. 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. g03fac provides the option for all eigenvalues to be computed so that the smallest eigenvalues can be checked.

4 References

Chatfield C and Collins A J (1980) Introduction to Multivariate Analysis Chapman and Hall

Gower J C (1966) Some distance properties of latent root and vector methods used in multivariate analysis Biometrika 53 325–338

Krzanowski W J (1990) Principles of Multivariate Analysis Oxford University Press

5 Arguments

1: $roots$ – Nag_Eigenvalues Input

On entry: indicates if all the eigenvalues are to be computed or just the ndim largest.

$roots = Nag_AllEigVals$: All the eigenvalues are computed.
$roots = Nag_LargeEigVals$: Only the largest ndim eigenvalues are computed.

Constraint:

roots = Nag_AllEigVals

Nag_LargeEigVals

2: $n$ – Integer Input

On entry: the number of objects in the distance matrix,

n

Constraint:

n > ndim

3: $d [n \times (n - 1) / 2]$ – const double Input

On entry: the lower triangle of the distance matrix

D

stored packed by rows. That is,

d [(i - 1) \times (i - 2) / 2 + j - 1]

must contain

d_{i j}

, for

i = 2, 3, \dots, n

and

j = 1, 2, \dots, i - 1

Constraint:

d [i - 1] \geq 0.0

, for

i = 1, 2, \dots, n (n - 1) / 2

4: $ndim$ – Integer Input

On entry: the number of dimensions used to represent the data,

k

Constraint:

ndim \geq 1

5: $x [n \times tdx]$ – double Output

Note: the

(i, j)

th element of the matrix

X

is stored in

x [(i - 1) \times tdx + j - 1]

On exit: the

i

th row contains

k

coordinates for the

i

th point,

i = 1, 2, \dots, n

6: $tdx$ – Integer Input

On entry: the stride separating matrix column elements in the array x.

Constraint:

tdx \geq ndim

7: $eval [n]$ – double Output

On exit: 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

8: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_2_INT_ARG_LE: On entry, $n = ⟨ value ⟩$ while $ndim = ⟨ value ⟩$ . These arguments must satisfy $n > ndim$ .
NE_2_INT_ARG_LT: On entry, $tdx = ⟨ value ⟩$ while $ndim = ⟨ value ⟩$ . These arguments must satisfy $tdx \geq ndim$ .
NE_ALLOC_FAIL: Dynamic memory allocation failed.
NE_BAD_PARAM: On entry, argument roots had an illegal value.
NE_EIGVAL: The computation of eigenvalues or eigenvectors has failed.
NE_INT_ARG_LT: On entry, $ndim = ⟨ value ⟩$ .
Constraint: $ndim \geq 1$ .
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 contact NAG for assistance.
NE_NEG_ELEMENT: At least one element of the array d < 0.0.
Constraint: $d [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n \times (n - 1) / 2$ .
NE_NONZERO_EIGVALS: There are fewer than ndim eigenvalues greater than zero. Try a smaller number of dimensions (ndim) or use non-metric scaling (g03fcc).
NE_REALARR: On entry, $d [⟨ value ⟩] = ⟨ value ⟩$ .
Constraint: $d [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n \times (n - 1) / 2$ .

7 Accuracy

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

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

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.

g03fac is not threaded in any implementation.

9 Further Comments

None.

10 Example

The data, given by Krzanowski (1990), are dissimilarities between water vole populations in Europe. The first two principal coordinates are computed.

g03fa: FL CL CPP AD PY MB

NAG CL Interfaceg03fac (multidimscal_​metric)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
g03fac (multidimscal_metric)