Algorithmic Lovász local lemma: Difference between revisions
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In [[linear algebra]] and [[statistics]], the '''pseudo-determinant'''<ref name="minka">{{cite web | author = Minka, T.P. | title = Inferring a Gaussian Distribution | url = http://research.microsoft.com/en-us/um/people/minka/papers/gaussian.html | year = 2001}} [http://research.microsoft.com/en-us/um/people/minka/papers/minka-gaussian.pdf PDF]</ref> is the product of all non-zero [[eigendecomposition (matrix)|eigenvalues]] of a [[square matrix]]. It coincides with the regular [[determinant]] when the matrix is [[invertible matrix|non-singular]]. | |||
== Definition == | |||
The pseudo-determinant of a square ''n''-by-''n'' matrix '''A''' may be defined as: | |||
:<math> | |||
|\mathbf{A}|_+ = \lim_{\alpha\to 0} \frac{|\mathbf{A} + \alpha \mathbf{I}|}{\alpha^{n-\operatorname{rank}(\mathbf{A})}} | |||
</math> | |||
where |'''A'''| denotes the usual [[determinant]], '''I''' denotes the [[identity matrix]] and rank('''A''') denotes the [[rank (linear algebra)| rank]] of '''A'''. | |||
==Definition of pseudo determinant using Vahlen Matrix== | |||
The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. <math>(ax+b)(cx+d)^{-1}</math> for <math>a,b,c,d\in \mathcal {G}(p,q)</math>)) is defined as <math>[f]=::\begin{bmatrix}a & b \\c & d \end{bmatrix}</math>. By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean | |||
<math> pdet:: \begin{bmatrix}a & b\\ c& d\end{bmatrix} =ad^\dagger -bc^\dagger</math> | |||
If <math>pdet[f]>0</math>, the transformation is sense-preserving (rotation) whereas if the <math>pdet[f]<0</math>, the transformation is sense-preserving (reflection). | |||
== Computation for positive semi-definite case == | |||
If <math>A</math> is [[positive-definite matrix | positive semi-definite]], then the [[singular value decomposition | singular values]] and [[eigendecomposition (matrix)|eigenvalues]] of <math>A</math> coincide. In this case, if the [[singular value decomposition]] ('''SVD''') is available, then <math>|\mathbf | |||
{A}|_+</math> may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1. | |||
==Application in statistics== | |||
If a statistical procedure ordinarily compares distributions in terms of the determinants of variance-covariance matrices then, in the case of singular matrices, this comparison can be undertaken by using a combination of the ranks of the matrices and their pseudo-determinants, with the matrix of higher rank being counted as "largest" and the pseudo-determinants only being used if the ranks are equal.<ref>[http://support.sas.com/documentation/cdl/en/statug/63347/HTML/default/viewer.htm#statug_rreg_sect021.htm SAS documentation on "Robust Distance"]</ref> Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.<ref>Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", ''Computers & Geosciences'', 23 (7), 739–761 {{DOI| 10.1016/S0098-3004(97)00050-2}}</ref> | |||
== See also == | |||
*[[Matrix determinant]] | |||
*[[Moore-Penrose pseudoinverse]], which can also be obtained in terms of the non-zero singular values. | |||
== References == | |||
<references/> | |||
[[Category:Multivariate statistics]] | |||
[[Category:Matrices]] | |||
{{statistics-stub}} | |||
Revision as of 03:14, 21 December 2013
In linear algebra and statistics, the pseudo-determinant[1] is the product of all non-zero eigenvalues of a square matrix. It coincides with the regular determinant when the matrix is non-singular.
Definition
The pseudo-determinant of a square n-by-n matrix A may be defined as:
where |A| denotes the usual determinant, I denotes the identity matrix and rank(A) denotes the rank of A.
Definition of pseudo determinant using Vahlen Matrix
The Vahlen matrix of a conformal transformation, the Möbius transformation (i.e. for )) is defined as . By the pseudo determinant of the Vahlen matrix for the conformal transformation, we mean
![{\displaystyle [f]=::{\begin{bmatrix}a&b\\c&d\end{bmatrix}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e1ed04ff2d704d2a6d3eef0d060dd594b9feb4d)
If , the transformation is sense-preserving (rotation) whereas if the , the transformation is sense-preserving (reflection).
![{\displaystyle pdet[f]>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0bdf447e15e65e00a0bb1464a4eb50c573c37a68)
![{\displaystyle pdet[f]<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c34bcce728a1175c0b8845e803cf1085dcf2ab7b)
Computation for positive semi-definite case
If is positive semi-definite, then the singular values and eigenvalues of coincide. In this case, if the singular value decomposition (SVD) is available, then may be computed as the product of the non-zero singular values. If all singular values are zero, then the pseudo-determinant is 1.
Application in statistics
If a statistical procedure ordinarily compares distributions in terms of the determinants of variance-covariance matrices then, in the case of singular matrices, this comparison can be undertaken by using a combination of the ranks of the matrices and their pseudo-determinants, with the matrix of higher rank being counted as "largest" and the pseudo-determinants only being used if the ranks are equal.[2] Thus pseudo-determinants are sometime presented in the outputs of statistical programs in cases where covariance matrices are singular.[3]
See also
- Matrix determinant
- Moore-Penrose pseudoinverse, which can also be obtained in terms of the non-zero singular values.
References
- ↑ Template:Cite web PDF
- ↑ SAS documentation on "Robust Distance"
- ↑ Bohling, Geoffrey C. (1997) "GSLIB-style programs for discriminant analysis and regionalized classification", Computers & Geosciences, 23 (7), 739–761 Electronic Instrument Positions Staff (Standard ) Cameron from Clarence Creek, usually spends time with hobbies and interests which include knotting, property developers in singapore apartment For sale and boomerangs. Has enrolled in a world contiki journey. Is extremely thrilled specifically about visiting .
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