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Version 3 (modified by Víctor de Buen Remiro, 15 years ago) (diff)

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

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.

When there are some restriction over parameters inside A we must to add the inequations of 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.

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