NAG CL Interface
g02buc (ssqmat)

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1 Purpose

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.

2 Specification

#include <nag.h>
void  g02buc (Nag_OrderType order, Nag_SumSquare mean, Integer n, Integer m, const double x[], Integer pdx, const double wt[], double *sw, double wmean[], double c[], NagError *fail)
The function may be called by the names: g02buc, nag_correg_ssqmat or nag_sum_sqs.

3 Description

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 n observations on m variables Xj, for j=1,2,,m. The algorithm makes a single pass through the data.
For the first i-1 observations let the mean of the jth variable be x¯j(i-1), the cross-product about the mean for the jth and kth variables be cjk(i-1) and the sum of weights be Wi-1. These are updated by the ith observation, xij, for j=1,2,,m, with weight wi as follows:
Wi = Wi-1 + wi x¯j (i) = x¯j (i-1) + wiWi (xj-x¯j(i-1)) ,   j=1,2,,m  
and
cjk (i) = cjk (i-1) + wi Wi (xj-x¯j(i-1)) (xk-x¯k(i-1)) Wi-1 ,   j=1,2,,m ​ and ​ k=j,j+ 1,,m .  
The algorithm is initialized by taking x¯j(1)=x1j, the first observation, and cij(1)=0.0.
For the unweighted case wi=1 and Wi=i for all i.
Note that only the upper triangle of the matrix is calculated and returned packed by column.

4 References

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. ACM 22 532–555

5 Arguments

1: order Nag_OrderType Input
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 order=Nag_RowMajor. 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.
Constraint: order=Nag_RowMajor or Nag_ColMajor.
2: mean Nag_SumSquare Input
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.
mean=Nag_AboutMean
The sums of squares and cross-products of deviations about the mean are calculated.
mean=Nag_AboutZero
The sums of squares and cross-products are calculated.
Constraint: mean=Nag_AboutMean or Nag_AboutZero.
3: n Integer Input
On entry: n, the number of observations in the dataset.
Constraint: n1.
4: m Integer Input
On entry: m, the number of variables.
Constraint: m1.
5: x[dim] const double Input
Note: the dimension, dim, of the array x must be at least
  • max(1,pdx×m) when order=Nag_ColMajor;
  • max(1,n×pdx) when order=Nag_RowMajor.
where X(i,j) appears in this document, it refers to the array element
  • x[(j-1)×pdx+i-1] when order=Nag_ColMajor;
  • x[(i-1)×pdx+j-1] when order=Nag_RowMajor.
On entry: X(i,j) must contain the ith observation on the jth variable, for i=1,2,,n and j=1,2,,m.
6: pdx Integer Input
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
  • if order=Nag_ColMajor, pdxn;
  • if order=Nag_RowMajor, pdxm.
7: wt[dim] const double Input
Note: the dimension, dim, of the array wt must be at least n.
On entry: the optional weights of each observation. If weights are not provided then wt must be set to NULL, otherwise wt[i-1] must contain the weight for the ith observation.
Constraint: if wtis notNULL, wt[i-1]0.0, for i=1,2,,n.
8: sw double * Output
On exit: the sum of weights.
If wtisNULL, sw contains the number of observations, n.
9: wmean[m] double Output
On exit: the sample means. wmean[j-1] contains the mean for the jth variable.
10: c[(m×m+m)/2] double Output
On exit: the cross-products.
If mean=Nag_AboutMean, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.
If mean=Nag_AboutZero, 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 jth and kth variable, kj, is stored in c[k×(k-1)/2+j-1].
11: 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_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument value had an illegal value.
NE_INT
On entry, m=value.
Constraint: m1.
On entry, n=value.
Constraint: n1.
NE_INT_2
On entry, pdx=value and n=value.
Constraint: pdxn.
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.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_NO_LICENCE
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.
NE_REAL_ARRAY_ELEM_CONS
On entry, wt[value]<0.0.
Constraint: wt[i-1]0.0, for i=1,2,,n.

7 Accuracy

For a detailed discussion of the accuracy of this algorithm see Chan et al. (1982) or West (1979).

8 Parallelism and Performance

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.

9 Further Comments

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.

10 Example

A program to calculate the means, the required sums of squares and cross-products matrix, and the variance matrix for a set of 3 observations of 3 variables.

10.1 Program Text

Program Text (g02buce.c)

10.2 Program Data

Program Data (g02buce.d)

10.3 Program Results

Program Results (g02buce.r)