nag_mv_prin_comp (g03aac) performs a principal component analysis on a data matrix; both the principal component loadings and the principal component scores are returned.

Let  be an  by  data matrix of  observations on  variables  and let the  by  variance-covariance matrix of  be . A vector  of length  is found such that:

is maximized subject to

The variable  is known as the first principal component and gives the linear combination of the variables that gives the maximum variation. A second principal component, , is found such that:

is maximized subject to

and

This gives the linear combination of variables that is orthogonal to the first principal component that gives the maximum variation. Further principal components are derived in a similar way.

The vectors , are the eigenvectors of the matrix  and associated with each eigenvector is the eigenvalue, . The value of  gives the proportion of variation explained by the th principal component. Alternatively, the 's can be considered as the right singular vectors in a singular value decomposition with singular values  of the data matrix centred about its mean and scaled by , . This latter approach is used in nag_mv_prin_comp (g03aac), with

where  is a diagonal matrix with elements ,  is the  by  matrix with columns  and  is an  by  matrix with , which gives the principal component scores.

Principal component analysis is often used to reduce the dimension of a data set, replacing a large number of correlated variables with a smaller number of orthogonal variables that still contain most of the information in the original data set.

The choice of the number of dimensions required is usually based on the amount of variation accounted for by the leading principal components. If  principal components are selected, then a test of the equality of the remaining  eigenvalues is

which has, asymptotically, a  distribution with  degrees of freedom.

Equality of the remaining eigenvalues indicates that if any more principal components are to be considered then they all should be considered.

Instead of the variance-covariance matrix the correlation matrix, the sums of squares and cross-products matrix or a standardized sums of squares and cross-products matrix may be used. In the last case  is replaced by  for a diagonal matrix  with positive elements. If the correlation matrix is used, the  approximation for the statistic given above is not valid.

The principal component scores, , are the values of the principal component variables for the observations. These can be standardized so that the variance of these scores for each principal component is 1.0 or equal to the corresponding eigenvalue.

Weights can be used with the analysis, in which case the matrix  is first centred about the weighted means then each row is scaled by an amount , where  is the weight for the th observation.

"NE_2_INT_ARG_GE"

On entry,  while . These arguments must satisfy .

"NE_2_INT_ARG_GT"

On entry,  while . These arguments must satisfy .

"NE_ALLOC_FAIL"

Dynamic memory allocation failed.

"NE_BAD_PARAM"

On entry, argument pcmatrix had an illegal value.

"NE_INT_ARG_LT"

On entry, m 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_WEIGHT_ELEMENT"

On entry, . 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 , while the number of variables included in the analysis, . Constraint: effective number of observations .

"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, nvar in the analysis , while the number of variables included in the analysis via array . Constraint: these two numbers must be the same.

"NE_VAR_INCL_STANDARD"

On entry, the standardization element , while the variable to be included . Constraint: when a variable is to included, the standardization element must be positive.

"NE_ZERO_EIGVALS"

 All eigenvalues/singular values are zero. This will be caused by all the variables being constant.

As nag_mv_prin_comp (g03aac) uses a singular value decomposition of the data matrix, it will be less affected by ill-conditioned problems than traditional methods using the eigenvalue decomposition of the variance-covariance matrix.

None.