1. Pádraig Cunningham University College Dublin Matrix Tutorial Transition Matrices Graphs Random Walks

2. 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. 3 Stochastic Matrix: Markov process From IFrom IIFrom III To I0.80.10 To II0.10.70.1 To III0.10.20.9 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. 4 Stochastic Matrix: Markov process 262252 0.80.10 0.70.1 0.20.9 30 20 50 = Land Use after 5 years v 1 = Av 0 v 2 = A 2 v 0 similarly and so on… http://kinetigram.com/mck/LinearAlgebra/JPaisMatrixMult04/classes/JPaisMatrixMult04.html

5. 5 Stochastic Matrix: Markov process When this converges:  v n = Av n  i.e. it converges to v n an eigenvector of A corresponding to an eigenvalue 1.  v n = [12.5 25 62.5]

6. 6 Brief Review of Eigenvectors The eigenvectors v and eigenvalues of a matrix A are the ones satisfying Av i = i v i i.e. v i is a vector that:  Pre-multiplying by matrix A is the same as  Multiplying by the corresponding eigenvalue i

7. 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 http://mathworld.wolfram.com/Eigenvector.html What has this got to do with Random Walks?...

8. 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. 9 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 http://kinetigram.com/mck/LinearAlgebra/JPaisEVD04/classes/JPaisEVD04.html A D C B E ABCDE A10001 B11000 C01101 D00110 E11111 From To

10. 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 http://mathworld.wolfram.com/Eigenvector.html  The Matrix Cookbook http://www.imm.dtu.dk/pubdb/views/edoc_download.php/3274/pdf/imm3274.pdf

11. 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. 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. 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