# Impact point selection with PVS This function implements the Partitioning Variable Selection (PVS) algorithm. This algorithm is specifically designed for estimating multivarite linear models, where the scalar covariates are derived from the discretisation of a curve. PVS is a two-stage procedure that selects the impact points of the discretised curve and estimates the model. The algorithm employs a penalised least-squares regularisation procedure. Additionally, it utilises an objective criterion (`criterion`) to determine the initial number of covariates in the reduced model (`w.opt`) of the first stage, and the penalisation parameter (`lambda.opt`). ```r PVS.fit(z, y, train.1 = NULL, train.2 = NULL, lambda.min = NULL, lambda.min.h = NULL, lambda.min.l = NULL, factor.pn = 1, nlambda = 100, vn = ncol(z), nfolds = 10, seed = 123, wn = c(10, 15, 20), range.grid = NULL, criterion = "GCV", penalty = "grSCAD", max.iter = 1000) ``` ## Arguments - `z`: Matrix containing the observations of the functional covariate collected by row (linear component). - `y`: Vector containing the scalar response. - `train.1`: Positions of the data that are used as the training sample in the 1st step. The default setting is `train.1<-1:ceiling(n/2)`. - `train.2`: Positions of the data that are used as the training sample in the 2nd step. The default setting is `train.2<-(ceiling(n/2)+1):n`. - `lambda.min`: The smallest value for lambda (i. e., the lower endpoint of the sequence in which `lambda.opt` is selected), as fraction of `lambda.max`. The defaults is `lambda.min.l` if the sample size is larger than `factor.pn` times the number of linear covariates and `lambda.min.h` otherwise. - `lambda.min.h`: The lower endpoint of the sequence in which `lambda.opt` is selected if the sample size is smaller than `factor.pn` times the number of linear covariates. The default is 0.05. - `lambda.min.l`: The lower endpoint of the sequence in which `lambda.opt` is selected if the sample size is larger than `factor.pn` times the number of linear covariates. The default is 0.0001. - `factor.pn`: Positive integer used to set `lambda.min`. The default value is 1. - `nlambda`: Number of values in the sequence from which `lambda.opt` is selected. The default is 100. - `vn`: Positive integer or vector of positive integers indicating the number of groups of consecutive variables to be penalised together. The default value is `vn=ncol(z)`, resulting in the individual penalization of each scalar covariate. - `nfolds`: Number of cross-validation folds (used when `criterion="k-fold-CV"`). Default is 10. - `seed`: You may set the seed for the random number generator to ensure reproducible results (applicable when `criterion="k-fold-CV"` is used). The default seed value is 123. - `wn`: A vector of positive integers indicating the eligible number of covariates in the reduced model. For more information, refer to the section `Details`. The default is `c(10,15,20)`. - `range.grid`: Vector of length 2 containing the endpoints of the grid at which the observations of the functional covariate `x` are evaluated (i.e. the range of the discretisation). If `range.grid=NULL`, then `range.grid=c(1,p)` is considered, where `p` is the discretisation size of `x` (i.e. `ncol(x))`. - `criterion`: The criterion used to select the tuning and regularisation parameters: `wn.opt` and `lambda.opt` (also `vn.opt` if needed). Options include `"GCV"`, `"BIC"`, `"AIC"`, or `"k-fold-CV"`. The default setting is `"GCV"`. - `penalty`: The penalty function applied in the penalised least-squares procedure. Currently, only "grLasso" and "grSCAD" are implemented. The default is "grSCAD". - `max.iter`: Maximum number of iterations allowed across the entire path. The default value is 1000. ## Details The sparse linear model with covariates coming from the discretization of a curve is given by the expression $$ Y_i=\sum_{j=1}^{p_n}\beta_{0j}\zeta_i(t_j)+\varepsilon_i,\ \ \ (i=1,\dots,n) $$ where * $Y_i$ is a real random response and $\zeta_i$ is assumed to be a random curve defined on some interval $[a,b]$, which is observed at the points $a\leq t_1\<\dots\ 25 index.atip <- index.y.25 (1:268)[index.atip] #Dataset to model z.sug<- z[!index.atip,] y.sug <- y[!index.atip] train<-1:216 ptm=proc.time() fit<- PVS.fit(z=z.sug[train,], y=y.sug[train],train.1=1:108,train.2=109:216, lambda.min.h=0.2,criterion="BIC", max.iter=5000) proc.time()-ptm fit names(fit) ```

PVS.fit function