felmdir {Renvlp}

Fit the functional envelope linear model

Description

Fit the response and predictor envelope model in function-on-function linear regression with dimensions ux and uy, using the direct estimation.

Usage

felmdir(X, Y, ux, uy, t1, t2, knots = c(0, 0.25, 0.5, 0.75, 1))

Arguments

X

Predictor function. An n by T1 matrix, T1 is number of observed time points, which is the length of t1. Here we assume that each function is observed at the same time points.

Y

Response function. An n by T2 matrix, T2 is number of observed time points, which is the length of t2. Here we assume that each function is observed at the same time points.

ux

Dimension of the predictor envelope. An integer between 0 and number of knots +2.

uy

Dimension of the response envelope. An integer between 0 and number of knots +2.

t1

The observed time points for the predictor functions.

t2

The observed time points for the response functions.

knots

The location of knots of the cubic splines used for estimation. Locations should be positive. The default location of the knots are 0, 0.25, 0.5, 0.75, 1.

Details

This function fits the envelope model to the function-on-function linear regression,

Y = \alpha + B X + \epsilon

, where X and Y are random functions in Hilbert spaces H_X and H_Y, \alpha is a fixed member in H_Y, \epsilon is a random member of H_Y, and B: H_X -> H_Y is a linear operator. We use cubic splines as the basis for both H_X and H_Y. The coefficients [X] and [Y] with respect to the basis are computed. The predictor and response envelope model is fitted on the linear regression model of [Y] on [X]. In this method, we do not need to estimate the eigenfunctions of Sigma_X and Sigma_\epsilon. Based on the estimation result, the fitted value of Y is calculated. The standard function-on-function regression model also works through the linear regression model of [Y] on [X]. But instead of fitting an envelope model, it fits a standard linear regression model, based on which the fitted value of [Y] is calculated. The details are elaborated in Section 5, direct estimation, in the reference of Su et al. (2022).

Value

The output is a list that contains the following components:

beta

The envelope estimator of the regression coefficients in the regression of [Y] on [X].

betafull

The standard estimator, i.e., the OLS estimator of the regression coefficients in the regression of [Y] on [X].

alpha

The envelope estimator of the intercept in the regression of [Y] on [X].

alphafull

The standard estimator of the intercept in the regression of [Y] on [X].

fitted.env

The fitted value of Y computed from the functional envelope linear model.

fitted.full

The fitted value of Y computed from the standard function-to-function linear model.

References

Su, Z., Li, B. and Cook, R. D. (2022+) Envelope model for function-on-function linear regression.

Examples

data(NJdata)
dataX <- matrix(NJdata[,6], nrow = 21)
X <- as.matrix(dataX[, 32:61])
dataY <- matrix(NJdata[,3], nrow = 21)
Y <- as.matrix(dataY[, 32:61])
t1 <- 0:29
t2 <- t1

m <- felmdir(X, Y, 3, 1, t1, t2)
head(m$fitted.env)
head(m$fitted.full)