R/tsal-param.R

In tsallisqexp: Tsallis q-Exp Distribution

# R functions for use with the q-exponential distribution of Tsallis

# Copyright (c) 2007, Cosma Shalizi, cshalizi@cmu.edu or cosma.shalizi@gmail.com

#####  This is free software; you can redistribute it and/or modify
#####  it under the terms of the GNU General Public License as published by
#####  the Free Software Foundation; either version 2 of the License, or
#####  (at your option) any later version.
#####
#####  This software is distributed in the hope that it will be useful,
#####  but WITHOUT ANY WARRANTY; without even the implied warranty of
#####  MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
#####  GNU General Public License for more details.
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#####  You should have received a copy of the GNU General Public License
#####  along with this software; if not, write to the Free Software
#####  Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA  02111-1307  USA

# Please contact me at the above e-mail addresses for bug reports, questions,
# etc.  Please also contact me if you use this code in a scientific paper!

# This is a special case of the generalized Pareto distribution (type II), but
# arguably of independent interest.
# The distribution is defined through the upper cumulative or complementary
# distribution function, a.k.a. survival function,
### Pr(X>=x) = (1-(1-q)x/kappa)^(1/(1-q))
# It is convenient to introduce a re-parameterization
### shape = -1/(1-q)
### scale = shape*kappa
# which makes the relationship to the Pareto clearer, and eases estimation.
# Users can employ either parameterization, with shape/scale as the default,
# but q/kappa over-riding them.  (No warning is given if a user provides both
# sets of parameters.)
# For the derivation of the maximum likelihood estimators, please see the
# manuscript "Maximum Likelihood Estimation for q-Exponential (Tsallis)
# Distributions", which should accompany this file; if not it is available from
# http://bactra.org/research/tsallis-MLE/, along with the latest version of
# this code.  The manuscript is also available from
# http://arxiv.org/abs/math.ST/0701854

# Functions are divided into four parts:
# functions for the distribution itself, conforming to the R standards;
# functions to estimate the parameters;
# functions which illustrate the numerical accuracy of the implementation;
# functions to go from one parametrization to the other.


# Functions in this file:
# Distribution-related (per R standards):
# dtsal				Probability density
# ptsal				Cumulative probability
# qtsal				Quantiles
# rtsal				Random variate generation
# Parameter Estimation:
# tsal.loglik			Log-likelihood calculation
# tsal.fit			Estimate parameters; wrapper for detailed
#				methods.  Call this, not its subroutines
# tsal.mle.direct		Direct maximization of likelihood
# tsal.mle.equation		Find MLE by solving estimating equation; default
# tsal.est.shape.from.scale	MLE of shape parameter given scale parameter
# tsal.est.scale.from.shape	MLE of scale parameter given shape parameter
# tsal.curvefit			Find parameters by fitting a curve to the
#				empirical distribution function; avoid
# tsal.bootstrap.errors		Bootstrap standard errors, biases for MLE
# tsal.fisher			Fisher information matrix, for asymptotics
# tsal.mean			Calculate the expectation value
# tsal.total.magnitude		Total magnitude of a population (estimated)
# Implementation Testing:
# plot.tsal.quantile.transform	Illustrates relative numerical inaccuracy in
#				ptsal and qtsal, which should be inverses
# plot.tsal.LR.distribution	Calculates the log likelihood ratio for
#				estimated vs. fixed true parameters, and plots
#				it against the theoretical asymptotic
#				distribution (chi^2 with 2 d.f.).
# Censored Data:
# dtsal.tail			Probability density (tail-conditional)
# ptsal.tail			Cumulative probability (tail-conditional)
# qtsal.tail			Quantiles (tail-conditional)
# rtsal.tail			Random variate generation (from the tail)
# Parameter Conversion:
# tsal.shape.from.q		Get shape parameter from q
# tsal.scale.from.qk		Get scale parameter from q, kappa
# tsal.q.from.shape		Get q from from shape parameter
# tsal.kappa.from.ss		Get kappa from shape, scale
# tsal.ss.from.qk		Get shape, scale from q, kappa (as pairs)
# tsal.qk.from.ss		Get q, kappa from shape, scale (as pairs




#################################################################
########        PARAMETER CONVERSION FUNCTIONS          #########
#################################################################

# Users will generally have little reason to invoke these functions

# Calculate generalized-Pareto shape parameter from q parameter
# Input: a real value
# Output: a real value
tsal.shape.from.q <- function(q) {
  shape <- -1/(1-q)
  return(shape)
}

# Calculate generalized-Pareto scale parameter from q and kappa parameters
# Input: two real values
# Output: a real value
tsal.scale.from.qk <- function(q,kappa) {
  shape <- tsal.shape.from.q(q)
  scale <- shape*kappa
  return(scale)
}

# Calculate q parameter from shape parameter
# Input: a real value
# Output: a real value
tsal.q.from.shape <- function(shape) {
  q <- 1+1/shape
  return(q)
}

# Calculate kappa parameter from shape and scale parameters
# Input: two real values
# Output: a real value
tsal.kappa.from.ss <- function(shape,scale) {
  kappa <- scale/shape
  return(kappa)
}

# Calculate shape & scale parameters from q & kappa parameters
# Input: two real values
# Output: vector of two real values
tsal.ss.from.qk <- function(q,kappa) {
  ss <- c(tsal.shape.from.q(q),tsal.scale.from.qk(q,kappa))
  return(ss)
}

# Calculate q & kappa parameters from shape & scale parameters
# Input: two real values
# Output: vector of two real values
tsal.qk.from.ss <- function(shape,scale) {
  qk <- c(tsal.q.from.shape(shape),tsal.kappa.from.ss(shape,scale))
  return(qk)
}