Advanced Probability Distributions for AI and Statistics
Project description
Advanced Distributions
Advanced probability distributions for:
- AI Research
- Machine Learning
- Heavy-tail Modeling
- Diffusion Models
- Statistical Research
- Copula Theory
- Quantum Probability
- Information Theory
Install
pip install advanced_distributions
# =============================================================
# EXAMPLE USAGE: slash distribution
# =============================================================
if __name__ == "__main__":
dist = SlashDistribution(
mu=0,
sigma=1
)
print("Mean:", dist.mean())
print(
"Variance:",
dist.var()
)
print(
"Standard Deviation:",
dist.std()
)
print(
"Skewness:",
dist.skewness()
)
print(
"Kurtosis:",
dist.kurtosis()
)
print(
"Entropy:",
dist.entropy()
)
print(
"Median:",
dist.median()
)
print(
"Mode:",
dist.mode()
)
print(
"Moment(2):",
dist.moment(2)
)
print(
"\nPDF at x=1:",
dist.pdf(1)
)
print(
"CDF at x=1:",
dist.cdf(
np.array([1])
)[0]
)
print(
"Hazard at x=1:",
dist.hazard(1)
)
print(
"MGF at t=0.01:",
dist.mgf(0.01)
)
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate samples
samples = dist.rvs(
size=5000
)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = (
SlashDistribution.fit(
samples
)
)
print("\nFitted Parameters:")
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: Fractal distribution
# =============================================================
if __name__ == "__main__":
dist = FractalDistribution(
D=1.3,
lambda_=0.8
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("\n3rd Moment:", dist.moment(3))
print("\nPDF at x=2:", dist.pdf(2))
print("CDF at x=2:", dist.cdf(np.array([2]))[0])
print("Hazard at x=2:", dist.hazard(np.array([2]))[0])
# Generate Samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted_params = FractalDistribution.fit(samples)
print("\nFitted Parameters:")
print("D =", fitted_params[0])
print("lambda =", fitted_params[1])
# Plotting
dist.plot_pdf()
dist.plot_cdf()
# =============================================================
# EXAMPLE USAGE: Fractional distribution
# =============================================================
if __name__ == "__main__":
dist = FractionalDistribution(
alpha=1.7,
beta=2.0,
gamma=0.9
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("3rd Moment:", dist.moment(3))
print("\nPDF at x=2:", dist.pdf(2))
print("CDF at x=2:", dist.cdf(np.array([2]))[0])
print("Hazard at x=2:", dist.hazard(np.array([2]))[0])
print("MGF at t=0.1:", dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted_params = FractionalDistribution.fit(samples)
print("\nFitted Parameters:")
print("alpha =", fitted_params[0])
print("beta =", fitted_params[1])
print("gamma =", fitted_params[2])
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: GeneralizedParetoDistribution
# =============================================================
if __name__ == "__main__":
dist = GeneralizedParetoDistribution(
xi=0.3,
sigma=2.0,
mu=0.0
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("3rd Moment:", dist.moment(3))
print("\nPDF at x=2:", dist.pdf(2))
print("CDF at x=2:", dist.cdf(2))
print("Hazard at x=2:", dist.hazard(2))
print("MGF at t=0.1:", dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = GeneralizedParetoDistribution.fit(samples)
print("\nFitted Parameters:")
print("xi =", fitted.xi)
print("sigma =", fitted.sigma)
print("mu =", fitted.mu)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: HyperbolicSecantDistribution
# =============================================================
if __name__ == "__main__":
dist = HyperbolicSecantDistribution(
mu=0,
sigma=1
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("3rd Moment:", dist.moment(3))
print("\nPDF at x=1:", dist.pdf(1))
print("CDF at x=1:", dist.cdf(1))
print("Hazard at x=1:", dist.hazard(1))
print("MGF at t=0.2:", dist.mgf(0.2))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = HyperbolicSecantDistribution.fit(samples)
print("\nFitted Parameters:")
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# # =============================================================
# EXAMPLE USAGE: LevyStableDistribution
# =============================================================
if __name__ == "__main__":
dist = LevyStableDistribution(
alpha=1.7,
beta=0.2,
loc=0,
scale=1
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("Moment(0.5):", dist.moment(0.5))
print("\nPDF at x=1:", dist.pdf(1))
print("CDF at x=1:", dist.cdf(1))
print("Hazard at x=1:", dist.hazard(1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = LevyStableDistribution.fit(samples)
print("\nFitted Parameters:")
print("alpha =", fitted.alpha)
print("beta =", fitted.beta)
print("loc =", fitted.loc)
print("scale =", fitted.scale)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE:NeuralSplineDistribution
# =============================================================
if __name__ == "__main__":
dist = NeuralSplineDistribution()
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("Moment(3):", dist.moment(3))
print("\nPDF at x=1:", dist.pdf(1))
print("CDF at x=1:", dist.cdf(np.array([1]))[0])
print("Hazard at x=1:",
dist.hazard(np.array([1]))[0])
print("MGF at t=0.1:",
dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = NeuralSplineDistribution.fit(samples)
print("\nFitted Knot Positions:")
print("knots_x =", fitted.knots_x)
print("knots_y =", fitted.knots_y)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: QuantileDistribution
# =============================================================
if __name__ == "__main__":
dist = QuantileDistribution(
mu=0,
sigma=1,
alpha=1.5,
beta=2.0
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("Moment(3):", dist.moment(3))
print("\nPDF at x=1:",
dist.pdf(np.array([1]))[0])
print("CDF at x=1:",
dist.cdf(np.array([1]))[0])
print("Hazard at x=1:",
dist.hazard(np.array([1]))[0])
print("MGF at t=0.1:",
dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = QuantileDistribution.fit(samples)
print("\nFitted Parameters:")
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
print("alpha =", fitted.alpha)
print("beta =", fitted.beta)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: QuantumProbabilityDistribution
# =============================================================
if __name__ == "__main__":
dist = QuantumProbabilityDistribution(
sigma=1.0,
k=2.0,
mu=0.0
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("Moment(4):", dist.moment(4))
print(
"Energy Expectation:",
dist.energy_expectation()
)
print("\nPDF at x=1:",
dist.pdf(1))
print("CDF at x=1:",
dist.cdf(np.array([1]))[0])
print("Hazard at x=1:",
dist.hazard(np.array([1]))[0])
print("MGF at t=0.1:",
dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = QuantumProbabilityDistribution.fit(
samples
)
print("\nFitted Parameters:")
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
print("k =", fitted.k)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_wave_function()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: csinhDistribution
# =============================================================
if __name__ == "__main__":
dist = SinhArcsinhDistribution(
epsilon=1.0,
delta=0.8,
mu=0,
sigma=1
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print("Standard Deviation:", dist.std())
print("Skewness:", dist.skewness())
print("Kurtosis:", dist.kurtosis())
print("Entropy:", dist.entropy())
print("Median:", dist.median())
print("Mode:", dist.mode())
print("Moment(4):", dist.moment(4))
print("\nPDF at x=1:",
dist.pdf(1))
print("CDF at x=1:",
dist.cdf(1))
print("Hazard at x=1:",
dist.hazard(1))
print("MGF at t=0.1:",
dist.mgf(0.1))
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(size=5000)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = SinhArcsinhDistribution.fit(
samples
)
print("\nFitted Parameters:")
print("epsilon =", fitted.epsilon)
print("delta =", fitted.delta)
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
# =============================================================
# EXAMPLE USAGE: SkewedStudentTDistribution
# =============================================================
if __name__ == "__main__":
dist = SkewedStudentTDistribution(
nu=8,
lam=0.3,
mu=0,
sigma=1
)
print("Mean:", dist.mean())
print("Variance:", dist.var())
print(
"Standard Deviation:",
dist.std()
)
print(
"Skewness:",
dist.skewness()
)
print(
"Kurtosis:",
dist.kurtosis()
)
print(
"Entropy:",
dist.entropy()
)
print(
"Median:",
dist.median()
)
print(
"Mode:",
dist.mode()
)
print(
"Moment(4):",
dist.moment(4)
)
print(
"\nPDF at x=1:",
dist.pdf(1)
)
print(
"CDF at x=1:",
dist.cdf(
np.array([1])
)[0]
)
print(
"Hazard at x=1:",
dist.hazard(1)
)
print(
"MGF at t=0.1:",
dist.mgf(0.1)
)
print(
"Characteristic Function at t=1:",
dist.characteristic_function(1)
)
# Generate random samples
samples = dist.rvs(
size=5000
)
print("\nRandom Samples:")
print(samples[:10])
# Fit distribution
fitted = (
SkewedStudentTDistribution.fit(
samples
)
)
print("\nFitted Parameters:")
print("nu =", fitted.nu)
print("lam =", fitted.lam)
print("mu =", fitted.mu)
print("sigma =", fitted.sigma)
# Plotting
dist.plot_pdf()
dist.plot_cdf()
dist.plot_histogram()
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