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Version 6 (modified by Víctor de Buen Remiro, 16 years ago) (diff)

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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 A and inverse prior over V

Vectorial linear regression equations are

Y=A \cdot X + E

where

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

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

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

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

Minka defines also the known data pair 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 A by means of adding a set of r inequations defining a feasible region

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

being

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 X must be full-range m <= N because X W X ^ T must be nonsingular with W = \alpha I_{m}, where \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.