R: Parameter Estimation for Stable Distributions and Their Mixtures

# Parameter Estimation for Stable Distributions and Their Mixtures

## Documentation for package ‘MixStable’ version 0.1.0

- DESCRIPTION file

## Help Pages

A B C D E F G I K L M N P Q R S T U V W

## -- A --

aic | Akaike Information Criterion (AIC)
analyse_stable_distribution | Perform full stability analysis and export results

## -- B --

bayesian_mixture_model | Bayesian mixture model using normal components (simplified)
bic | Bayesian Information Criterion (BIC)
build_mcculloch_interpolators | Build interpolation functions from McCulloch table

## -- C --

calculate_log_likelihood | Calculate simplified log-likelihood
CDF | Estimate stable distribution parameters using classical ECF regression
clip | Clip values between lower and upper bounds
compare_em_vs_em_gibbs | Compare standard EM and EM with Gibbs sampling using kernel ECF
compare_estimators_on_simulations | Compare MLE, ECF, and McCulloch estimators on simulated data
compare_methods_across_configs | Compare McCulloch, ECF, and MLE methods across parameter configurations
compare_methods_with_gibbs | Compare estimation methods with and without Gibbs sampling
compute_model_metrics | Compute log-likelihood, AIC, and BIC for alpha-stable model
compute_quantile_ratios | Compute McCulloch quantile ratios from sample data
compute_serial_interval | Compute serial interval from CSV file
cosine_exp_ralpha | Cosine exponential function
cosine_log_weighted_exp_ralpha | Cosine-log-weighted exponential with r^(-alpha) term

## -- D --

DONNEE_with_serial_interval | Example serial interval data

## -- E --

ecf_components | Extract magnitude and phase components from ECF
ecf_empirical | Compute empirical characteristic function
ecf_estimate_all | Estimate all stable parameters from empirical characteristic function
ecf_fn | Empirical Characteristic Function
ecf_regression | Estimate stable parameters using weighted ECF regression
empirical_r0 | Empirical R0 estimation using growth model
em_alpha_stable | EM algorithm for alpha-stable mixture
em_estimate_stable_from_cdf | EM algorithm for mixture of alpha-stable distributions using CDF-based ECF
em_estimate_stable_from_cdf_with_gibbs | EM algorithm for alpha-stable mixture using CDF-based ECF and Gibbs M-step
em_estimate_stable_kernel_ecf | EM algorithm for mixture of alpha-stable distributions using kernel ECF
em_estimate_stable_kernel_ecf_with_gibbs | EM algorithm for alpha-stable mixture using kernel ECF and Gibbs M-step
em_estimate_stable_recursive_ecf | EM algorithm for mixture of alpha-stable distributions using recursive ECF
em_estimate_stable_recursive_ecf_with_gibbs | EM algorithm for alpha-stable mixture using recursive ECF and Gibbs M-step
em_estimate_stable_weighted_ols | EM algorithm for mixture of alpha-stable distributions using weighted OLS
em_estimate_stable_weighted_ols_with_gibbs | EM algorithm for alpha-stable mixture using weighted OLS and Gibbs M-step
em_estimation_mixture | EM algorithm for two-component Gaussian mixture
em_fit_alpha_stable_mixture | EM algorithm for two-component alpha-stable mixture using MLE
em_stable_mixture | EM algorithm for alpha-stable mixture using a custom estimator
ensure_positive_scale | Ensure positive scale parameter
estimate_alpha_gamma | Estimate alpha and gamma from ECF modulus
estimate_beta_delta | Estimate beta and delta from ECF phase
estimate_mixture_params | Estimate mixture of two stable distributions
estimate_stable_from_cdf | Estimate stable parameters using CDF-based ECF regression
estimate_stable_kernel_ecf | Estimate stable parameters using kernel-based ECF method
estimate_stable_params | Estimate single stable distribution parameters
estimate_stable_r | Estimate stable parameters using method of moments
estimate_stable_recursive_ecf | Estimate stable parameters using recursive ECF method
estimate_stable_weighted_ols | Estimate stable parameters using weighted OLS on recursive ECF
est_r0_ml | Estimate R0 using maximum likelihood
est_r0_mle | MLE estimation of R0 using generation time
eta0 | Helper function for eta0 computation
eta_func | General eta function
evaluate_estimation_method | Evaluate estimation method using MSE over multiple trials
evaluate_fit | Evaluate fit quality using RMSE and log-likelihood
export_analysis_report | Export analysis report to JSON and Excel

## -- F --

false_position_update | False position method update step
fast_integrate | Fast numerical integration using trapezoidal rule
fit_alpha_stable_mle | Fit Alpha-Stable Distribution using MLE (L-BFGS-B)
fit_mle_mixture | Fit MLE Mixture of Two Stable Distributions
fit_stable_ecf | Estimate stable parameters using filtered and weighted ECF regression

## -- G --

generate_alpha_stable_mixture | Generate samples from a predefined alpha-stable mixture
generate_mcculloch_table | Generate McCulloch lookup table from simulated stable samples
generate_mixture_data | Simulates a mixture of alpha-stable distributions with randomly sampled parameters.
generate_synthetic_data | Generate synthetic data from two alpha-stable components
gibbs_sampler | Gibbs sampler for Gaussian mixture model
grad_loglik_alpha | Log-likelihood gradient with respect to alpha
grad_loglik_beta | Log-likelihood gradient with respect to beta
grad_loglik_delta | Log-likelihood gradient with respect to delta (scale)
grad_loglik_omega | Log-likelihood gradient with respect to omega (location)

## -- I --

Im | Imaginary part of the ECF integral
integrate_cosine | Integrate cosine exponential
integrate_cosine_log_weighted | Integrate cosine-log-weighted exponential
integrate_function | Robust integration helper function
integrate_sine | Integrate sin exponential
integrate_sine_log_weighted | Integrate sine-log-weighted exponential
integrate_sine_r_weighted | Integrate sine-r-weighted exponential
integrate_sine_weighted | Integration wrappers for specific integrands
Int_Im | Integrate imaginary component over \mathbb{R}
Int_Re | Integrate real component over \mathbb{R}

## -- K --

kde_bandwidth_plugin | KDE bandwidth selection using plugin method

## -- L --

log_likelihood_mixture | Log-likelihood for mixture of stable distributions
L_stable | Negative log-likelihood for stable distribution using dstable

## -- M --

Max_vrai | Maximum likelihood estimation using Nelder-Mead
mcculloch_lookup_estimate | Estimate stable parameters using McCulloch lookup
mcculloch_quantile_init | Initialization using McCulloch quantile method
metropolis_hastings | Metropolis-Hastings MCMC for stable mixture clustering
mixture_stable_pdf | Mixture of two stable PDFs
mle_estimate | Simple MLE estimation with default starting values
mock_gibbs_sampling | Mock Gibbs sampling for alpha-stable mixture estimation
mock_lookup_alpha_beta | Mock lookup for alpha and beta (fallback)

## -- N --

negative_log_likelihood | Negative log-likelihood for single stable distribution
normalized_grad_alpha | Normalized gradient for alpha parameter Computes the normalized gradient of the log-likelihood with respect to the alpha parameter over a set of observations. This is useful for optimization routines where scale-invariant updates are preferred.
normalized_objective_beta | Normalized objective for beta parameter
normalized_objective_delta | Normalized objective for delta parameter
normalized_objective_omega | Normalized objective for omega parameter
N_epanechnikov | Epanechnikov kernel
N_gaussian | Gaussian kernel
N_uniform | Uniform kernel

## -- P --

plot_comparison | Compare EM-estimated mixture with a non-optimized reference model
plot_distributions | Plot histogram with normal and stable PDF overlays
plot_effective_reproduction_number | Plot effective reproduction number (Re) over time
plot_final_mixture_fit | Plot final fitted mixture of alpha-stable distributions
plot_fit_vs_true | Plot true vs estimated mixture density
plot_fit_vs_true_methods | Compare estimated mixture densities from two methods against the true density
plot_method_comparison | Plot RMSE and Log-Likelihood comparison across methods
plot_mixture | Plot mixture of two alpha-stable distributions
plot_mixture_fit | Plot mixture fit with individual components
plot_real_mixture_fit | Plot fitted mixture on real dataset
plot_results | Plot posterior mixture density from MCMC samples
plot_trace | Plot trace of a parameter across MCMC iterations
plot_vs_normal_stable | Plot comparison between normal and stable distributions

## -- Q --

qcv_stat | QCV statistic for tail heaviness

## -- R --

Re | Real part of the ECF integral
recursive_weight | Recursive weight function
robust_ecf_regression | Estimate stable parameters using robust ECF regression
robust_mle_estimate | Robust MLE estimation with multiple starting points
rstable | Generate random samples from stable distribution
RT | Compute effective reproduction number Rt
run_all_estimations | Run all EM-based estimations without Gibbs sampling (CRAN-safe)
run_estimations_with_gibbs | Run all EM-based estimations with Gibbs sampling (CRAN-safe)
r_stable_pdf | Robust stable PDF computation

## -- S --

safe_integrate | Safe integration wrapper with multiple fallback strategies
simple_em_real | Simple 2-component EM using ECF initialization
simulate_mixture | Simulate mixture data from alpha-stable components
sine_exp_ralpha | Sine exponential function
sine_log_weighted_exp_ralpha | Sine-log-weighted exponential with r^(-alpha) term
sine_r_weighted_exp_ralpha | Sine-r-weighted exponential function
sine_weighted_exp_ralpha | Sine-weighted exponential with r^alpha term
skew_kurtosis | Calculate skewness and kurtosis
stable_fit_init | Initialize stable distribution parameters

## -- T --

TableS2_serial_interval_mean_ | Example transmission pair data with mean serial interval
test_normality | Test normality using multiple statistical tests

## -- U --

unpack_params | Helper function to unpack parameters

## -- V --

validate_params | Validate and clip parameters for stable distribution

## -- W --

wasserstein_distance_mixture | Wasserstein distance between two mixture distributions