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Matrix Tutorial Transition Matrices Graphs Random Walks Pádraig Cunningham University College Dublin 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: From I From II From III To I 0.8 0.1 0 To II 0.1 0.7 0.1 To III 0.1 0.2 0.9 4 Stochastic Matrix: Markov process Land Use after 5 years 26 22 = 52 0.8 0.1 0 30 0.1 0.7 0.1 20 0.1 0.2 0.9 50 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 vn an 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 lim A y b v n n n 1 1 1 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 Web Pages Example From A B C D E A 1 0 0 0 1 B 1 1 0 0 0 C 0 1 1 0 1 D 0 0 1 1 0 E 1 1 1 1 1 B C To A E Eigenvector corresponding to largest Eigenvalue D 0.38 0.20 0.49 0.26 0.71 EVD: http://kinetigram.com/mck/LinearAlgebra/JPaisEVD04/classes/JPaisEVD04.html 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