A Plot for Visualizing Multivariate Data Rida E. A. Moustafa - - PowerPoint PPT Presentation

A Plot for Visualizing Multivariate Data

Rida E. A. Moustafa George Mason University ADM Group,AAL rmoustaf@galaxy.gmu.edu rmustafa@aalcpas.com

Talk Outline

The Theory of MV-Plot. Detecting Linear Structures with MV-plot. Detecting Non-Linear Structures with MV-plot. Comparisons with other methods and application on real data.

MV-Plot Theory

∑ ∑

= =

− = = = =

d j j d d j j d

x f x x f x g v x x f m

1 2 1 1 1

| ) ( | )) ( , ( | | ) ( Given an observation x=(x1,x2,…,xd) We define m and v as follows: Computing m and v for every observation produces vector of m and v. What is the relationship between m and v?

MV-Relationship in 2-d

2 1 2 1 2 2 1 2 1 2 1 2 1 2 1 2 1

| | |) | | (| | |

i i i j ij i i i j ij i

x x m x v x x x m − = − = + = =

∑ ∑

= =

• Normalizing the data in range (0,1) avoid the abs-value in computing m.
• Close to the PC in 2-d

MV- detects linear structure(s)

1 1 1 1 1 1 1 1 1 1 2 1 1 1 2 1 1 1 2

; ; ) 1 ( ) 1 ( if ) 1 ( ; ) 1 ( a x a v a x a m a w a w w w w x w v w x w m w x w x

i i i i i i i i i i

+ = + = ⇒ = = − ≈ + ⇒ + + + − = + + = ⇒ + =

If the data is linear in the original space It will be linear in the MV-space!!

MV- detects linear structure(s)

      − + − − =       + + =

∑ ∑

− = − − = 1 1 1 1 1 1

) 1 ( ) 1 ) 1 ( ) 1 (

2

d j ij j d d j d j ij j d j

w d x w d v w x w m

      + ≈       + =

∑ ∑

− = − = 1 1 1 1 d j ij j j d j ij j j

a x a v a x a m

Detecting Linear structure(s) Example I

Detecting Linear structure(s) Example II

Detecting Linear structure(s) Example III

Detecting nonlinear data with MV-plot

MV- plot can detect nonlinear structure

in the data set without any changes in the equations.

Detecting nonlinear structure

| ) sin( | ), sin( ) sin( , | ) cos( | ), cos( ) cos( , x x v x x m x x x x v x x m x x − = + = → − = + = →

Detecting Sphere(s)

( ) ( )

.

2

2 2 1 2 2 1 2 1 1 2 d R i i d j i ij d d j i ij d i

m v dm x m x v = + ⇒ − = − =

∑ ∑

= =

Case I:

• The sphere radius R
• The sphere center is the origin

Detecting Sphere(s)

( ) ( )

. ) ( ) (

2

2 2 1 2 2 1 2 1 1 2 d R i i d j i c j c j ij d d j i c j c j ij d i

m v m x d x x m x x x v = + ⇒ − − − = = − + − =

∑ ∑

= =

Case II:

• The sphere radius R
• The sphere center is not the origin

Detecting Sphere(s)

Fisher’s IRIS data (150x4)

3-classes of( 50 point each)

Process control data (600x60)

6-classes of (100 points each)

Pollen data (3,848x5) (Wegman’s data)

2-classes (linear and nonlinear)

Application on Real data

Multidimensional Scaling Fisher Discriminate Analysis Principal Component

Related Dimensional Reduction Methods

IRIS (R. A. Fisher) Dataset 150-cases in 4-dim

Time Series Dataset

600-cases in 60-dim

Pollen dataset

3,848-points in 5-dim

Other methods: Require more storage and speed. Even if it work, we expect bad results on this particular data. (Wegman2002)

Pollen dataset

Linear and Nonlinear mixed structures.

The linear structure in the Pollen data set

17+16+18+17+14+16=98 Linear, 3750 nonlinear

Summary

MV-algorithm can discover the linear and

nonlinear pattern at the same time.

MV-algorithm can discover symmetric data. MV-algorithm deals with large multivariate

data.