TOL Package BysVecLinReg

BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with arbitrary constraining inequations.

The method used in this package is based on ​Bayesian linear regression Thomas Minka (2001) using invariant scale prior over LatexEquation(A) and inverse prior over LatexEquation(V)

Vectorial linear regression equations are

LatexEquation(Y=A \cdot X + E)

where

• LatexEquation(Y\in\mathbb{R}^{d\times N} ) is the multivariant known output matrix, where each row is a different output vector LatexEquation(y_{n}\in\mathbb{R}^{N} )
  
• LatexEquation(X\in\mathbb{R}^{m\times N} ) is the known and full rank input matrix, where each row is a different input vector LatexEquation(x_{n}\in\mathbb{R}^{N} )
  
• LatexEquation(A\in\mathbb{R}^{d\times m} ) has the unknown regression coefficients that we want to estimate
  
• LatexEquation(E\in\mathbb{R}^{d\times N} ) is the multivariant residuals, where each row is the residuals vector LatexEquation(e_{n}\in\mathbb{R}^{N} ) corresponding to output LatexEquation(y_{n} )

All residuals inside the same row are incorrelated normal, but resiudals in the same column LatexEquation(j) are

LatexEquation(e_{.,j} \sim N\left(0,V\right) E\in\mathbb{R}^{d\times d} \forall j=1 \ldots d )

where LatexEquation(V) is symmetric positive definite and unknown, but the same for each column.

Minka defines also the known data pair LatexEquation(D = left(Y,Xright)) that will be used just to get more compact conditioninig expressions.

We will extend the model scope with arbitrary non null meassured restrictions over parameters inside LatexEquation(A) by means of adding a set of LatexEquation(r) inequations defining a feasible region

LatexEquation(\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\})

being

LatexEquation( F\left(A\right):\mathbb{R}^{d\times m}\longrightarrow\mathbb{R}^{r} )

the arbitrary constraining function.

Although Minka not explicitly stated in any place, under the invariant prior follows that LatexEquation(X) must be full-range LatexEquation(m <= N) because LatexEquation(X W X ^ T) must be nonsingular with LatexEquation(W = \alpha I_{m}), where LatexEquation(\alpha) is the scale-invariant parameter governing the prior and estimated more forward to maximize the evidence of the data, which depends on the assumptions the model.