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Version 11 (modified by lcereceda, 14 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 to solve it in this package is based on ​https://www.tol-project.org/export/HEAD/tolp/OfficialTolArchiveNetwork/BysVecLinReg/doc/bayes-linear-minka.pdf Bayesian linear regression Thomas Minka (2001)] using invariant scale prior over A and inverse prior over V

Vectorial linear regression

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 residuals 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,X \right) that will be used just to get more compact conditioninig expressions.

Arbitrary constraining inequations

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\}

with

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

being the arbitrary constraining function.

Invariant-scale prior over coefficient matrix

Although Minka does not explicitly state this, from the invariant prior it follows that X must be full-rank 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 of the model.

Inverse Wishart prior over covariance matrix

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