incrmvmr

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

For a window of size W, the unbiased sample variance is defined as

s squared equals StartFraction 1 Over upper W minus 1 EndFraction sigma-summation Underscript i equals 0 Overscript upper W minus 1 Endscripts left-parenthesis x Subscript i Baseline minus x overbar right-parenthesis squared

and the arithmetic mean is defined as

x overbar equals StartFraction 1 Over upper W EndFraction sigma-summation Underscript i equals 0 Overscript upper W minus 1 Endscripts x Subscript i

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

upper F equals StartFraction s squared Over x overbar EndFraction

Usage

var incrmvmr = require( '@stdlib/stats/incr/mvmr' );

incrmvmr( window[, mean] )

Returns an accumulator function which incrementally computes a moving variance-to-mean ratio. The window parameter defines the number of values over which to compute the moving variance-to-mean ratio.

var accumulator = incrmvmr( 3 );

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

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.

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

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:

    VMRDescriptionExample Distribution
    0not dispersedconstant
    0 < VMR < 1under-dispersedbinomial
    1--Poisson
    >1over-dispersedgeometric, 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, relative variance, and the Fano factor.

Examples

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() );