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g03 – Multivariate Methods g03fac
nag mv prin coord analysis (g03fac)
1. Purpose
nag mv prin coord analysis (g03fac) performs a principal coordinate analysis also known as classical
metric scaling.
2. Speciﬁcation
#include <nag.h>
#include <nagg03.h>
void nag_mv_prin_coord_analysis(Nag_Eigenvalues roots, Integer n,
double d[], Integer ndim, double x[], Integer tdx,
double eval[], NagError *fail)
3. Description
For a set of n objects, a distance matrix D can be calculated such that dij is a measure of how ‘far
apart’ objects i and j are (see nag mv distance mat (g03eac) for example). Principal coordinate
analysis or metric scaling starts with a distance matrix and ﬁnds points X in Euclidean space such
that those points have the same distance matrix. The aim is to ﬁnd a small number of dimensions,
k< <(n − 1), that provide an adequate representation of the distances.
The principal co-ordinates of the points are computed from the eigenvectors of the matrix E where
eij = −1
2(d2
ij − d2
i. − d2
.j − d2
..)w i t hd2
i denoting the average of d2
ij over the suﬃx j etc.. The
eigenvectors are then scaled by multiplying by the square root of the value of the corresponding
eigenvalue.
Provided that the ordered eigenvalues, λi,o ft h em a t r i xE are all positive,
k
i=1 λi/
n−1
i=1 λi
shows how well the data is represented in k dimensions. The eigenvalues will be non-negative if E
is positive semi-deﬁnite. This will be true provided dij satisﬁes the inequality: dij ≤ dik + djk for
all i,j,k. If this is not the case the size of the negative eigenvalue reﬂects the amount of deviation
from this condition and the results should be treated cautiously in the presence of large negative
eigenvalues. See Krzanowski (1990) for further discussion. nag mv prin coord analysis provides the
option for all eigenvalues to be computed so that the smallest eigenvalues can be checked.
4. Parameters
roots
Input: indicates if all the eigenvalues are to be computed or just the ndim largest.
If roots = Nag AllEigVals, all the eigenvalues are computed.
If roots = Nag LargeEigVals, only the largest ndim eigenvalues are computed.
Constraint: roots = Nag AllEigVals or Nag LargeEigVals.
n
Input: the number of objects in the distance matrix, n.
Constraint: n > ndim.
d[n∗(n−1)/2]
Input: the lower triangle of the distance matrix D stored packed by rows. That is,
d[(i − 1) ∗ (i − 2)/2+j − 1] must contain dij for i =2 ,3,...,n; j =1 ,2,...,i− 1.
Constraint: d[i − 1] ≥ 0.0, i =1 ,2,...,n(n − 1)/2.
ndim
Input: the number of dimensions used to represent the data, k.
Constraint: ndim ≥ 1.
x[n][tdx]
Output: the ith row contains k coordinates for the ith point, i =1 ,2,...,n.
[NP3275/5/pdf] 3.g03fac.1nag mv prin coord analysis NAG C Library Manual
tdx
Input: the last dimension of the array x as declared in the calling program.
Constraint: tdx ≥ ndim.
eval[n]
Output: if roots = Nag AllEigVals, eval contains the n scaled eigenvalues of the matrix E.
If roots = Nag LargeEigVals, eval contains the largest k scaled eigenvalues of the matrix E.
In both cases the eigenvalues are divided by the sum of the eigenvalues (that is, the trace of
E).
fail
The NAG error parameter, see the Essential Introduction to the NAG C Library.
5. Error Indications and Warnings
NE BAD PARAM
On entry, parameter roots had an illegal value.
NE INT ARG LT
On entry, ndim must not be less than 1: ndim =  value .
NE 2 INT ARG LE
On entry, n =  value  while ndim =  value .
These parameters must satisfy n > ndim.
NE 2 INT ARG LT
On entry, tdx =  value  while ndim =  value .
These parameters must satisfy tdx ≥ ndim.
NE REALARR
On entry, d[ value ]= value .
Constraint: d[i − 1] ≥ 0.0, i =1 ,2,...,n∗ (n − 1)/2.
NE NONZERO EIGVALS
There are fewer than ndim eigenvalues greater than zero.
Try a smaller number of dimensions (ndim) or use non-metric scaling (nag mv ordinal multidimscale
(g03fcc)).
NE EIGVAL
The computation of eigenvalues or eigenvectors has failed.
NE ALLOC FAIL
Memory allocation failed.
NE INTERNAL ERROR
An internal error has occurred in this function. Check the function call and any array sizes.
If the call is correct then please consult NAG for assistance.
6. Further Comments
6.1. Accuracy
Alternative, non-metric, methods of scaling are provided by nag mv ordinal multidimscale (g03fcc).
The relationship between principal coordinates and principal components
(see nag mv ordinal multidimscale (g03fcc)), is discussed in Krzanowski (1990) and Gower (1966).
6.2. References
Gower J C (1966) Some distance properties of latent root and vector methods used in multivariate
analysis Biometrika 53 325–338.
Chatﬁeld C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall.
Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press.
7. See Also
nag mv ordinal multidimscale (g03fcc)
nag mv distance mat (g03eac)
3.g03fac.2 [NP3275/5/pdf]g03 – Multivariate Methods g03fac
8. Example
The data, given by Krzanowski (1990), are dissimilarities between water vole populations in Europe.
The ﬁrst two principal co-ordinates are computed by nag mv prin coord analysis.
8.1. Program Text
/* nag_mv_prin_coord_analysis (g03fac) Example Program.
*
* Copyright 1998 Numerical Algorithms Group.
*
* Mark 5, 1998.
*
*/
#include <nag.h>
#include <stdio.h>
#include <nag_stdlib.h>
#include <nagg01.h>
#include <nagg03.h>
#define NMAX 14
#define NNMAX NMAX*(NMAX-1)/2
#define XTMP(I) xtmp[(I)-1]
#define YTMP(I) ytmp[(I)-1]
main()
{
double d[NNMAX], e[NMAX], x[NMAX][NMAX];
Integer ndim;
Integer tdx=NMAX;
Integer i, j, n;
Integer nn;
char char_roots[2];
Nag_Eigenvalues roots;
Vprintf("g03fac Example Program Results\n\n");
/* Skip heading in data file */
Vscanf("%*[^\n]");
Vscanf("%ld",&n);
Vscanf("%ld",&ndim);
Vscanf("%s",char_roots);
if (n <= NMAX)
{
n n=n*( n-1 )/2 ;
for (i = 0; i < nn; ++i)
Vscanf("%lf",&d[i]);
if (*char_roots == ’L’)
roots = Nag_LargeEigVals;
else
roots = Nag_AllEigVals;
g03fac(roots, n, d, ndim, (double *)x, tdx, e, NAGERR_DEFAULT);
Vprintf("\nScaled Eigenvalues\n\n");
if (*char_roots == ’L’)
{
for (i = 0; i < ndim; ++i)
Vprintf("%10.4f",e[i]);
}
else
[NP3275/5/pdf] 3.g03fac.3nag mv prin coord analysis NAG C Library Manual
{
f o r( i=0 ;i<n ;++i)
Vprintf("%10.4f",e[i]);
Vprintf("\n");
}
Vprintf("\nCo-ordinates\n\n");
f o r( i=0 ;i<n ;++i)
{
for (j = 0; j < ndim; ++j)
Vprintf("%10.4f",x[i][j]);
Vprintf("\n");
}
exit(EXIT_SUCCESS);
}
else
{
Vprintf("Incorrect input value of n or m.\n");
exit(EXIT_FAILURE);
}
}
8.2. Program Data
g03fac Example Program Data
1 42L
0.099
0.033 0.022
0.183 0.114 0.042
0.148 0.224 0.059 0.068
0.198 0.039 0.053 0.085 0.051
0.462 0.266 0.322 0.435 0.268 0.025
0.628 0.442 0.444 0.406 0.240 0.129 0.014
0.113 0.070 0.046 0.047 0.034 0.002 0.106 0.129
0.173 0.119 0.162 0.331 0.177 0.039 0.089 0.237 0.071
0.434 0.419 0.339 0.505 0.469 0.390 0.315 0.349 0.151 0.430
0.762 0.633 0.781 0.700 0.758 0.625 0.469 0.618 0.440 0.538 0.607
0.530 0.389 0.482 0.579 0.597 0.498 0.374 0.562 0.247 0.383 0.387 0.084
0.586 0.435 0.550 0.530 0.552 0.509 0.369 0.471 0.234 0.346 0.456 0.090 0.038
8.3. Program Results
g03fac Example Program Results
Scaled Eigenvalues
0.7871 0.2808
Co-ordinates
0.2408 0.2337
0.1137 0.1168
0.2394 0.0760
0.2129 0.0605
0.2495 -0.0693
0.1487 -0.0778
-0.0514 -0.1623
0.0115 -0.3446
-0.0039 0.0059
0.0386 -0.0089
-0.0421 -0.0566
-0.5158 0.0291
-0.3180 0.1501
-0.3238 0.0475
3.g03fac.4 [NP3275/5/pdf]

void nag_mv_prin_coord_analysis(Nag_Eigenvalues roots, Integer n, double d[], Integer ndim, double x[], Integer tdx,

http://www.nag.com/numeric/CL/manual/pdf/G03/g03fac.pdf

void nag_mv_prin_coord_analysis(Nag_Eigenvalues roots, Integer n, double d[], Integer ndim, double x[], Integer tdx,

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