# Smooth signal (multivariate calibration) regression using P-splines. ```r psSignal( y, x_signal, x_index = c(1:ncol(x_signal)), nseg = 10, bdeg = 3, pord = 3, lambda = 1, wts = 1 + 0 * y, family = "gaussian", link = "default", m_binomial = 1 + 0 * y, r_gamma = wts, y_predicted = NULL, x_predicted = x_signal, ridge_adj = 0, int = TRUE ) ``` ## Arguments - `y`: a (glm) response vector, usually continuous, binomial or count data. - `x_signal`: a matrix of continuous regressor with `nrow(x_signal) == length(y)`, often a discrete digitization of a signal or histogram or time series. - `x_index`: a vector to of length `ncol(x_signal) == p`, associated with the ordering index of the signal. Default is `1:ncol(x_signal)`. - `nseg`: the number of evenly spaced segments between `xl` and `xr` (default 10). - `bdeg`: the degree of the basis, usually 1, 2, or 3 (defalult). - `pord`: the order of the difference penalty, usually 1, 2, or 3 (defalult). - `lambda`: the (positive) tuning parameter for the penalty (default 1). - `wts`: the weight vector of `length(y)`; default is 1. - `family`: the response distribution, e.g. `"gaussian", "binomial", "poisson", "Gamma"` distribution; quotes are needed. Default is `"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. - `r_gamma`: a vector of gamma shape parameters. Default is 1 vector for `family = "Gamma"`, NULL otherwise. - `y_predicted`: a vector of responses associated with `x_predicted` which are used to calculate standard error of external prediction. Default is NULL. - `x_predicted`: a matrix of external signals to yield external prediction. - `ridge_adj`: A ridge penalty tuning parameter, which can be set to small value, e.g. `1e-8` to stabilize estimation, (default 0). - `int`: set to TRUE or FALSE to include intercept term in linear predictor (default TRUE). ## Returns - **coef**: a vector with `length(n)` of estimated P-spline coefficients. - **mu**: a vector with `length(m)` of estimated means. - **eta**: a vector of `length(m)` of estimated linear predictors. - **B**: the B-spline basis (for the coefficients), with dimension `p` by `n`. - **deviance**: the deviance of fit. - **eff_df**: the approximate effective dimension of fit. - **aic**: AIC. - **df_resid**: approximate df residual. - **beta**: a vector of length `p`, containing estimated smooth signal coefficients. - **std_beta**: a vector of length `p`, containing standard errors of smooth signal coefficients. - **cv**: leave-one-out standard error prediction, when `family = "gaussian"`. - **cv_predicted**: standard error prediction for `y_predict`, when `family = "gaussian"`, NULL otherwise. - **nseg**: the number of evenly spaced B-spline segments. - **bdeg**: the degree of B-splines. - **pord**: the order of the difference penalty. - **lambda**: the positive tuning parameter. - **family**: the family of the response. - **link**: the link function. - **y_intercept**: the estimated y-intercept (when `int = TRUE`.) - **int**: a logical variable related to use of y-intercept in model. - **dispersion_param**: 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. - **x_index**: a vector to of length `ncol(x_signal) == p`, associated with the ordering of the signal. ## Details Support functions needed: `pspline_fitter`, `bbase` and `pspline_checker`. ## 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]) # percent fat oout <- 23 dX <- t(dX[, -oout]) y <- y[-oout] fit1 <- psSignal(y, dX, diindex, nseg = 25, bdeg = 3, lambda = 0.0001, pord = 2, family = "gaussian", link = "identity", x_predicted = dX, int = TRUE) plot(fit1, xlab = "Coefficient Index", ylab = "ps Smooth Coeff") title(main = "25 B-spline segments with tuning = 0.0001") names(fit1) ``` ## References Marx, B.D. and Eilers, P.H.C. (1999). Generalized linear regression for sampled signals and curves: A P-spline approach. **Technometrics**, 41(1): 1-13. Eilers, P.H.C. and Marx, B.D. (2021). **Practical Smoothing, The Joys of P-splines.** Cambridge University Press. ## Author(s) Brian Marx

psSignal function