Entropy

Cauchy distribution differential entropy.

The differential entropy for a Cauchy random variable with location parameter x0 and scale parameter Ɣ > 0 is

h left-parenthesis upper X right-parenthesis equals log left-parenthesis gamma right-parenthesis plus log left-parenthesis 4 pi right-parenthesis

Usage

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

entropy( x0, gamma )

Returns the differential entropy of a Cauchy distribution with location parameter x0 and scale parameter gamma (in nats).

var v = entropy( 10.0, 5.0 );
// returns ~4.14

v = entropy( 7.0, 2.0 );
// returns ~3.224

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

var v = entropy( NaN, 5.0 );
// returns NaN

v = entropy( 20.0, NaN );
// returns NaN

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

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

v = entropy( 1.0, 0.0 );
// 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/cauchy/entropy' );

var gamma;
var x0;
var v;
var i;

for ( i = 0; i < 10; i++ ) {
    x0 = randu() * 100.0;
    gamma = ( randu()*10.0 ) + EPS;
    v = entropy( x0, gamma );
    console.log( 'x0: %d, γ: %d, h(X;x0,γ): %d', x0.toFixed( 4 ), gamma.toFixed( 4 ), v.toFixed( 4 ) );
}