# Single-Index signal regression using P-splines `sim_psr` is a single-index signal regression model that estimates both the signal coefficients vector and the unknown link function using P-splines. ```r sim_psr( y, X, x_index = c(1:ncol(X)), nsegs = rep(10, 2), bdegs = rep(3, 3), lambdas = rep(1, 2), pords = rep(2, 2), max_iter = 100 ) ``` ## Arguments - `y`: a response vector of length `m`, usually continuous. - `X`: The signal regressors with dimension `m` by `p`. - `x_index`: an index of length `p` for columns of signal matrix; default is simple sequence, `c(1: ncol(X))`. - `nsegs`: a vector of length 2 containing the number of evenly spaced segments between min and max, for each the coefficient vector and the (unknown) link function, resp. (default `c(10, 10)`). - `bdegs`: a vector of length 2 containing the degree of B-splines, for the coefficient vector and the (unknown) link function, resp. (default cubic or `c(3, 3)`). - `lambdas`: a vector of length 2 containing the positive tuning parameters, for each the coefficient vector and the (unknown) link function, resp. (default `c(1, 1)`). - `pords`: a vector of length 2 containing the difference penalty order, for each the coefficient vector and the (unknown) link function, resp. (default`c(2, 2)` ). - `max_iter`: a scalar for the maximum number of iterations (default 100). ## Returns - **y**: the response vector of length `m`. - **alpha**: the P-spline coefficient vector of length `(nsegs[1]+bdeg[1])`. - **iter**: the number of iterations used for the single-index fit. - **yint**: the estimated y-intercept for the single-index model. - **B**: the B-spline matrix built along the signal index, using `nsegs[1]`, used for the coefficient vector. - **Q**: the effective regressors from the `psVCSignal` portion of the single-index fit with dimension `m` by `length(alpha)`. - **nsegs**: a vector of length 2 containing the number of evenly spaced segments between min and max, for each the coefficient vector and the link function, resp. - **bdegs**: a vector of length 2 containing the degree of B-splines, for each the coefficient vector and the link function, resp. - **lambdas**: a vector of length 2 containing the positive tuning parameters, for each the coefficient vector and the link function, resp. - **pords**: a vector of length 2 containing the difference penalty order, for each the coefficient vector and the link function, resp. - **eta**: the estimated linear predictor for the single-index fit. - **cv**: the leave-one-out cross-validation statistic or the standard error of prediction for the single-index fit. - **delta_alpha**: change measure in signal-coefficent parameters at convervence. - **x_index**: the index of length `p` for columns of signal matrix. - **f_fit**: the `psNormal` object, fitting link function f(`eta`). - **f_eta**: the predicted values of the link function estimated with `f_fit` or estimated f(`eta`), at `x = eta`. ## Examples ```r library(JOPS) # Get the data library(fds) data(nirc) iindex <- nirc$x X <- nirc$y sel <- 50:650 # 1200 <= x & x<= 2400 X <- X[sel, ] iindex <- iindex[sel] dX <- diff(X) diindex <- iindex[-1] y <- as.vector(labc[1, 1:40]) oout <- 23 dX <- t(dX[, -oout]) y <- y[-oout] pords <- c(2, 2) nsegs <- c(27, 7) bdegs = c(3, 3) lambdas <- c(1e-6, .1) max_iter <- 100 # Single-index model fit <- sim_psr(y, dX, diindex, nsegs, bdegs, lambdas, pords, max_iter) plot(fit, xlab = "Wavelength (nm)", ylab = " ") ``` ## References Eilers, P.H.C., B. Li, B.D. Marx (2009). Multivariate calibration with single-index signal regression, **Chemometrics and Intellegent Laboratory Systems**, 96(2), 196-202. Eilers, P.H.C. and Marx, B.D. (2021). **Practical Smoothing, The Joys of P-splines.** Cambridge University Press. ## Author(s) Paul Eilers, Brian Marx, and Bin Li

sim_psr function