g03fa:: Multivariate Methods (NAG Toolbox)

Description

For a set of

n

objects a distance matrix

D

can be calculated such that

d_{i j}

is a measure of how ‘far apart’ are objects

i

and

j

(see nag_mv_distance_mat (g03ea) 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 coordinates of the points are computed from the eigenvectors of the matrix

E

where

e_{i j} = - 1 / 2 (d_{i j}^{2} - d_{i .}^{2} - d_{. j}^{2} + d_{. .}^{2})

with

d_{i .}^{2}

denoting the average of

d_{i j}^{2}

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,

λ_{i}

, of the matrix

E

are all positive,

\sum_{i = 1}^{k} λ_{i} / \sum_{i = 1}^{n - 1} λ_{i}

shows how well the data is represented in

k

dimensions. The eigenvalues will be non-negative if

E

is positive semidefinite. This will be true provided

d_{i j}

satisfies the inequality:

d_{i j} \leq d_{i k} + d_{j k}

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_multidimscal_metric (g03fa) provides the option for all eigenvalues to be computed so that the smallest eigenvalues can be checked.

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

nag_mv_multidimscal_metric (g03fa) uses nag_lapack_dsterf (f08jf) or nag_lapack_dstebz (f08jj) to compute the eigenvalues and nag_lapack_dstein (f08jk) to compute the eigenvectors. These functions should be consulted for a discussion of the accuracy of the computations involved.

Further Comments

Example

function g03fa_example


fprintf('g03fa example results\n\n');

roots = 'l';
n = int64(14);
% Distance matrix (lower part)
d = [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];
ndim = int64(2);

[x, eval, ifail] = g03fa( ...
			  roots, n, d, ndim);

% Display results
disp(' Scaled Eigenvalues');
disp(eval(1:ndim)');

mtitle = 'Co-ordinates';
matrix = 'General';
diag = ' ';
[ifail] = x04ca( ...
		 matrix,diag,x,mtitle);

fig1 = figure;
hold on;
xlabel('PC 1');
ylabel('PC 2');
title({'Observation numbers', 'for PC 1 and PC 2'});
axis([-0.6 0.3 -0.4 0.3]);
for j = 1:size(x,1)
  ch = sprintf('%d',j);
  text(x(j,1),x(j,2),ch);
end
plot([-0.6 0.3], [0 0], ':');
plot([0 0], [-0.4 0.3], ':');
hold off;

g03fa example results

 Scaled Eigenvalues
    0.7871    0.2808

 Co-ordinates
           1       2
  1   0.2408  0.2337
  2   0.1137  0.1168
  3   0.2394  0.0760
  4   0.2129  0.0605
  5   0.2495 -0.0693
  6   0.1487 -0.0778
  7  -0.0514 -0.1623
  8   0.0115 -0.3446
  9  -0.0039  0.0059
 10   0.0386 -0.0089
 11  -0.0421 -0.0566
 12  -0.5158  0.0291
 13  -0.3180  0.1501
 14  -0.3238  0.0475

On entry,	$ndim < 1$ ,
or	$n < ndim$ ,
or	$roots \neq'A'$ or $'L'$ ,
or	$ldx < n$ .

On entry,	$d (i) < 0.0$ for some $i$ , $i = 1, 2, \dots, n (n - 1) / 2$ ,
or	all elements of $d = 0.0$ .

NAG Toolbox: nag_mv_multidimscal_metric (g03fa)

▸▿ Contents

Purpose

Syntax