Type: Package
Title: Parameter Estimation for Stable Distributions and Their Mixtures
Version: 0.1.0
Description: Provides various functions for parameter estimation of one-dimensional stable distributions and their mixtures. It implements a diverse set of estimation methods, including quantile-based approaches, regression methods based on the empirical characteristic function (empirical, kernel, and recursive), and maximum likelihood estimation. For mixture models, it provides stochastic expectation–maximization (SEM) algorithms and Bayesian estimation methods using sampling and importance sampling to overcome the long burn-in period of Markov Chain Monte Carlo (MCMC) strategies. The package also includes tools and statistical tests for analyzing whether a dataset follows a stable distribution. Some of the implemented methods are described in Hajjaji, O., Manou-Abi, S. M., and Slaoui, Y. (2024) <doi:10.1080/02664763.2024.2434627>.
License: GPL-3
Encoding: UTF-8
RoxygenNote: 7.3.2
Imports: stabledist, mixtools, nortest, openxlsx, e1071, jsonlite, libstable4u, stats, graphics, MASS, utils
Suggests: ggplot2, grDevices, moments, readxl, testthat (≥ 3.0.0)
Config/testthat/edition: 3
NeedsCompilation: no
Packaged: 2025-10-30 00:19:17 UTC; FX506
Author: Solym Manou-Abi [aut, cre], Adam Najib [aut], Yousri Slaoui [aut]
Maintainer: Solym Manou-Abi <solym.manou.abi@univ-poitiers.fr>
Repository: CRAN
Date/Publication: 2025-11-03 19:20:18 UTC

Estimate stable distribution parameters using classical ECF regression

Description

Estimate stable distribution parameters using classical ECF regression

Usage

CDF(x, u)

Arguments

x

Numeric vector of data.

u

Vector of frequency values.

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

r

Integration variable.

u

Frequency.

x

Data point.

bn

Bandwidth.

Value

Imaginary component value.

Integrate imaginary component over \mathbb{R}

Description

Integrate imaginary component over \mathbb{R}

Usage

Int_Im(u, x, bn)

Arguments

u

Frequency.

x

Data point.

bn

Bandwidth.

Value

Integrated imaginary value.

Integrate real component over \mathbb{R}

Description

Integrate real component over \mathbb{R}

Usage

Int_Re(u, x, bn)

Arguments

u

Frequency.

x

Data point.

bn

Bandwidth.

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

param

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

obs

Numeric vector of observations.

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

x

Numeric vector of observations.

Value

List with estimated alpha, beta, gamma, delta.

Epanechnikov kernel

Description

Epanechnikov kernel

Usage

N_epanechnikov(z)

Arguments

z

Numeric input.

Value

Kernel value.

Gaussian kernel

Description

Gaussian kernel

Usage

N_gaussian(z)

Arguments

z

Numeric input.

Value

Kernel value.

Uniform kernel

Description

Uniform kernel

Usage

N_uniform(z)

Arguments

z

Numeric input.

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

inc

Numeric vector of incidence counts.

G

Numeric vector representing the generation time distribution.

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

r

Integration variable.

u

Frequency.

x

Data point.

bn

Bandwidth.

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

log_likelihood

Log-likelihood value.

num_params

Number of estimated parameters.

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

x

Numeric vector of data.

filename

Base name for output files (JSON and Excel). Written to tempdir().

qcv_threshold

Threshold for QCV to consider distribution heavy-tailed.

fig_path

Optional path to save the plot (PNG). If NULL, uses tempdir().

verbose

Logical; if TRUE, prints progress messages. Default is FALSE.

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

data

Numeric vector of observations.

draws

Number of iterations per chain.

chains

Number of chains to simulate.

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

log_likelihood

Log-likelihood value.

num_params

Number of estimated parameters.

n

Sample size.

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

table

Lookup table generated by generate_mcculloch_table.

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

data

Numeric vector of observations.

params

Vector of parameters (not used in this placeholder).

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

x

Numeric input.

lower

Minimum allowed value.

upper

Maximum allowed value.

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

data

Numeric vector. Observed data.

n_runs

Integer. Number of runs for comparison.

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

n_samples

Integer. Number of samples per simulation.

n_runs

Integer. Number of simulation runs.

interp_alpha

Optional interpolation function for alpha (McCulloch).

interp_beta

Optional interpolation function for beta (McCulloch).

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

parameter_configs

Named list of parameter sets (each a list with alpha, beta, gamma, delta).

trials

Integer. Number of trials per configuration.

n

Integer. Number of samples per trial.

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

data

Numeric vector of observations.

em_params

List of EM-estimated parameters.

ecf_kernel_params

List of kernel ECF-estimated parameters.

ecf_empirical_params

List of empirical ECF-estimated parameters.

fig_dir

Optional path to save plots. Defaults to tempdir().

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

data

Numeric vector of observations.

params

Vector of parameters: alpha, beta, gamma, delta.

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

X

Numeric vector of sample data.

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

filepath

Character. Path to the CSV file.

Value

Numeric vector of serial intervals.

Cosine exponential function

Description

Cosine exponential function

Usage

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

Arguments

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

phi_vals

Complex vector of ECF values

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

X

Numeric vector of data

t_grid

Vector of frequency values

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

X

Numeric vector of data.

t_grid

Optional vector of frequencies (default: seq(0.1, 1.0, length.out = 50)).

weights

Optional vector of weights.

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

x

Numeric vector of data.

u

Numeric vector of frequencies.

method

Method: "simple", "kernel", or "recursive".

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

x

Numeric vector of data.

u

Vector of frequency values.

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

data

Numeric vector of observations.

n_components

Integer. Number of mixture components.

max_iter

Integer. Maximum number of EM iterations.

tol

Numeric. Convergence tolerance.

random_init

Logical. Whether to use random initialization.

debug

Logical. Whether to print debug information.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Maximum number of EM iterations.

tol

Convergence tolerance on log-likelihood.

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

data

Numeric vector of observations.

max_iter

Integer. Maximum number of EM iterations.

tol

Numeric. Convergence tolerance.

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

data

Numeric vector of observations.

max_iter

Integer. Maximum number of EM iterations.

tol

Numeric. Convergence tolerance on log-likelihood.

return_trace

Logical. Whether to return trace of responsibilities and log-likelihoods.

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

data

Numeric vector of observations.

u

Numeric vector of frequency values for ECF.

estimator_func

Function to estimate stable parameters.

max_iter

Integer. Maximum number of EM iterations.

epsilon

Numeric. Convergence threshold on log-likelihood.

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

incidence

Numeric vector of incidence counts.

serial_interval

Mean serial interval.

growth_model

Type of growth model: "exponential" or "logistic".

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

val

Numeric input.

min_scale

Minimum scale.

max_scale

Maximum scale.

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

W

Generation time distribution.

N

Secondary case counts.

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

incidence

Numeric vector of incidence counts.

gen_time

Mean generation time.

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

t_grid

Vector of frequency values.

phi_vals

Complex vector of ECF values.

weights

Optional vector of weights.

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

t_grid

Vector of frequency values.

phi_vals

Complex vector of ECF values.

alpha_hat

Estimated alpha.

gamma_hat

Estimated gamma.

weights

Optional vector of weights.

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

data

Numeric vector of observations.

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

data

Numeric vector of observations

u

Vector of frequency values

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

data

Numeric vector of observations

u

Vector of frequency values

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

data

Numeric vector of observations.

method

Estimation method: "basic", "robust", or "lbfgs".

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

x

Numeric vector of data.

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

data

Numeric vector of observations.

u

Vector of frequency values.

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

data

Numeric vector of observations

u

Vector of frequency values

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

u

Frequency vector.

alpha

Stability parameter.

gamma

Scale parameter.

eps

Tolerance for \alpha \approx 1.

Value

Numeric vector of eta0 values.

General eta function

Description

General eta function

Usage

eta_func(t, alpha, gamma)

Arguments

t

Frequency vector.

alpha

Stability parameter.

gamma

Scale parameter.

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

estimator_fn

Function. Estimation function to evaluate.

true_params

List. True parameters: alpha, beta, gamma, delta.

n

Integer. Number of samples per trial.

trials

Integer. Number of trials to run.

seed

Integer. Random seed for reproducibility.

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

data

Numeric vector of observations.

methods

Named list of parameter sets (each with two components and a weight).

x_vals

Numeric vector. Evaluation grid.

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

data

Numeric vector of data.

stable_params

List of estimated stable parameters.

qcv

QCV statistic.

skew_kurt

List with skewness and kurtosis.

normality

List of normality test results.

verdict

Final verdict string.

filename

Base name for output files.

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

a

Numeric scalar. Left bound of the interval.

b_n

Numeric scalar. Right bound of the interval (current step).

f_a

Numeric scalar. Value of the objective function evaluated at 'a'.

f_b

Numeric scalar. Value of the objective function evaluated at 'b_n'.

objective_func

Function. The objective function whose root is being sought.

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

func

Function to integrate.

a

Lower bound.

b

Upper bound.

N

Number of points.

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

data

Numeric vector of observations.

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

data

Numeric vector of observations.

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

data

Numeric vector of observations.

frequencies

Vector of frequency values.

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

weights

Numeric vector of mixture weights.

alphas

Numeric vector of alpha parameters.

betas

Numeric vector of beta parameters.

gammas

Numeric vector of gamma parameters.

deltas

Numeric vector of delta parameters.

size

Integer. Number of samples to generate.

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

alpha_grid

Vector of alpha values to simulate.

beta_grid

Vector of beta values to simulate.

size

Number of samples per simulation.

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

K

Integer. Number of mixture components.

N

Integer. Total number of samples to generate.

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

n

Integer. Total number of samples to generate.

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

data

Numeric vector of observations.

iterations

Number of Gibbs iterations.

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

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

x

Numeric vector. Observations at which the integral is computed.

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

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

x

Numeric vector. Observations at which the integral is computed.

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

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

x

Numeric vector. Observations at which the integral is computed.

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

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

x

Numeric vector. Observations at which the integral is computed.

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

f

Function. The integrand function to be evaluated.

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

upper

Numeric scalar. Upper limit of integration (default = 50).

eps

Numeric scalar. Tolerance for numerical integration (default = 1e-8).

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

x

Numeric vector. Observations at which the integral is computed.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

X

Numeric vector of data.

alpha

Stability parameter.

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

params

Numeric vector of parameters.

data

Numeric vector of observations.

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

X

Numeric vector of data.

interpolators

Optional list with interp_alpha and interp_beta functions.

interp_alpha

Optional function to interpolate alpha.

interp_beta

Optional function to interpolate beta.

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

X

Numeric vector of data.

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

fct

Function to estimate initial parameters.

iterations

Number of MCMC iterations.

lok

Numeric vector of observations.

aa

Prior parameters for Dirichlet distribution.

proposal_std

Standard deviation for proposal distributions.

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

x

Numeric vector.

p1

List of parameters for first stable distribution.

p2

List of parameters for second stable distribution.

w

Mixture weight (0 < w < 1).

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

X

Numeric vector of observations.

x0

Optional starting values.

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

data

Numeric vector of observations.

n_samples

Number of Gibbs samples to draw.

verbose

Logical, whether to print best log-likelihood.

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

v_alpha

Quantile-based shape ratio.

v_beta

Quantile-based skewness ratio.

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

params

Numeric vector (alpha, beta, gamma, delta).

data

Numeric vector of observations.

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

alpha

Stability parameter.

beta

Skewness parameter.

delta

Scale parameter.

omega

Location parameter.

x

Numeric vector of observations.

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

beta

Skewness parameter.

X1

Numeric vector of observations.

L_alpha

Vector of alpha values (global memory).

L_delta

Vector of delta values (global memory).

L_omega

Vector of omega values (global memory).

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

delta

Location parameter.

X1

Numeric vector of observations.

L_alpha

Vector of alpha values.

L_beta

Vector of beta values.

L_omega

Vector of omega values.

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

omega

Location parameter.

X1

Numeric vector of observations.

L_alpha

Vector of alpha values.

L_beta

Vector of beta values.

L_delta

Vector of delta values.

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

data

Numeric vector of observations.

p1

Vector of parameters for component 1 (alpha, beta, gamma, delta).

p2

Vector of parameters for component 2.

w

Mixing weight for component 1.

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

x

Numeric vector of data.

params_stable

List with alpha, beta, gamma, delta.

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

GT

Numeric vector. Generation time distribution.

S

Numeric vector. Secondary cases.

inc

Numeric vector. Incidence time series.

dates

Date vector corresponding to observations.

est_r0_ml

Function to estimate R0.

RT

Function to compute Rt.

output_file

Output PDF filename.

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

data

Numeric vector of observations.

p1

Vector of parameters for component 1.

p2

Vector of parameters for component 2.

w

Mixing weight for component 1.

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

true_params

List of true mixture components with alpha, beta, gamma, delta, pi.

est_params

List of estimated mixture components.

data

Numeric vector of observations.

bins

Number of histogram bins.

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

data

Numeric vector of observations.

true_params

List of true mixture components.

method1

First estimation method ("MLE", "ECF", etc.).

method2

Second estimation method.

bins

Number of histogram bins.

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

scores

Named list of score objects with RMSE and LogLikelihood.

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

data

Numeric vector of observations.

params1

Vector of parameters for component 1 (alpha, beta, gamma, delta).

params2

Vector of parameters for component 2.

w

Mixing weight for component 1.

label

Optional label for output file.

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

data

Numeric vector of observations.

estimated_params

List with weights, alphas, betas, gammas, deltas.

bins

Number of histogram bins.

plot_components

Logical, whether to show individual components.

save_path

Optional PNG filename.

show_plot

Logical, whether to display the plot.

title

Plot title.

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

X

Numeric vector of observations.

result

List containing params1, params2, and lambda1.

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

M2_w1

Numeric scalar. Mixture weight of first component.

M2_alpha1

Numeric vector. Stability parameter samples of first component.

M2_beta1

Numeric vector. Skewness parameter samples of first component.

M2_delta1

Numeric vector. Location parameter samples of first component.

M2_omega1

Numeric vector. Scale parameter samples of first component.

M2_w2

Numeric scalar. Mixture weight of second component.

M2_alpha2

Numeric vector. Stability parameter samples of second component.

M2_beta2

Numeric vector. Skewness parameter samples of second component.

M2_delta2

Numeric vector. Location parameter samples of second component.

M2_omega2

Numeric vector. Scale parameter samples of second component.

xx

Numeric vector of grid values for density evaluation.

xx_true

Optional numeric vector for true density x-values.

yy_true

Optional numeric vector for true density y-values.

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

samples

List of sample objects from MCMC.

param_name

Name of the parameter to trace.

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

x

Numeric vector of data.

params_stable

List with alpha, beta, gamma, delta.

fig_path

Optional path to save the plot (PNG). If NULL, uses tempdir().

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

x

Numeric vector of data.

Value

Numeric: QCV value.

Robust stable PDF computation

Description

Robust stable PDF computation

Usage

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

Arguments

x

Points at which to evaluate the density.

alpha

Stability parameter (0,2].

beta

Skewness parameter [-1,1].

scale

Scale (>0).

location

Location parameter.

Value

Numeric vector of density values.

Recursive weight function

Description

Recursive weight function

Usage

recursive_weight(l)

Arguments

l

Index.

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

x

Numeric vector of data.

u

Vector of frequency values.

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

data

Numeric vector of observations.

n_starts

Number of random initializations.

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

n

Number of samples.

alpha

Stability parameter (0,2].

beta

Skewness parameter [-1,1].

gamma

Scale (>0).

delta

Location.

pm

Parameterization (0 or 1).

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

X1

Numeric vector of data.

bw_sj

Numeric. Bandwidth value (e.g., Silverman's rule).

max_iter

Integer. Maximum number of EM iterations.

tol

Numeric. Convergence tolerance.

save_plots

Logical. Whether to save PNG plots to tempdir().

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

data

Numeric vector of observations.

bw_sj

Numeric. Bandwidth value (e.g., Silverman's rule).

max_iter

Integer. Maximum number of EM iterations.

tol

Numeric. Convergence tolerance.

save_plots

Logical. Whether to save PNG plots to tempdir().

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

integrand_func

Function to integrate. Must accept a numeric vector as input and return numeric values.

lower

Numeric scalar. Lower bound of the integration interval (default = 1e-8 to avoid singularities at 0).

upper

Numeric scalar. Upper bound of the integration interval (default = 50).

max_attempts

Integer. Maximum number of fallback attempts with modified settings if the initial integration fails.

...

Additional arguments passed to integrand_func.

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

X

Numeric vector of observations.

max_iter

Integer. Maximum number of iterations.

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

n

Number of samples to generate.

weights

Vector of mixture weights.

params

List of parameter sets for each component (each with alpha, beta, gamma, delta).

Value

Numeric vector of simulated samples.

Sine exponential function

Description

Sine exponential function

Usage

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

Arguments

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution (0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

r

Integration variable.

x

Observation.

alpha

Numeric scalar. Stability parameter of the stable distribution ((0 < \alpha \le 2), controlling tail thickness.

beta

Numeric scalar. Skewness parameter of the stable distribution (-1 \le \beta \le 1).

delta

Numeric scalar. Location parameter of the stable distribution.

omega

Numeric scalar. Scale parameter of the stable distribution (omega > 0).

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

x

Numeric vector of data.

Value

List with skewness and kurtosis.

Initialize stable distribution parameters

Description

Initialize stable distribution parameters

Usage

stable_fit_init(x)

Arguments

x

Numeric vector of data.

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

x

Numeric vector of data.

Value

List of test statistics and p-values.

Helper function to unpack parameters

Description

Helper function to unpack parameters

Usage

unpack_params(p)

Arguments

p

A list containing alpha, beta, gamma, delta.

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

alpha

Stability parameter (must be > 0 and < 2).

beta

Skewness parameter (clipped to [-1, 1]).

delta

Scale parameter (must be > 0).

r

Optional numeric vector. Must be positive if provided.

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

params1

List of parameters for first mixture.

params2

List of parameters for second mixture.

size

Number of samples to approximate distance.

Value

Wasserstein distance.

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