Fit the response and predictor envelope model in function-on-function linear regression with dimensions ux and uy, using the direct estimation.
felmdir(X, Y, ux, uy, t1, t2, knots = c(0, 0.25, 0.5, 0.75, 1))The output is a list that contains the following components:
The envelope estimator of the regression coefficients in the regression of \([Y]\) on \([X]\).
The standard estimator, i.e., the OLS estimator of the regression coefficients in the regression of \([Y]\) on \([X]\).
The envelope estimator of the intercept in the regression of \([Y]\) on \([X]\).
The standard estimator of the intercept in the regression of \([Y]\) on \([X]\).
The fitted value of Y computed from the functional envelope linear model.
The fitted value of Y computed from the standard function-to-function linear model.
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.
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.
Dimension of the predictor envelope. An integer between 0 and number of knots +2.
Dimension of the response envelope. An integer between 0 and number of knots +2.
The observed time points for the predictor functions.
The observed time points for the response functions.
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.
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).
Su, Z., Li, B. and Cook, R. D. (2022+) Envelope model for function-on-function linear regression.
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)
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