# incrmcovariance

Compute a moving unbiased sample covariance incrementally.

For unknown population means, the unbiased sample 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 for a window `n`

of size `W`

is defined as

## Usage

```
var incrmcovariance = require( '@stdlib/stats/incr/mcovariance' );
```

#### incrmcovariance( window[, mx, my] )

Returns an accumulator `function`

which incrementally computes a moving unbiased sample covariance. The `window`

parameter defines the number of values over which to compute the moving unbiased sample covariance.

```
var accumulator = incrmcovariance( 3 );
```

If means are already known, provide `mx`

and `my`

arguments.

```
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. If not provided input values `x`

and `y`

, the accumulator function returns the current unbiased sample covariance.

```
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

```
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() );
```