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

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Matrix Tutorial

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Matrix Tutorial

Transition Matrices

Graphs

Random Walks


Objective

  • 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.


Stochastic Matrix: Markov process

  • 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:


Stochastic Matrix: Markov process

Land Use after 5 years

=

  • v1 = Av0

similarly

  • v2 = A2v0

and so on…

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


Stochastic Matrix: Markov process

  • 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]


Brief Review of Eigenvectors

  • 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


  • The important property…

    • 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?...


    Transition Matrices & 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

    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


    Review of Matrix Algebra

    • 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


    Brief Review of Eigenvectors

    • 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


    Matrices in JAVA - e.g. JAMA

    • Class EigenvalueDecomposition

      • Constructor EigenvalueDecomposition(Matrix Arg)

      • Methods

        • Matrix GetV()

        • Matrix GetD()

    • Where A is the original matrix and:

      • AV=VD


    Summary

    • 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


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