asa266

asa266, an Octave code which estimates the parameters of a Dirichlet probability density function (PDF).

This is a version of Applied Statistics Algorithm 266.

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.

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 MIT license

Languages:

asa266 is available in a C version and a C++ version and a Fortran90 version and a MATLAB version and an Octave version.

Related Data and Programs:

asa266_test

asa032, an Octave code which evaluates the incomplete Gamma integral.

asa066, an Octave code which evaluates the percentage points of the normal distribution.

asa091, an Octave code which evaluates the percentage points of the Chi-Squared distribution.

asa103, an Octave code which evaluates the digamma or psi function.

asa111, an Octave code which evaluates the percentage points of the normal distribution.

asa121, an Octave code which evaluates the trigamma function.

asa147, an Octave code which evaluates the incomplete Gamma function.

asa239, an Octave code which evaluates the percentage points of the Chi-Squared distribution and the incomplete Gamma function.

asa241, an Octave code which evaluates the percentage points of the normal distribution.

asa245, an Octave code which evaluates the logarithm of the Gamma function.

normal, an Octave code which samples the normal distribution.

prob, an Octave code which evaluates the PDF, CDF, mean and variance for a number of probability density functions.

test_values, an Octave code which contains sample values for a number of distributions.

toms291, an Octave code which evaluates the logarithm of the Gamma function.

uniform, an Octave code 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:

• 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;
• 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;
• trigamma.m, calculates trigamma(x) = d**2 log(gamma(x)) / dx**2;