Download Escalogramas multidimensionales

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
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?
• 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
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
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
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)
1. Method to recover Q given D
2. Obtain X given Q
Note that:
We cannot find exactly X because there will be many solutions to this
problem.
IF Q=XX’ also
Q=X A A-1 X’ for any orthogonal matrix A. Thus B=XA is also a solution
The standard solution: Make the spectral decomposition of the matrix Q
Q=ABA’
Where A and B contain the non zero eigenvectors and eigenvalues of the
matrix and take as solution
X=AB1/2
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)
Summary of the procedure
Example 1.Cities
(Note that they add up to zero by rows and columns. The matrix
10000)
has been divided by
Example 1
Eigenstructure of Q :
Final coordinates for the cities taking two dimensions:
Example 1. Plot
Similarities matrix
Example 2: similarity between
products
Example 2
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
Biplots
Representar conjuntamente los observaciones por las filas de V2 y
Las variables mediante las coordenadas D2/2 A’2
Se denimina biplots porque se hace una aproximación de dos
dimensiones a la matriz de datos
Biplot
Non metric MS
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