neutonsevero
e780bb956e
Add Log-Rayleigh and Log-Rice continuous distributions as scipy rv_continuous subclasses with PDF, CDF, SF, PPF, ISF, moments, entropy, and RVS methods. Log-Rice reduces to Log-Rayleigh when nu=0. Both are derived via the change-of-variable Y = ln X on their respective parent distributions. Includes unit tests verifying numerical correctness and the change-of-variable identity.
433 lines
15 KiB
Python
433 lines
15 KiB
Python
from scipy.stats import rv_continuous, ncx2
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from scipy.special import kve, gammaln
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from scipy.integrate import quad
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import scipy.special as sc
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from scipy.stats._distn_infrastructure import _ShapeInfo
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import numpy as np
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class k_gen(rv_continuous):
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"""Generalized K distribution for radar clutter modeling.
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The probability density function is:
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f(x; mu, alpha, beta) = 2 / (Gamma(alpha) * Gamma(beta))
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* (alpha*beta/mu)^((alpha+beta)/2)
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* x^((alpha+beta)/2 - 1)
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* K_{alpha-beta}(2*sqrt(alpha*beta*x/mu))
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for x > 0, where K_{alpha-beta} is the modified Bessel function of the
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second kind of order alpha-beta, mu > 0 is the mean, and alpha > 0,
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beta > 0 are shape parameters.
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The standard (2-parameter) K distribution is the special case beta=1,
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with nu=alpha and b=alpha/mu.
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"""
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def _shape_info(self):
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return [
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_ShapeInfo("mu", domain=(0, np.inf), inclusive=(False, True)),
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_ShapeInfo("alpha", domain=(0, np.inf), inclusive=(True, True)),
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_ShapeInfo("beta", domain=(0, np.inf), inclusive=(True, True)),
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]
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def _argcheck(self, mu, alpha, beta):
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return (mu > 0) & (alpha > 0) & (beta > 0)
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def _pdf(self, x, mu, alpha, beta):
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return np.exp(self._logpdf(x, mu, alpha, beta))
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def _logpdf(self, x, mu, alpha, beta):
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half_sum = (alpha + beta) / 2.0
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log_ab_over_mu = np.log(alpha) + np.log(beta) - np.log(mu)
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z = 2.0 * np.sqrt(alpha * beta * x / mu)
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return (
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np.log(2.0)
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- gammaln(alpha)
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- gammaln(beta)
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+ half_sum * log_ab_over_mu
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+ (half_sum - 1.0) * np.log(x)
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+ np.log(kve(alpha - beta, z))
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- z
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)
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def _stats(self, mu, alpha, beta, moments="mv"):
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mean = mu if "m" in moments or "v" in moments or "s" in moments or "k" in moments else None
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variance = None
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if "v" in moments or "s" in moments or "k" in moments:
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second_raw = (mu ** 2) * sc.poch(alpha, 2) * sc.poch(beta, 2) / ((alpha * beta) ** 2)
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variance = second_raw - mean ** 2
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skewness = None
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if "s" in moments or "k" in moments:
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third_raw = (mu ** 3) * sc.poch(alpha, 3) * sc.poch(beta, 3) / ((alpha * beta) ** 3)
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third_central = third_raw - 3.0 * mean * second_raw + 2.0 * mean ** 3
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skewness = third_central / np.power(variance, 1.5)
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kurtosis = None
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if "k" in moments:
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fourth_raw = (mu ** 4) * sc.poch(alpha, 4) * sc.poch(beta, 4) / ((alpha * beta) ** 4)
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fourth_central = fourth_raw - 4.0 * mean * third_raw + 6.0 * mean ** 2 * second_raw - 3.0 * mean ** 4
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kurtosis = fourth_central / (variance ** 2) - 3.0
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return mean, variance, skewness, kurtosis
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def _cdf(self, x, mu, alpha, beta):
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# scipy broadcasts params to match x's shape before calling _cdf,
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# so mu/alpha/beta may be arrays. Pass all four to np.vectorize so
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# each argument arrives as a scalar inside _scalar.
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# Substitution u = sqrt(x) regularises the integrand near u=0:
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# f(u²)*2u ~ u^(alpha+beta-1), smooth for alpha,beta > 0.
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def _scalar(xi, mui, ai, bi):
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val, _ = quad(
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lambda u: float(self._pdf(float(u * u), float(mui), float(ai), float(bi))) * 2.0 * u,
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0.0, float(np.sqrt(xi)),
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limit=200, epsabs=1.49e-10, epsrel=1.49e-8,
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)
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return val
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return np.vectorize(_scalar)(x, mu, alpha, beta)
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def fit(self, data, *args, **kwds):
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if ("loc" in kwds and kwds["loc"] != 0.0) or ("floc" in kwds and kwds["floc"] != 0.0):
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raise ValueError("k_distribution uses a fixed loc=0; use mu to control the mean/scale.")
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if ("scale" in kwds and kwds["scale"] != 1.0) or ("fscale" in kwds and kwds["fscale"] != 1.0):
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raise ValueError("k_distribution uses a fixed scale=1; use mu to control the mean/scale.")
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kwds.pop("loc", None)
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kwds.pop("scale", None)
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return super().fit(data, *args, floc=0.0, fscale=1.0, **kwds)
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k_dist = k_gen(a=0.0, name="k_distribution", shapes="mu, alpha, beta")
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class lognakagami_gen(rv_continuous):
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"""Log-Nakagami continuous random variable.
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The probability density function is:
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f(y; m, Omega) = 2*m^m / (Gamma(m)*Omega^m)
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* exp(2*m*y) * exp(-(m/Omega)*exp(2*y))
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for y in (-inf, +inf), m >= 0.5, Omega > 0. Derived via the change of
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variable Y = ln X on a Nakagami(m, Omega) variate; the Jacobian e^y
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combines with x^(2m-1) to give the exp(2*m*y) term.
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The mean and variance are:
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E[Y] = 0.5 * (psi(m) - ln(m) + ln(Omega))
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Var[Y] = 0.25 * psi_1(m)
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where psi and psi_1 are the digamma and trigamma functions. The spread
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Omega shifts the mean by 0.5*ln(Omega) but leaves all higher moments
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unchanged; entropy is also Omega-independent.
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`lognakagami` takes ``m`` and ``Omega`` as shape parameters.
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"""
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def _shape_info(self):
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return [
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_ShapeInfo("m", False, (0, np.inf), (False, False)),
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_ShapeInfo("Omega", False, (0, np.inf), (False, False)),
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]
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def _argcheck(self, m, Omega):
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return (m >= 0.5) & (Omega > 0)
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def _pdf(self, y, m, Omega):
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return np.exp(self._logpdf(y, m, Omega))
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def _logpdf(self, y, m, Omega):
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# ln f = ln2 + m*ln(m) - ln Γ(m) - m*ln(Ω) + 2m*y - (m/Ω)*exp(2y)
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return (np.log(2.0)
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+ sc.xlogy(m, m)
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- sc.gammaln(m)
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- m * np.log(Omega)
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+ 2.0 * m * y
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- (m / Omega) * np.exp(2.0 * y))
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def _cdf(self, y, m, Omega):
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# P(Y <= y) = P(X <= e^y) = gammainc(m, (m/Omega) * e^{2y})
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return sc.gammainc(m, (m / Omega) * np.exp(2.0 * y))
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def _sf(self, y, m, Omega):
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return sc.gammaincc(m, (m / Omega) * np.exp(2.0 * y))
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def _logcdf(self, y, m, Omega):
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return np.log(sc.gammainc(m, (m / Omega) * np.exp(2.0 * y)))
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def _logsf(self, y, m, Omega):
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return np.log(sc.gammaincc(m, (m / Omega) * np.exp(2.0 * y)))
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def _ppf(self, q, m, Omega):
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# q = gammainc(m, (m/Omega)*e^{2y})
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# => (m/Omega)*e^{2y} = gammaincinv(m, q)
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# => y = 0.5 * ln(Omega * gammaincinv(m, q) / m)
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return 0.5 * np.log(Omega * sc.gammaincinv(m, q) / m)
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def _isf(self, q, m, Omega):
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return 0.5 * np.log(Omega * sc.gammainccinv(m, q) / m)
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def _stats(self, m, Omega):
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# E[Y] = 0.5*(psi(m) - ln(m) + ln(Omega))
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mu = 0.5 * (sc.digamma(m) - np.log(m) + np.log(Omega))
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# Var[Y] = 0.25 * psi_1(m) (Omega cancels — pure shift)
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mu2 = 0.25 * sc.polygamma(1, m)
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g1 = sc.polygamma(2, m) / sc.polygamma(1, m) ** 1.5
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g2 = sc.polygamma(3, m) / sc.polygamma(1, m) ** 2
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return mu, mu2, g1, g2
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def _entropy(self, m, Omega):
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# H = -E[ln f(Y)] = -ln2 + ln Γ(m) - m*psi(m) + m
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# Omega cancels exactly (translation invariance of differential entropy)
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return -np.log(2.0) + sc.gammaln(m) - m * sc.digamma(m) + m
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def _rvs(self, m, Omega, size=None, random_state=None):
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# X ~ Nakagami(m, Omega) => X^2 ~ Gamma(m, Omega/m)
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g = random_state.gamma(m, Omega / m, size=size)
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return 0.5 * np.log(g)
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lognakagami = lognakagami_gen(name='lognakagami', shapes="m, Omega")
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class loggamma_gen(rv_continuous):
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"""Log-Gamma continuous random variable.
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The probability density function is:
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f(y; a) = 1 / Gamma(a) * exp(a*y - exp(y))
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for y in (-inf, +inf), a > 0. Derived via the change of variable
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Y = ln X on a Gamma(a, 1) variate; the Jacobian e^y combines with
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x^(a-1) to give the exp(a*y) term.
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The mean and variance are:
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E[Y] = psi(a)
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Var[Y] = psi_1(a)
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where psi and psi_1 are the digamma and trigamma functions.
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The non-unit scale beta is recovered through scipy's ``loc`` parameter:
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loc = ln(beta), since ln(beta * X) = ln(beta) + ln(X).
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`loggamma_dist` takes ``a`` as a shape parameter.
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"""
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def _shape_info(self):
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return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
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def _argcheck(self, a):
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return a > 0
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def _pdf(self, y, a):
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return np.exp(self._logpdf(y, a))
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def _logpdf(self, y, a):
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# ln f = a*y - e^y - ln Γ(a)
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return a * y - np.exp(y) - sc.gammaln(a)
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def _cdf(self, y, a):
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# P(Y <= y) = P(X <= e^y) = gammainc(a, e^y)
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return sc.gammainc(a, np.exp(y))
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def _sf(self, y, a):
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return sc.gammaincc(a, np.exp(y))
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def _logcdf(self, y, a):
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return np.log(sc.gammainc(a, np.exp(y)))
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def _logsf(self, y, a):
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return np.log(sc.gammaincc(a, np.exp(y)))
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def _ppf(self, q, a):
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# Invert CDF: q = gammainc(a, e^y)
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# => e^y = gammaincinv(a, q)
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# => y = ln(gammaincinv(a, q))
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return np.log(sc.gammaincinv(a, q))
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def _isf(self, q, a):
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return np.log(sc.gammainccinv(a, q))
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def _stats(self, a):
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# Y = ln(X), X ~ Gamma(a, 1)
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# E[Y] = psi(a) (digamma)
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# Var[Y] = psi_1(a) (trigamma)
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# skew = psi_2(a) / psi_1(a)^{3/2}
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# excess kurtosis = psi_3(a) / psi_1(a)^2
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mu = sc.digamma(a)
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mu2 = sc.polygamma(1, a)
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g1 = sc.polygamma(2, a) / mu2 ** 1.5
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g2 = sc.polygamma(3, a) / mu2 ** 2
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return mu, mu2, g1, g2
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def _entropy(self, a):
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# H(Y) = -E[ln f(Y)] = -E[a*Y - e^Y - ln Γ(a)]
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# = ln Γ(a) - a*psi(a) + E[e^Y]
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# E[e^Y] = E[X] = a (for Gamma(a, 1))
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return sc.gammaln(a) - a * sc.digamma(a) + a
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def _rvs(self, a, size=None, random_state=None):
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# Generate Gamma(a, 1) variates, then take log.
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# For small a, use Gamma(a) ~ Gamma(a+1) * U^(1/a)
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# so ln Gamma(a) ~ ln Gamma(a+1) + ln(U)/a
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# This avoids precision loss when a << 1.
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return (np.log(random_state.gamma(a + 1, size=size))
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+ np.log(random_state.uniform(size=size)) / a)
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loggamma_dist = loggamma_gen(name='loggamma_dist')
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class lograyleigh_gen(rv_continuous):
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"""Log-Rayleigh continuous random variable.
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The probability density function is:
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f(y; sigma) = (e^{2y} / sigma^2) * exp(-e^{2y} / (2*sigma^2))
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for y in (-inf, +inf), sigma > 0. Derived via the change of variable
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Y = ln X on a Rayleigh(sigma) variate; the Jacobian e^y combines with
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x to give the e^{2y} term.
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The mean and variance are:
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E[Y] = 0.5 * (ln(2*sigma^2) + psi(1))
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Var[Y] = pi^2 / 24
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where psi is the digamma function. The scale sigma shifts the mean by
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ln(sigma) but leaves all higher central moments unchanged; entropy is
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also sigma-independent.
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`lograyleigh` takes ``sigma`` as a shape parameter.
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"""
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def _shape_info(self):
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return [_ShapeInfo("sigma", False, (0, np.inf), (False, False))]
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def _argcheck(self, sigma):
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return sigma > 0
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def _pdf(self, y, sigma):
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return np.exp(self._logpdf(y, sigma))
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def _logpdf(self, y, sigma):
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# ln f = 2y - 2*ln(sigma) - e^{2y} / (2*sigma^2)
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return 2.0 * y - 2.0 * np.log(sigma) - np.exp(2.0 * y) / (2.0 * sigma**2)
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def _cdf(self, y, sigma):
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return 1.0 - np.exp(-np.exp(2.0 * y) / (2.0 * sigma**2))
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def _sf(self, y, sigma):
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return np.exp(-np.exp(2.0 * y) / (2.0 * sigma**2))
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def _logcdf(self, y, sigma):
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return np.log1p(-np.exp(-np.exp(2.0 * y) / (2.0 * sigma**2)))
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def _logsf(self, y, sigma):
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return -np.exp(2.0 * y) / (2.0 * sigma**2)
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def _ppf(self, q, sigma):
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# q = 1 - exp(-e^{2y}/(2*sigma^2))
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# => e^{2y} = -2*sigma^2 * ln(1-q)
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# => y = 0.5 * ln(-2*sigma^2 * ln(1-q))
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return 0.5 * np.log(-2.0 * sigma**2 * np.log1p(-q))
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def _isf(self, q, sigma):
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# q = exp(-e^{2y}/(2*sigma^2))
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# => y = 0.5 * ln(-2*sigma^2 * ln(q))
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return 0.5 * np.log(-2.0 * sigma**2 * np.log(q))
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def _stats(self, sigma):
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# Y = 0.5*ln(2) + 0.5*ln(W) + ln(sigma), W ~ Exp(1) = Gamma(1, 1)
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# E[Y] = 0.5*(ln(2*sigma^2) + psi(1))
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# Var[Y] = 0.25*psi_1(1) = pi^2/24
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# Standardised skewness/kurtosis are sigma-independent (pure shift)
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mu = 0.5 * (np.log(2.0 * sigma**2) + sc.digamma(1))
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mu2 = 0.25 * sc.polygamma(1, 1)
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g1 = sc.polygamma(2, 1) / sc.polygamma(1, 1)**1.5
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g2 = sc.polygamma(3, 1) / sc.polygamma(1, 1)**2
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return mu, mu2, g1, g2
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def _entropy(self, sigma):
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# H(Y) = 1 - ln(2) - psi(1) (sigma-independent; derived via H(X) - E[ln X])
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return 1.0 - np.log(2.0) - sc.digamma(1)
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def _rvs(self, sigma, size=None, random_state=None):
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# X ~ Rayleigh(sigma) = sigma*sqrt(-2*ln(U)), U ~ Uniform(0,1)
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u = random_state.uniform(size=size)
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return np.log(sigma) + 0.5 * np.log(-2.0 * np.log(u))
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lograyleigh = lograyleigh_gen(name='lograyleigh', shapes="sigma")
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class logrice_gen(rv_continuous):
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"""Log-Rice continuous random variable.
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The probability density function is:
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f(y; nu, sigma) = (e^{2y} / sigma^2)
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* exp(-(e^{2y} + nu^2) / (2*sigma^2))
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* I_0(e^y * nu / sigma^2)
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for y in (-inf, +inf), nu >= 0, sigma > 0. Derived via the change of
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variable Y = ln X on a Rice(nu, sigma) variate; the Jacobian e^y
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combines with x to give the e^{2y} factor. I_0 is the modified Bessel
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function of the first kind of order zero.
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The CDF involves the Marcum Q-function of order 1:
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F(y; nu, sigma) = 1 - Q_1(nu/sigma, e^y/sigma)
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When nu = 0, the distribution reduces to Log-Rayleigh.
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`logrice` takes ``nu`` and ``sigma`` as shape parameters.
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"""
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def _shape_info(self):
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return [
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_ShapeInfo("nu", False, (0, np.inf), (True, False)),
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_ShapeInfo("sigma", False, (0, np.inf), (False, False)),
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]
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def _argcheck(self, nu, sigma):
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return (nu >= 0) & (sigma > 0)
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def _pdf(self, y, nu, sigma):
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return np.exp(self._logpdf(y, nu, sigma))
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def _logpdf(self, y, nu, sigma):
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# ln f = 2y - 2*ln(sigma) - (e^{2y} + nu^2)/(2*sigma^2) + ln(I_0(z))
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# Use ive(0, z)*exp(z) = I_0(z) for stability: ln(I_0(z)) = ln(ive(0,z)) + z
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z = np.exp(y) * nu / sigma**2
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return (2.0 * y
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- 2.0 * np.log(sigma)
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- (np.exp(2.0 * y) + nu**2) / (2.0 * sigma**2)
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+ np.log(sc.ive(0, z)) + z)
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def _cdf(self, y, nu, sigma):
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# P(Y <= y) = 1 - Q_1(nu/sigma, e^y/sigma) = ncx2.cdf(e^{2y}/sigma^2, 2, nu^2/sigma^2)
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return ncx2.cdf(np.exp(2.0 * y) / sigma**2, 2, (nu / sigma)**2)
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def _sf(self, y, nu, sigma):
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return ncx2.sf(np.exp(2.0 * y) / sigma**2, 2, (nu / sigma)**2)
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def _logcdf(self, y, nu, sigma):
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return ncx2.logcdf(np.exp(2.0 * y) / sigma**2, 2, (nu / sigma)**2)
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def _logsf(self, y, nu, sigma):
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return ncx2.logsf(np.exp(2.0 * y) / sigma**2, 2, (nu / sigma)**2)
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def _rvs(self, nu, sigma, size=None, random_state=None):
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# X ~ Rice(nu, sigma): norm of 2D Gaussian with mean (nu, 0) and std sigma
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z1 = random_state.normal(nu, sigma, size=size)
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z2 = random_state.normal(0.0, sigma, size=size)
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return np.log(np.hypot(z1, z2))
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logrice = logrice_gen(name='logrice', shapes="nu, sigma") |