TPR.fit {TRES}

Tensor predictor regression

Description

This function is used for estimation of tensor predictor regression. The available method including standard OLS type estimation, PLS type of estimation as well as envelope estimation with FG, 1D and ECD approaches.

Usage

TPR.fit(x, y, u, method=c('standard', 'FG', '1D', 'ECD', 'PLS'), Gamma_init = NULL)

Arguments

x

The predictor tensor instance of dimension p_1\times p_2\times\cdots\times p_m \times n, where n is the sample size. Array with the same dimensions and matrix with dimension p\times n are acceptable. If y is missing, x should be a list or an environment consisting of predictor and response datasets.

y

The response matrix of dimension r \times n, where n is the sample size. Vector of length n is acceptable.

u

The dimension of envelope subspace. u=(u_1,\cdots, u_m). Used for methods "FG", "1D", "ECD" and "PLS". User can use TPRdim to select dimension.

method

The method used for estimation of tensor response regression. There are four possible choices.

• "standard": The standard OLS type estimation.

• "FG": Envelope estimation with full Grassmannian (FG) algorithm.

• "1D": Envelope estimation with one dimensional optimization approaches by 1D algorithm.

• "ECD": Envelope estimation with one dimensional optimization approaches by ECD algorithm.

• "PLS": The SIMPLS-type estimation without manifold optimization.

Gamma_init

A list specifying the initial envelope subspace basis for "FG" method. By default, the estimators given by "1D" algorithm is used.

Details

Please refer to Details part of TPRsim for the description of the tensor predictor regression model.

Value

TPR.fit returns an object of class "Tenv".

The function summary (i.e., summary.Tenv) is used to print the summary of the results, including additional information, e.g., the p-value and the standard error for coefficients, and the prediction mean squared error.

The functions coefficients, fitted.values and residuals can be used to extract different features returned from TPR.fit.

The function plot (i.e., plot.Tenv) plots the two-dimensional coefficients and p-value for object of class "Tenv".

The function predict (i.e., predict.Tenv) predicts response for the object returned from TPR.fit function.

x

The original predictor dataset.

y

The original response dataset.

call

The matched call.

method

The implemented method.

coefficients

The estimation of regression coefficient tensor.

Gamma

The estimation of envelope subspace basis.

Sigma

A lists of estimated covariance matrices at each mode for the tensor predictors.

fitted.values

The fitted response matrix.

residuals

The residuals matrix.

References

Zhang, X. and Li, L., 2017. Tensor envelope partial least-squares regression. Technometrics, 59(4), pp.426-436.

See Also

summary.Tenv for summaries, calculating mean squared error from the prediction.

plot.Tenv(via graphics::image) for drawing the two-dimensional coefficient plot.

predict.Tenv for prediction.

The generic functions coef, residuals, fitted.

TPRdim for selecting the dimension of envelope by cross-validation.

TPRsim for generating the simulated data used in tensor prediction regression.

The simulated data square used in tensor predictor regression.

Examples

# The dimension of predictor
p <- c(10, 10, 10)
# The envelope dimensions u.
u <- c(1, 1, 1)
# The dimension of response
r <- 5
# The sample size
n <- 200

# Simulate the data with TPRsim.
dat <- TPRsim(p = p, r = r, u = u, n = n)
x <- dat$x
y <- dat$y
B <- dat$coefficients

fit_std <- TPR.fit(x, y, method="standard")
fit_FG <- TPR.fit(x, y, u, method="FG")
fit_pls <- TPR.fit(x, y, u, method="PLS")

rTensor::fnorm(B-stats::coef(fit_std))
rTensor::fnorm(B-stats::coef(fit_FG))
rTensor::fnorm(B-stats::coef(fit_pls))

## ----------- Pass a list or an environment to x also works ------------- ##
# Pass a list to x
l <- dat[c("x", "y")]
fit_std_l <- TPR.fit(l, method="standard")

# Pass an environment to x
e <- new.env()
e$x <- dat$x
e$y <- dat$y
fit_std_e <- TPR.fit(e, method="standard")

## ----------- Use dataset "square" included in the package ------------- ##
data("square")
x <- square$x
y <- square$y
fit_std <- TPR.fit(x, y, method="standard")