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Matrix Tutorial

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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  1. Matrix Tutorial Transition Matrices Graphs Random Walks

  2. 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.

  3. 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:

  4. 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

  5. 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]

  6. 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

  7. 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?...

  8. 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.

  9. 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

  10. 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

  11. 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

  12. 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

  13. 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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