| Title: | Stable Distribution Functions...For You |
|---|---|
| Description: | Tools for fast and accurate evaluation of skew stable distributions (CDF, PDF and quantile functions), random number generation, and parameter estimation. This is 'libstableR' as per Royuela del Val, Simmross-Wattenberg, and Alberola López (2017) <doi:10.18637/jss.v078.i01> under a new maintainer. |
| Authors: | Javier Royuela del Val [aut], Federico Simmross-Wattenberg [aut], Carlos Alberola López [aut], Bob Rudis [ctb] (Several bugs fixed and added macOS compatibility), Bruce Swihart [ctb, cre] (Several bugs fixed and added macOS compatibility) |
| Maintainer: | Bruce Swihart <[email protected]> |
| License: | GPL-3 |
| Version: | 1.0.4 |
| Built: | 2024-11-12 06:48:47 UTC |
| Source: | CRAN |
libstable4u provides functions to work with skew stable distributions in a fast and accurate way [1]. It performs:
Fast and accurate evaluation of the probability density function (PDF) and cumulative density function (CDF).
Fast and accurate evaluation of the quantile function (inverse CDF).
Random numbers generation [2].
Skew stable parameter estimation with:
McCulloch's method of quantiles [3].
Koutrouvellis' method based on the characteristic function [4].
Maximum likelihood estimation.
Modified maximum likelihood estimation as described in [1]. *The evaluation of the PDF and CDF is based on the formulas provided by John P Nolan in [5].
Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López;
Maintainer: Javier Royuela del Val [email protected]
[1] Royuela-del-Val J, Simmross-Wattenberg F, Alberola López C (2017). libstable: Fast, Parallel and High-Precision Computation of alpha-stable Distributions in R, C/C++ and MATLAB. Journal of Statistical Software, 78(1), 1-25. doi:10.18637/jss.v078.i01
[2] Chambers JM, Mallows CL, Stuck BW (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association, 71(354), 340-344. doi:10.1080/01621459.1976.10480344
[3] McCulloch JH (1986). Simple Consistent Estimators of Stable Distribution Parameters. Communications in Statistics - Simulation and Computation, 15(4), 1109-1136. doi:10.1080/03610918608812563
[4] Koutrouvelis IA (1981). An Iterative Procedure for the Estimation of the Parameters of Stable Laws. Communications in Statistics - Simulation and Computation, 10(1), 17-28. doi:10.1080/03610918108812189
[5] Nolan JP (1997). Numerical Calculation of Stable Densities and Distribution Functions. Stochastic Models, 13(4), 759-774. doi:10.1080/15326349708807450
# Set alpha, beta, sigma and mu stable parameters in a vector pars <- c(1.5, 0.9, 1, 0) # Generate an abscissas axis and probabilities vector x <- seq(-5, 10, 0.05) p <- seq(0.01, 0.99, 0.01) # Calculate pdf, cdf and quantiles pdf <- stable_pdf(x, pars) cdf <- stable_cdf(x, pars) xq <- stable_q(p, pars) # Generate random values set.seed(1) rnd <- stable_rnd(100, pars) head(rnd) # Estimate the parameters of the skew stable distribution given # the generated sample: # Using the McCulloch's estimator: pars_init <- stable_fit_init(rnd) # Using the Koutrouvelis' estimator, with McCulloch estimation # as a starting point: pars_est_K <- stable_fit_koutrouvelis(rnd, pars_init) # Using maximum likelihood estimator: pars_est_ML <- stable_fit_mle(rnd, pars_est_K) # Using modified maximum likelihood estimator (see [1]): pars_est_ML2 <- stable_fit_mle2d(rnd, pars_est_K)# Set alpha, beta, sigma and mu stable parameters in a vector pars <- c(1.5, 0.9, 1, 0) # Generate an abscissas axis and probabilities vector x <- seq(-5, 10, 0.05) p <- seq(0.01, 0.99, 0.01) # Calculate pdf, cdf and quantiles pdf <- stable_pdf(x, pars) cdf <- stable_cdf(x, pars) xq <- stable_q(p, pars) # Generate random values set.seed(1) rnd <- stable_rnd(100, pars) head(rnd) # Estimate the parameters of the skew stable distribution given # the generated sample: # Using the McCulloch's estimator: pars_init <- stable_fit_init(rnd) # Using the Koutrouvelis' estimator, with McCulloch estimation # as a starting point: pars_est_K <- stable_fit_koutrouvelis(rnd, pars_init) # Using maximum likelihood estimator: pars_est_ML <- stable_fit_mle(rnd, pars_est_K) # Using modified maximum likelihood estimator (see [1]): pars_est_ML2 <- stable_fit_mle2d(rnd, pars_est_K)
A set of functions are provided that perform the parameter estimation of skew stable distributions with different methods.
stable_fit_init(rnd, parametrization = 0L) stable_fit_koutrouvelis(rnd, pars_init = as.numeric(c()), parametrization = 0L)stable_fit_init(rnd, parametrization = 0L) stable_fit_koutrouvelis(rnd, pars_init = as.numeric(c()), parametrization = 0L)
rnd |
Random sample |
parametrization |
Parametrization used for the skew stable distribution, as defined by JP Nolan (1997). By default, parametrization = 0. |
pars_init |
Vector with an initial estimation of the parameters.
|
stable_fit_init() uses McCulloch's method of quantiles [3]. This is usually a good initialization for the rest of the methods.
stable_fit_koutrouvelis() implements Koutrouvellis' method based on the characteristic function [4].
stable_fit_mle() implements a Maximum likelihood estimation.
stable_fit_mle2() implements a modified maximum likelihood estimation as described in [1].
A numeric vector.
Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López
Maintainer: Javier Royuela del Val [email protected]
[1] Royuela-del-Val J, Simmross-Wattenberg F, Alberola López C (2017). libstable: Fast, Parallel and High-Precision Computation of alpha-stable Distributions in R, C/C++ and MATLAB. Journal of Statistical Software, 78(1), 1-25. doi:10.18637/jss.v078.i01
[2] Chambers JM, Mallows CL, Stuck BW (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association, 71(354), 340-344. doi:10.1080/01621459.1976.10480344.
[3] McCulloch JH (1986). Simple Consistent Estimators of Stable Distribution Parameters. Communications in Statistics - Simulation and Computation, 15(4), 1109-1136. doi:10.1080/03610918608812563.
[4] Koutrouvelis IA (1981). An Iterative Procedure for the Estimation of the Parameters of Stable Laws. Communications in Statistics - Simulation and Computation, 10(1), 17-28. doi:10.1080/03610918108812189.
[5] Nolan JP (1997). Numerical Calculation of Stable Densities and Distribution Functions. Stochastic Models, 13(4) 759-774. doi:10.1080/15326349708807450.
# Set alpha, beta, sigma and mu stable parameters in a vector pars <- c(1.5, 0.9, 1, 0) # Generate random values set.seed(1) rnd <- stable_rnd(100, pars) head(rnd) # Estimate the parameters of the skew stable distribution given # the generated sample: # Using the McCulloch's estimator: pars_init <- stable_fit_init(rnd) # Using the Koutrouvelis' estimator, with McCulloch estimation # as a starting point: pars_est_K <- stable_fit_koutrouvelis(rnd, pars_init) # Using maximum likelihood estimator: pars_est_ML <- stable_fit_mle(rnd, pars_est_K) # Using modified maximum likelihood estimator (see [1]): pars_est_ML2 <- stable_fit_mle2d(rnd, pars_est_K)# Set alpha, beta, sigma and mu stable parameters in a vector pars <- c(1.5, 0.9, 1, 0) # Generate random values set.seed(1) rnd <- stable_rnd(100, pars) head(rnd) # Estimate the parameters of the skew stable distribution given # the generated sample: # Using the McCulloch's estimator: pars_init <- stable_fit_init(rnd) # Using the Koutrouvelis' estimator, with McCulloch estimation # as a starting point: pars_est_K <- stable_fit_koutrouvelis(rnd, pars_init) # Using maximum likelihood estimator: pars_est_ML <- stable_fit_mle(rnd, pars_est_K) # Using modified maximum likelihood estimator (see [1]): pars_est_ML2 <- stable_fit_mle2d(rnd, pars_est_K)
Evaluate the PDF or the CDF of the skew stable distribution with parameters
pars = c(alpha, beta, sigma, mu) at the points given in x.
parametrization argument specifies the parametrization used for the distribution
as described by JP Nolan (1997). The default value is parametrization = 0.
tol sets the relative error tolerance (precision) to tol. The default value is tol = 1e-12.
stable_pdf(x, pars, parametrization = 0L, tol = 1e-12)stable_pdf(x, pars, parametrization = 0L, tol = 1e-12)
x |
Vector of points where the pdf will be evaluated. |
pars |
Vector with an initial estimation of the parameters.
|
parametrization |
Parametrization used for the skew stable distribution, as defined by JP Nolan (1997). By default, parametrization = 0. |
tol |
Relative error tolerance (precission) of the calculated values. By default, tol = 1e-12. |
A numeric vector.
Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López
Maintainer: Javier Royuela del Val [email protected]
Nolan JP (1997). Numerical Calculation of Stable Densities and Distribution Functions. Stochastic Models, 13(4) 759-774.
pars <- c(1.5, 0.9, 1, 0) x <- seq(-5, 10, 0.001) pdf <- stable_pdf(x, pars) cdf <- stable_cdf(x, pars) plot(x, pdf, type = "l")pars <- c(1.5, 0.9, 1, 0) x <- seq(-5, 10, 0.001) pdf <- stable_pdf(x, pars) cdf <- stable_cdf(x, pars) plot(x, pdf, type = "l")
Evaluate the quantile function (CDF^-1) of the skew stable distribution
with parameters pars = c(alpha, beta, sigma, mu) at the points given in p.
parametrization argument specifies the parametrization used for the distribution
as described by JP Nolan (1997). The default value is parametrization = 0.
tol sets the relative error tolerance (precission) to tol. The default value is tol = 1e-12.
stable_q(p, pars, parametrization = 0L, tol = 1e-12)stable_q(p, pars, parametrization = 0L, tol = 1e-12)
p |
Vector of points where the quantile function will be evaluated, with 0 < p[i] < 1.0 |
pars |
Vector with an initial estimation of the parameters.
|
parametrization |
Parametrization used for the skew stable distribution, as defined by JP Nolan (1997). By default, parametrization = 0. |
tol |
Relative error tolerance (precission) of the calculated values. By default, tol = 1e-12. |
A numeric vector.
Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López
Maintainer: Javier Royuela del Val [email protected]
stable_rnd(N, pars) generates N random samples of a skew stable distribuiton
with parameters pars = c(alpha, beta, sigma, mu) using the Chambers, Mallows,
and Stuck (1976) method.
stable_rnd(N, pars, parametrization = 0L)stable_rnd(N, pars, parametrization = 0L)
N |
Number of values to generate. |
pars |
Vector with an initial estimation of the parameters.
|
parametrization |
Parametrization used for the skew stable distribution, as defined by JP Nolan (1997). By default, parametrization = 0. |
A numeric vector.
Javier Royuela del Val, Federico Simmross Wattenberg and Carlos Alberola López
Maintainer: Javier Royuela del Val [email protected]
Chambers JM, Mallows CL, Stuck BW (1976). A Method for Simulating Stable Random Variables. Journal of the American Statistical Association, 71(354), 340-344. doi:10.1080/01621459.1976.10480344.
N <- 1000 pars <- c(1.25, 0.95, 1.0, 0.0) rnd <- stable_rnd(N, pars) hist(rnd)N <- 1000 pars <- c(1.25, 0.95, 1.0, 0.0) rnd <- stable_rnd(N, pars) hist(rnd)