tsallis_entropy_normal.h

// Description: Tsallis entropy of a normal distribution
// Documentation: tsallis_entropy_analytic.txt

#ifndef TIM_TSALLIS_ENTROPY_NORMAL_H
#define TIM_TSALLIS_ENTROPY_NORMAL_H

#include "tim/core/mytypes.h"
#include "tim/core/differential_entropy_normal.h"

#include <pastel/sys/real/real_concept.h>

namespace Tim
{

    //! Tsallis entropy of a normal distribution.
    /*!
   Preconditions:
   dimension > 0
   covarianceDeterminant >= 0

   q:
   The power in the definition of Renyi entropy.
   If q = 1, differentialEntropyNormal() is returned
   instead.

   dimension:
   The dimension of the distribution.

   covarianceDeterminant:
   Determinant of the covariance matrix of the distribution.
   */
    template <typename Real>
    Real normalTsallisEntropy(
        const NoDeduction<Real>& q,
        integer dimension,
        const NoDeduction<Real>& covarianceDeterminant)
    {
        PENSURE_OP(q, >, 0);
        PENSURE_OP(dimension, >, 0);
        PENSURE_OP(covarianceDeterminant, >=, 0);

        if (q == 1)
        {
            return differentialEntropyNormal<Real>(
                dimension,
                covarianceDeterminant);
        }

        // A = 1 / ((2 pi)^(n / 2) det(C)^(1 / 2))

        Real A = inverse(
            std::pow(2 * constantPi<Real>(), (Real)dimension / 2) *
            std::sqrt(covarianceDeterminant));

        // I = A^(q - 1) / q^(n / 2)

        Real I = std::pow(A, q - 1) / 
            std::pow(q, (Real)dimension / 2);

        // H_q(X) = (1 - I) / (q - 1)

        return (1 - I) / (q - 1);
    }

}

#endif