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# incrmvmr

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

<section class="intro">

For a window of size `W`, the [unbiased sample variance][sample-variance] is defined as

<!-- <equation class="equation" label="eq:unbiased_sample_variance" align="center" raw="s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2" alt="Equation for the unbiased sample variance."> -->

<div class="equation" align="center" data-raw-text="s^2 = \frac{1}{W-1} \sum_{i=0}^{W-1} ( x_i - \bar{x} )^2" data-equation="eq:unbiased_sample_variance">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/164b8f1010c4535340eed9ad0b2af32c4a19863c/lib/node_modules/@stdlib/stats/incr/mvmr/docs/img/equation_unbiased_sample_variance.svg" alt="Equation for the unbiased sample variance.">
    <br>
</div>

<!-- </equation> -->

and the [arithmetic mean][arithmetic-mean] is defined as

<!-- <equation class="equation" label="eq:arithmetic_mean" align="center" raw="\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i" alt="Equation for the arithmetic mean."> -->

<div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{W} \sum_{i=0}^{W-1} x_i" data-equation="eq:arithmetic_mean">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/164b8f1010c4535340eed9ad0b2af32c4a19863c/lib/node_modules/@stdlib/stats/incr/mvmr/docs/img/equation_arithmetic_mean.svg" alt="Equation for the arithmetic mean.">
    <br>
</div>

<!-- </equation> -->

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

<!-- <equation class="equation" label="eq:variance_to_mean_ratio" align="center" raw="F = \frac{s^2}{\bar{x}}" alt="Equation for the variance-to-mean ratio (VMR)."> -->

<div class="equation" align="center" data-raw-text="F = \frac{s^2}{\bar{x}}" data-equation="eq:variance_to_mean_ratio">
    <img src="https://cdn.rawgit.com/stdlib-js/stdlib/164b8f1010c4535340eed9ad0b2af32c4a19863c/lib/node_modules/@stdlib/stats/incr/mvmr/docs/img/equation_variance_to_mean_ratio.svg" alt="Equation for the variance-to-mean ratio (VMR).">
    <br>
</div>

<!-- </equation> -->

</section>

<!-- /.intro -->

<section class="usage">

## 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
```

</section>

<!-- /.usage -->

<section class="notes">

## 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 &lt; VMR &lt; 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].

</section>

<!-- /.notes -->

<section class="examples">

## Examples

<!-- eslint no-undef: "error" -->

```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() );
```

</section>

<!-- /.examples -->

<section class="links">

[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

</section>

<!-- /.links -->
