time-to-botec/js/node_modules/@stdlib/stats/incr/vmr/README.md
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incrvmr

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

The unbiased sample variance is defined as

Equation for the unbiased sample variance.

and the arithmetic mean is defined as

Equation for the arithmetic mean.

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

Equation for the variance-to-mean ratio (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() );