Escalogramas multidimensionales

1
(No Transcript)
2
Introduction

• Given a Matrix of distances D, (which contains
  zeros in the main diagonal and is squared and
  symmetric), find variables which could be able,
  approximately, to generate, these distances.
• The matrix can also be a similarities matrix,
  squared and symmetric but with ones in the main
  diagonal and values between zero and one
  elsewhere.
• Broadly Distance (0 d 1) 1- similarity

3
(No Transcript)
4
(No Transcript)
5
Principal Coordinates (Metric Multidimensional
Scaling)

• Given the D matrix of distances, Can we find a
  set of variables able to generate it ?
• Can we find a data matrix X able to generate D?

6

• Main idea of the procedure
• (1) To understand how to obtain D when X is
  known and given,
• (2) Then work backwards to build the matrix X
  given D

7
Procedure
Remember that given a data matrix we have a zero
mean data matrix by the transformation
With this matrix we can compute two squared and
symmetric matrices
The first is the covariance matrix S The
second is the Q matrix of scalar products among
observations
8
The matrix of products Q is closely related to
the distance matrix , D, we are interested in.
The relation between D and Q is as follows
Elements of Q
Elements of D
Main result Given the matrix Q we can obtain the
matrix D
9
How to recover Q given D?
Note that as we have zero mean variables the sum
of any row in Q must be zero
t trace(Q)
10
1. Method to recover Q given D
11
2. Obtain X given Q
Note that

• We cannot find exactly X because there will be
  many solutions to this problem.
• IF QXX also
• QX A A-1 X for any orthogonal matrix A. Thus
  BXA is also a solution
• The standard solution Make the spectral
  decomposition of the matrix Q
• QABA
• Where A and B contain the non zero eigenvectors
  and eigenvalues of the matrix and take as
  solution
• XAB1/2

12
Conclusion

• We say that D is compatible with an euclidean
  metric if Q obtained as
• Q-(1/2)PDP
• is nonnegative (all eigenvalues non negative)

13
Summary of the procedure
14
Example 1.Cities
15
(Note that they add up to zero by rows and
columns. The matrix has been divided by 10000)
16
Example 1
Eigenstructure of Q
17
Final coordinates for the cities taking two
dimensions
18
Example 1. Plot
19
(No Transcript)
20
Similarities matrix
21
Example 2 similarity between products
22
Example 2
23
Relationship with PC

• PC eigenvalues and vectors of S
• PCoordinates eigenvalues and vectors of Q

If the data are matric both are identical. P
Coordinates generalizes PC for non exactly
metric data
24
(No Transcript)
25
Biplots
Representar conjuntamente los observaciones por
las filas de V2 y Las variables mediante las
coordenadas D2/2 A2
Se denimina biplots porque se hace una
aproximación de dos dimensiones a la matriz de
datos
26
(No Transcript)
27
Biplot
28
(No Transcript)
29
Non metric MS
30
(No Transcript)
31
(No Transcript)
32
(No Transcript)
33
A common method

• Idea if we have a monotone relation between x
  and y it must be a linear exact relationship
  between the ranks of both variables
• Ordered regression or assign ranks and make a
  regression between ranks iterating