nag_mv_canon_corr (g03adc) : NAG Library, Mark 24

2 Specification

#include <nag.h>

#include <nagg03.h>

void	nag_mv_canon_corr (Integer n, Integer m, const double z[], Integer tdz, const Integer isz[], Integer nx, Integer ny, const double wt[], double e[], Integer tde, Integer ncv, double cvx[], Integer tdcvx, double cvy[], Integer tdcvy, double tol, NagError fail)

3 Description

Let there be two sets of variables,

x

and

y

. For a sample of

n

observations on

n_{x}

variables in a data matrix

X

and

n_{y}

variables in a data matrix

Y

, canonical correlation analysis seeks to find a small number of linear combinations of each set of variables in order to explain or summarise the relationships between them. The variables thus formed are known as canonical variates.

Let the variance-covariance matrix of the two datasets be

(\begin{matrix} S_{x x} & S_{x y} \\ S_{y x} & S_{y y} \end{matrix})

and let

Σ = S_{y y}^{- 1} S_{y x} S_{x x}^{- 1} S_{x y}

then the canonical correlations can be calculated from the eigenvalues of the matrix

Σ

. However, nag_mv_canon_corr (g03adc) calculates the canonical correlations by means of a singular value decomposition (SVD) of a matrix

V

. If the rank of the data matrix

X

k_{x}

and the rank of the data matrix

Y

k_{y}

, and both

X

and

Y

have had variable (column) means subtracted, then the

k_{x}

k_{y}

matrix

V

is given by:

V = Q_{x}^{T} Q_{y},

where

Q_{x}

is the first

k_{x}

rows of the orthogonal matrix

Q

either from the

Q R

decomposition of

X

X

is of full column rank, i.e.,

k_{x} = n_{x}

X = Q_{x} R_{x}

or from the SVD of

X

k_{x} < n_{x}

X = Q_{x} D_{x} P_{x}^{T} .

Similarly

Q_{y}

is the first

k_{y}

rows of the orthogonal matrix

Q

either from the

Q R

decomposition of

Y

Y

is of full column rank, i.e.,

k_{y} = n_{y}

Y = Q_{y} R_{y}

or from the SVD of

Y

k_{y} < n_{y}

Y = Q_{y} D_{y} P_{y}^{T} .

Let the SVD of

V

be:

V = U_{x} Δ U_{y}^{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

canonical variates, where

l = \min (k_{x}, k_{y})

The eigenvalues,

λ_{i}^{2}

, of the matrix

Σ

are given by:

λ_{i}^{2} = \frac{δ_{i}^{2}}{1 + δ_{i}^{2}} .

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}

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 - \frac{1}{2} (k_{x} + k_{y} + 3)) \sum_{j = i + 1}^{l} \log (1 + λ_{j}^{2})

can be used. This is asymptotically distributed as a

χ^{2}

distribution with

(k_{x} - i) (k_{y} - i)

degrees of freedom. If the test for

i = k_{\min}

is not significant, then the remaining tests for

i > k_{\min}

should be ignored.

The loadings for the canonical variates are calculated from the matrices

U_{x}

and

U_{y}

respectively. These matrices are scaled so that the canonical variates have unit variance.

5 Arguments

1: n – IntegerInput

On entry: the number of observations,

n

Constraint:

n > nx + ny

2: m – IntegerInput

On entry: the total number of variables,

m

Constraint:

m \geq nx + ny

3: z[

n \times tdz

] – const doubleInput

On entry:

z [(i - 1) \times tdz + j - 1]

must contain the

i

th observation for the

j

th variable, for

i = 1, 2, \dots, n

and

j = 1, 2, \dots, m

Both

x

and

y

variables are to be included in z, the indicator array, isz, being used to assign the variables in z to the

x

y

sets as appropriate.

4: tdz – IntegerInput

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

Constraint:

tdz \geq m

5: isz[m] – const IntegerInput

On entry:

isz [j - 1]

indicates whether or not the

j

th variable is to be included in the analysis and to which set of variables it belongs.

isz [j - 1] > 0

, then the variable contained in the

j

th column of z is included as an

x

variable in the analysis.

isz [j - 1] < 0

, then the variable contained in the

j

th column of z is included as a

y

variable in the analysis.

isz [j - 1] = 0

, then the variable contained in the

j

th column of z is not included in the analysis.

Constraint: only nx elements of isz can be

> 0

and only ny elements of isz can be

< 0

6: nx – IntegerInput

On entry: the number of

x

variables in the analysis,

n_{x}

Constraint:

nx \geq 1

7: ny – IntegerInput

On entry: the number of

y

variables in the analysis,

n_{y}

Constraint:

ny \geq 1

8: wt[n] – const doubleInput

On entry: the elements of wt must contain the weights to be used in the analysis. The effective number of observations is the sum of the weights. If

wt [i - 1] = 0.0

then the

i

th observation is not included in the analysis.

If weights are not provided then wt must be set to NULL and the effective number of observations is n.

Constraints:

if wt is not NULL, $wt [i - 1] \geq 0.0$ , for $i = 1, 2, \dots, n$ ;
$\sum_{i = 1}^{n} wt [i - 1] \geq nx + ny + 1$ .

9: e[

\min (nx, ny) \times tde

] – doubleOutput

On exit: the statistics of the canonical variate analysis.

e [(i - 1) \times tde]

, the canonical correlations,

δ_{i}

, for

i = 1, 2, \dots, l

e [(i - 1) \times tde + 1]

, the eigenvalues of

Σ

λ_{i}^{2}

, for

i = 1, 2, \dots, l

e [(i - 1) \times tde + 2]

, the proportion of variation explained by the

i

th canonical variate, for

i = 1, 2, \dots, l

e [(i - 1) \times tde + 3]

, the

χ^{2}

statistic for the

i

th canonical variate, for

i = 1, 2, \dots, l

e [(i - 1) \times tde + 4]

, the degrees of freedom for

χ^{2}

statistic for the

i

th canonical variate, for

i = 1, 2, \dots, l

e [(i - 1) \times tde + 5]

, the significance level for the

χ^{2}

statistic for the

i

th canonical variate, for

i = 1, 2, \dots, l

10: tde – IntegerInput

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

Constraint:

tde \geq 6

11: ncv – Integer *Output

On exit: the number of canonical correlations,

l

. This will be the minimum of the rank of

X

and the rank of

Y

12: cvx[

nx \times tdcvx

] – doubleOutput

On exit: the canonical variate loadings for the

x

variables.

cvx [(i - 1) \times tdcvx + j - 1]

contains the loading coefficient for the

i

x

variable on the

j

th canonical variate.

13: tdcvx – IntegerInput

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

Constraint:

tdcvx \geq \min

(nx,ny).

14: cvy[

ny \times tdcvy

] – doubleOutput

On exit: the canonical variate loadings for the

y

variables.

cvy [(i - 1) \times tdcvy + j - 1]

contains the loading coefficient for the

i

y

variable on the

j

th canonical variate.

15: tdcvy – IntegerInput

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

Constraint:

tdcvy \geq \min

(nx,ny).

16: tol – doubleInput

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, then the square root of machine precision is used instead.

Constraint:

tol \geq 0.0

17: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_2_INT_ARG_LT

On entry,

tdz = ⟨value⟩

while

m = ⟨value⟩

. These arguments must satisfy

tdz \geq m

NE_3_INT_ARG_CONS

On entry,

m = ⟨value⟩

nx = ⟨value⟩

and

ny = ⟨value⟩

. These arguments must satisfy

m \geq nx + ny

On entry,

n = ⟨value⟩

nx = ⟨value⟩

and

ny = ⟨value⟩

. These arguments must satisfy

n > nx + ny

On entry,

tdcvx = ⟨value⟩

nx = ⟨value⟩

and

ny = ⟨value⟩

. These arguments must satisfy

tdcvx \geq \min

(nx,ny).

On entry,

tdcvy = ⟨value⟩

nx = ⟨value⟩

and

ny = ⟨value⟩

. These arguments must satisfy

tdcvy \geq \min

(nx,ny).

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_CANON_CORR_1

A canonical correlation is equal to one. This will happen if the

x

and

y

variables are perfectly correlated.

NE_INT_ARG_LT

On entry,

nx = ⟨value⟩

.
Constraint:

nx \geq 1

On entry,

ny = ⟨value⟩

.
Constraint:

ny \geq 1

On entry,

tde = ⟨value⟩

.
Constraint:

tde \geq 6

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_MAT_RANK_ZERO

The rank of the

X

matrix or the rank of the

Y

matrix is zero. This will happen if all the

x

and

y

variables are constants.

NE_NEG_WEIGHT_ELEMENT

On entry,

wt [⟨value⟩] = ⟨value⟩

.
Constraint: when referenced, all elements of wt must be non-negative.

NE_OBSERV_LT_VAR

With weighted data, the effective number of observations given by the sum of weights

= ⟨value⟩

, while number of variables included in the analysis,

nx + ny = ⟨value⟩

.
Constraint: Effective number of observations

\geq nx + ny + 1

NE_REAL_ARG_LT

On entry, tol must not be less than

0.0

tol = ⟨value⟩

NE_SVD_NOT_CONV

The singular value decomposition has failed to converge. This is an unlikely error exit.

NE_VAR_INCL_INDICATED

The number of variables, nx in the analysis

= ⟨value⟩

, while the number of

x

variables included in the analysis via array

isz = ⟨value⟩

.
Constraint: these two numbers must be the same.

The number of variables, ny in the analysis

= ⟨value⟩

, while the number of

y

variables included in the analysis via array

isz = ⟨value⟩

.
Constraint: these two numbers must be the same.

NAG Library Function Document

nag_mv_canon_corr (g03adc)

+− 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 Library Function Documentnag_mv_canon_corr (g03adc)

+− 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 Library Function Document

nag_mv_canon_corr (g03adc)