Estimate stable distribution parameters using classical ECF regression

Description

Estimate stable distribution parameters using classical ECF regression

Usage

CDF(x, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Example serial interval data

Description

A data frame containing serial interval information for transmission pairs.

Usage

data(DONNEE_with_serial_interval)

Format

A data frame with columns:

ID

Integer. Unique identifier.

x.lb

Date. Lower bound of infector symptom onset (YYYY-MM-DD).

x.ub

Date. Upper bound of infector symptom onset (YYYY-MM-DD).

y

Date. Infectee symptom onset (YYYY-MM-DD).

serial_interval

Integer. Serial interval in days.

Source

Simulated or real data for demonstration.

Imaginary part of the ECF integral

Description

Imaginary part of the ECF integral

Usage

Im(r, u, x, bn)

Arguments

Value

Imaginary component value.

Integrate imaginary component over \mathbb{R}

Description

Integrate imaginary component over \mathbb{R}

Usage

Int_Im(u, x, bn)

Arguments

Value

Integrated imaginary value.

Integrate real component over \mathbb{R}

Description

Integrate real component over \mathbb{R}

Usage

Int_Re(u, x, bn)

Arguments

Value

Integrated real value.

Negative log-likelihood for stable distribution using dstable

Description

Computes the negative log-likelihood of a stable distribution given parameters and data.

Usage

L_stable(param, obs)

Arguments

Value

Scalar value of negative log-likelihood.

Maximum likelihood estimation using Nelder-Mead

Description

Estimates parameters of a stable distribution using the Nelder-Mead method with penalty constraints.

Usage

Max_vrai(x)

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Epanechnikov kernel

Description

Epanechnikov kernel

Usage

N_epanechnikov(z)

Arguments

Value

Kernel value.

Gaussian kernel

Description

Gaussian kernel

Usage

N_gaussian(z)

Arguments

Value

Kernel value.

Uniform kernel

Description

Uniform kernel

Usage

N_uniform(z)

Arguments

Value

Kernel value.

Compute effective reproduction number Rt

Description

Estimates the effective reproduction number (Rt) over time using incidence data and a generation time distribution.

Usage

RT(inc, G)

Arguments

Value

Numeric vector of Rt values.

Real part of the ECF integral

Description

Real part of the ECF integral

Usage

Re(r, u, x, bn)

Arguments

Value

Real component value.

Example transmission pair data with mean serial interval

Description

A data frame of infector-infectee pairs with demographic and serial interval information.

Usage

data(TableS2_serial_interval_mean_)

Format

A data frame with columns:

infector_id

Character. Infector ID.

infector_age

Numeric. Age of infector.

infector_sex

Character. Sex of infector.

infector_onsetDate

Date. Infector symptom onset.

infector_labConfirmDate

Date. Infector lab confirmation.

infectee_id

Character. Infectee ID.

infectee_age

Numeric. Age of infectee.

infectee_sex

Character. Sex of infectee.

infectee_onsetDate

Date. Infectee symptom onset.

infectee_labConfirmDate

Date. Infectee lab confirmation.

serial_interval_mean_based

Numeric. Mean-based serial interval.

Source

Simulated or real data for demonstration.

Akaike Information Criterion (AIC)

Description

Computes the AIC value given a log-likelihood and number of parameters.

Usage

aic(log_likelihood, num_params)

Arguments

Value

AIC value.

Perform full stability analysis and export results

Description

Executes the full pipeline for stability analysis: normality tests, QCV computation, parameter estimation, plotting, and report export.

Usage

analyse_stable_distribution(
  x,
  filename = "interval_analysis",
  qcv_threshold = 1.8,
  fig_path = NULL,
  verbose = FALSE
)

Arguments

Value

Invisibly returns a character string containing a Markdown-formatted summary report.

Bayesian mixture model using normal components (simplified)

Description

Bayesian mixture model using normal components (simplified)

Usage

bayesian_mixture_model(data, draws = 1000, chains = 2)

Arguments

Value

List containing model fit, chains, weights, means, and standard deviations.

Bayesian Information Criterion (BIC)

Description

Computes the BIC value given a log-likelihood, number of parameters, and sample size.

Usage

bic(log_likelihood, num_params, n)

Arguments

Value

BIC value.

Build interpolation functions from McCulloch table

Description

Constructs nearest-neighbor interpolation functions for alpha and beta based on quantile ratios (v_alpha, v_beta) from the lookup table.

Usage

build_mcculloch_interpolators(table)

Arguments

Value

List with functions interp_alpha and interp_beta.

Calculate simplified log-likelihood

Description

Computes a placeholder log-likelihood value based on squared deviations from the mean. This is not a true likelihood function and is used for testing or comparison purposes.

Usage

calculate_log_likelihood(data, params)

Arguments

Value

Scalar value representing the pseudo log-likelihood.

Clip values between lower and upper bounds

Description

Clip values between lower and upper bounds

Usage

clip(x, lower, upper)

Arguments

Value

Clipped numeric vector.

Compare standard EM and EM with Gibbs sampling using kernel ECF

Description

Compares log-likelihood and weight estimates between standard EM and EM with Gibbs sampling.

Usage

compare_em_vs_em_gibbs(data, n_runs = 20)

Arguments

Value

Data frame with log-likelihoods and weights for each method across runs.

Compare MLE, ECF, and McCulloch estimators on simulated data

Description

Runs multiple simulations and compares the mean squared error (MSE) of parameter estimates from MLE, kernel ECF, and McCulloch methods.

Usage

compare_estimators_on_simulations(
  n_samples = 1000,
  n_runs = 30,
  interp_alpha = NULL,
  interp_beta = NULL
)

Arguments

Value

Data frame of simulation results with MSE and estimated parameters.

Compare McCulloch, ECF, and MLE methods across parameter configurations

Description

Evaluates and compares three estimation methods (McCulloch, ECF, MLE) over a set of parameter configurations.

Usage

compare_methods_across_configs(parameter_configs, trials = 20, n = 1000)

Arguments

Value

A named list of results with MSE values for each method and configuration.

Compare estimation methods with and without Gibbs sampling

Description

Visualizes and compares mixture fits from EM, ECF (kernel and empirical), and Gibbs sampling.

Usage

compare_methods_with_gibbs(
  data,
  em_params,
  ecf_kernel_params,
  ecf_empirical_params,
  fig_dir = tempdir()
)

Arguments

Value

NULL (plots are saved to fig_dir)

Compute log-likelihood, AIC, and BIC for alpha-stable model

Description

Evaluates the model fit by computing the log-likelihood, AIC, and BIC for a given set of alpha-stable parameters.

Usage

compute_model_metrics(data, params)

Arguments

Value

List with log_likelihood, AIC, BIC, and optional error message.

Compute McCulloch quantile ratios from sample data

Description

This function calculates four key ratios used in McCulloch's method for estimating stable distribution parameters: - v_alpha: shape indicator - v_beta: skewness indicator - v_gamma: scale proxy - v_delta: location proxy

Usage

compute_quantile_ratios(X)

Arguments

Value

A list containing v_alpha, v_beta, v_gamma, and v_delta.

Compute serial interval from CSV file

Description

Calculates the serial interval from a CSV file containing date columns.

Usage

compute_serial_interval(filepath)

Arguments

Value

Numeric vector of serial intervals.

Cosine exponential function

Description

Cosine exponential function

Usage

cosine_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Cosine-log-weighted exponential with r^(-alpha) term

Description

Cosine-log-weighted exponential with r^(-alpha) term

Usage

cosine_log_weighted_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Extract magnitude and phase components from ECF

Description

Extract magnitude and phase components from ECF

Usage

ecf_components(phi_vals)

Arguments

Value

A list with two numeric vectors: y_vals (log-log magnitude) and arg_vals (phase).

Compute empirical characteristic function

Description

Compute empirical characteristic function

Usage

ecf_empirical(X, t_grid)

Arguments

Value

Complex vector of empirical characteristic function values.

Estimate all stable parameters from empirical characteristic function

Description

Estimate all stable parameters from empirical characteristic function

Usage

ecf_estimate_all(X, t_grid = NULL, weights = NULL)

Arguments

Value

A list containing estimated parameters: alpha, beta, gamma, and delta.

Empirical Characteristic Function

Description

Empirical Characteristic Function

Usage

ecf_fn(x, u, method = "simple")

Arguments

Value

List with magnitude and phase of ECF.

Estimate stable parameters using weighted ECF regression

Description

Estimate stable parameters using weighted ECF regression

Usage

ecf_regression(x, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

EM algorithm for alpha-stable mixture

Description

Estimates parameters of an alpha-stable mixture using EM with optional random initialization.

Usage

em_alpha_stable(
  data,
  n_components = 2,
  max_iter = 100,
  tol = 1e-04,
  random_init = TRUE,
  debug = TRUE
)

Arguments

Value

List with estimated weights and parameters (alpha, beta, gamma, delta).

EM algorithm for mixture of alpha-stable distributions using CDF-based ECF

Description

Performs EM estimation of a two-component alpha-stable mixture using a simplified empirical characteristic function derived from the cumulative distribution function (CDF).

Usage

em_estimate_stable_from_cdf(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for alpha-stable mixture using CDF-based ECF and Gibbs M-step

Description

Performs EM estimation of a two-component alpha-stable mixture using a simplified empirical characteristic function derived from the cumulative distribution function (CDF), with the M-step replaced by Gibbs sampling.

Usage

em_estimate_stable_from_cdf_with_gibbs(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for mixture of alpha-stable distributions using kernel ECF

Description

Performs EM estimation of a two-component alpha-stable mixture using kernel-smoothed empirical characteristic function (ECF).

Usage

em_estimate_stable_kernel_ecf(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for alpha-stable mixture using kernel ECF and Gibbs M-step

Description

Performs EM estimation of a two-component alpha-stable mixture using kernel-smoothed empirical characteristic function (ECF), with the M-step replaced by Gibbs sampling.

Usage

em_estimate_stable_kernel_ecf_with_gibbs(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for mixture of alpha-stable distributions using recursive ECF

Description

Performs Expectation-Maximization (EM) to estimate parameters of a two-component alpha-stable mixture model using recursive kernel smoothing on the empirical characteristic function.

Usage

em_estimate_stable_recursive_ecf(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for alpha-stable mixture using recursive ECF and Gibbs M-step

Description

Performs EM estimation of a two-component alpha-stable mixture using recursive kernel smoothing on the empirical characteristic function (ECF), with the M-step replaced by Gibbs sampling.

Usage

em_estimate_stable_recursive_ecf_with_gibbs(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for mixture of alpha-stable distributions using weighted OLS

Description

Performs EM estimation of a two-component alpha-stable mixture using weighted least squares regression on the ECF.

Usage

em_estimate_stable_weighted_ols(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for alpha-stable mixture using weighted OLS and Gibbs M-step

Description

Performs EM estimation of a two-component alpha-stable mixture using weighted least squares regression on the ECF, with the M-step replaced by Gibbs sampling.

Usage

em_estimate_stable_weighted_ols_with_gibbs(data, max_iter = 100, tol = 1e-04)

Arguments

Value

A list with estimated component parameters (alpha, beta, gamma, delta) and mixture weight (w).

EM algorithm for two-component Gaussian mixture

Description

Estimates parameters of a Gaussian mixture using the EM algorithm.

Usage

em_estimation_mixture(data, max_iter = 100, tol = 1e-06)

Arguments

Value

A list containing estimated mixture parameters: pi, mu1, mu2, sigma1, sigma2.

EM algorithm for two-component alpha-stable mixture using MLE

Description

Estimates parameters of a two-component alpha-stable mixture using MLE and EM.

Usage

em_fit_alpha_stable_mixture(
  data,
  max_iter = 200,
  tol = 1e-04,
  return_trace = FALSE
)

Arguments

Value

List with estimated parameters and optional trace.

EM algorithm for alpha-stable mixture using a custom estimator

Description

Performs EM estimation using a user-defined parameter estimator and ECF frequencies.

Usage

em_stable_mixture(data, u, estimator_func, max_iter = 300, epsilon = 0.001)

Arguments

Value

List with estimated weights, parameters, and log-likelihood.

Empirical R0 estimation using growth model

Description

Estimates the basic reproduction number (R0) using a growth model (exponential or logistic) fitted to incidence data.

Usage

empirical_r0(incidence, serial_interval, growth_model = "exponential")

Arguments

Value

Estimated R0 value.

Ensure positive scale parameter

Description

Ensure positive scale parameter

Usage

ensure_positive_scale(val, min_scale = 1e-06, max_scale = 1e+06)

Arguments

Value

A finite positive scale value.

Estimate R0 using maximum likelihood

Description

Computes the basic reproduction number (R0) using a maximum likelihood approach based on secondary cases and generation time weights.

Usage

est_r0_ml(W, N)

Arguments

Value

Estimated R0 value.

MLE estimation of R0 using generation time

Description

Estimates the basic reproduction number (R0) using maximum likelihood and a parametric model based on generation time and incidence data.

Usage

est_r0_mle(incidence, gen_time)

Arguments

Value

Estimated R0 value.

Estimate alpha and gamma from ECF modulus

Description

Estimate alpha and gamma from ECF modulus

Usage

estimate_alpha_gamma(t_grid, phi_vals, weights = NULL)

Arguments

Value

A list with elements alpha_hat and gamma_hat.

Estimate beta and delta from ECF phase

Description

Estimate beta and delta from ECF phase

Usage

estimate_beta_delta(t_grid, phi_vals, alpha_hat, gamma_hat, weights = NULL)

Arguments

Value

A list with elements beta_hat and delta_hat.

Estimate mixture of two stable distributions

Description

Wrapper function to estimate parameters of a two-component alpha-stable mixture using MLE.

Usage

estimate_mixture_params(data)

Arguments

Value

List with estimated weight and parameters for both components.

Estimate stable parameters using CDF-based ECF regression

Description

Estimate stable parameters using CDF-based ECF regression

Usage

estimate_stable_from_cdf(data, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Estimate stable parameters using kernel-based ECF method

Description

Estimate stable parameters using kernel-based ECF method

Usage

estimate_stable_kernel_ecf(data, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Estimate single stable distribution parameters

Description

Wrapper function to estimate stable parameters using one of three methods: "basic", "robust", or "lbfgs".

Usage

estimate_stable_params(data, method = "robust")

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Estimate stable parameters using method of moments

Description

Provides a rough estimation of alpha-stable parameters using empirical moments.

Usage

estimate_stable_r(x)

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Estimate stable parameters using recursive ECF method

Description

Estimate stable parameters using recursive ECF method

Usage

estimate_stable_recursive_ecf(data, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Estimate stable parameters using weighted OLS on recursive ECF

Description

Estimate stable parameters using weighted OLS on recursive ECF

Usage

estimate_stable_weighted_ols(data, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Helper function for eta0 computation

Description

Helper function for eta0 computation

Usage

eta0(u, alpha, gamma, eps = 0.05)

Arguments

Value

Numeric vector of eta0 values.

General eta function

Description

General eta function

Usage

eta_func(t, alpha, gamma)

Arguments

Value

Numeric vector of eta values.

Evaluate estimation method using MSE over multiple trials

Description

Computes the mean squared error (MSE) of a parameter estimation function over several trials.

Usage

evaluate_estimation_method(
  estimator_fn,
  true_params,
  n = 1000,
  trials = 20,
  seed = 42
)

Arguments

Value

Numeric value representing the average MSE across trials.

Evaluate fit quality using RMSE and log-likelihood

Description

Computes RMSE and log-likelihood scores for different mixture models compared to histogram data.

Usage

evaluate_fit(data, methods, x_vals)

Arguments

Value

Named list of scores per method (RMSE and LogLikelihood).

Export analysis report to JSON and Excel

Description

Saves the results of a stability analysis to a JSON file and an Excel workbook.

Usage

export_analysis_report(
  data,
  stable_params,
  qcv,
  skew_kurt,
  normality,
  verdict,
  filename = "stable_report"
)

Arguments

Value

Invisibly returns a list containing the paths of the exported JSON and Excel files.

False position method update step

Description

Performs a single iteration of the false position (regula falsi) method for root-finding. This method approximates the root of a function by using a linear interpolation between two points where the function changes sign.

Usage

false_position_update(a, b_n, f_a, f_b, objective_func)

Arguments

Value

Numeric scalar. Updated estimate for the root after one iteration.

Fast numerical integration using trapezoidal rule

Description

Fast numerical integration using trapezoidal rule

Usage

fast_integrate(func, a = -6, b = 6, N = 100)

Arguments

Value

Approximated integral value.

Fit Alpha-Stable Distribution using MLE (L-BFGS-B)

Description

Estimates the parameters of a single alpha-stable distribution using maximum likelihood and the L-BFGS-B optimization method.

Usage

fit_alpha_stable_mle(data)

Arguments

Value

Numeric vector of estimated parameters: alpha, beta, gamma, delta.

Fit MLE Mixture of Two Stable Distributions

Description

Estimates parameters of a two-component alpha-stable mixture using maximum likelihood and the L-BFGS-B optimization method.

Usage

fit_mle_mixture(data)

Arguments

Value

Numeric vector of estimated parameters: weight, alpha1, beta1, gamma1, delta1, alpha2, beta2, gamma2, delta2.

Estimate stable parameters using filtered and weighted ECF regression

Description

Estimate stable parameters using filtered and weighted ECF regression

Usage

fit_stable_ecf(data, frequencies)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Generate samples from a predefined alpha-stable mixture

Description

Simulates samples from a mixture of alpha-stable distributions using specified parameters.

Usage

generate_alpha_stable_mixture(
  weights,
  alphas,
  betas,
  gammas,
  deltas,
  size = 1000
)

Arguments

Value

A list with:

samples

Numeric vector of generated samples.

labels

Integer vector indicating the component each sample came from.

Generate McCulloch lookup table from simulated stable samples

Description

Simulates alpha-stable samples across a grid of alpha and beta values, and computes quantile-based ratios used for McCulloch estimation. The result is a lookup table indexed by (alpha, beta) keys.

Usage

generate_mcculloch_table(alpha_grid, beta_grid, size = 1e+05)

Arguments

Value

Named list of quantile ratios indexed by "alpha_beta".

Simulates a mixture of alpha-stable distributions with randomly sampled parameters.

Description

Simulates a mixture of alpha-stable distributions with randomly sampled parameters.

Usage

generate_mixture_data(K = 2, N = 1000)

Arguments

Value

A list containing:

data

Numeric vector of generated samples.

params

List of parameters for each component (alpha, beta, gamma, delta, pi).

Generate synthetic data from two alpha-stable components

Description

Creates a synthetic dataset composed of two alpha-stable distributions.

Usage

generate_synthetic_data(n = 1000)

Arguments

Value

Numeric vector of shuffled synthetic data.

Gibbs sampler for Gaussian mixture model

Description

Performs Gibbs sampling for a two-component Gaussian mixture model. Updates latent assignments, component means, variances, and mixing proportions.

Usage

gibbs_sampler(data, iterations = 1000)

Arguments

Value

List of sampled parameters per iteration.

Log-likelihood gradient with respect to alpha

Description

Log-likelihood gradient with respect to alpha

Usage

grad_loglik_alpha(alpha, beta, delta, omega, x)

Arguments

Value

Numeric scalar: estimated gradient (or NA_real_ on invalid input).

Log-likelihood gradient with respect to beta

Description

Log-likelihood gradient with respect to beta

Usage

grad_loglik_beta(alpha, beta, delta, omega, x)

Arguments

Value

Numeric scalar: estimated gradient (or NA_real_ on invalid input).

Log-likelihood gradient with respect to delta (scale)

Description

Log-likelihood gradient with respect to delta (scale)

Usage

grad_loglik_delta(alpha, beta, delta, omega, x)

Arguments

Value

Numeric scalar: estimated gradient (or NA_real_ on invalid input).

Log-likelihood gradient with respect to omega (location)

Description

Log-likelihood gradient with respect to omega (location)

Usage

grad_loglik_omega(alpha, beta, delta, omega, x)

Arguments

Value

Numeric scalar: estimated gradient (or NA_real_ on invalid input).

Integrate cosine exponential

Description

Computes the integral of the cosine exponential function for stable distributions.

Usage

integrate_cosine(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

Integrate cosine-log-weighted exponential

Description

Computes the integral of the cosine-log-weighted exponential function for stable distributions.

Usage

integrate_cosine_log_weighted(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

Robust integration helper function

Description

Applies 'safe_integrate' to a given integrand over a vector of x values.

Usage

integrate_function(f, x, alpha, beta, delta, omega, upper = 50, eps = 1e-08)

Arguments

Value

Numeric vector of integral values evaluated at each element of x.

Integrate sin exponential

Description

Computes the integral of the sin exponential function for stable distributions.

Usage

integrate_sine(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

Integrate sine-log-weighted exponential

Description

Computes the integral of the sine-log-weighted exponential function for stable distributions.

Usage

integrate_sine_log_weighted(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

Integrate sine-r-weighted exponential

Description

Computes the integral of the sine-r-weighted exponential function for stable distributions.

Usage

integrate_sine_r_weighted(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

Integration wrappers for specific integrands

Description

Integration wrappers for specific integrands

Usage

integrate_sine_weighted(x, alpha, beta, delta, omega)

Arguments

Value

Numeric vector of integral values.

KDE bandwidth selection using plugin method

Description

KDE bandwidth selection using plugin method

Usage

kde_bandwidth_plugin(X, alpha)

Arguments

Value

Bandwidth value.

Log-likelihood for mixture of stable distributions

Description

Log-likelihood for mixture of stable distributions

Usage

log_likelihood_mixture(params, data)

Arguments

Value

Negative log-likelihood (for minimization).

Estimate stable parameters using McCulloch lookup

Description

Estimates alpha and beta using quantile ratios and interpolation functions. Gamma and delta are derived directly from quantiles.

Usage

mcculloch_lookup_estimate(
  X,
  interpolators = NULL,
  interp_alpha = NULL,
  interp_beta = NULL
)

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Initialization using McCulloch quantile method

Description

Estimates all four stable parameters using quantile ratios and mock lookup. Useful for initializing optimization algorithms.

Usage

mcculloch_quantile_init(X)

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Metropolis-Hastings MCMC for stable mixture clustering

Description

Performs Metropolis-Hastings sampling to estimate parameters of a two-component alpha-stable mixture model. Proposals are generated for each parameter and accepted based on likelihood ratios. Cluster assignments are updated at each iteration.

Usage

metropolis_hastings(fct, iterations, lok, aa = c(1, 1), proposal_std = 0.1)

Arguments

Value

List of sampled parameters and weights across iterations.

Mixture of two stable PDFs

Description

Mixture of two stable PDFs

Usage

mixture_stable_pdf(x, p1, p2, w)

Arguments

Value

Numeric vector of mixture PDF values.

Simple MLE estimation with default starting values

Description

Estimates stable distribution parameters using Nelder-Mead optimization from default or user-provided starting values.

Usage

mle_estimate(X, x0 = NULL)

Arguments

Value

List with estimated alpha, beta, gamma, delta.

Mock Gibbs sampling for alpha-stable mixture estimation

Description

Performs a simplified Gibbs sampling procedure to estimate parameters of a two-component alpha-stable mixture. Samples are drawn from prior distributions and evaluated using log-likelihood. The best parameter set is selected based on likelihood.

Usage

mock_gibbs_sampling(data, n_samples = 500, verbose = FALSE)

Arguments

Value

List containing best_params and all sampled parameter sets.

Mock lookup for alpha and beta (fallback)

Description

Provides a fallback estimation of alpha and beta from quantile ratios using simple linear approximations.

Usage

mock_lookup_alpha_beta(v_alpha, v_beta)

Arguments

Value

List with estimated alpha and beta.

Negative log-likelihood for single stable distribution

Description

Negative log-likelihood for single stable distribution

Usage

negative_log_likelihood(params, data)

Arguments

Value

Negative log-likelihood.

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.

Description

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.

Usage

normalized_grad_alpha(alpha, beta, delta, omega, x)

Arguments

Value

Scalar value representing the normalized gradient.

Normalized objective for beta parameter

Description

Computes the normalized gradient of the log-likelihood with respect to the beta parameter using the latest values of alpha, delta, and omega stored in global vectors.

Usage

normalized_objective_beta(beta, X1, L_alpha, L_delta, L_omega)

Arguments

Value

Scalar value representing the normalized gradient.

Normalized objective for delta parameter

Description

Computes the normalized gradient of the log-likelihood with respect to the delta parameter using the latest values of alpha, beta, and omega stored in global vectors.

Usage

normalized_objective_delta(delta, X1, L_alpha, L_beta, L_omega)

Arguments

Value

Scalar value representing the normalized gra

Normalized objective for omega parameter

Description

Computes the normalized gradient of the log-likelihood with respect to the omega parameter using the latest values of alpha, beta, and delta stored in global vectors.

Usage

normalized_objective_omega(omega, X1, L_alpha, L_beta, L_delta)

Arguments

Value

Scalar value representing the normalized gradient.

Compare EM-estimated mixture with a non-optimized reference model

Description

Plots the fitted EM mixture density alongside a manually defined reference mixture to visually compare estimation quality.

Usage

plot_comparison(data, p1, p2, w)

Arguments

Value

Invisibly returns the file path to the saved PNG image of the comparison plot.

Plot histogram with normal and stable PDF overlays

Description

Displays a histogram of the data with overlaid normal and alpha-stable probability density functions.

Usage

plot_distributions(x, params_stable)

Arguments

Value

NULL (plot is displayed as a side effect)

Plot effective reproduction number (Re) over time

Description

Computes and visualizes the effective reproduction number (Rt) over time using incidence data and generation time distribution. Optionally overlays a smoothed spline curve.

Usage

plot_effective_reproduction_number(
  GT,
  S,
  inc,
  dates,
  est_r0_ml,
  RT,
  output_file = "RN_avec_dates_EM-ML.pdf"
)

Arguments

Value

NULL (plot is saved as PDF)

Plot final fitted mixture of alpha-stable distributions

Description

Plots the final fitted mixture PDF over the data histogram.

Usage

plot_final_mixture_fit(data, p1, p2, w)

Arguments

Value

NULL (plot is saved as PNG)

Plot true vs estimated mixture density

Description

Compares the true mixture density with the estimated one by overlaying both on the histogram of the observed data.

Usage

plot_fit_vs_true(true_params, est_params, data, bins = 100)

Arguments

Value

NULL (plot is displayed)

Compare estimated mixture densities from two methods against the true density

Description

Plots the true mixture density and compares it with two estimated densities obtained from different methods (e.g., MLE vs ECF).

Usage

plot_fit_vs_true_methods(
  data,
  true_params,
  method1 = "MLE",
  method2 = "ECF",
  bins = 100
)

Arguments

Value

NULL (plot is displayed)

Plot RMSE and Log-Likelihood comparison across methods

Description

Generates bar plots comparing RMSE and log-likelihood values across different estimation methods.

Usage

plot_method_comparison(scores)

Arguments

Value

NULL (plot is saved as PNG)

Plot mixture of two alpha-stable distributions

Description

Plots the histogram of data and overlays the estimated mixture density and its components.

Usage

plot_mixture(data, params1, params2, w, label = "EM")

Arguments

Value

NULL (plot is saved as PNG)

Plot mixture fit with individual components

Description

Displays the estimated mixture density and optionally its individual components over a histogram of the data.

Usage

plot_mixture_fit(
  data,
  estimated_params,
  bins = 200,
  plot_components = TRUE,
  save_path = NULL,
  show_plot = TRUE,
  title = "Mixture of Alpha-Stable Distributions"
)

Arguments

Value

Invisibly returns the file path to the saved plot (if save_path is provided), or NULL otherwise.

Plot fitted mixture on real dataset

Description

Plots the fitted mixture model over a histogram of real-world data, showing both components and the overall mixture.

Usage

plot_real_mixture_fit(X, result)

Arguments

Value

NULL (plot is saved as PNG)

Plot posterior mixture density from MCMC samples

Description

Visualizes the estimated two-component alpha-stable mixture density using parameters obtained from MCMC sampling. Optionally overlays the true density for comparison.

Usage

plot_results(
  M2_w1,
  M2_alpha1,
  M2_beta1,
  M2_delta1,
  M2_omega1,
  M2_w2,
  M2_alpha2,
  M2_beta2,
  M2_delta2,
  M2_omega2,
  xx,
  xx_true = NULL,
  yy_true = NULL
)

Arguments

Value

Invisibly returns the file path to the saved PNG image of the posterior mixture density plot.

Plot trace of a parameter across MCMC iterations

Description

Displays the evolution of a parameter across MCMC iterations to assess convergence.

Usage

plot_trace(samples, param_name)

Arguments

Value

NULL (plot is displayed as a side effect)

Plot comparison between normal and stable distributions

Description

Displays a histogram of the data with overlaid normal and alpha-stable PDFs.

Usage

plot_vs_normal_stable(x, params_stable, fig_path = NULL)

Arguments

Value

NULL (saves plot to file)

QCV statistic for tail heaviness

Description

Computes the QCV (Quantile Conditional Variance) statistic to assess tail heaviness of a distribution.

Usage

qcv_stat(x)

Arguments

Value

Numeric: QCV value.

Robust stable PDF computation

Description

Robust stable PDF computation

Usage

r_stable_pdf(x, alpha, beta, scale, location)

Arguments

Value

Numeric vector of density values.

Recursive weight function

Description

Recursive weight function

Usage

recursive_weight(l)

Arguments

Value

Weight value.

Estimate stable parameters using robust ECF regression

Description

Estimate stable parameters using robust ECF regression

Usage

robust_ecf_regression(x, u)

Arguments

Value

A list with estimated parameters: alpha, beta, gamma, and delta.

Robust MLE estimation with multiple starting points

Description

Performs multiple MLE estimations with randomized starting points and selects the best result based on log-likelihood.

Usage

robust_mle_estimate(data, n_starts = 5)

Arguments

Value

List with best estimated alpha, beta, gamma, delta.

Generate random samples from stable distribution

Description

Generate random samples from stable distribution

Usage

rstable(n, alpha, beta, gamma = 1, delta = 0, pm = 1)

Arguments

Value

Numeric vector of samples.

Run all EM-based estimations without Gibbs sampling (CRAN-safe)

Description

Executes multiple EM algorithms (recursive ECF, kernel ECF, weighted OLS, CDF-based) on the input data. Optionally saves mixture plots to a temporary directory.

Usage

run_all_estimations(X1, bw_sj, max_iter = 200, tol = 1e-04, save_plots = FALSE)

Arguments

Value

Invisibly returns a list containing fitted parameters for each EM method.

Run all EM-based estimations with Gibbs sampling (CRAN-safe)

Description

Executes EM algorithms with Gibbs sampling on the input data. Optionally saves mixture plots to a temporary directory.

Usage

run_estimations_with_gibbs(
  data,
  bw_sj,
  max_iter = 100,
  tol = 1e-04,
  save_plots = FALSE
)

Arguments

Value

Invisibly returns a list containing fitted parameters for each EM method with Gibbs.

Safe integration wrapper with multiple fallback strategies

Description

Performs numerical integration with progressively more conservative settings to improve robustness. Useful when standard integration may fail due to oscillatory or heavy-tailed functions.

Usage

safe_integrate(
  integrand_func,
  lower = 1e-08,
  upper = 50,
  max_attempts = 3,
  ...
)

Arguments

Value

Numeric scalar. The estimated value of the integral, or 0 if all attempts fail.

Simple 2-component EM using ECF initialization

Description

Initializes EM using k-means clustering and ECF-based parameter estimation.

Usage

simple_em_real(X, max_iter = 10)

Arguments

Value

List with lambda1 and estimated parameters for both components.

Simulate mixture data from alpha-stable components

Description

Generates synthetic data from a mixture of alpha-stable distributions using specified weights and parameters.

Usage

simulate_mixture(n, weights, params)

Arguments

Value

Numeric vector of simulated samples.

Sine exponential function

Description

Sine exponential function

Usage

sine_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Sine-log-weighted exponential with r^(-alpha) term

Description

Sine-log-weighted exponential with r^(-alpha) term

Usage

sine_log_weighted_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Sine-r-weighted exponential function

Description

Sine-r-weighted exponential function

Usage

sine_r_weighted_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Sine-weighted exponential with r^alpha term

Description

Sine-weighted exponential with r^alpha term

Usage

sine_weighted_exp_ralpha(r, x, alpha, beta, delta, omega)

Arguments

Value

Numeric value of the integrand.

Calculate skewness and kurtosis

Description

Computes the skewness and kurtosis of a numeric vector.

Usage

skew_kurtosis(x)

Arguments

Value

List with skewness and kurtosis.

Initialize stable distribution parameters

Description

Initialize stable distribution parameters

Usage

stable_fit_init(x)

Arguments

Value

A list with estimated parameters (alpha, beta, gamma, delta).

Test normality using multiple statistical tests

Description

Applies Shapiro-Wilk, Jarque-Bera, Anderson-Darling, and Kolmogorov-Smirnov tests to assess normality of a dataset.

Usage

test_normality(x)

Arguments

Value

List of test statistics and p-values.

Helper function to unpack parameters

Description

Helper function to unpack parameters

Usage

unpack_params(p)

Arguments

Value

A numeric vector of parameters.

Validate and clip parameters for stable distribution

Description

Ensures parameters are within valid bounds for stable distribution computations.

Usage

validate_params(alpha, beta, delta, r = NULL)

Arguments

Value

Clipped beta value.

Wasserstein distance between two mixture distributions

Description

Wasserstein distance between two mixture distributions

Usage

wasserstein_distance_mixture(params1, params2, size = 5000)

Arguments

Value

Wasserstein distance.