Entropy

F distribution differential entropy.

The differential entropy (in nats) for a F random variable is

h left-parenthesis upper X right-parenthesis equals ln left-parenthesis StartFraction d 2 Over d 1 EndFraction normal upper Gamma left-parenthesis StartFraction d 1 Over 2 EndFraction right-parenthesis normal upper Gamma left-parenthesis StartFraction d 2 Over 2 EndFraction right-parenthesis normal upper Gamma left-parenthesis StartFraction d 1 plus d 2 Over 2 EndFraction right-parenthesis right-parenthesis plus left-parenthesis 1 minus StartFraction d 1 Over 2 EndFraction right-parenthesis normal upper Psi left-parenthesis StartFraction d 1 Over 2 EndFraction right-parenthesis plus left-parenthesis 1 minus StartFraction d 2 Over 2 EndFraction right-parenthesis normal upper Psi left-parenthesis StartFraction d 2 Over 2 EndFraction right-parenthesis plus StartFraction d 1 plus d 2 Over 2 EndFraction normal upper Psi left-parenthesis StartFraction d 1 plus d 2 Over 2 EndFraction right-parenthesis

where d1 is the numerator degrees of freedom, d2 is the denominator degrees of freedom, and Γ and Ψ denote the gamma and digamma functions, respectively.

Usage

var entropy = require( '@stdlib/math/base/dists/f/entropy' );

entropy( d1, d2 )

Returns the differential entropy of a F distribution with numerator degrees of freedom d1 and denominator degrees of freedom d2 (in nats).

var v = entropy( 4.0, 7.0 );
// returns ~1.277

v = entropy( 4.0, 12.0 );
// returns ~1.12

v = entropy( 8.0, 2.0 );
// returns ~2.144

If provided NaN as any argument, the function returns NaN.

var v = entropy( NaN, 7.0 );
// returns NaN

v = entropy( 3.0, NaN );
// returns NaN

If provided d1 <= 0, the function returns NaN.

var v = entropy( 0.0, 2.0 );
// returns NaN

v = entropy( -1.0, 1.0 );
// returns NaN

If provided d2 <= 0, the function returns NaN.

var v = entropy( 3.0, 0.0 );
// returns NaN

v = entropy( 3.0, -1.0 );
// returns NaN

Examples

var randu = require( '@stdlib/random/base/randu' );
var EPS = require( '@stdlib/constants/math/float64-eps' );
var entropy = require( '@stdlib/math/base/dists/f/entropy' );

var d1;
var d2;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    d1 = ( randu()*10.0 ) + EPS;
    d2 = ( randu()*10.0 ) + EPS;
    v = entropy( d1, d2 );
    console.log( 'd1: %d, d2: %d, h(X;d1,d2): %d', d1.toFixed( 4 ), d2.toFixed( 4 ), v.toFixed( 4 ) );
}