L0TFinv (version 0.1.0)

XMat: Generate an artificial design matrix

Description

This matrix corresponds to the difference matrix, transforming the L0 trend filtering model into an inverse statistical problem.

Usage

XMat(n, q)

Value

A matrix with dimensions \(n\) by \(n\), whose elements correspond to the difference matrix.

Arguments

n

The number of data points

q

The order of the difference

Details

Noticing the correspondence between \(\boldsymbol{D}^{(q+1)}\) and \(\boldsymbol{X}^{(q+1)}\), the result of their matrix multiplication is a combination of a zero matrix and an identity matrix. Expressed as \(\boldsymbol{D}^{(q+1)} \boldsymbol{X}^{(q+1)}=(\boldsymbol{O}_{(n-q-1)\times(q+1)},\quad \boldsymbol{I}_{(n-q-1)\times(n-q-1)})\). The result is advantageous for the invertible processing of the original L0 trend filtering problem.

Examples

mat1 <- XMat(n = 10, q = 0)
print(mat1)

mat2 <- XMat(n = 15, q = 1)
print(mat2)

mat3 <- XMat(n = 15, q = 2)
print(mat3)

Mat1 <- DiffMat(n = 10, q = 0)
Mat2 <- DiffMat(n = 15, q = 1)
Mat3 <- DiffMat(n = 15, q = 2)
print(Mat1%*%mat1)
print(Mat2%*%mat2)
print(Mat3%*%mat3)