# NAG FL Interfaceg02baf (coeffs_​pearson)

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## 1Purpose

g02baf computes means and standard deviations of variables, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for a set of data.

## 2Specification

Fortran Interface
 Subroutine g02baf ( n, m, x, ldx, xbar, std, ssp, r, ldr,
 Integer, Intent (In) :: n, m, ldx, ldssp, ldr Integer, Intent (Inout) :: ifail Real (Kind=nag_wp), Intent (In) :: x(ldx,m) Real (Kind=nag_wp), Intent (Inout) :: ssp(ldssp,m), r(ldr,m) Real (Kind=nag_wp), Intent (Out) :: xbar(m), std(m)
#include <nag.h>
 void g02baf_ (const Integer *n, const Integer *m, const double x[], const Integer *ldx, double xbar[], double std[], double ssp[], const Integer *ldssp, double r[], const Integer *ldr, Integer *ifail)
The routine may be called by the names g02baf or nagf_correg_coeffs_pearson.

## 3Description

The input data consist of $n$ observations for each of $m$ variables, given as an array
 $[xij], i=1,2,…,n(n≥2),j=1,2,…,m(m≥2),$
where ${x}_{ij}$ is the $i$th observation on the $j$th variable.
The quantities calculated are:
1. (a)Means:
 $x¯j=1n∑i=1nxij, j=1,2,…,m.$
2. (b)Standard deviations:
 $sj=1n- 1 ∑i= 1n (xij-x¯j) 2, j= 1,2,…,m.$
3. (c)Sums of squares and cross-products of deviations from means:
 $Sjk=∑i=1n (xij-x¯j) (xik-x¯k) , j,k=1,2,…,m.$
4. (d)Pearson product-moment correlation coefficients:
 $Rjk=SjkSjjSkk , j,k= 1,2,…,m.$
If ${S}_{jj}$ or ${S}_{kk}$ is zero, ${R}_{jk}$ is set to zero.

None.

## 5Arguments

1: $\mathbf{n}$Integer Input
On entry: $n$, the number of observations or cases.
Constraint: ${\mathbf{n}}\ge 2$.
2: $\mathbf{m}$Integer Input
On entry: $m$, the number of variables.
Constraint: ${\mathbf{m}}\ge 2$.
3: $\mathbf{x}\left({\mathbf{ldx}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Input
On entry: ${\mathbf{x}}\left(\mathit{i},\mathit{j}\right)$ must be set to ${x}_{\mathit{i}\mathit{j}}$, the $\mathit{i}$th observation on the $\mathit{j}$th variable, for $\mathit{i}=1,2,\dots ,n$ and $\mathit{j}=1,2,\dots ,m$.
4: $\mathbf{ldx}$Integer Input
On entry: the first dimension of the array x as declared in the (sub)program from which g02baf is called.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
5: $\mathbf{xbar}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the mean value, ${\overline{x}}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
6: $\mathbf{std}\left({\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: the standard deviation, ${s}_{\mathit{j}}$, of the $\mathit{j}$th variable, for $\mathit{j}=1,2,\dots ,m$.
7: $\mathbf{ssp}\left({\mathbf{ldssp}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{ssp}}\left(\mathit{j},\mathit{k}\right)$ is the cross-product of deviations ${S}_{\mathit{j}\mathit{k}}$, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
8: $\mathbf{ldssp}$Integer Input
On entry: the first dimension of the array ssp as declared in the (sub)program from which g02baf is called.
Constraint: ${\mathbf{ldssp}}\ge {\mathbf{m}}$.
9: $\mathbf{r}\left({\mathbf{ldr}},{\mathbf{m}}\right)$Real (Kind=nag_wp) array Output
On exit: ${\mathbf{r}}\left(\mathit{j},\mathit{k}\right)$ is the product-moment correlation coefficient ${R}_{\mathit{j}\mathit{k}}$ between the $\mathit{j}$th and $\mathit{k}$th variables, for $\mathit{j}=1,2,\dots ,m$ and $\mathit{k}=1,2,\dots ,m$.
10: $\mathbf{ldr}$Integer Input
On entry: the first dimension of the array r as declared in the (sub)program from which g02baf is called.
Constraint: ${\mathbf{ldr}}\ge {\mathbf{m}}$.
11: $\mathbf{ifail}$Integer Input/Output
On entry: ifail must be set to $0$, $-1$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $-1$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $-1$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ or $\mathbf{1}$ is used it is essential to test the value of ifail on exit.
On exit: ${\mathbf{ifail}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see Section 6).

## 6Error Indicators and Warnings

If on entry ${\mathbf{ifail}}=0$ or $-1$, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
${\mathbf{ifail}}=1$
On entry, ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{n}}\ge 2$.
${\mathbf{ifail}}=2$
On entry, ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{m}}\ge 2$.
${\mathbf{ifail}}=3$
On entry, ${\mathbf{ldr}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldr}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldssp}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{m}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldssp}}\ge {\mathbf{m}}$.
On entry, ${\mathbf{ldx}}=⟨\mathit{\text{value}}⟩$ and ${\mathbf{n}}=⟨\mathit{\text{value}}⟩$.
Constraint: ${\mathbf{ldx}}\ge {\mathbf{n}}$.
${\mathbf{ifail}}=-99$
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

## 7Accuracy

g02baf does not use additional precision arithmetic for the accumulation of scalar products, so there may be a loss of significant figures for large $n$.

## 8Parallelism and Performance

g02baf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.
g02baf 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 routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

The time taken by g02baf depends on $n$ and $m$.
The routine uses a two-pass algorithm.

### 9.1Internal Changes

Internal changes have been made to this routine as follows:
• At Mark 27: The algorithm underlying this routine has been altered to improve efficiency for large problem sizes on a multi-threaded system.
For details of all known issues which have been reported for the NAG Library please refer to the Known Issues.

## 10Example

This example reads in a set of data consisting of five observations on each of three variables. The means, standard deviations, sums of squares and cross-products of deviations from means, and Pearson product-moment correlation coefficients for all three variables are then calculated and printed.

### 10.1Program Text

Program Text (g02bafe.f90)

### 10.2Program Data

Program Data (g02bafe.d)

### 10.3Program Results

Program Results (g02bafe.r)