# incrmcovariance
> Compute a moving [unbiased sample covariance][covariance] incrementally.
For unknown population means, the [unbiased sample covariance][covariance] for a window `n` of size `W` is defined as
where `j` specifies the index of the value at which the window begins. For example, for a trailing (i.e., non-centered) window using zero-based indexing and `j` greater than or equal to `W`, `j` is the `n-W`th value with `n` being the number of values thus analyzed.
For known population means, the [unbiased sample covariance][covariance] for a window `n` of size `W` is defined as
## Usage
```javascript
var incrmcovariance = require( '@stdlib/stats/incr/mcovariance' );
```
#### incrmcovariance( window\[, mx, my] )
Returns an accumulator `function` which incrementally computes a moving [unbiased sample covariance][covariance]. The `window` parameter defines the number of values over which to compute the moving [unbiased sample covariance][covariance].
```javascript
var accumulator = incrmcovariance( 3 );
```
If means are already known, provide `mx` and `my` arguments.
```javascript
var accumulator = incrmcovariance( 3, 5.0, -3.14 );
```
#### accumulator( \[x, y] )
If provided input values `x` and `y`, the accumulator function returns an updated [unbiased sample covariance][covariance]. If not provided input values `x` and `y`, the accumulator function returns the current [unbiased sample covariance][covariance].
```javascript
var accumulator = incrmcovariance( 3 );
var v = accumulator();
// returns null
// Fill the window...
v = accumulator( 2.0, 1.0 ); // [(2.0, 1.0)]
// returns 0.0
v = accumulator( -5.0, 3.14 ); // [(2.0, 1.0), (-5.0, 3.14)]
// returns ~-7.49
v = accumulator( 3.0, -1.0 ); // [(2.0, 1.0), (-5.0, 3.14), (3.0, -1.0)]
// returns -8.35
// Window begins sliding...
v = accumulator( 5.0, -9.5 ); // [(-5.0, 3.14), (3.0, -1.0), (5.0, -9.5)]
// returns -29.42
v = accumulator( -5.0, 1.5 ); // [(3.0, -1.0), (5.0, -9.5), (-5.0, 1.5)]
// returns -24.5
v = accumulator();
// returns -24.5
```
## 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.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var incrmcovariance = require( '@stdlib/stats/incr/mcovariance' );
var accumulator;
var x;
var y;
var i;
// Initialize an accumulator:
accumulator = incrmcovariance( 5 );
// For each simulated datum, update the moving unbiased sample covariance...
for ( i = 0; i < 100; i++ ) {
x = randu() * 100.0;
y = randu() * 100.0;
accumulator( x, y );
}
console.log( accumulator() );
```
[covariance]: https://en.wikipedia.org/wiki/Covariance
