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Add pkg to incrementally compute the Fano factor
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‎lib/node_modules/@stdlib/stats/incr/mvmr/README.md‎

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<!--
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@license Apache-2.0
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Copyright (c) 2018 The Stdlib Authors.
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Licensed under the Apache License, Version 2.0 (the "License");
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you may not use this file except in compliance with the License.
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You may obtain a copy of the License at
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http://www.apache.org/licenses/LICENSE-2.0
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Unless required by applicable law or agreed to in writing, software
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distributed under the License is distributed on an "AS IS" BASIS,
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WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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See the License for the specific language governing permissions and
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limitations under the License.
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-->
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# incrmvmr
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> Compute a moving [variance-to-mean ratio][variance-to-mean-ratio] (VMR) incrementally.
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<section class="intro">
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For a window of size `W`, the [unbiased sample variance][sample-variance] is defined as
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<!-- <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."> -->
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<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">
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<img src="" alt="Equation for the unbiased sample variance.">
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<br>
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</div>
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<!-- </equation> -->
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and the [arithmetic mean][arithmetic-mean] is defined as
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<!-- <equation class="equation" label="eq:arithmetic_mean" align="center" raw="\bar{x} = \frac{1}{n} \sum_{i=0}^{W-1} x_i" alt="Equation for the arithmetic mean."> -->
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<div class="equation" align="center" data-raw-text="\bar{x} = \frac{1}{n} \sum_{i=0}^{W-1} x_i" data-equation="eq:arithmetic_mean">
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<img src="" alt="Equation for the arithmetic mean.">
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<br>
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</div>
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<!-- </equation> -->
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The [variance-to-mean ratio][variance-to-mean-ratio] (VMR) is thus defined as
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<!-- <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)."> -->
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<div class="equation" align="center" data-raw-text="F = \frac{s^2}{\bar{x}}" data-equation="eq:variance_to_mean_ratio">
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<img src="" alt="Equation for the variance-to-mean ratio (VMR).">
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<br>
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</div>
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<!-- </equation> -->
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</section>
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<!-- /.intro -->
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<section class="usage">
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## Usage
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```javascript
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var incrmvmr = require( '@stdlib/stats/incr/mvmr' );
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```
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#### incrmvmr( window\[, mean] )
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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].
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```javascript
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var accumulator = incrmvmr( 3 );
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```
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If the mean is already known, provide a `mean` argument.
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```javascript
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var accumulator = incrmvmr( 3, 5.0 );
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```
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#### accumulator( \[x] )
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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.
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```javascript
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var accumulator = incrmvmr( 3 );
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var F = accumulator();
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// returns null
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// Fill the window...
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F = accumulator( 2.0 ); // [2.0]
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// returns 0.0
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F = accumulator( 1.0 ); // [2.0, 1.0]
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// returns ~0.33
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F = accumulator( 3.0 ); // [2.0, 1.0, 3.0]
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// returns 0.5
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// Window begins sliding...
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F = accumulator( -7.0 ); // [1.0, 3.0, -7.0]
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// returns -28.0
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F = accumulator( -5.0 ); // [3.0, -7.0, -5.0]
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// returns ~-9.33
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F = accumulator();
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// returns ~-9.33
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```
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</section>
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<!-- /.usage -->
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<section class="notes">
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## Notes
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- 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.
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- 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.
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- The following table summarizes how to interpret the [variance-to-mean ratio][variance-to-mean-ratio]:
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| VMR | Description | Example Distribution |
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| :---------------: | :-------------: | :--------------------------: |
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| 0 | not dispersed | constant |
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| 0 &lt; VMR &lt; 1 | under-dispersed | binomial |
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| 1 | -- | Poisson |
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| >1 | over-dispersed | geometric, negative-binomial |
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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).
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- 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].
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</section>
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<!-- /.notes -->
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<section class="examples">
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## Examples
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<!-- eslint no-undef: "error" -->
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```javascript
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var randu = require( '@stdlib/random/base/randu' );
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var incrmvmr = require( '@stdlib/stats/incr/mvmr' );
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var accumulator;
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var v;
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var i;
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// Initialize an accumulator:
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accumulator = incrmvmr( 5 );
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// For each simulated datum, update the moving variance-to-mean ratio...
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for ( i = 0; i < 100; i++ ) {
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v = randu() * 100.0;
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accumulator( v );
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}
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console.log( accumulator() );
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```
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</section>
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<!-- /.examples -->
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<section class="links">
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[variance-to-mean-ratio]: https://en.wikipedia.org/wiki/Index_of_dispersion
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[arithmetic-mean]: https://en.wikipedia.org/wiki/Arithmetic_mean
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[sample-variance]: https://en.wikipedia.org/wiki/Variance
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[fano-factor]: https://en.wikipedia.org/wiki/Fano_factor
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</section>
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<!-- /.links -->

‎lib/node_modules/@stdlib/stats/incr/mvmr/benchmark/benchmark.js‎

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/**
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* @license Apache-2.0
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*
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* Copyright (c) 2018 The Stdlib Authors.
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*
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* Licensed under the Apache License, Version 2.0 (the "License");
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* you may not use this file except in compliance with the License.
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* You may obtain a copy of the License at
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*
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* http://www.apache.org/licenses/LICENSE-2.0
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*
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* Unless required by applicable law or agreed to in writing, software
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* distributed under the License is distributed on an "AS IS" BASIS,
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* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
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* See the License for the specific language governing permissions and
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* limitations under the License.
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*/
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'use strict';
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// MODULES //
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var bench = require( '@stdlib/bench' );
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var randu = require( '@stdlib/random/base/randu' );
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var pkg = require( './../package.json' ).name;
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var incrmvmr = require( './../lib' );
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// MAIN //
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bench( pkg, function benchmark( b ) {
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var f;
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var i;
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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f = incrmvmr( (i%5)+1 );
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if ( typeof f !== 'function' ) {
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b.fail( 'should return a function' );
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}
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}
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b.toc();
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if ( typeof f !== 'function' ) {
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b.fail( 'should return a function' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});
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bench( pkg+'::accumulator', function benchmark( b ) {
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var acc;
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var v;
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var i;
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acc = incrmvmr( 5 );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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v = acc( randu() );
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if ( v !== v ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( v !== v ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});
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bench( pkg+'::accumulator,known_mean', function benchmark( b ) {
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var acc;
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var v;
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var i;
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acc = incrmvmr( 5, 0.5 );
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b.tic();
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for ( i = 0; i < b.iterations; i++ ) {
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v = acc( randu() );
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if ( v !== v ) {
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b.fail( 'should not return NaN' );
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}
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}
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b.toc();
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if ( v !== v ) {
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b.fail( 'should not return NaN' );
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}
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b.pass( 'benchmark finished' );
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b.end();
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});

‎lib/node_modules/@stdlib/stats/incr/mvmr/docs/img/equation_arithmetic_mean.svg‎

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