# Impact point selection with PVS and kernel estimation This function computes the partitioning variable selection (PVS) algorithm for multi-functional partial linear models (MFPLM). 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, integrated with kernel estimation with Nadaraya-Watson weights. Additionally, it utilises an objective criterion (`criterion`) to select the number of covariates in the reduced model (`w.opt`), the bandwidth (`h.opt`) and the penalisation parameter (`lambda.opt`). ```r PVS.kernel.fit(x, z, y, train.1 = NULL, train.2 = NULL, semimetric = "deriv", q = NULL, min.q.h = 0.05, max.q.h = 0.5, h.seq = NULL, num.h = 10, range.grid = NULL, kind.of.kernel = "quad", nknot = 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), criterion = "GCV", penalty = "grSCAD", max.iter = 1000) ``` ## Arguments - `x`: Matrix containing the observations of the functional covariate (functional nonparametric component), collected by row. - `z`: Matrix containing the observations of the functional covariate that is discretised (linear component), collected by row. - `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`. - `semimetric`: Semi-metric function. Only `"deriv"` and `"pca"` are implemented. By default `semimetric="deriv"`. - `q`: Order of the derivative (if `semimetric="deriv"`) or number of principal components (if `semimetric="pca"`). The default values are 0 and 2, respectively. - `min.q.h`: Minimum quantile order of the distances between curves, which are computed using the projection semi-metric. This value determines the lower endpoint of the range from which the bandwidth is selected. The default is 0.05. - `max.q.h`: Maximum quantile order of the distances between curves, which are computed using the projection semi-metric. This value determines the upper endpoint of the range from which the bandwidth is selected. The default is 0.5. - `h.seq`: Vector containing the sequence of bandwidths. The default is a sequence of `num.h` equispaced bandwidths in the range constructed using `min.q.h` and `max.q.h`. - `num.h`: Positive integer indicating the number of bandwidths in the grid. The default is 10. - `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))`. - `kind.of.kernel`: The type of kernel function used. Currently, only Epanechnikov kernel (`"quad"`) is available. - `nknot`: Positive integer indicating the number of interior knots for the B-spline expansion of the functional covariate. The default value is `(p - order.Bspline - 1)%/%2`. - `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`: Positive integer indicating the 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)`. - `criterion`: The criterion used to select the tuning and regularisation parameters: `wn.opt`, `lambda.opt` and `h.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 multi-functional partial linear model (MFPLM) is given by the expression $$ Y_i=\sum_{j=1}^{p_n}\beta_{0j}\zeta_i(t_j)+m\left(X_i\right)+\varepsilon_i,\ \ \ (i=1,\dots,n), $$ where: * $Y_i$ is a real random response and $X_i$ denotes a random element belonging to some semi-metric space $\mathcal{H}$. The second functional predictor $\zeta_i$ is assumed to be a curve defined on some interval $[a,b]$, observed at the points $a\leq t_1\<\dots\ 25 index.atip <- index.y.25 (1:268)[index.atip] #Dataset to model x.sug <- x[!index.atip,] z.sug<- z[!index.atip,] y.sug <- y[!index.atip] train<-1:216 ptm=proc.time() fit<- PVS.kernel.fit(x=x.sug[train,],z=z.sug[train,], y=y.sug[train], train.1=1:108,train.2=109:216,lambda.min.h=0.03, lambda.min.l=0.03, max.q.h=0.35, nknot=20, criterion="BIC", max.iter=5000) proc.time()-ptm fit names(fit) ```

PVS.kernel.fit function