g02buc calculates the sample means and sums of squares and cross-products, or sums of squares and cross-products of deviations from the mean, in a single pass for a set of data. The data may be weighted.
The function may be called by the names: g02buc, nag_correg_ssqmat or nag_sum_sqs.
g02buc is an adaptation of West's WV2 algorithm; see West (1979). This function calculates the (optionally weighted) sample means and (optionally weighted) sums of squares and cross-products or sums of squares and cross-products of deviations from the (weighted) mean for a sample of observations on variables , for . The algorithm makes a single pass through the data.
For the first observations let the mean of the th variable be , the cross-product about the mean for the th and th variables be and the sum of weights be . These are updated by the th observation, , for , with weight as follows:
The algorithm is initialized by taking , the first observation, and .
For the unweighted case and for all .
Note that only the upper triangle of the matrix is calculated and returned packed by column.
Chan T F, Golub G H and Leveque R J (1982) Updating Formulae and a Pairwise Algorithm for Computing Sample Variances Compstat, Physica-Verlag
West D H D (1979) Updating mean and variance estimates: An improved method Comm. ACM22 532–555
1: – Nag_OrderTypeInput
On entry: the order argument specifies the two-dimensional storage scheme being used, i.e., row-major ordering or column-major ordering. C language defined storage is specified by . See Section 3.1.3 in the Introduction to the NAG Library CL Interface for a more detailed explanation of the use of this argument.
2: – Nag_SumSquareInput
On entry: indicates whether g02buc is to calculate sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.
The sums of squares and cross-products of deviations about the mean are calculated.
The sums of squares and cross-products are calculated.
3: – IntegerInput
On entry: , the number of observations in the dataset.
4: – IntegerInput
On entry: , the number of variables.
5: – const doubleInput
Note: the dimension, dim, of the array
must be at least
where appears in this document, it refers to the array element
On entry: must contain the th observation on the th variable, for and .
6: – IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
if , .
7: – const doubleInput
Note: the dimension, dim, of the array wt
must be at least
On entry: the optional weights of each observation. If weights are not provided then wt must be set to NULL, otherwise must contain the weight for the th observation.
On exit: the sample means. contains the mean for the th variable.
10: – doubleOutput
On exit: the cross-products.
If , c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.
If , c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products.
These are stored packed by columns, i.e., the cross-product between the th and th variable, , is stored in .
11: – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, and .
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
Background information to multithreading can be found in the Multithreading documentation.
g02buc 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 function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
g02bwc may be used to calculate the correlation coefficients from the cross-products of deviations about the mean. The cross-products of deviations about the mean may be scaled
to give a variance-covariance matrix.
The means and cross-products produced by g02buc may be updated by adding or removing observations using g02btc.
Two sets of means and cross-products, as produced by g02buc, can be combined using g02bzc.
A program to calculate the means, the required sums of squares and cross-products matrix, and the variance matrix for a set of observations of variables.