# incrvmr
> Compute a [variance-to-mean ratio][variance-to-mean-ratio] (VMR) incrementally.
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 incrvmr = require( '@stdlib/stats/incr/vmr' );
```
#### incrvmr( \[mean] )
Returns an accumulator `function` which incrementally computes a [variance-to-mean ratio][variance-to-mean-ratio].
```javascript
var accumulator = incrvmr();
```
If the mean is already known, provide a `mean` argument.
```javascript
var accumulator = incrvmr( 3.0 );
```
#### 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 = incrvmr();
var D = accumulator( 2.0 );
// returns 0.0
D = accumulator( 1.0 ); // => s^2 = ((2-1.5)^2+(1-1.5)^2) / (2-1)
// returns ~0.33
D = accumulator( 3.0 ); // => s^2 = ((2-2)^2+(1-2)^2+(3-2)^2) / (3-1)
// returns 0.5
D = accumulator();
// returns 0.5
```
## 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 **all** 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.
- 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**, and **relative variance**.
## Examples
```javascript
var randu = require( '@stdlib/random/base/randu' );
var incrvmr = require( '@stdlib/stats/incr/vmr' );
var accumulator;
var v;
var i;
// Initialize an accumulator:
accumulator = incrvmr();
// For each simulated datum, update the 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
