nag_sum_sqs_combine (g02bzc) combines two sets of sample means and sums of squares and cross-products matrices. It is designed to be used in conjunction with
nag_sum_sqs (g02buc) to allow large datasets to be summarised.
Let
and
denote two sets of data, each with
variables and
and
observations respectively. Let
denote the (optionally weighted) vector of
means for the first dataset and
denote either the sums of squares and cross-products of deviations from
or the sums of squares and cross-products, in which case
where
is a vector of
ones and
is a diagonal matrix of (optional) weights and
is defined as the sum of the diagonal elements of
. Similarly, let
,
and
denote the same quantities for the second dataset.
Given
and
nag_sum_sqs_combine (g02bzc) calculates
,
and
as if a dataset
, with
variables and
observations were supplied to
nag_sum_sqs (g02buc), with
constructed as
nag_sum_sqs_combine (g02bzc) has been designed to combine the results from two calls to
nag_sum_sqs (g02buc) allowing large datasets, or cases where all the data is not available at the same time, to be summarised.
Bennett J, Pebay P, Roe D and Thompson D (2009) Numerically stable, single-pass, parallel statistics algorithms Proceedings of IEEE International Conference on Cluster Computing
- 1:
– Nag_SumSquareInput
-
On entry: indicates whether the matrices supplied in
xc and
yc are sums of squares and cross-products, or sums of squares and cross-products of deviations about the mean.
- Sums of squares and cross-products of deviations about the mean have been supplied.
- Sums of squares and cross-products have been supplied.
Constraint:
or .
- 2:
– IntegerInput
-
On entry: , the number of variables.
Constraint:
.
- 3:
– double *Input/Output
-
On entry: , the sum of weights, from the first set of data, . If the data is unweighted then this will be the number of observations in the first dataset.
On exit: , the sum of weights, from the combined dataset, . If both datasets are unweighted then this will be the number of observations in the combined dataset.
Constraint:
.
- 4:
– doubleInput/Output
-
On entry: , the sample means for the first set of data, .
On exit: , the sample means for the combined data, .
- 5:
– doubleInput/Output
-
On entry:
, the sums of squares and cross-products matrix for the first set of data,
, as returned by
nag_sum_sqs (g02buc).
nag_sum_sqs (g02buc), returns this matrix packed by columns, i.e., the cross-product between the
th and
th variable,
, is stored in
.
No check is made that is a valid cross-products matrix.
On exit:
, the sums of squares and cross-products matrix for the combined dataset,
.
This matrix is again stored packed by columns.
- 6:
– doubleInput
-
On entry: , the sum of weights, from the second set of data, . If the data is unweighted then this will be the number of observations in the second dataset.
Constraint:
.
- 7:
– const doubleInput
-
On entry: , the sample means for the second set of data, .
- 8:
– const doubleInput
-
On entry:
, the sums of squares and cross-products matrix for the second set of data,
, as returned by
nag_sum_sqs (g02buc).
nag_sum_sqs (g02buc), returns this matrix packed by columns, i.e., the cross-product between the
th and
th variable,
, is stored in
.
No check is made that is a valid cross-products matrix.
- 9:
– NagError *Input/Output
-
The NAG error argument (see
Section 3.7 in How to Use the NAG Library and its Documentation).
Not applicable.
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.
None.
This example illustrates the use of
nag_sum_sqs_combine (g02bzc) by dividing a dataset into three blocks of
,
and
observations respectively. Each block of data is summarised using
nag_sum_sqs (g02buc) and then the three summaries combined using
nag_sum_sqs_combine (g02bzc).
The resulting sums of squares and cross-products matrix is then scaled to obtain the covariance matrix for the whole dataset.