MATLAB Function Reference |

**Syntax**

R = corrcoef(X) R = corrcoef(x,y) [R,P]=corrcoef(...) [R,P,RLO,RUP]=corrcoef(...) [...]=corrcoef(...,'param1',val1,'param2',val2,...)

**Description**

```
R = corrcoef(X)
```

returns a matrix `R`

of correlation coefficients calculated from an input matrix `X`

whose rows are observations and whose columns are variables. The matrix `R`

`=`

`corrcoef(X)`

is related to the covariance matrix `C`

`=`

`cov(X)`

by

`corrcoef(X)`

is the zeroth lag of the normalized covariance function, that is, the zeroth lag of `xcov(x,'coeff')`

packed into a square array.

```
R = corrcoef(x,y)
```

where `x`

and `y`

are column vectors is the same as `corrcoef([x y])`

.

```
[R,P]=corrcoef(...)
```

also returns `P`

, a matrix of p-values for testing the hypothesis of no correlation. Each p-value is the probability of getting a correlation as large as the observed value by random chance, when the true correlation is zero. If `P(i,j)`

is small, say less than `0.05`

, then the correlation `R(i,j)`

is significant.

```
[R,P,RLO,RUP]=corrcoef(...)
```

also returns matrices `RLO`

and `RUP`

, of the same size as `R`

, containing lower and upper bounds for a 95% confidence interval for each coefficient.

```
[...]=corrcoef(...,'param1',val1,'param2',val2,...)
```

specifies additional parameters and their values. Valid parameters are the following.

The p-value is computed by transforming the correlation to create a t statistic having `n`

-2 degrees of freedom, where `n`

is the number of rows of `X`

. The confidence bounds are based on an asymptotic normal distribution of `0.5*log((1+R)/(1-R))`

, with an approximate variance equal to `1/(n-3)`

. These bounds are accurate for large samples when `X`

has a multivariate normal distribution. The `'pairwise'`

option can produce an `R`

matrix that is not positive definite.

**Examples**

Generate random data having correlation between column 4 and the other columns.

x = randn(30,4); % Uncorrelated data x(:,4) = sum(x,2); % Introduce correlation. [r,p] = corrcoef(x) % Compute sample correlation and p-values. [i,j] = find(p<0.05); % Find significant correlations. [i,j] % Display their (row,col) indices. r = 1.0000 -0.3566 0.1929 0.3457 -0.3566 1.0000 -0.1429 0.4461 0.1929 -0.1429 1.0000 0.5183 0.3457 0.4461 0.5183 1.0000 p = 1.0000 0.0531 0.3072 0.0613 0.0531 1.0000 0.4511 0.0135 0.3072 0.4511 1.0000 0.0033 0.0613 0.0135 0.0033 1.0000 ans = 4 2 4 3 2 4 3 4

**See Also**

`xcorr`

, `xcov`

in the Signal Processing Toolbox

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