-0.5*x)*cos(3.5*pi*x)-1) # true probability function eta_x <- eta_fun(xp) yp <- rep(0,np) for(i in 1:np) yp[i] <- rbinom(1,1, eta_x[i]) x.test <- seq(0,1,0.001) etahat <- KLR(xp,yp,x.test) plot(xp,yp) curve(eta_fun, col = "blue", lty = 2, add = TRUE) lines(x.test, etahat, col = 2) ##### cross-validation with K=5 ##### ##### to determine the parameter rho ##### cv.out <- cv.KLR(xp,yp,K=5) print(cv.out) etahat.cv <- KLR(xp,yp,x.test,lambda=cv.out$lambda,rho=cv.out$rho) plot(xp,yp) curve(eta_fun, col = "blue", lty = 2, add = TRUE) lines(x.test, etahat, col = 2) lines(x.test, etahat.cv, col = 3) ``` ## Author(s) Chih-Li Sung ## References Zhu, J. and Hastie, T. (2005). Kernel logistic regression and the import vector machine. Journal of Computational and Graphical Statistics, 14(1), 185-205. ## See Also `cv.KLR` for performing cross-validation to choose the tuning parameters.

KLR function