NAG Library Function Document
nag_sum_sqs (g02buc)
1
Purpose
nag_sum_sqs (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> |
#include <nagg02.h> |
void |
nag_sum_sqs (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) |
|
3
Description
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:
and
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.
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:
– 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 3.3.1.3 in How to Use the NAG Library and its Documentation for a more detailed explanation of the use of this argument.
Constraint:
or .
- 2:
– Nag_SumSquareInput
-
On entry: indicates whether
nag_sum_sqs (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.
Constraint:
or .
- 3:
– IntegerInput
-
On entry: , the number of observations in the dataset.
Constraint:
.
- 4:
– IntegerInput
-
On entry: , the number of variables.
Constraint:
.
- 5:
– const doubleInput
-
Note: the dimension,
dim, of the array
x
must be at least
- when ;
- when .
Where
appears in this document, it refers to the array element
- when ;
- when .
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.
Constraints:
- if ,
;
- 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.
Constraint:
if , , for .
- 8:
– double *Output
-
On exit: the sum of weights.
If
,
sw contains the number of observations,
.
- 9:
– doubleOutput
-
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 3.7 in How to Use the NAG Library and its Documentation).
6
Error Indicators and Warnings
- NE_ALLOC_FAIL
-
Dynamic memory allocation failed.
See
Section 2.3.1.2 in How to Use the NAG Library and its Documentation for further information.
- NE_BAD_PARAM
-
On entry, argument had an illegal value.
- NE_INT
-
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
- NE_INT_2
-
On entry, and .
Constraint: .
On entry, and .
Constraint: .
- 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 2.7.6 in How to Use the NAG Library and its Documentation for further information.
- NE_NO_LICENCE
-
Your licence key may have expired or may not have been installed correctly.
See
Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
- NE_REAL_ARRAY_ELEM_CONS
-
On entry, .
Constraint: , for .
7
Accuracy
For a detailed discussion of the accuracy of this algorithm see
Chan et al. (1982) or
West (1979).
8
Parallelism and Performance
nag_sum_sqs (g02buc) is not threaded in any implementation.
nag_cov_to_corr (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
nag_sum_sqs (g02buc) may be updated by adding or removing observations using
nag_sum_sqs_update (g02btc).
Two sets of means and cross-products, as produced by
nag_sum_sqs (g02buc), can be combined using
nag_sum_sqs_combine (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 observations of 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)