Download Matrix Tutorial - University College Dublin

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