The unbiased sample variance is defined as

and the arithmetic mean is defined as

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

Installation

npm install @stdlib/stats-incr-vmr

Usage

var incrvmr = require( '@stdlib/stats-incr-vmr' );

incrvmr( [mean] )

Returns an accumulator function which incrementally computes a variance-to-mean ratio.

var accumulator = incrvmr();

If the mean is already known, provide a mean argument.

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.

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:

  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 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 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 is also known as the index of dispersion, dispersion index, coefficient of dispersion, and relative variance.

Examples

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() );

See Also

• @stdlib/stats-incr/mean: compute an arithmetic mean incrementally.
• @stdlib/stats-incr/mvmr: compute a moving variance-to-mean ratio (VMR) incrementally.
• @stdlib/stats-incr/variance: compute an unbiased sample variance incrementally.

Notice

This package is part of stdlib, a standard library for JavaScript and Node.js, with an emphasis on numerical and scientific computing. The library provides a collection of robust, high performance libraries for mathematics, statistics, streams, utilities, and more.

For more information on the project, filing bug reports and feature requests, and guidance on how to develop stdlib, see the main project repository.

Community

License

See LICENSE.

Copyright

Copyright © 2016-2024. The Stdlib Authors.