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# incrpcorrmat

> Compute a [sample Pearson product-moment correlation matrix][pearson-correlation] incrementally.

<section class="intro">

A [Pearson product-moment correlation matrix][pearson-correlation] is an M-by-M matrix whose elements specified by indices `j` and `k` are the [Pearson product-moment correlation coefficients][pearson-correlation] between the jth and kth data variables. The [Pearson product-moment correlation coefficient][pearson-correlation] between random variables `X` and `Y` is defined as

<!-- <equation class="equation" label="eq:pearson_correlation_coefficient" align="center" raw="\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}" alt="Equation for the Pearson product-moment correlation coefficient."> -->

<div class="equation" align="center" data-raw-text="\rho_{X,Y} = \frac{\operatorname{cov}(X,Y)}{\sigma_X \sigma_Y}" data-equation="eq:pearson_correlation_coefficient">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/27e2a43c70db648bb5bbc3fd0cdee050c25adc0b/lib/node_modules/@stdlib/stats/incr/pcorrmat/docs/img/equation_pearson_correlation_coefficient.svg" alt="Equation for the Pearson product-moment correlation coefficient.">
    <br>
</div>

<!-- </equation> -->

where the numerator is the [covariance][covariance] and the denominator is the product of the respective standard deviations.

For a sample of size `n`, the [sample Pearson product-moment correlation coefficient][pearson-correlation] is defined as

<!-- <equation class="equation" label="eq:sample_pearson_correlation_coefficient" align="center" raw="r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}}" alt="Equation for the sample Pearson product-moment correlation coefficient."> -->

<div class="equation" align="center" data-raw-text="r = \frac{\sum_{i=0}^{n-1} (x_i - \bar{x})(y_i - \bar{y})}{\sqrt{\sum_{i=0}^{n-1} (x_i - \bar{x})^2} \sqrt{\sum_{i=0}^{n-1} (y_i - \bar{y})^2}}" data-equation="eq:sample_pearson_correlation_coefficient">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/27e2a43c70db648bb5bbc3fd0cdee050c25adc0b/lib/node_modules/@stdlib/stats/incr/pcorrmat/docs/img/equation_sample_pearson_correlation_coefficient.svg" alt="Equation for the sample Pearson product-moment correlation coefficient.">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var incrpcorrmat = require( '@stdlib/stats/incr/pcorrmat' );
```

#### incrpcorrmat( out\[, means] )

Returns an accumulator `function` which incrementally computes a [sample Pearson product-moment correlation matrix][pearson-correlation].

```javascript
// Create an accumulator for computing a 2-dimensional correlation matrix:
var accumulator = incrpcorrmat( 2 );
```

The `out` argument may be either the order of the [correlation matrix][pearson-correlation] or a square 2-dimensional [`ndarray`][@stdlib/ndarray/ctor] for storing the [correlation matrix][pearson-correlation].

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var ndarray = require( '@stdlib/ndarray/ctor' );

var buffer = new Float64Array( 4 );
var shape = [ 2, 2 ];
var strides = [ 2, 1 ];

// Create a 2-dimensional output correlation matrix:
var corr = ndarray( 'float64', buffer, shape, strides, 0, 'row-major' );

var accumulator = incrpcorrmat( corr );
```

When means are known, the function supports providing a 1-dimensional [`ndarray`][@stdlib/ndarray/ctor] containing mean values.

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var ndarray = require( '@stdlib/ndarray/ctor' );

var buffer = new Float64Array( 2 );
var shape = [ 2 ];
var strides = [ 1 ];

var means = ndarray( 'float64', buffer, shape, strides, 0, 'row-major' );

means.set( 0, 3.0 );
means.set( 1, -5.5 );

var accumulator = incrpcorrmat( 2, means );
```

#### accumulator( \[vector] )

If provided a data vector, the accumulator function returns an updated [sample Pearson product-moment correlation matrix][pearson-correlation]. If not provided a data vector, the accumulator function returns the current [sample Pearson product-moment correlation matrix][pearson-correlation].

```javascript
var Float64Array = require( '@stdlib/array/float64' );
var ndarray = require( '@stdlib/ndarray/ctor' );

var buffer = new Float64Array( 4 );
var shape = [ 2, 2 ];
var strides = [ 2, 1 ];
var corr = ndarray( 'float64', buffer, shape, strides, 0, 'row-major' );

buffer = new Float64Array( 2 );
shape = [ 2 ];
strides = [ 1 ];
var vec = ndarray( 'float64', buffer, shape, strides, 0, 'row-major' );

var accumulator = incrpcorrmat( corr );

vec.set( 0, 2.0 );
vec.set( 1, 1.0 );

var out = accumulator( vec );
// returns <ndarray>

var bool = ( out === corr );
// returns true

vec.set( 0, 1.0 );
vec.set( 1, -5.0 );

out = accumulator( vec );
// returns <ndarray>

vec.set( 0, 3.0 );
vec.set( 1, 3.14 );

out = accumulator( vec );
// returns <ndarray>

out = accumulator();
// returns <ndarray>
```

</section>

<!-- /.usage -->

<section class="notes">

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```javascript
var randu = require( '@stdlib/random/base/randu' );
var ndarray = require( '@stdlib/ndarray/ctor' );
var Float64Array = require( '@stdlib/array/float64' );
var incrpcorrmat = require( '@stdlib/stats/incr/pcorrmat' );

var corr;
var rxy;
var ryx;
var rx;
var ry;
var i;

// Initialize an accumulator for a 2-dimensional correlation matrix:
var accumulator = incrpcorrmat( 2 );

// Create a 1-dimensional data vector:
var buffer = new Float64Array( 2 );
var shape = [ 2 ];
var strides = [ 1 ];

var vec = ndarray( 'float64', buffer, shape, strides, 0, 'row-major' );

// For each simulated data vector, update the sample correlation matrix...
for ( i = 0; i < 100; i++ ) {
    vec.set( 0, randu()*100.0 );
    vec.set( 1, randu()*100.0 );
    corr = accumulator( vec );

    rx = corr.get( 0, 0 ).toFixed( 4 );
    ry = corr.get( 1, 1 ).toFixed( 4 );
    rxy = corr.get( 0, 1 ).toFixed( 4 );
    ryx = corr.get( 1, 0 ).toFixed( 4 );

    console.log( '[ %d, %d\n  %d, %d ]', rx, rxy, ryx, ry );
}
```

</section>

<!-- /.examples -->

<section class="links">

[pearson-correlation]: https://en.wikipedia.org/wiki/Pearson_correlation_coefficient

[covariance]: https://en.wikipedia.org/wiki/Covariance

[@stdlib/ndarray/ctor]: https://github.com/stdlib-js/ndarray-ctor

</section>

<!-- /.links -->
