Changeset fa6ea84d in tol for tol_pkg/BysVecLinReg/doc/BysVecLinReg.wiki
- Timestamp:
- May 7, 2010, 8:11:44 AM (16 years ago)
- Branches:
- 1941-docker-image-to-test, devel, master
- Children:
- e1746f4
- Parents:
- 1ea1fbe
- File:
-
- 1 edited
-
tol_pkg/BysVecLinReg/doc/BysVecLinReg.wiki (modified) (3 diffs)
Legend:
- Unmodified
- Added
- Removed
-
tol_pkg/BysVecLinReg/doc/BysVecLinReg.wiki
r1ea1fbe rfa6ea84d 1 = TOL Package BysVecLinReg = 1 2 2 BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with arbitrary constraining inequations 3 4 BysVecLinReg yields for Bayesian simulator of Vectorial Linear Regression with 5 arbitrary constraining inequations. 6 7 The method used to solve it in this package is based on 8 [https://www.tol-project.org/export/HEAD/tolp/trunk/tol_pkg/BysVecLinReg/doc/bayes-linear-minka.pdf 9 Bayesian linear regression Thomas Minka (2001)] using invariant scale prior over 10 [[LatexEquation(A)]] and inverse prior over [[LatexEquation(V)]] 11 12 == Vectorial linear regression == 3 13 4 14 Vectorial linear regression equations are [[BR]] … … 8 18 where [[BR]] 9 19 10 * [[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]] 11 * [[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]] 12 * [[LatexEquation(A\in\mathbb{R}^{d\times m} )]] has the unknown regression coefficients that we want to estimate [[BR]] 13 * [[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} )]] 20 * [[LatexEquation(Y\in\mathbb{R}^{d\times N} )]] is the multivariant known 21 output matrix, where each row is a different output vector 22 [[LatexEquation(y_{n}\in\mathbb{R}^{N} )]][[BR]] 23 * [[LatexEquation(X\in\mathbb{R}^{m\times N} )]] is the known and full rank 24 input matrix, where each row is a different input vector 25 [[LatexEquation(x_{n}\in\mathbb{R}^{N} )]] [[BR]] 26 * [[LatexEquation(A\in\mathbb{R}^{d\times m} )]] has the unknown regression 27 coefficients that we want to estimate [[BR]] 28 * [[LatexEquation(E\in\mathbb{R}^{d\times N} )]] is the multivariant 29 residuals, where each row is the residuals vector 30 [[LatexEquation(e_{n}\in\mathbb{R}^{N} )]] corresponding to output 31 [[LatexEquation(y_{n} )]] 14 32 15 All residuals inside the same row are incorrelated normal, but resiudals in the same column [[LatexEquation(j)]] are [[BR]] 33 All residuals inside the same row are incorrelated normal, but resiudals in 34 the same column [[LatexEquation(j)]] are [[BR]] 16 35 17 36 [[LatexEquation(e_{.,j} \sim N\left(0,V\right) E\in\mathbb{R}^{d\times d} \forall j=1 \ldots d )]][[BR]] 18 37 19 where [[LatexEquation(V)]] is symmetric positive definite and unknown, but the same for each column[[BR]] 38 where [[LatexEquation(V)]] is symmetric positive definite and unknown, but the 39 same for each column.[[BR]] 20 40 21 When there are some restriction over parameters inside [[LatexEquation(A)]] we must to add the inequations of feasible region [[BR]] 41 Minka defines also the known data pair [[LatexEquation(D = left(Y,Xright))]] 42 that will be used just to get more compact conditioninig expressions. 43 44 == Arbitrary constraining inequations == 45 46 We will extend the model scope with arbitrary non null meassured restrictions 47 over parameters inside [[LatexEquation(A)]] by means of adding a set of 48 [[LatexEquation(r)]] inequations defining a feasible region [[BR]] 22 49 23 50 [[LatexEquation(\Omega = \left\{ A\in\mathbb{R}^{d\times m} \mid F\left(A\right) \le 0 \right\})]] [[BR]] … … 27 54 [[LatexEquation( F\left(A\right):\mathbb{R}^{d\times m}\longrightarrow\mathbb{R}^{r} )]] [[BR]] 28 55 29 the constraining function. [[BR]]56 the arbitrary constraining function. [[BR]] 30 57 31 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)] 58 == Invariant-scale prior over coefficient matrix == 59 60 Although Minka not explicitly stated in any place, under the invariant prior 61 follows that [[LatexEquation(X)]] must be full-range [[LatexEquation(m <= N)]] 62 because [[LatexEquation(X W X ^ T)]] must be nonsingular with 63 [[LatexEquation(W = \alpha I_{m})]], where [[LatexEquation(\alpha)]] is the 64 scale-invariant parameter governing the prior and estimated more 65 forward to maximize the evidence of the data, which depends on the assumptions 66 the model. 67 68 == Inverse Wishart prior over covariance matrix == 69 70 ...
Note: See TracChangeset
for help on using the changeset viewer.
