!!!Boost: Nakagami distribution

!!!Boost: Nakagami distribution#

The following functions return the pdf, cdf, qtf or boost class of the Nakagami distribution with shape \(m > 0\), scale \(\omega > 0\), and the support interval \((0,+\infty)\).

See also Wikipedia [1279], MathWorld [262], Dharmawansa P and Ahmed [248], Hauberg [382], Ehrhardt [309] (3.9.22).

See also: https://reference.wolfram.com/language/ref/NakagamiDistribution.html

See also: https://mathworld.wolfram.com/PochhammerSymbol.html

Ctx.nakagami_pdf(x, m, omega)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{pdf}(x)\), the value of the probability density function (Pdf) of the Nakagami distribution:

\[\text{pdf}(x) = \frac{2m^m x^{2m-1}}{\omega^m \Gamma(m)} \exp\left( -\frac{m}{\omega} x^2 \right).\]

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("NakagamiPdf(x, a, b): ", NakagamiPdf(x, a, b))
>>> print ("dist_nakagami(a, b).pdf(x): ", dist_nakagami(a, b).pdf(x))
6.3563523462564525615615615614561356E+00

Ctx.nakagami_cdf(x, m, omega)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{cdf}(x)\), the value of the cumulative distribution function (Cdf) of the Nakagami distribution:

\[\text{cdf}(x) = P\left( \frac{m}{\omega} x^2 \right).\]

Here \(P(\cdot)\) denotes the lower regularized incomplete gamma function (RealGammaP).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; x = 0.6;
>>> print ("NakagamiCdf(x, a, b): ", NakagamiCdf(x, a, b))
>>> print ("dist_nakagami(a, b).cdf(x): ", dist_nakagami(a, b).cdf(x))
6.3563523462564525615615615614561356E+00

Ctx.nakagami_qtf(q, m, omega)#

where Ctx is Math53 or CtxBoost.

Returns \(\text{qtf}(q)\), the value of the quantile function (Qtf) of the Nakagami distribution:

\[\text{qtf}(q) = \sqrt{ \frac{\omega}{m} P^{-1}(m, q)}.\]

Here \(P^{-1}(\cdot)\) denotes the inverse of the lower regularized incomplete gamma function (RealGammaPInv).

The following example shows both forms of the syntax:

>>> from mpfebnet import *
>>> a = 0; b = 1; t = 0.3; q = 0.6;
>>> print ("NakagamiQtf(q, a, b): ", NakagamiQtf(q, a, b))
>>> print ("dist_nakagami(a, b).qtf(q): ", dist_nakagami(a, b).qtf(q))
6.3563523462564525615615615614561356E+00

class ctx.dist_nakagami(m, omega)#

where ctx is fpm, mpm, ipm, dec, gmp or apm.

The Nakagami distribution is a continuous probability distribution with shape \(m > 0\), scale \(\omega > 0\), and the support interval \((0,+\infty)\). See also Wikipedia [1279], MathWorld [262], Witkovský [1640], Dharmawansa P and Ahmed [248], Hauberg [382].

Dharmawansa 2007: Characteristic function Hauberg 2005: Moments

dist_nakagami.pdf(x)#

Returns \(\text{pdf}_X(x)\), the probability density function (pdf) of a random variable \(X\), following an Nakagami distribution:

\[\text{pdf}_X(x) = \frac{2m^m x^{2m-1}}{\omega^m \Gamma(m)} \exp\left( -\frac{m}{\omega} x^2 \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("pdf: ", nakagami(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nakagami.cdf(x)#

Returns \(\text{cdf}_X(x)\), the cumulative distribution function (cdf) of a random variable \(X\), following an Nakagami distribution:

\[\text{cdf}_X(x) = P\left( \frac{m}{\omega} x^2 \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print ("cdf: ", nakagami(mu, sigma).pdf(x))
6.3563523462564525615615615614561356E-20

dist_nakagami.sf(x)#

Returns \(\text{sf}_X(x)\), the survival function (sf) of a random variable \(X\), following an Nakagami distribution:

\[\text{sf}_X(x) = Q\left( \frac{m}{\omega} x^2 \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; x = 3;
>>> print (" sf: ", nakagami(mu, sigma).pdf(x))
sf: 6.3563523462564525615615615614561356E-20

dist_nakagami.qtf(q)#

Returns \(\text{qtf}_X(x)\), the quantile function (qtf) of a random variable \(X\), following an Nakagami distribution:

\[\text{qtf}_X(q) = \sqrt{ \frac{\omega}{m} P^{-1}(m, q) }\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("qtf: ", nakagami(mu, sigma).qtf(q))
qtf: 6.3563523462564525615615615614561356E+00

dist_nakagami.isf(q)#

Returns \(\text{isf}_X(q)\), the inverse survival function (isf) of a random variable \(X\), following an Nakagami distribution:

\[\text{isf}_X(q) = \sqrt{ \frac{\omega}{m} Q^{-1}(m, q) }\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; q = 0.3;
>>> print ("isf: ", nakagami(mu, sigma).isf(q))
6.3563523462564525615615615614561356E+00

dist_nakagami.c_x(t)#

Returns \(C_X(t)\), the characteristic function of a random variable \(X\), following an Nakagami distribution:

\[C_X(t) = \frac{\Gamma(2m)}{2^{m-1}\Gamma(m)} \exp\left( -\frac{\omega}{8m} t^2 \right) D_{-2m} \left( -it \sqrt{\frac{\omega}{2m}} \right).\]

Dharmawansa 2007: Characteristic function

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("c_x: ", nakagami(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_nakagami.m_x(t)#

Returns \(M_X(t)\), the moment generating function of a random variable \(X\), following an Nakagami distribution:

\[M_X(t) = \frac{\Gamma(2m)}{2^{m-1}\Gamma(m)} \exp\left( -\frac{\omega}{8m} t^2 \right) D_{-2m} \left( -t \sqrt{\frac{\omega}{2m}} \right).\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3;
>>> print ("m_x: ", nakagami(mu, sigma).c_x(t))
6.3563523462564525615615615614561356E+00

dist_nakagami.k_x(t, k=0)#

Returns \(K_X(t)\), the cumulant generating function of a random variable \(X\), following an Nakagami distribution:

\[K_X(t) = \log \left[ \frac{\Gamma(2m)}{2^{m-1}\Gamma(m)} \exp\left( -\frac{\omega}{8m} t^2 \right) D_{-2m} \left( -t \sqrt{\frac{\omega}{2m}} \right) \right].\]
>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; t = 0.3; k = 6;
>>> print ("c_x: ", nakagami(mu, sigma).k_x(t, k))
6.3563523462564525615615615614561356E+00

dist_nakagami.moments(k)#

Returns the first \(j\) central moments, \(\mu_j, j = 1 \ldots k\), of a random variable \(X\), following an Nakagami distribution (Wikipedia). The raw moments are calculated from the central moments.

\[\mu_{X}(n) = tbd.\]

Hauberg 2005: Moments

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", nakagami(mu, sigma).moments(k))
6.3563523462564525615615615614561356E+00

dist_nakagami.cumulants(k)#

Returns the first \(j\) cumulants, \(\kappa_j, j = 1 \ldots k\), of a random variable \(X\), following an Nakagami distribution. The cumulants are calculated from the moments.

>>> from mpfunlab import *
>>> mp.dps = 30
>>> mu = 0; sigma = 1; k = 6;
>>> print ("saddlepoint: ", nakagami(mu, sigma).cumulants(k))
6.3563523462564525615615615614561356E+00
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