@stdlib/stats/incr/vmr/README.md

Standard library statistical functions.

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# incrvmr

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

<section class="intro">

The [unbiased sample variance][sample-variance] is defined as

<!-- <equation class="equation" label="eq:unbiased_sample_variance" align="center" raw="s^2 = \frac{1}{n-1} \sum_{i=0}^{n-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}{n-1} \sum_{i=0}^{n-1} ( x_i - \bar{x} )^2" data-equation="eq:unbiased_sample_variance">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@7fe559e94716008fb414ec7c6b3d0e3e1194f2ba/lib/node_modules/@stdlib/stats/incr/vmr/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}{n} \sum_{i=0}^{n-1} x_i" alt="Equation for the arithmetic mean."> -->

<!-- <div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{n} \sum_{i=0}^{n-1} x_i" data-equation="eq:arithmetic_mean">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@86f8c49b0e95ee794f0b098b8d17444c0cbeea0a/lib/node_modules/@stdlib/stats/incr/vmr/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="D = \frac{s^2}{\bar{x}}" alt="Equation for the variance-to-mean ratio (VMR)."> -->

<!-- <div class="equation" align="center" data-raw-text="D = \frac{s^2}{\bar{x}}" data-equation="eq:variance_to_mean_ratio">
    <img src="https://cdn.jsdelivr.net/gh/stdlib-js/stdlib@86f8c49b0e95ee794f0b098b8d17444c0cbeea0a/lib/node_modules/@stdlib/stats/incr/vmr/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 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
```

</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 **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 &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**, and **relative variance**.

</section>

<!-- /.notes -->

<section class="examples">

## Examples

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

```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() );
```

</section>

<!-- /.examples -->

<!-- Section for related `stdlib` packages. Do not manually edit this section, as it is automatically populated. -->

<section class="related">

* * *

## See Also

-   <span class="package-name">[`@stdlib/stats/incr/mean`][@stdlib/stats/incr/mean]</span><span class="delimiter">: </span><span class="description">compute an arithmetic mean incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/mvmr`][@stdlib/stats/incr/mvmr]</span><span class="delimiter">: </span><span class="description">compute a moving variance-to-mean ratio (VMR) incrementally.</span>
-   <span class="package-name">[`@stdlib/stats/incr/variance`][@stdlib/stats/incr/variance]</span><span class="delimiter">: </span><span class="description">compute an unbiased sample variance incrementally.</span>

</section>

<!-- /.related -->

<!-- Section for all links. Make sure to keep an empty line after the `section` element and another before the `/section` close. -->

<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

<!-- <related-links> -->

[@stdlib/stats/incr/mean]: https://github.com/stdlib-js/stats/tree/main/incr/mean

[@stdlib/stats/incr/mvmr]: https://github.com/stdlib-js/stats/tree/main/incr/mvmr

[@stdlib/stats/incr/variance]: https://github.com/stdlib-js/stats/tree/main/incr/variance

<!-- </related-links> -->

</section>

<!-- /.links -->