Changes between Version 4 and Version 5 of BysVecLinReg

Timestamp:

May 6, 2010, 5:10:01 PM (15 years ago)

Author:

Víctor de Buen Remiro

Comment:

--

BysVecLinReg

@@ -1 +1 @@
 = TOL Package BysVecLinReg = 
 
-BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with arbitrary constraining inequations
+BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with 
+arbitrary constraining inequations.
+
+The method used in this package is based on [https://www.tol-project.org/export/HEAD/tolp/trunk/tol_pkg/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 equations are [[BR]]
@@ -9 +12 @@
 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} )]] 
+ * [[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]] 
+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]]
+where [[LatexEquation(V)]] is symmetric positive definite and unknown, but the 
+same for each column.[[BR]]
 
-When there are some restriction over parameters inside [[LatexEquation(A)]] we must to add the inequations of feasible region [[BR]]
+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 [[BR]]
 
 [[LatexEquation(\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\})]] [[BR]] 
@@ -30 +48 @@
 the arbitrary constraining function. [[BR]]
 
-The method used in this package is based on [https://www.tol-project.org/export/HEAD/tolp/trunk/tol_pkg/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)]]
+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.