Module uncertainty_quantification 

Uncertainty Quantification (UQ) methods for physics simulation.

Provides:

• Monte Carlo uncertainty propagation
• Latin hypercube sampling (LHS)
• Sensitivity analysis (Sobol indices, Morris method)
• Polynomial chaos expansion (PCE)
• Stochastic collocation
• Gaussian process regression for UQ
• Reliability analysis (FORM / SORM)
• Bootstrap confidence intervals
• Bayesian inference via Metropolis-Hastings MCMC
• Probability of failure estimation

Structs

CollocationNode

A collocation node (quadrature point + weight) for one dimension.

FormResult

Result of a FORM reliability analysis.

GaussianProcessSurrogate

A fitted Gaussian Process regression model.

LatinHypercubeSample

A Latin Hypercube Sample: rows = samples, columns = dimensions.

McUqResult

Results from a Monte Carlo uncertainty propagation run.

MhConfig

Configuration for the Metropolis-Hastings sampler.

MhResult

Result of Metropolis-Hastings sampling.

MorrisResult

Result of a Morris (elementary effects) sensitivity screening.

PolynomialChaosExpansion

Polynomial Chaos Expansion surrogate model.

SobolIndices

First-order and total-order Sobol sensitivity indices.

UncertainParameter

Description of a single uncertain input parameter.

Functions

bootstrap_ci

Bootstrap percentile confidence interval for a statistic.

coefficient_of_variation

Coefficient of variation CoV = σ / |μ|.

delta_sensitivity

Estimate the moment-independent delta sensitivity index for dimension dim.

erf

Error function (Horner’s method approximation).

excess_kurtosis

Excess kurtosis = E[(X−μ)⁴] / σ⁴ − 3.

fit_gp

Fit a Gaussian Process surrogate using exact GP regression.

fit_pce

Fit a PCE by non-intrusive (regression) approach using LHS-sampled training data.

form_analysis

First-Order Reliability Method (FORM) via Hasofer-Lind iteration.

gauss_hermite_nodes

Gauss-Hermite quadrature nodes and weights for n-point rule.

hermite_poly

Computes the n-th Hermite polynomial He_n(x) (probabilists’ convention).

importance_sampling_pf

Importance sampling estimate of P_f with a shifted Gaussian importance density.

latin_hypercube_sample

Generate a Latin Hypercube Sample with n_samples points in n_dims dimensions.

mean

Compute the mean of a slice of values.

metropolis_hastings

Metropolis-Hastings MCMC sampler.

monte_carlo_propagation

Run Monte Carlo uncertainty propagation.

morris_sensitivity

Run the Morris method (elementary effects) for n_dims inputs.

normal_cdf

Standard normal CDF (error function approximation).

pce_multi_indices

Generate all multi-indices of total degree ≤ max_degree for n_dims dimensions.

percentile

Sort a slice in-place and return the p-th percentile (linear interpolation).

probability_box

Compute a P-box (probability box) from a set of CDFs.

probability_of_failure_mc

Estimate the probability of failure P_f = P(g(X) ≤ 0) via crude Monte Carlo.

probit

Inverse CDF of the standard normal (rational approximation).

rbf_kernel

Squared-exponential (RBF) kernel: k(x, x’) = σ² exp(-‖x−x’‖² / (2ℓ²)).

reliability_index

Compute the reliability index β from a probability of failure estimate.

scale_lhs

Scale LHS samples from [0,1] to [low, high] for each dimension.

skewness

Skewness of a dataset = E[(X−μ)³] / σ³.

sobol_indices

Estimate Sobol sensitivity indices using the Saltelli (2010) estimator.

sorm_breitung

Second-Order Reliability Method (SORM) correction factor.

std_dev

Compute the standard deviation (population) of a slice.

stochastic_collocation

Compute mean and variance of a model via tensor-product stochastic collocation.

variance

Compute the variance (population) of a slice.