R: Functional single-index model fit using kNN estimation and...

fsim.kNN.fit {fsemipar}

R Documentation

Functional single-index model fit using kNN estimation and joint LOOCV minimisation

Description

This function fits a functional single-index model (FSIM) between a functional covariate and a scalar response. It employs kNN estimation with Nadaraya-Watson weights and uses B-spline expansions to represent curves and eligible functional indexes.

The function also utilises the leave-one-out cross-validation (LOOCV) criterion to select the number of neighbours (k.opt) and the coefficients of the functional index in the spline basis (theta.est). It performs a joint minimisation of the LOOCV objective function in both the number of neighbours and the functional index.

Usage

fsim.kNN.fit(x, y, seed.coeff = c(-1, 0, 1), order.Bspline = 3, nknot.theta = 3,
knearest = NULL, min.knn = 2, max.knn = NULL,  step = NULL, 
kind.of.kernel = "quad", range.grid = NULL, nknot = NULL, n.core = NULL)

Arguments

`x`	Matrix containing the observations of the functional covariate (i.e. curves) collected by row.
`y`	Vector containing the scalar response.
`seed.coeff`	Vector of initial values used to build the set `\Theta_n` (see section `Details`). The coefficients for the B-spline representation of each eligible functional index `\theta \in \Theta_n` are obtained from `seed.coeff`. The default is `c(-1,0,1)`.
`order.Bspline`	Positive integer giving the order of the B-spline basis functions. This is the number of coefficients in each piecewise polynomial segment. The default is 3
`nknot.theta`	Positive integer indicating the number of regularly spaced interior knots in the B-spline expansion of `\theta_0`. The default is 3.
`knearest`	Vector of positive integers that defines the sequence within which the optimal number of nearest neighbours `k.opt` is selected. If `knearest=NULL`, then `knearest <- seq(from =min.knn, to = max.knn, by = step)`.
`min.knn`	A positive integer that represents the minimum value in the sequence for selecting the number of nearest neighbours `k.opt`. This value should be less than the sample size. The default is 2.
`max.knn`	A positive integer that represents the maximum value in the sequence for selecting number of nearest neighbours `k.opt`. This value should be less than the sample size. The default is `max.knn <- n%/%5`.
`step`	A positive integer used to construct the sequence of k-nearest neighbours as follows: `min.knn, min.knn + step, min.knn + 2step, min.knn + 3step,...`. The default value for `step` is `step<-ceiling(n/100)`.
`kind.of.kernel`	The type of kernel function used. Currently, only Epanechnikov kernel (`"quad"`) is available.
`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))`.
`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`.
`n.core`	Number of CPU cores designated for parallel execution.The default is `n.core<-availableCores(omit=1)`.

Details

The functional single-index model (FSIM) is given by the expression:

Y_i=r(\langle\theta_0,X_i\rangle)+\varepsilon_i, \quad i=1,\dots,n,

where Y_i denotes a scalar response, X_i is a functional covariate valued in a separable Hilbert space \mathcal{H} with an inner product \langle \cdot, \cdot\rangle. The term \varepsilon denotes the random error, \theta_0 \in \mathcal{H} is the unknown functional index and r(\cdot) denotes the unknown smooth link function.

The FSIM is fitted using the kNN estimator

\widehat{r}_{k,\hat{\theta}}(x)=\sum_{i=1}^nw_{n,k,\hat{\theta}}(x,X_i)Y_i, \quad \forall x\in\mathcal{H},

with Nadaraya-Watson weights

w_{n,k,\hat{\theta}}(x,X_i)=\frac{K\left(H_{k,x,\hat{\theta}}^{-1}d_{\hat{\theta}}\left(X_i,x\right)\right)}{\sum_{i=1}^nK\left(H_{k,x,\hat{\theta}}^{-1}d_{\hat{\theta}}\left(X_i,x\right)\right)},

where

the positive integer k is a smoothing factor, representing the number of nearest neighbours.
K is a kernel function (see the argument kind.of.kernel).
d_{\hat{\theta}}(x_1,x_2)=|\langle\hat{\theta},x_1-x_2\rangle| is the projection semi-metric, computed using semimetric.projec and \hat{\theta} is an estimate of \theta_0.
H_{k,x,\hat{\theta}}=\min\{h\in R^+ \text{ such that } \sum_{i=1}^n1_{B_{\hat{\theta}}(x,h)}(X_i)=k\}, where 1_{B_{\hat{\theta}}(x,h)}(\cdot) is the indicator function of the open ball defined by the projection semi-metric, with centre x\in\mathcal{H} and radius h.

The procedure requires the estimation of the function-parameter \theta_0. Therefore, we use B-spline expansions to represent curves (dimension nknot+order.Bspline) and eligible functional indexes (dimension nknot.theta+order.Bspline). Then, we build a set \Theta_n of eligible functional indexes by calibrating (to ensure the identifiability of the model) the set of initial coefficients given in seed.coeff. The larger this set is, the greater the size of \Theta_n. Since our approach requires intensive computation, a trade-off between the size of \Theta_n and the performance of the estimator is necessary. For that, Ait-Saidi et al. (2008) suggested considering order.Bspline=3 and seed.coeff=c(-1,0,1). For details on the construction of \Theta_n, see Novo et al. (2019).

We obtain the estimated coefficients of \theta_0 in the spline basis (theta.est) and the selected number of neighbours (k.opt) by minimising the LOOCV criterion. This function performs a joint minimisation in both parameters, the number of neighbours and the functional index, and supports parallel computation. To avoid parallel computation, we can set n.core=1.

Value

`call`	The matched call.
`fitted.values`	Estimated scalar response.
`residuals`	Differences between `y` and the `fitted.values`
`theta.est`	Coefficients of `\hat{\theta}` in the B-spline basis: a vector of `length(order.Bspline+nknot.theta)`.
`k.opt`	Selected number of nearest neighbours.
`r.squared`	Coefficient of determination.
`var.res`	Redidual variance.
`df`	Residual degrees of freedom.
`yhat.cv`	Predicted values for the scalar response using leave-one-out samples.
`CV.opt`	Minimum value of the CV function, i.e. the value of CV for `theta.est` and `k.opt`.
`CV.values`	Vector containing CV values for each functional index in `\Theta_n` and the value of `k` that minimises the CV for such index (i.e. `CV.values[j]` contains the value of the CV function corresponding to `theta.seq.norm[j,]` and the best value of the `k.seq` for this functional index according to the CV criterion).
`H`	Hat matrix.
`m.opt`	Index of `\hat{\theta}` in the set `\Theta_n`.
`theta.seq.norm`	The vector `theta.seq.norm[j,]` contains the coefficientes in the B-spline basis of the jth functional index in `\Theta_n`.
`k.seq`	Sequence of eligible values for `k`.
`...`

Author(s)

German Aneiros Perez german.aneiros@udc.es

Silvia Novo Diaz snovo@est-econ.uc3m.es

References

Ait-Saidi, A., Ferraty, F., Kassa, R., and Vieu, P. (2008) Cross-validated estimations in the single-functional index model, Statistics, 42(6), 475–494, doi:10.1080/02331880801980377.

Novo S., Aneiros, G., and Vieu, P., (2019) Automatic and location-adaptive estimation in functional single–index regression, Journal of Nonparametric Statistics, 31(2), 364–392, doi:10.1080/10485252.2019.1567726.

Examples


data(Tecator)
y<-Tecator$fat
X<-Tecator$absor.spectra2

#FSIM fit.
ptm<-proc.time()
fit<-fsim.kNN.fit(y=y[1:160],x=X[1:160,],max.knn=20,nknot.theta=4,nknot=20,
range.grid=c(850,1050))
proc.time()-ptm
fit
names(fit)

[Package fsemipar version 1.1.1 Index]