Entropy

Erlang distribution differential entropy.

The differential entropy for an Erlang random variable with shape k and rate λ is

h left-parenthesis upper X right-parenthesis equals left-parenthesis 1 minus k right-parenthesis psi left-parenthesis k right-parenthesis plus ln left-bracket StartFraction normal upper Gamma left-parenthesis k right-parenthesis Over lamda EndFraction right-bracket plus k

where Γ and Ψ denote the gamma and digamma function, respectively.

Usage

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

entropy( k, lambda )

Returns the differential entropy of an Erlang distribution with parameters shape k and rate lambda (in nats).

var v = entropy( 1, 1.0 );
// returns ~-0.154

v = entropy( 4, 12.0 );
// returns ~9.587

v = entropy( 8, 2.0 );
// returns ~33.973

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

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

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

If not provided a positive integer for k, the function returns NaN.

var v = entropy( 1.8, 1.0 );
// returns NaN

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

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

var v = entropy( 2, 0.0 );
// returns NaN

v = entropy( 2, -1.0 );
// returns NaN

Examples

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

var lambda;
var k;
var v;
var i;

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