ASA266
Estimating the Parameters of a Dirichlet PDF

ASA266 is a MATLAB library which estimates the parameters of a Dirichlet probability density function.

ASA266 is Applied Statistics Algorithm 266. Source code for many Applied Statistics Algorithms is available through STATLIB.

The assumption is that a given process is governed by a Dirichlet distribution with parameters ALPHA(I), I = 1 to N, positive quantities which are required to sum to 1. Each observation of the process yields a vector of N data values. After a number of observations of this sort, it is desired to estimate the the underlying parameters ALPHA of the Dirichlet distribution.

There are a considerable number of routines required to get DIRICH to work. In some cases, there are several versions of the routines, and they all were included, in order to provide a way to check results. Most of the routines are themselves Applied Statistics Algorithms, and their source code is available through STATLIB.

Also included is a routine DIRICHLET_SAMPLE, with which experiments can be carried out. Values for the parameters ALPHA can be chosen, and data generated by DIRICHLET_SAMPLE. Then DIRICH can analyze this data and attempt to determine the values of ALPHA.

Another routine, DIRICHLET_MIX_SAMPLE, allows you to sample a probability distribution that is a weighted mixture of Dirichlet distributions.

Licensing:

The computer code and data files described and made available on this web page are distributed under the GNU LGPL license.

Languages:

ASA266 is available in a C version and a C++ version and a FORTRAN90 version and a MATLAB version.

Related Data and Programs:

ASA032, a MATLAB library which evaluates the incomplete Gamma integral.

ASA066, a MATLAB library which evaluates the percentage points of the normal distribution.

ASA091, a MATLAB library which evaluates the percentage points of the Chi-Squared distribution.

ASA103, a MATLAB library which evaluates the digamma or psi function.

ASA111, a MATLAB library which evaluates the percentage points of the normal distribution.

ASA121, a MATLAB library which evaluates the trigamma function.

ASA147, a MATLAB library which evaluates the incomplete Gamma function.

ASA239, a MATLAB library which evaluates the percentage points of the Chi-Squared distribution and the incomplete Gamma function.

ASA241, a MATLAB library which evaluates the percentage points of the normal distribution.

ASA245, a MATLAB library which evaluates the logarithm of the Gamma function.

asa266_test

NORMAL, a MATLAB library which samples the normal distribution.

PROB, a MATLAB library which evaluates the PDF, CDF, mean and variance for a number of probability density functions.

TEST_VALUES, a MATLAB library which contains sample values for a number of distributions.

TOMS291, a MATLAB library which evaluates the logarithm of the Gamma function.

UNIFORM, a MATLAB library which samples the uniform distribution.

Reference:

AG Adams,
Algorithm 39: Areas Under the Normal Curve,
Computer Journal,
Volume 12, Number 2, May 1969, pages 197-198.
Joachim Ahrens, Ulrich Dieter,
Computer Methods for Sampling from Gamma, Beta, Poisson and Binomial Distributions,
Computing,
Volume 12, Number 3, September 1974, pages 223-246.
Joachim Ahrens, Ulrich Dieter,
Generating Gamma Variates by a Modified Rejection Technique,
Communications of the ACM,
Volume 25, Number 1, January 1982, pages 47-54.
Jerry Banks, editor,
Handbook of Simulation,
Wiley, 1998,
ISBN: 0471134031,
LC: T57.62.H37.
JD Beasley, SG Springer,
Algorithm AS 111: The Percentage Points of the Normal Distribution,
Applied Statistics,
Volume 26, Number 1, 1977, pages 118-121.
Jose Bernardo,
Algorithm AS 103: Psi ( Digamma ) Function,
Applied Statistics,
Volume 25, Number 3, 1976, pages 315-317.
Donald Best, DE Roberts,
Algorithm AS 91: The Percentage Points of the Chi-Squared Distribution,
Applied Statistics,
Volume 24, Number 3, 1975, pages 385-390.
G Bhattacharjee,
Algorithm AS 32: The Incomplete Gamma Integral,
Applied Statistics,
Volume 19, Number 3, 1970, pages 285-287.
William Cody, Kenneth Hillstrom,
Chebyshev Approximations for the Natural Logarithm of the Gamma Function,
Mathematics of Computation,
Volume 21, Number 98, April 1967, pages 198-203.
William Cody, Anthony Strecok, Henry Thacher,
Chebyshev Approximations for the Psi Function,
Mathematics of Computation,
Volume 27, Number 121, January 1973, pages 123-127.
John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thacher, Christoph Witzgall,
Computer Approximations,
Wiley, 1968,
LC: QA297.C64.
David Hill, Algorithm AS 66: The Normal Integral,
Applied Statistics,
Volume 22, Number 3, 1973, pages 424-427.
Cornelius Lanczos,
A precision approximation of the gamma function,
SIAM Journal on Numerical Analysis, B,
Volume 1, 1964, pages 86-96.
Chi Leung Lau,
Algorithm AS 147: A Simple Series for the Incomplete Gamma Integral,
Applied Statistics,
Volume 29, Number 1, 1980, pages 113-114.
Allan Mcleod,
Algorithm AS 245: A Robust and Reliable Algorithm for the Logarithm of the Gamma Function,
Applied Statistics,
Volume 38, Number 2, 1989, pages 397-402.
A. Naryanan,
Algorithm AS 266: Maximum Likelihood Estimation of the Parameters of the Dirichlet Distribution,
Applied Statistics,
Volume 40, Number 2, 1991, pages 365-374.
Malcolm Pike, David Hill,
Algorithm 291: Logarithm of Gamma Function,
Communications of the ACM,
Volume 9, Number 9, September 1966, page 684.
BE Schneider,
Algorithm AS 121: Trigamma Function,
Applied Statistics,
Volume 27, Number 1, 1978, pages 97-99.
BL Shea,
Algorithm AS 239: Chi-squared and Incomplete Gamma Integral,
Applied Statistics,
Volume 37, Number 3, 1988, pages 466-473.
Michael Wichura,
Algorithm AS 241: The Percentage Points of the Normal Distribution,
Applied Statistics,
Volume 37, Number 3, 1988, pages 477-484.

Source Code:

alngam.m, computes the logarithm of the gamma function;
alnorm.m, computes the cumulative density of the standard normal distribution.;
alogam.m, computes the logarithm of the Gamma function;
digamma.m, calculates DIGAMMA ( X ) = d ( LOG ( GAMMA ( X ) ) ) / dX;
dirichlet_check.m, checks the parameters of the Dirichlet PDF;
dirichlet.m, estimates the parameters of a Dirichlet distribution;
dirichlet_mean.m, returns the means of the Dirichlet PDF;
dirichlet_mix_mean.m, returns the means of a Dirichlet mixture PDF;
dirichlet_mix_sample.m, samples a Dirichlet mixture PDF;
dirichlet_sample.m, samples the Dirichlet PDF;
dirichlet_variance.m, returns the variances of the Dirichlet PDF;
exponential_01_sample.m, samples the Exponential PDF with parameter 1;
exponential_cdf_inv.m, inverts the Exponential CDF;
gamain.m, computes the incomplete gamma ratio;
gamma_sample.m, samples the gamma PDF;
gammad.m, computes the Incomplete Gamma Integral;
gammds.m, computes the incomplete Gamma integral;
lngamma.m, computes Log(Gamma(X)) using a Lanczos approximation;
normal_01_sample.m, samples the standard normal PDF;
normp.m, computes the CDF of the standard normal distribution;
nprob.m, computes the CDF of the standard normal distribution;
ppchi2.m, evaluates the percentage points of the Chi-squared PDF;
ppnd.m, produces the normal deviate value corresponding to lower tail area = P;
r4_normal_01_cdf_inverse.m, inverts the standard normal CDF;
r4poly_val_horner.m, evaluates a polynomial in standard form.;
r8_gamma_log.m, calculates Log(Gamma(X));
r8_normal_01.m, returns a unit pseudonormal R8;
r8_normal_01_cdf_inverse.m, inverts the standard normal CDF;
r8_psi.m, evaluates the function Psi(X);
r8_uniform_01.m, returns a unit pseudorandom R8;
r8col_mean.m, returns the column means of an R8COL;
r8col_variance.m, returns the column variances of an R8COL;
r8poly_val_horner.m, evaluates a polynomial in standard form.;
r8vec_unit_sum.m, normalizes an R8VEC to have unit norm;
timestamp.m, returns the current YMDHMS date as a timestamp;
trigamma.m, calculates trigamma(x) = d**2 log(gamma(x)) / dx**2;