TRES (version 1.1.5)

square: Square simulated data

Description

Synthetic data generated from tensor predictor regression (TPR) model. Each response observation is univariate, and each predictor observation is a matrix.

Usage

data("square")

Format

A list consisting of four components:

x

A $32\times 32\times 200$ tensor, each matrix x@data[,,i] represents a predictor observation.

y

A $1\times 200$ matrix, each entry represents a response observation.

coefficients

A $32\times 32\times 1$ tensor with a square pattern.

Gamma

A list consisting of two $32\times 2$ envelope basis.

Details

The dataset is generated from the tensor predictor regression (TPR) model: $$Y_i=B_{(m+1)}vec(X_i)+{ϵ}_i,i=1,\dots ,n,$$ where $n=200$ and the regression coefficient $B\in R^{32\times 32}$ is a given image with rank 2, which has a square pattern. All the elements of the coefficient matrix $B$ are either 0.1 or 1. To make the model conform to the envelope structure, we construct the envelope basis ${\mathrm{\Gamma }}_k$ and the covariance matrices ${\mathrm{\Sigma }}_k,k=1,2$, of predictor $X$ as following. With the singular value decomposition of $B$, namely $B={\mathrm{\Gamma }}_1\mathrm{\Lambda }{\mathrm{\Gamma }}_2^T$, we choose the envelope basis as ${\mathrm{\Gamma }}_k\in R^{32\times 2},k=1,2$. Then the envelope dimensions are $u_1=u_2=2$. We set matrices ${\mathrm{\Omega }}_k=I_2$ and ${\mathrm{\Omega }}_{0k}=0.01I_{30}$, $k=1,2$. Then we generate the covariance matrices ${\mathrm{\Sigma }}_k={\mathrm{\Gamma }}_k{\mathrm{\Omega }}_k{\mathrm{\Gamma }}_k^T+{\mathrm{\Gamma }}_{0k}{\mathrm{\Omega }}_{0k}{\mathrm{\Gamma }}_{0k}^T$, followed by normalization with their Frobenius norms. The predictor $X_i$ is then generated from two-way tensor (matrix) normal distribution $TN(0;{\mathrm{\Sigma }}_1,{\mathrm{\Sigma }}_2)$. And the error term ${ϵ}_i$ is generated from standard normal distribution.

References

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

Examples

# NOT RUN {
## Fit square dataset with the tensor predictor regression model
data("square")
x <- square$x
y <- square$y
# Model fitting with ordinary least square.
fit_std <- TPR.fit(x, y, method="standard")
# Draw the coefficient plot.
plot(fit_std)

# }