# incrmvmr

Compute a moving variance-to-mean ratio (VMR) incrementally.

For a window of size `W`

, the unbiased sample variance is defined as

and the arithmetic mean is defined as

The variance-to-mean ratio (VMR) is thus defined as

## Usage

```
var incrmvmr = require( '@stdlib/stats/incr/mvmr' );
```

#### incrmvmr( window[, mean] )

Returns an accumulator `function`

which incrementally computes a moving variance-to-mean ratio. The `window`

parameter defines the number of values over which to compute the moving variance-to-mean ratio.

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

If the mean is already known, provide a `mean`

argument.

```
var accumulator = incrmvmr( 3, 5.0 );
```

#### accumulator( [x] )

If provided an input value `x`

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

, the accumulator function returns the current accumulated value.

```
var accumulator = incrmvmr( 3 );
var F = accumulator();
// returns null
// Fill the window...
F = accumulator( 2.0 ); // [2.0]
// returns 0.0
F = accumulator( 1.0 ); // [2.0, 1.0]
// returns ~0.33
F = accumulator( 3.0 ); // [2.0, 1.0, 3.0]
// returns 0.5
// Window begins sliding...
F = accumulator( 7.0 ); // [1.0, 3.0, 7.0]
// returns ~2.55
F = accumulator( 5.0 ); // [3.0, 7.0, 5.0]
// returns ~0.80
F = accumulator();
// returns ~0.80
```

## 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`

values 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.The following table summarizes how to interpret the variance-to-mean ratio:

VMR Description Example Distribution 0 not dispersed constant 0 < VMR < 1 under-dispersed binomial 1 -- Poisson >1 over-dispersed geometric, negative-binomial Accordingly, one can use the variance-to-mean ratio to assess whether observed data can be modeled as a Poisson process. When observed data is "under-dispersed", observed data may be more regular than as would be the case for a Poisson process. When observed data is "over-dispersed", observed data may contain clusters (i.e., clumped, concentrated data).

The variance-to-mean ratio is typically computed on nonnegative values. The measure may lack meaning for data which can assume both positive and negative values.

The variance-to-mean ratio is also known as the

**index of dispersion**,**dispersion index**,**coefficient of dispersion**,**relative variance**, and the**Fano factor**.

## Examples

```
var randu = require( '@stdlib/random/base/randu' );
var incrmvmr = require( '@stdlib/stats/incr/mvmr' );
var accumulator;
var v;
var i;
// Initialize an accumulator:
accumulator = incrmvmr( 5 );
// For each simulated datum, update the moving variance-to-mean ratio...
for ( i = 0; i < 100; i++ ) {
v = randu() * 100.0;
accumulator( v );
}
console.log( accumulator() );
```