incrvmr

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

The unbiased sample variance is defined as

s squared equals StartFraction 1 Over n minus 1 EndFraction sigma-summation Underscript i equals 0 Overscript n 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 n EndFraction sigma-summation Underscript i equals 0 Overscript n minus 1 Endscripts x Subscript i

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

upper D equals StartFraction s squared Over x overbar EndFraction

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:

    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, 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() );