Changes between Initial Version and Version 1 of OfficialTolArchiveNetworkBysVecLinReg

Timestamp:

Oct 19, 2010, 10:30:53 AM (15 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

OfficialTolArchiveNetworkBysVecLinReg

@@ +1 @@
+= BysVecLinReg = 
+
+BysVecLinReg is an open source [wiki:TolPackageRulesAndComments TOL Package] published as partt of [wiki:OfficialTolArchiveNetwork Official Tol Archive Network]
+
+BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with 
+arbitrary constraining inequations and lineal constraining equations.
+
+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 
+[[LatexEquation(A)]] and inverse prior over [[LatexEquation(V)]]
+
+== Vectorial linear regression ==
+
+Vectorial linear regression equations are [[BR]]
+
+[[LatexEquation(Y=A \cdot X + E)]] [[BR]]
+
+where [[BR]]
+
+ * [[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} )]][[BR]]
+ * [[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} )]] [[BR]]
+ * [[LatexEquation(A\in\mathbb{R}^{d\times m} )]] has the unknown regression 
+   coefficients that we want to estimate [[BR]]
+ * [[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 [[BR]] 
+
+[[LatexEquation(e_{.,j} \sim N\left(0,V\right) E\in\mathbb{R}^{d\times d} \forall j=1 \ldots d )]][[BR]]
+
+where [[LatexEquation(V)]] is symmetric positive definite and unknown, but the 
+same for each column.[[BR]]
+
+Minka defines also the known data pair [[LatexEquation(D = left(Y,Xright))]] 
+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 [[LatexEquation(A)]] by means of adding a set of 
+[[LatexEquation(r)]] inequations defining a feasible region [[BR]]
+
+[[LatexEquation(\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\})]] [[BR]] 
+
+being [[BR]]
+
+[[LatexEquation( F\left(A\right):\mathbb{R}^{d\times m}\longrightarrow\mathbb{R}^{r} )]]  [[BR]]
+
+the arbitrary constraining function. [[BR]]
+
+== Invariant-scale prior over coefficient matrix ==
+
+Although Minka not explicitly stated in any place, under the invariant prior
+follows that [[LatexEquation(X)]] must be full-rank [[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.
+