# incrmvmr
> Compute a moving [variance-to-mean ratio][variance-to-mean-ratio] (VMR) incrementally.
For a window of size `W`, the [unbiased sample variance][sample-variance] is defined as
and the [arithmetic mean][arithmetic-mean] is defined as
The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as
## Usage
```javascript
var incrmvmr = require( '@stdlib/stats/incr/mvmr' );
```
#### incrmvmr( window\[, mean] )
Returns an accumulator `function` which incrementally computes a moving [variance-to-mean ratio][variance-to-mean-ratio]. The `window` parameter defines the number of values over which to compute the moving [variance-to-mean ratio][variance-to-mean-ratio].
```javascript
var accumulator = incrmvmr( 3 );
```
If the mean is already known, provide a `mean` argument.
```javascript
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.
```javascript
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][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][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][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][variance-to-mean-ratio] is also known as the **index of dispersion**, **dispersion index**, **coefficient of dispersion**, **relative variance**, and the [**Fano factor**][fano-factor].
## Examples
```javascript
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() );
```
[variance-to-mean-ratio]: https://en.wikipedia.org/wiki/Index_of_dispersion
[arithmetic-mean]: https://en.wikipedia.org/wiki/Arithmetic_mean
[sample-variance]: https://en.wikipedia.org/wiki/Variance
[fano-factor]: https://en.wikipedia.org/wiki/Fano_factor
