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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

Matrix Tutorial

Transition Matrices

Graphs

Random Walks


Objective
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
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 process1
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 process2
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
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
    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
    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.


    Web pages example

    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
    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 eigenvectors1
    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
    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
    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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