NAG Library Routine Document

G03ACF

INTEGER	N, M, LDX, ISX(M), NX, ING(N), NG, NIG(NG), LDCVM, LDE, NCV, LDCVX, IRANKX, IWK, IFAIL
REAL (KIND=nag_wp)	X(LDX,M), WT(*), CVM(LDCVM,NX), E(LDE,6), CVX(LDCVX,NG-1), TOL, WK(IWK)
CHARACTER(1)	WEIGHT

3 Description

Let a sample of

n

observations on

n_{x}

variables in a data matrix come from

n_{g}

groups with

n_{1}, n_{2}, \dots, n_{n_{g}}

observations in each group,

\sum n_{i} = n

. Canonical variate analysis finds the linear combination of the

n_{x}

variables that maximizes the ratio of between-group to within-group variation. The variables formed, the canonical variates can then be used to discriminate between groups.

The canonical variates can be calculated from the eigenvectors of the within-group sums of squares and cross-products matrix. However, G03ACF calculates the canonical variates by means of a singular value decomposition (SVD) of a matrix

V

. Let the data matrix with variable (column) means subtracted be

X

, and let its rank be

k

; then the

k

by (

n_{g} - 1

) matrix

V

is given by:

V = Q_{X}^{T} Q_{g},

where

Q_{g}

is an

n

(n_{g} - 1)

orthogonal matrix that defines the groups and

Q_{X}

is the first

k

rows of the orthogonal matrix

Q

either from the

Q R

decomposition of

X

X = Q R

X

is of full column rank, i.e.,

k = n_{x}

, else from the SVD of

X

X = Q D P^{T} .

Let the SVD of

V

be:

V = U_{x} Δ U_{g}^{T}

then the nonzero elements of the diagonal matrix

Δ

δ_{i}

, for

i = 1, 2, \dots, l

, are the

l

canonical correlations associated with the

l = \min (k, n_{g} - 1)

canonical variates, where

l = \min (k, n_{g})

The eigenvalues,

λ_{i}^{2}

, of the within-group sums of squares matrix are given by:

λ_{i}^{2} = \frac{δ_{i}^{2}}{1 - δ_{i}^{2}}

and the value of

π_{i} = λ_{i}^{2} / \sum λ_{i}^{2}

gives the proportion of variation explained by the

i

th canonical variate. The values of the

π_{i}

's give an indication as to how many canonical variates are needed to adequately describe the data, i.e., the dimensionality of the problem.

To test for a significant dimensionality greater than

i

the

χ^{2}

statistic:

(n - 1 - n_{g} - \frac{1}{2} (k - n_{g})) \sum_{j = i + 1}^{l} \log (1 + λ_{j}^{2})

can be used. This is asymptotically distributed as a

χ^{2}

-distribution with

(k - i) (n_{g} - 1 - i)

degrees of freedom. If the test for

i = h

is not significant, then the remaining tests for

i > h

should be ignored.

The loadings for the canonical variates are calculated from the matrix

U_{x}

. This matrix is scaled so that the canonical variates have unit within-group variance.

In addition to the canonical variates loadings the means for each canonical variate are calculated for each group.

Weights can be used with the analysis, in which case the weighted means are subtracted from each column and then each row is scaled by an amount

\sqrt{w_{i}}

, where

w_{i}

is the weight for the

i

th observation (row).

4 References

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

Gnanadesikan R (1977) Methods for Statistical Data Analysis of Multivariate Observations Wiley

Hammarling S (1985) The singular value decomposition in multivariate statistics SIGNUM Newsl. 20(3) 2–25

Kendall M G and Stuart A (1969) The Advanced Theory of Statistics (Volume 1) (3rd Edition) Griffin

5 Parameters

1: $WEIGHT$ – CHARACTER(1)Input

On entry: indicates if weights are to be used.

$WEIGHT ='U'$: No weights are used.
$WEIGHT ='W'$ or $'V'$: Weights are used and must be supplied in WT.

WEIGHT ='W'

, the weights are treated as frequencies and the effective number of observations is the sum of the weights.

WEIGHT ='V'

, the weights are treated as being inversely proportional to the variance of the observations and the effective number of observations is the number of observations with nonzero weights.

Constraint:

WEIGHT ='U'

'W'

'V'

2: $N$ – INTEGERInput

On entry:

n

, the number of observations.

Constraint:

N \geq NX + NG

3: $M$ – INTEGERInput

On entry:

m

, the total number of variables.

Constraint:

M \geq NX

4: $X (LDX, M)$ – REAL (KIND=nag_wp) arrayInput

On entry:

X (i, j)

must contain the

i

th observation for the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

5: $LDX$ – INTEGERInput

On entry: the first dimension of the array X as declared in the (sub)program from which G03ACF is called.

Constraint:

LDX \geq N

6: $ISX (M)$ – INTEGER arrayInput

On entry:

ISX (j)

indicates whether or not the

j

th variable is to be included in the analysis.

ISX (j) > 0

, the variables contained in the

j

th column of X is included in the canonical variate analysis, for

j = 1, 2, \dots, m

Constraint:

ISX (j) > 0

for NX values of

j

7: $NX$ – INTEGERInput

On entry: the number of variables in the analysis,

n_{x}

Constraint:

NX \geq 1

8: $ING (N)$ – INTEGER arrayInput

On entry:

ING (i)

indicates which group the

i

th observation is in, for

i = 1, 2, \dots, n

. The effective number of groups is the number of groups with nonzero membership.

Constraint:

1 \leq ING (i) \leq NG

, for

i = 1, 2, \dots, n

9: $NG$ – INTEGERInput

On entry: the number of groups,

n_{g}

Constraint:

NG \geq 2

10: $WT (*)$ – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array WT must be at least

N

WEIGHT ='W'

'V'

, and at least

1

otherwise.

On entry: if

WEIGHT ='W'

'V'

, the first

n

elements of WT must contain the weights to be used in the analysis.

WT (i) = 0.0

, the

i

th observation is not included in the analysis.

WEIGHT ='U'

, WT is not referenced.

Constraints:

$WT (i) \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$\sum_{1}^{n} WT (i) \geq NX + effective number of groups$ .

11: $NIG (NG)$ – INTEGER arrayOutput

On exit:

NIG (j)

gives the number of observations in group

j

, for

j = 1, 2, \dots, n_{g}

12: $CVM (LDCVM, NX)$ – REAL (KIND=nag_wp) arrayOutput

On exit:

CVM (i, j)

contains the mean of the

j

th canonical variate for the

i

th group, for

i = 1, 2, \dots, n_{g}

and

j = 1, 2, \dots, l

; the remaining columns, if any, are used as workspace.

13: $LDCVM$ – INTEGERInput

On entry: the first dimension of the array CVM as declared in the (sub)program from which G03ACF is called.

Constraint:

LDCVM \geq NG

14: $E (LDE, 6)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the statistics of the canonical variate analysis.

$E (i, 1)$: The canonical correlations, $δ_{i}$ , for $i = 1, 2, \dots, l$ .
$E (i, 2)$: The eigenvalues of the within-group sum of squares matrix, $λ_{i}^{2}$ , for $i = 1, 2, \dots, l$ .
$E (i, 3)$: The proportion of variation explained by the $i$ th canonical variate, for $i = 1, 2, \dots, l$ .
$E (i, 4)$: The $χ^{2}$ statistic for the $i$ th canonical variate, for $i = 1, 2, \dots, l$ .
$E (i, 5)$: The degrees of freedom for $χ^{2}$ statistic for the $i$ th canonical variate, for $i = 1, 2, \dots, l$ .
$E (i, 6)$: The significance level for the $χ^{2}$ statistic for the $i$ th canonical variate, for $i = 1, 2, \dots, l$ .

15: $LDE$ – INTEGERInput

On entry: the first dimension of the array E as declared in the (sub)program from which G03ACF is called.

Constraint:

LDE \geq \min (NX, NG - 1)

16: $NCV$ – INTEGEROutput

On exit: the number of canonical variates,

l

. This will be the minimum of

n_{g} - 1

and the rank of X.

17: $CVX (LDCVX, NG - 1)$ – REAL (KIND=nag_wp) arrayOutput

On exit: the canonical variate loadings.

CVX (i, j)

contains the loading coefficient for the

i

th variable on the

j

th canonical variate, for

i = 1, 2, \dots, n_{x}

and

j = 1, 2, \dots, l

; the remaining columns, if any, are used as workspace.

18: $LDCVX$ – INTEGERInput

On entry: the first dimension of the array CVX as declared in the (sub)program from which G03ACF is called.

Constraint:

LDCVX \geq NX

19: $TOL$ – REAL (KIND=nag_wp)Input

On entry: the value of TOL is used to decide if the variables are of full rank and, if not, what is the rank of the variables. The smaller the value of TOL the stricter the criterion for selecting the singular value decomposition. If a non-negative value of TOL less than machine precision is entered, the square root of machine precision is used instead.

Constraint:

TOL \geq 0.0

20: $IRANKX$ – INTEGEROutput

On exit: the rank of the dependent variables.

If the variables are of full rank then

IRANKX = NX

If the variables are not of full rank then IRANKX is an estimate of the rank of the dependent variables. IRANKX is calculated as the number of singular values greater than

TOL \times (largest singular value)

21: $WK (IWK)$ – REAL (KIND=nag_wp) arrayWorkspace

22: $IWK$ – INTEGERInput

On entry: the dimension of the array WK as declared in the (sub)program from which G03ACF is called.

Constraints:

if $NX \geq NG - 1$ , $IWK \geq N \times NX + \max (5 \times (NX - 1) + (NX + 1) \times NX, N) + 1$ ;
if $NX < NG - 1$ , $IWK \geq N \times NX + \max (5 \times (NX - 1) + (NG - 1) \times NX, N) + 1$ .

23: $IFAIL$ – INTEGERInput/Output

On entry: IFAIL must be set to

0

- 1 ​ or ​ 1

. If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.

For environments where it might be inappropriate to halt program execution when an error is detected, the value

- 1 ​ or ​ 1

is recommended. If the output of error messages is undesirable, then the value

1

is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is

0

. When the value $- 1 or 1$ is used it is essential to test the value of IFAIL on exit.

On exit:

IFAIL = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

IFAIL = 0

- 1

, explanatory error messages are output on the current error message unit (as defined by X04AAF).

Errors or warnings detected by the routine:

$IFAIL = 1$

On entry,	$NX < 1$ ,
or	$NG < 2$ ,
or	$M < NX$ ,
or	$N < NX + NG$ ,
or	$LDX < N$ ,
or	$LDCVX < NX$ ,
or	$LDCVM < NG$ ,
or	$LDE < \min (NX, NG - 1)$ ,
or	$NX \geq NG - 1$ and $IWK < N \times NX + \max (5 \times (NX - 1) + (NX + 1) \times NX, N)$ ,
or	$NX < NG - 1$ and $IWK < N \times NX + \max (5 \times (NX - 1) + (NG - 1) \times NX, N)$ ,
or	$WEIGHT \neq'U'$ , $'W'$ or $'V'$ ,
or	$TOL < 0.0$ .

$IFAIL = 2$

On entry,

WEIGHT ='W'

'V'

and a value of

WT < 0.0

$IFAIL = 3$

On entry,	a value of $ING < 1$ ,
or	a value of $ING > NG$ .

$IFAIL = 4$: On entry, the number of variables to be included in the analysis as indicated by ISX is not equal to NX.

$IFAIL = 5$: A singular value decomposition has failed to converge. This is an unlikely error exit.

$IFAIL = 6$: A canonical correlation is equal to $1$ . This will happen if the variables provide an exact indication as to which group every observation is allocated.

$IFAIL = 7$

On entry,	less than two groups have nonzero membership, i.e., the effective number of groups is less than $2$ ,
or	the effective number of groups plus the number of variables, NX, is greater than the effective number of observations.

$IFAIL = 8$: The rank of the variables is $0$ . This will happen if all the variables are constants.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction for further information.

7 Accuracy

As the computation involves the use of orthogonal matrices and a singular value decomposition rather than the traditional computing of a sum of squares matrix and the use of an eigenvalue decomposition, G03ACF should be less affected by ill-conditioned problems.

8 Parallelism and Performance

G03ACF is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

G03ACF makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

This example uses a sample of nine observations, each consisting of three variables plus a group indicator. There are three groups. An unweighted canonical variate analysis is performed and the results printed.

NAG Library Routine DocumentG03ACF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

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 Library Routine Document

G03ACF