# The MV Plot by alicejenny

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```									A Plot for Visualizing
Multivariate Data
Rida E. A. Moustafa

George Mason University

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

Given an observation x=(x1,x2,…,xd)
We define m and v as follows:

d
m  f ( x)    1
d   | x
j 1
j   |

d
v  g ( x, f ( x))       1
d       | x
j 1
j    f ( x ) |2

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
mi     1
2    | xij |  1 (| xi1 |  | xi 2 |)
j 1
2

2
vi    1
2      | xij  mi |2 
j 1
1
2   xi1  xi 2

• 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)

If the data is linear in the original space

xi 2  w1 xi1  w0  mi                  1
2   ( w1  1) xi1  w0 ;
vi     1
2   ( w1  1) xi1  w0
if    w1   
( w1  1)  ( w1  1)  a1 ; w0  a0
 mi  a1 xi1  a0 ; vi  a1 xi1  a0

It will be linear in the MV-space!!
MV- detects linear structure(s)
 d 1               
m j  d  ( w j  1) xij  w0 
1

 j 1               
 d 1                              
v j  d 2  (d  1) w j  1) xij  (d  1) w0 
d 1

 j 1                              

 d 1          
m j   a j xij  a0 
 j 1          
 d 1          
v j   a j xij  a0 
 j 1          
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
x, cos(x)  m  x  cos(x), v | x  cos(x) |
x, sin( x)  m  x  sin( x), v | x  sin( x) |
Detecting Sphere(s)
Case I:
• The sphere center is the origin

2

vi2  d  xij  mi   d  xij  dmi2      
d                       d
1                 1     2

j 1                    j 1

v m 
2
i
2
i
R2
d    .
Detecting Sphere(s)
Case II:
• The sphere center is not the origin

2

                
d
vi2  d  xij  x cj  x cj  mi 
1

j 1

                                
d
 d  ( xij  x cj ) 2  d ( x cj  mi ) 2
1

j 1

v m 
2
i
2
i
R2
d    .
Detecting Sphere(s)
Application on Real data

   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)
Related Dimensional
Reduction Methods

   Multidimensional Scaling
   Fisher Discriminate Analysis
   Principal Component
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
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.

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