# NAG C Library Function Document

## 1Purpose

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.

## 2Specification

 #include #include
 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)

## 3Description

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 $n$ observations on $m$ variables ${X}_{j}$, for $\mathit{j}=1,2,\dots ,m$. The algorithm makes a single pass through the data.
For the first $i-1$ observations let the mean of the $j$th variable be ${\stackrel{-}{x}}_{j}\left(i-1\right)$, the cross-product about the mean for the $j$th and $k$th variables be ${c}_{jk}\left(i-1\right)$ and the sum of weights be ${W}_{i-1}$. These are updated by the $i$th observation, ${x}_{ij}$, for $\mathit{j}=1,2,\dots ,m$, with weight ${w}_{i}$ 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 ${\stackrel{-}{x}}_{j}\left(1\right)={x}_{1j}$, the first observation, and ${c}_{ij}\left(1\right)=0.0$.
For the unweighted case ${w}_{i}=1$ and ${W}_{i}=i$ for all $i$.
Note that only the upper triangle of the matrix is calculated and returned packed by column.

## 4References

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

## 5Arguments

1:    $\mathbf{order}$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 ${\mathbf{order}}=\mathrm{Nag_RowMajor}$. 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: ${\mathbf{order}}=\mathrm{Nag_RowMajor}$ or $\mathrm{Nag_ColMajor}$.
2:    $\mathbf{mean}$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.
${\mathbf{mean}}=\mathrm{Nag_AboutMean}$
The sums of squares and cross-products of deviations about the mean are calculated.
${\mathbf{mean}}=\mathrm{Nag_AboutZero}$
The sums of squares and cross-products are calculated.
Constraint: ${\mathbf{mean}}=\mathrm{Nag_AboutMean}$ or $\mathrm{Nag_AboutZero}$.
3:    $\mathbf{n}$IntegerInput
On entry: $n$, the number of observations in the dataset.
Constraint: ${\mathbf{n}}\ge 1$.
4:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
5:    $\mathbf{x}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array x must be at least
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{pdx}}×{\mathbf{m}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{n}}×{\mathbf{pdx}}\right)$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
Where ${\mathbf{X}}\left(i,j\right)$ appears in this document, it refers to the array element
• ${\mathbf{x}}\left[\left(j-1\right)×{\mathbf{pdx}}+i-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_ColMajor}$;
• ${\mathbf{x}}\left[\left(i-1\right)×{\mathbf{pdx}}+j-1\right]$ when ${\mathbf{order}}=\mathrm{Nag_RowMajor}$.
On entry: ${\mathbf{X}}\left(\mathit{i},\mathit{j}\right)$ must contain the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
6:    $\mathbf{pdx}$IntegerInput
On entry: the stride separating row or column elements (depending on the value of order) in the array x.
Constraints:
• if ${\mathbf{order}}=\mathrm{Nag_ColMajor}$, ${\mathbf{pdx}}\ge {\mathbf{n}}$;
• if ${\mathbf{order}}=\mathrm{Nag_RowMajor}$, ${\mathbf{pdx}}\ge {\mathbf{m}}$.
7:    $\mathbf{wt}\left[\mathit{dim}\right]$const doubleInput
Note: the dimension, dim, of the array wt must be at least ${\mathbf{n}}$.
On entry: the optional weights of each observation. If weights are not provided then wt must be set to NULL, otherwise ${\mathbf{wt}}\left[i-1\right]$ must contain the weight for the $i$th observation.
Constraint: if ${\mathbf{wt}}\phantom{\rule{0.25em}{0ex}}\text{is not}\phantom{\rule{0.25em}{0ex}}\mathbf{NULL}$, ${\mathbf{wt}}\left[\mathit{i}-1\right]\ge 0.0$, for $\mathit{i}=1,2,\dots ,n$.
8:    $\mathbf{sw}$double *Output
On exit: the sum of weights.
If ${\mathbf{wt}}\phantom{\rule{0.25em}{0ex}}\text{is}\phantom{\rule{0.25em}{0ex}}\mathbf{NULL}$, sw contains the number of observations, $n$.
9:    $\mathbf{wmean}\left[{\mathbf{m}}\right]$doubleOutput
On exit: the sample means. ${\mathbf{wmean}}\left[j-1\right]$ contains the mean for the $j$th variable.
10:  $\mathbf{c}\left[\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right]$doubleOutput
On exit: the cross-products.
If ${\mathbf{mean}}=\mathrm{Nag_AboutMean}$, c contains the upper triangular part of the matrix of (weighted) sums of squares and cross-products of deviations about the mean.
If ${\mathbf{mean}}=\mathrm{Nag_AboutZero}$, 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 $j$th and $k$th variable, $k\ge j$, is stored in ${\mathbf{c}}\left[k×\left(k-1\right)/2+j-1\right]$.
11:  $\mathbf{fail}$NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

## 6Error 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.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{m}}\ge 1$.
On entry, ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{n}}\ge 1$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}>0$.
NE_INT_2
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{m}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{pdx}}=〈\mathit{\text{value}}〉$ and ${\mathbf{n}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{pdx}}\ge$ or ${\mathbf{n}}$.
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, ${\mathbf{wt}}\left[〈\mathit{\text{value}}〉\right]<0.0$.
Constraint: ${\mathbf{wt}}\left[i-1\right]\ge 0.0$, for $i=1,2,\dots ,n$.

## 7Accuracy

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

## 8Parallelism 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).

## 10Example

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.

### 10.1Program Text

Program Text (g02buce.c)

### 10.2Program Data

Program Data (g02buce.d)

### 10.3Program Results

Program Results (g02buce.r)