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 FORTRAN77 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.

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:

1. AG Adams,
   Algorithm 39: Areas Under the Normal Curve,
   Computer Journal,
   Volume 12, Number 2, May 1969, pages 197-198.
2. 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.
3. 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.
4. Jerry Banks, editor,
   Handbook of Simulation,
   Wiley, 1998,
   ISBN: 0471134031,
   LC: T57.62.H37.
5. JD Beasley, SG Springer,
   Algorithm AS 111: The Percentage Points of the Normal Distribution,
   Applied Statistics,
   Volume 26, Number 1, 1977, pages 118-121.
6. Jose Bernardo,
   Algorithm AS 103: Psi ( Digamma ) Function,
   Applied Statistics,
   Volume 25, Number 3, 1976, pages 315-317.
7. 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.
8. G Bhattacharjee,
   Algorithm AS 32: The Incomplete Gamma Integral,
   Applied Statistics,
   Volume 19, Number 3, 1970, pages 285-287.
9. 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.
10. William Cody, Anthony Strecok, Henry Thacher,
    Chebyshev Approximations for the Psi Function,
    Mathematics of Computation,
    Volume 27, Number 121, January 1973, pages 123-127.
11. John Hart, Ward Cheney, Charles Lawson, Hans Maehly, Charles Mesztenyi, John Rice, Henry Thacher, Christoph Witzgall,
    Computer Approximations,
    Wiley, 1968,
    LC: QA297.C64.
12. David Hill, Algorithm AS 66: The Normal Integral,
    Applied Statistics,
    Volume 22, Number 3, 1973, pages 424-427.
13. Cornelius Lanczos,
    A precision approximation of the gamma function,
    SIAM Journal on Numerical Analysis, B,
    Volume 1, 1964, pages 86-96.
14. Chi Leung Lau,
    Algorithm AS 147: A Simple Series for the Incomplete Gamma Integral,
    Applied Statistics,
    Volume 29, Number 1, 1980, pages 113-114.
15. 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.
16. 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.
17. Malcolm Pike, David Hill,
    Algorithm 291: Logarithm of Gamma Function,
    Communications of the ACM,
    Volume 9, Number 9, September 1966, page 684.
18. BE Schneider,
    Algorithm AS 121: Trigamma Function,
    Applied Statistics,
    Volume 27, Number 1, 1978, pages 97-99.
19. BL Shea,
    Algorithm AS 239: Chi-squared and Incomplete Gamma Integral,
    Applied Statistics,
    Volume 37, Number 3, 1988, pages 466-473.
20. 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_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;

Examples and Tests:

• asa266_test.m, calls all the tests.
• asa266_test_output.txt, the output file.
• asa266_test01.m, tests ALNORM, NORMP, NPROB.
• asa266_test02.m, tests PPND, R4_NORMAL_01_CDF_INVERSE, R8_NORMAL_01_CDF_INVERSE.
• asa266_test03.m, tests DIGAMMA, R8_PSI.
• asa266_test04.m, tests TRIGAMMA.
• asa266_test05.m, tests ALNGAM, ALOGAM, R8_GAMMA_LOG, LNGAMMA;
• asa266_test06.m, tests GAMAIN, GAMMDS, GAMMAD.
• asa266_test07.m, tests PPCHI2.
• asa266_test08.m, tests DIRICHLET_ESTIMATE, DIRICHLET_MEAN, DIRICHLET_VARIANCE.
• asa266_test09.m, tests DIRICHLET_ESTIMATE, DIRICHLET_MEAN, DIRICHLET_VARIANCE, DIRICHLET_SAMPLE.
• asa266_test10.m, tests DIRICHLET_MIX_SAMPLE, DIRICHLET_MIX_MEAN, DIRICHLET_MIX_VARIANCE.

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