# Varying-coefficient penalized signal regression using P-splines. `psVCSignal` is used to regress a (glm) response onto a signal such that the signal coefficients can vary over another covariate `t`. Anisotripic penalization of tensor product B-splines produces a 2D coefficient surface that can be sliced at `t`. @details Support functions needed: `pspline_fitter`, `pspline_2dchecker`, and `bbase`. @import stats ```r psVCSignal( y, X, x_index, t_var, Pars = rbind(c(min(x_index), max(x_index), 10, 3, 1, 2), c(min(t_var), max(t_var), 10, 3, 1, 2)), family = "gaussian", link = "default", m_binomial = 1 + 0 * y, wts = 1 + 0 * y, r_gamma = 1 + 0 * y, X_pred = X, t_pred = t_var, y_predicted = NULL, ridge_adj = 1e-08, int = TRUE ) ``` ## Arguments - `y`: a glm response vector of length `m`, usually continuous, binary/bimomial or counts. - `X`: a `m` by `p1` Signal matrix of regressors. - `x_index`: `p1`-vector for index of Signal (e.g. wavelength). - `t_var`: `p2`-vector with other (indexing) variable in coefficient surface (e.g. temperature, depth, time). - `Pars`: a matrix with 2 rows, each with P-spline parameters: `min max nseg bdeg lambda pord`, for row and columns of tensor product surface; defaults are min and max for `x_index` and `t_var` (resp.), `nseg` = 10, `bdeg` =3, `lambda` = 1, `pord` = 2. - `family`: the response distribution, e.g. `"gaussian", "binomial", "poisson", "Gamma"` distribution; quotes are needed (default `"gaussian"`. - `link`: the link function, one of `"identity"`, `"log"`, `"sqrt"`, `"logit"`, `"probit"`, `"cloglog"`, `"loglog"`, `"reciprocal"`); quotes are needed (default `"identity"`. - `m_binomial`: a vector of binomial trials having `length(y)`. Default is 1 vector for `family = "binomial"`, NULL otherwise. - `wts`: a `m` vector of weights (default 1). - `r_gamma`: a vector of gamma shape parameters. Default is 1 vector for `family = "Gamma"`, NULL otherwise. - `X_pred`: a matrix of signals with `ncol(X)` columns for prediction, default is `X`. - `t_pred`: a vector for the VC indexing variable with length `nrow(X_pred)`, default is `t_var`. - `y_predicted`: a vector for the responses associated with `X_pred` with length `nrow(X_pred)` useful for CV when `family = "binomial"`, default is `NULL`. - `ridge_adj`: a small ridge penalty tuning parameter to regularize estimation (default `1e-8`). - `int`: intercept set to TRUE or FALSE for intercept term. ## Returns - **pcoef**: a vector of length `(Pars[1,3]+Pars[1,4])`*`(Pars[2,3]+Pars[2,4])` of estimated P-spline coefficients for tensor surface. - **summary_predicted**: inverse link prediction vectors, and twice se bands. - **dev**: the deviance of fit. - **eff_dim**: the approximate effective dimension of fit. - **family**: the family of the response. - **link**: the link function. - **aic**: AIC. - **df_resid**: approximate df residual. - **cv**: leave-one-out standard error prediction when `family = "gaussian"`, NULL otherwise. - **cv_predicted**: standard error prediction for `y_predict` when `family = "gaussian"`, NULL otherwise. - **Pars**: design and tuning parameters; see arguments above. - **dispersion_parm**: estimate of dispersion, `Dev/df_resid`. - **summary_predicted**: inverse link prediction vectors, and twice se bands. - **eta_predicted**: estimated linear predictor of `length(y)`. - **press_mu**: leave-one-out prediction of mean when `family = "gaussian"`, NULL otherwise. - **bin_percent_correct**: percent correct classification based on 0.5 cut-off when `family = "binomial"`, NULL otherwise. - **Bx**: B-spline basis matrix of dimension `p1` by `n1`, along `x_index`. - **By**: B-spline basis matrix of dimension `p2` by `n2`, along `t_var`. - **Q**: Modified tensor basis (`m` by `(n1*n2)`) for VC signal regression. - **yint**: the estimated y-intercept (when `int = TRUE`.) - **int**: a logical variable related to use of y-intercept in model. ## Examples ```r 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]) # percent fat t_var <- as.vector(labc[4, 1:40]) # percent flour oout <- 23 dX <- t(dX[, -oout]) y <- y[-oout] t_var = t_var[-oout] Pars = rbind(c(min(diindex), max(diindex), 25, 3, 1e-7, 2), c(min(t_var), max(t_var), 20, 3, 0.0001, 2)) fit1 <- psVCSignal(y, dX, diindex, t_var, Pars = Pars, family = "gaussian", link = "identity", int = TRUE) plot(fit1, xlab = "Coefficient Index", ylab = "VC: % Flour") names(fit1) ``` ## References Eilers, P.H.C. and Marx, B.D. (2003). Multivariate calibration with temperature interaction using two-dimensional penalized signal regression. **Chemometrics and Intellegent Laboratory Systems**, 66, 159–174. Eilers, P.H.C. and Marx, B.D. (2021). **Practical Smoothing, The Joys of P-splines.** Cambridge University Press. ## Author(s) Paul Eilers and Brian Marx

psVCSignal function