Transcript Basic principles of probability theory

Proximity matrices and scaling
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Purpose of scaling
Similarities and dissimilarities
Classical Euclidean scaling
Non-Euclidean scaling
Horseshoe effect
Non-Metric Scaling
Example
Proximity matrices
There are many situations when dissimilarities (or similarities) between individuals are
measured. Purpose of multidimensional scaling is to find relative orientation (in some
sense) of the individuals. If dissimilarities are ordinary Euclidean distances then purpose
could be finding relative coordinates of the individuals. Scaling is also used for seriation
purposes. E.g. in archeology, history. In this case only one dimensional configuration is
sought.
In general dissimilarity matrix is called proximity matrix. I.e. how close are individuals to
each other.
Euclidean distances are simplest of all and they relatively easy to work with. Distances are
metric if for all triples (triangular inequality) (i,j,k):
1)dik  dij  d jk 2)dij  0 3)dij  d ji
Matrix with elements of these distances is denoted as D. This distance matrix is said to be
Euclidean if n points corresponding to this matrix can be embedded in an Euclidean
space so that distances between the points in the Euclidean space is equal to the
corresponding elements of the matrix (it can be modified to add weights).
n
d   ( xik  x jk )2
2
ij
k 1
If distances correspond to Euclidean distances then there is an elegant solution to the problem
of finding configuration from the distance matrix.
Similarity and dissimilarity
There are situations when instead of distances (similarities) between of object
similarities are given. Similarities can be defined as a matrix with elements with
conditions:
1)crs  csr 2)crs  crr for any r and s
For example correlation matrix can be considered as a similarity matrix. We can
calculated dissimilarities (distances) from the similarities using:
drs  (crr  css  2crs )1/ 2
This matrix obeys the conditions required by the metric distances
Metric scaling
Suppose that we have an nxn matrix of square of pairwise distances that are Euclidean. Denote
this matrix and its elements with D and dij2. We want to find n points in k dimensional
Euclidean space so that distances between these points are equal to the elements of D.
Denote nxk matrix of points by X. Let us define Q=XXT. Then for distances and elements
of Q we can write:
k
k
d rs2   ( xrj  xsj )2 , qrs   xrj xsj
j 1
j 1
D is the matrix with elements equal to square of drs. Then between elements of D and Q the
following relation can be written:
drs2  qrr  qss  2qrs
We can find the positions of the points if we assume that centroid of the elements of X is 0, i.e.
n
x
rj
0
r 1
Then we can write the following relations:
n
n
n
n




rr
ss
rs
rr
ss
Here we used the fact that centroid
the points
we can write:
r 1
r of
1
r 1is 0. Furthermore
r 1
d 
q  nq  2
2
rs
n
n
n
q 
n
q  nq
n


rr
ss
Using these identities we can 
express diagonal
elements
of
therr Q using elements of D:
r 1 s 1
r 1
s 1
s 1
d n
2
rs
n
1 n n 2
qrr 

 d rs ,
2n r 1 s 1
r 1
q n
q  2n
q
1 n 2
1 n n 2
qss   d rs  2  d rt
n r 1
2n r 1 t 1
Metric scaling: Cont.
Let us denote matrix of diagonal elements of Q by E=diag(q11,q22,,,,qnn). Then relation between
elements of D and Q could be written in a matrix form:
D  11T E  E11T  2Q
If we will use the relation between diagonal terms of Q and elements of D we can write:
11T E  11T D/n  11T D11T /(2n2 )
E11T  D11T /n  11T D11T /(2n2 )
Then we can write relation between Q and D:
(I  11T / n)D(I  11T / n)  2Q  2XXT
Thus we see that if we have the matrix of dissimilarities D with elements of dij we can find matrix
Q using the relation:
1
Q  (I  11T / n )Γ(I  11T /n ), Γ   D
2
Since D is Euclidean then matrix Q is positive semi-definite and symmetric. Then we can use
spectral decomposition of Q (decomposition using eigenvalues and eigenvectors):
Q  TΛ TT
If Q is positive semi-definite then all eigenvalues are non-negative. We can write (recall the form
of the Q=XXT):
Q  TΛ1/2 (TΛ1/ 2 )T  X  TΛ1/2
This gives matrix of n coordinates (principal coordinates). Of course this configuration is not
unique.
Metric scaling: Cont.
Algorithm for metric scaling (finding principal coordinates) can be written as
follows:
1)
Given matrix of dissimilarities form the matrix  with elements of –1/2dij
2)
Subtract from each elements of this matrix average of raw and column elements
where it is located. Denote it by 
3)
Find the eigenvalues of this matrix and corresponding eigenvectors of . The
dimensionality of the representation corresponds to the number of non-zero
eigenvalues of this matrix.
4)
Normalise eigenvectors so that:
νTi νi  i , where - s are eigenvalues and ν - s are eigenvectors
Dimensionality
“Goodness of fit” of k dimensional configuration to the original matrix of dissimilarity is
measured using “per cent trace” defined as:
k
k 

i 1
n
i

i 1
100%
i
Note that principal coordinate and principal component analysis are similar in some sense. If X is
nxp data matrix and we have nxn dissimilarity matrix then principal component scores
coincide with the coordinates calculated using scaling.
It might happen that dissimilarity matrix is not Euclidean. Then some of the calculated
eigenvalues may become negative and the coordinate representation may include
imaginary coordinates. If these eigenvalues are small then they can be ignored and set to 0.
Another way of avoiding this problem is finding best possible approximation of the
dissimilarity matrix by Euclidean dissimilarity matrix.
If some eigenvalues are negative then goodness of fit can be modified in two different ways:
k


i 1
n
i
|  |
i 1
i
k
100%, or  

i 1
k
2
i

i 1
2
i
100%
Horseshoe effect
In many situations close distances between objects are measured more accurately than long
distances. As a result scaling techniques pulls objects further from each other closer to each
other. Relative positions of objects close to each other are defined better than for object far
from each other. This is called horseshoe effect. This effect is present in many other
dimension reduction techniques also. Define similarity matrix 51x51 between objects:

8 1 | r  s | 3

1 22 | r  s | 24
| r  s | 0
0

ot herwise
9
Then distance matrix is derived from this matrix using the relation give above. For this matrix the
result of metric scaling is:
Other types of scaling
Classical scaling is one of the techniques used to find configuration from the dissimilarity matrix.
There are several other techniques. Although they do not have direct algebraic solution
with modern computer they can be implemented and used. One of these techniques
minmises the function:
n
n
 ' '   ( ij  dij )2
i 1 j 1
Where -s are calculated distances.
There is no algebraic solution exists. In this case initial
dimension is chosen then starting configuration is postulated and then this function is
minimisied. If dimension is changed then whole procedure is repeated again. Then values
of the function at the minimum for different dimensions can be used for scree-plot to
decide dimensionality.
One of the techniques uses following modification of the calculated distances:
dˆij  a  b ij
Then it used in two types of functions. First of them is the standardised residual sum of squares:
1/ 2


STRESS   (d ij  dˆij )2 /  d ij2 
i
j
 i j

Another one is the modified version of this:
1/ 2


SSTRESS  (d ij2  dˆij2 )2 /  d ij4 
i
j
 i j

Both these functions must be minimised iteratively using one of the optimisation techniques.
For this technique we do not have to use Euclidean distances.
Non-metric scaling
Although Euclidean and metric scaling can be applied for wide range of the cases there might be
cases then requirement of these techniques might not be satisfied. For example:
1)
Dissimilarity of the objects may not be true distance but only order of similarity. I.e. we
can only say that for a certain objects we should have (M=1/2 n(n-1))
di1 j1  di2 j2  ...  diM jM
2)
3)
Measurement may have such large error that we can be sure only on order of distances
When we use metric scaling we assume that true configuration exists and what we measure
is an approximation for interpoint distances for this configurations. It may happen that we
measure some function of the interpoint distances.
These cases are usually handled using non-metric scaling. One of the techniques is to use the
function STRESS with constraints:
dij  drs ,  ij  rs , for all i, j, r, s
This technique is called non-metric to distinguish it from the previously described techniques.
If the observed distances are equal then usual technique is to exclude them from constraints.
Non-metric scaling techniques can handle missing distances simply by ignoring them.
Example: U.K. distances
Here is table of some of the intercity distances:
Bristol Cardiff Dover Exeter Hull Leeds London
Bristol
0
Cardiff 47
0
Dover 210
245
0
Exeter 84
121
250
0
Hull
231
251
266
305
0
Leeds 220
240
271
294
61
0
London 120
155
80
200 218 199
0
Example:
Result of the metric scaling. Two dimensional coordinates:
[,1]
[,2]
[1,] -66.96555 42.04003
[2,] -76.31011 67.37324
[3,] -21.27937 -162.65703
[4,] -136.59765 58.13424
[5,] 163.00725 31.67896
[6,] 152.04793 40.75884
[7,] -13.90250 -77.32827
Example: Plot
Two dimensional plot gives. In this case we see that it gives representation of the UK
map.
50
Cardiff
Exeter
Bristol
-50
-100
London
-150
y
0
Leeds
Hull
Dover
-100
-50
0
50
100
150
R commands for scaling
Classical metric scaling is in the library mva
library(mva)
cc = cmdscale(d, k = 2, eig = FALSE, add = FALSE, x.ret = FALSE)
d is a distance matrix, k is the number of the dimensions required.
You can plot using
plot(cc)
or using the following set of commands:
x = cc[,1] (or x = cc$points[,1])
y = cc[,2] (or y = cc$points[,2])
plot(x,y,main(“Metric scaling results”)
text(x,y,names(cc))
It is a good idea to have a look if number of the dimension requested is
sufficient. It can be done by requesting eigenvalues and comparing
them.
R commands for scaling
Non-metric scaling can be done using isoMDS from the library(MASS)
library(MASS)
?isoMDS
then you can use
cc1 = isoMDS(a,cmdscale(a,k),k=2).
The second argument is for the initial configuration. Then we can plot using
x = cc1$points[,1]
y = cc1$points[,2]
plot(x,y,main=“isoMDS scaling”)
text(x,y,names(a))
Another non-metric scaling command is
sammon(a,cmdscale(a,k),k=2)
I
f you have a data matrix X then you can calculate distances using the command dist
dist(X,method=“euclidean”)
then result of this command can be used for analysis with cmdscale, isoMDS or sammon.
References
1)
2)
Krzanowski WJ and Marriout FHC. (1994) Multivatiate analysis. Kendall’s
library of statistics
Mardia, K.V. Kent, J.T. and Bibby, J.M. (2003) Multivariate analysis