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Matrix Tutorial

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Matrix Tutorial

Transition Matrices

Graphs

Random Walks

- To show how some advanced mathematics has practical application in data mining / information retrieval.
- To show how some practical problems in data mining / information retrieval can be solved using matrix decomposition.
- To give you a flavour of some aspects of the course.

- In 1998 (in some state) Land use is:
- 30% I (Res), 20% II (Com), 50% III (Ind)

- Over 5 year period, the probabilities for change of use are:

Land Use after 5 years

=

- v1 = Av0

similarly

- v2 = A2v0

and so on…

http://kinetigram.com/mck/LinearAlgebra/JPaisMatrixMult04/classes/JPaisMatrixMult04.html

- When this converges:
- vn = Avn
- i.e. it converges to vnan eigenvector of A corresponding to an eigenvalue 1.
- vn= [12.5 25 62.5]

- The eigenvectors v and eigenvalues of a matrix A are the ones satisfying
- Avi = ivi

- Pre-multiplying by matrix A
is the same as

- Multiplying by the corresponding eigenvalue i

- Repeated application of the matrix to an arbitrary vector results in a vector proportional to the eigenvector with largest eigenvalue
- http://mathworld.wolfram.com/Eigenvector.html

- What has this got to do with Random Walks?...

- Consider a random walk over a set of linked web pages.
- The situation is defined by a transition (links) matrix.
- The eigenvector corresponding to the largest eigenvalue of the transition matrix tells us the probabilities of the walk ending on the various pages.

From

To

- Eigenvector corresponding to largest Eigenvalue
- 0.38
- 0.20
- 0.49
- 0.26
- 0.71

- EVD: http://kinetigram.com/mck/LinearAlgebra/JPaisEVD04/classes/JPaisEVD04.html

B

C

A

D

E

- Why matrix algebra now?
- The Google PageRank algorithm uses Eigenvectors in ranking relevant pages.

- Resources
- http://mathworld.wolfram.com/Eigenvector.html
- The Matrix Cookbook
- http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf

- Eigenvectors are a special set of vectors associated with a linear system of equations (i.e., a matrix equation).
- Each eigenvector is paired with a corresponding so-called eigenvalue.
- The decomposition of a square matrix into eigenvalues and eigenvectors is known as eigen decomposition

http://mathworld.wolfram.com/Eigenvector.html

- Class EigenvalueDecomposition
- Constructor EigenvalueDecomposition(Matrix Arg)
- Methods
- Matrix GetV()
- Matrix GetD()

- Where A is the original matrix and:
- AV=VD

- Data describing connections between objects can be described as a graph
- This graph can be represented as a matrix
- Interesting structure can be discovered in this data using Matrix Eigen-decomposition