# NAG C Library Function Document

## 1Purpose

nag_cov_to_corr (g02bwc) calculates a matrix of Pearson product-moment correlation coefficients from sums of squares and cross-products of deviations about the mean.

## 2Specification

 #include #include
 void nag_cov_to_corr (Integer m, double r[], NagError *fail)

## 3Description

nag_cov_to_corr (g02bwc) calculates a matrix of Pearson product-moment correlation coefficients from sums of squares and cross-products about the mean for observations on $m$ variables which can be computed by a single call to nag_sum_sqs (g02buc) or a series of calls to nag_sum_sqs_update (g02btc). The sums of squares and cross-products are stored in an array packed by column and are overwritten by the correlation coefficients.
Let ${c}_{jk}$ be the cross-product of deviations from the mean, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=j,\dots ,m$, then the product-moment correlation coefficient, ${r}_{jk}$ is given by
 $rjk=cjkcjjckk .$

None.

## 5Arguments

1:    $\mathbf{m}$IntegerInput
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 1$.
2:    $\mathbf{r}\left[\left({\mathbf{m}}×{\mathbf{m}}+{\mathbf{m}}\right)/2\right]$doubleInput/Output
On entry: contains the upper triangular part of the sums of squares and cross-products matrix of deviations from the mean. These are stored packed by column, i.e., the cross-product between variable $j$ and $k$, $k\ge j$, is stored in ${\mathbf{r}}\left[\left(k×\left(k-1\right)/2+j\right)-1\right]$.
On exit: the Pearson product-moment correlation coefficients.
These are stored packed by column corresponding to the input cross-products.
3:    $\mathbf{fail}$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: ${\mathbf{m}}\ge 1$.
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_ZERO_VARIANCE
A variable has a zero variance. All correlations involving the variable with zero variance will be returned as zero.

## 7Accuracy

The accuracy of nag_cov_to_corr (g02bwc) is entirely dependent upon the accuracy of the elements of array r.

## 8Parallelism and Performance

nag_cov_to_corr (g02bwc) is not threaded in any implementation.

nag_cov_to_corr (g02bwc) may also be used to calculate the correlations between parameter estimates from the variance-covariance matrix of the parameter estimates as is given by several functions in this chapter.

## 10Example

A program to calculate the correlation matrix from raw data. The sum of squares and cross-products about the mean are calculated from the raw data by a call to nag_sum_sqs (g02buc). The correlation matrix is then calculated from these values.

### 10.1Program Text

Program Text (g02bwce.c)

### 10.2Program Data

Program Data (g02bwce.d)

### 10.3Program Results

Program Results (g02bwce.r)