Cupid

Object-oriented MATLAB toolbox for working with probability distributions

[1] Overview

This toolbox can be helpful for working with univariate probability distributions. For any given distribution, it can compute all of the standard quantities: density and cumulative density functions, hazard function, mean, variance, skewness, kurtosis, etc, and it can generate random numbers. It can also estimate parameter values (i.e., by maximum likelihood and several other methods). Parameters estimates can be constrained within a single distribution, and it is possible to estimate the parameters of multiple distributions jointly while specifying constraints among the parameters of different distributions. More than 40 standard distributions (e.g., normal, exponential, gamma, ...) have already been implemented, and it is easy to add new ones. (Any distribution for which you know either the PDF or the CDF can be added; everything else can be computed numerically from either of those functions.)

A somewhat novel feature of CUPID is the ability to create what might be called "derived" distributions. For example, you can form a new distribution by truncating one of the standard ones (e.g., a standard normal distribution restricted to the range of -2 to +3). Within the object-oriented framework, this truncated distribution is simply a new distribution object, so its functions can be computed and parameters estimated just like those of the standard distributions. In addition to truncation, you can form derived distributions by transformations of standard distributions (e.g., linear, inverse, exponential, power, or log). You can also form derived distribution that involve independent random variables from two or more standard distributions. For example, this can be done by convolution (i.e., the distribution of the sum of random variables from two or more standard distributions), by a probability mixture of two or more distributions, or by an order statistic of two or more random variables [e.g., the minimum a standard normal and a uniform(0,1)].

Moreover, the object-oriented framework is fully recursive, so you can form derived distributions from other derived distributions rather than from standard ones (e.g., form the distribution of 1/sqrt(x), where x is uniform over some range, or form the convolution of a mixture distribution and a log-transformed distribution). All of the same functions (e.g., PDF) and operations (e.g., parameter estimation) that are available for the standard distributions are also available for the derived distributions, although the required numerical computations might be much slower and less accurate.

[2] Standard Distributions

Here are most of the standard probability distributions that have been implemented so far (several alternative parameterizations are available for some of them):

Beta
Cauchy
Chi
ChiSq
ChiSqNoncentral
DblMon (double monomial)
ExGaussian (sum of exponential and normal)
Exponential
ExpSum (sum of exponentials)
ExpSumT (truncated sum of exponentials)
ExtrVal1 (extreme value, type 1)
ExtrVal2 (extreme value, type 2)
ExWald (sum of exponential and Wald)
F
FNoncentral
Gamma
GenNor1 (general error distribution, type 1)
GenNor2 (general error distribution, type 2)
HyperbolicTan
JohnsonSB
JohnsonSU
Laplace
Logistic
Lognormal
NakaRush (Naka-Rushton)
Normal
Pareto
Quantal
Quick
r
Rayleigh
Recinormal (1/normal)
RNGamma
rNoncentral
Rosin
SkewNor (skew normal)
t
tNoncentral
Triangular
Uniform
VonMises
Wald
Weibull

[3] Derived Distributions

Here are most of the types of derived distributions that have been implemented so far: (RVs are the distributions from which these are derived).

AddTrans (add a constant to 1 RV)
ArcsinTrans (arcsin transformation of 1 RV)
Convolution (sum of 2+ RVs)
Difference (difference of 2 RVs)
ExpTrans (exponential transformation of 1 RV)
InfMix (infinite mixture; i.e., a parameter of one distribution is not fixed but instead varies according to some other distribution)
InverseTrans (inverse transformation of 1 RV)
LinearTrans (linear transformation of 1 RV)
LogTrans (log transformation of 1 RV)
MinBound (distribution implied by Frechet's bound)
Mixture (probability mixture of 2+ RVs)
MonotoneTrans (any monotonic transformation of 1 RV)
MultTrans (multiplicative transformation of 1 RV)
Order (k'th order statistic of any 2+ RVs)
OrderIID (k'th order statistic of K IID RVs)
PowerTrans (power transformation of 1 RV)
Product (product of 2 RVs)
Ratio (ratio of 2 RVs)
TruncatedX (truncated version of 1 RV)

[4] Functions and Operations Available for Any Distribution

Here are most of the values that can be computed for any distribution, standard or derived:

Probability density and cumulative distribution functions
Spline approximations of PDF and CDF, sometimes useful for speed
Inverse cumulative probability
Mean, standard deviation, variance, skewness, kurtosis, coefficient of variation
Generate random numbers
Log likelihood of a set of observations
Moments: Raw and central; unconditional and conditional
Hazard function
Moment generating function
Chi-square and Kolmogorov-Smirnov one-sample tests
Maximum likelihood parameter estimates & standard errors based on Fisher information
Parameter estimates based on method of moments
Parameter estimates based on minimum chi-square for bin frequencies
Parameter estimates based on closest match to estimated percentiles
Parameter estimates to produce a desired probability within a certain range
Parameter estimates for probit models with yes/no and m-AFC tasks

[5] Additional requirements

The special-purpose multivariate random number generator class RandGen also requires the ExtractNameVal package from https://github.com/milleratotago/ExtractNameVal.

[6] Installation

The easiest way to install Cupid is to download the latest version of the MATLAB toolbox file Cupid*.mltbx and then let MATLAB itself manage the installation. In a Windows environment, simply double-click the toolbox file, and a MATLAB will start and guide you through the installation.

  This program 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 3 of the License, or
  any later version.

  This program 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.

  You should have received a copy of the GNU General Public License
  along with this program (00License.txt). If not, see 
  <http://www.gnu.org/licenses/>.