Matrix tutorial
This presentation is the property of its rightful owner.
Sponsored Links
1 / 13

Matrix Tutorial PowerPoint PPT Presentation


  • 91 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

Matrix Tutorial

An Image/Link below is provided (as is) to download presentation

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

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


  • Login