1 / 13

# Matrix Tutorial - PowerPoint PPT Presentation

Matrix Tutorial. Transition Matrices Graphs Random Walks. Objective . To show how some advanced mathematics has practical application in data mining / information retrieval.

I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.

## PowerPoint Slideshow about ' Matrix Tutorial' - nirav

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.

- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

### 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

• i.e. vi is a vector that:

• 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.

To

Web Pages Example

• 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

• 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