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# incrmpcorr2

> Compute a moving squared sample [Pearson product-moment correlation coefficient][pearson-correlation] incrementally.

<section class="intro">

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/e6bc812ab63010afd0f25418c0c6954c3a680357/lib/node_modules/@stdlib/stats/incr/mpcorr2/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 `W`, 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/e6bc812ab63010afd0f25418c0c6954c3a680357/lib/node_modules/@stdlib/stats/incr/mpcorr2/docs/img/equation_sample_pearson_correlation_coefficient.svg" alt="Equation for the sample Pearson product-moment correlation coefficient.">
    <br>
</div>

<!-- </equation> -->

The squared sample [Pearson product-moment correlation coefficient][pearson-correlation] is thus defined as the square of the sample [Pearson product-moment correlation coefficient][pearson-correlation].

</section>

<!-- /.intro -->

<section class="usage">

## Usage

```javascript
var incrmpcorr2 = require( '@stdlib/stats/incr/mpcorr2' );
```

#### incrmpcorr2( window\[, mx, my] )

Returns an accumulator `function` which incrementally computes a moving squared sample [Pearson product-moment correlation coefficient][pearson-correlation]. The `window` parameter defines the number of values over which to compute the moving squared sample [Pearson product-moment correlation coefficient][pearson-correlation].

```javascript
var accumulator = incrmpcorr2( 3 );
```

If means are already known, provide `mx` and `my` arguments.

```javascript
var accumulator = incrmpcorr2( 3, 5.0, -3.14 );
```

#### accumulator( \[x, y] )

If provided input values `x` and `y`, the accumulator function returns an updated accumulated value. If not provided input values `x` and `y`, the accumulator function returns the current accumulated value.

```javascript
var accumulator = incrmpcorr2( 3 );

var r2 = accumulator();
// returns null

// Fill the window...
r2 = accumulator( 2.0, 1.0 ); // [(2.0, 1.0)]
// returns 0.0

r2 = accumulator( -5.0, 3.14 ); // [(2.0, 1.0), (-5.0, 3.14)]
// returns ~1.0

r2 = accumulator( 3.0, -1.0 ); // [(2.0, 1.0), (-5.0, 3.14), (3.0, -1.0)]
// returns ~0.86

// Window begins sliding...
r2 = accumulator( 5.0, -9.5 ); // [(-5.0, 3.14), (3.0, -1.0), (5.0, -9.5)]
// returns ~0.74

r2 = accumulator( -5.0, 1.5 ); // [(3.0, -1.0), (5.0, -9.5), (-5.0, 1.5)]
// returns ~0.64

r2 = accumulator();
// returns ~0.64
```

</section>

<!-- /.usage -->

<section class="notes">

## Notes

-   Input values are **not** type checked. If provided `NaN` or a value which, when used in computations, results in `NaN`, the accumulated value is `NaN` for **at least** `W-1` future invocations. If non-numeric inputs are possible, you are advised to type check and handle accordingly **before** passing the value to the accumulator function.
-   As `W` (x,y) pairs are needed to fill the window buffer, the first `W-1` returned values are calculated from smaller sample sizes. Until the window is full, each returned value is calculated from all provided values.
-   In comparison to the sample [Pearson product-moment correlation coefficient][pearson-correlation], the squared sample [Pearson product-moment correlation coefficient][pearson-correlation] is useful for emphasizing strong correlations.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```javascript
var randu = require( '@stdlib/random/base/randu' );
var incrmpcorr2 = require( '@stdlib/stats/incr/mpcorr2' );

var accumulator;
var x;
var y;
var i;

// Initialize an accumulator:
accumulator = incrmpcorr2( 5 );

// For each simulated datum, update the moving squared sample correlation coefficient...
for ( i = 0; i < 100; i++ ) {
    x = randu() * 100.0;
    y = randu() * 100.0;
    accumulator( x, y );
}
console.log( accumulator() );
```

</section>

<!-- /.examples -->

<section class="links">

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

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

</section>

<!-- /.links -->
