nag_mv_prin_coord_analysis (g03fac) performs a principal coordinate analysis also known as classical metric scaling.

For a set of  objects, a distance matrix  can be calculated such that  is a measure of how "far apart" objects  and  are (see g03eac (nag_mv_distance_mat) for example). Principal coordinate analysis or metric scaling starts with a distance matrix and finds points  in Euclidean space such that those points have the same distance matrix. The aim is to find a small number of dimensions, , that provide an adequate representation of the distances.

The principal co-ordinates of the points are computed from the eigenvectors of the matrix  where  with  denoting the average of  over the suffix  etc.. The eigenvectors are then scaled by multiplying by the square root of the value of the corresponding eigenvalue.

Provided that the ordered eigenvalues, , of the matrix  are all positive,  shows how well the data is represented in  dimensions. The eigenvalues will be non-negative if  is positive semi-definite. This will be true provided  satisfies the inequality:  for all . 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 (g03fac) provides the option for all eigenvalues to be computed so that the smallest eigenvalues can be checked.

"NE_2_INT_ARG_LE"

On entry,  while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument compute_roots had an illegal value.

"NE_EIGVAL"

The computation of eigenvalues or eigenvectors has failed.

"NE_INT_ARG_LT"

On entry, ndim must not be less than 1: .

"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.

"NE_NEG_ELEMENT"

At least one element of the array d < 0.0.Constraint: , .

"NE_NONZERO_EIGVALS"

There are fewer than ndim eigenvalues greater than zero. Try a smaller number of dimensions (ndim) or use non-metric scaling (g03fcc (nag_mv_ordinal_multidimscale)).

"NE_REALARR"

On entry, . Constraint: , .

Alternative, non-metric, methods of scaling are provided by g03fcc (nag_mv_ordinal_multidimscale).

The relationship between principal coordinates and principal components (see g03fcc (nag_mv_ordinal_multidimscale)), is discussed in Krzanowski (1990) and Gower (1966).

None.