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randraw.m
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randraw.m
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function varargout = randraw(distribName, distribParams, varargin)
%
% EFFICIENT RANDOM VARIATES GENERATOR
%
% See alphabetical list of the supported distributions below (over 50 distributions)
%
% 1) randraw
% presents general help.
% 2) randraw( distribName )
% presents help for the specific distribution defined
% by usage string distribName (see table below).
% 3) Y = randraw( distribName, distribParams, sampleSize );
% returns array Y of size = sampleSize of random variates from distribName
% distribution with parameters distribParams
%
% ALPHABETICAL LIST OF THE SUPPORTED DISTRIBUTIONS:
% ____________________________________________________________________
% | DISTRIBUTION NAME | USAGE STRING |
% |___________________________________________|________________________|
% | Alpha | 'alpha' |
% | Anglit | 'anglit' |
% | Antilognormal | 'lognorm' |
% | Arcsin | 'arcsin' |
% | Bernoulli | 'bern' |
% | Bessel | 'bessel' |
% | Beta | 'beta' |
% | Binomial | 'binom' |
% | Bradford | 'bradford' |
% | Burr | 'burr' |
% | Cauchy | 'cauchy' |
% | Chi | 'chi' |
% | Chi-Square (Non-Central) | 'chisqnc' |
% | Chi-Square (Central) | 'chisq' |
% | Cobb-Douglas | 'lognorm' |
% | Cosine | 'cosine' |
% | Double-Exponential | 'laplace' |
% | Erlang | 'erlang' |
% | Exponential | 'exp' |
% | Extreme-Value | 'extrval' |
% | F (Central) | 'f' |
% | F (Non-Central) | 'fnc' |
% | Fisher-Tippett | 'extrval' |
% | Fisk | 'fisk' |
% | Frechet | 'frechet' |
% | Furry | 'furry' |
% | Gamma | 'gamma' |
% | Generalized Inverse Gaussian | 'gig' |
% | Generalized Hyperbolic | 'gh' |
% | Geometric | 'geom' |
% | Gompertz | 'gompertz' |
% | Gumbel | 'gumbel' |
% | Half-Cosine | 'hcos' |
% | Hyperbolic Secant | 'hsec' |
% | Hypergeometric | 'hypergeom' |
% | Inverse Gaussian | 'ig' |
% | Laplace | 'laplace' |
% | Logistic | 'logistic' |
% | Lognormal | 'lognorm' |
% | Lomax | 'lomax' |
% | Lorentz | 'lorentz' |
% | Maxwell | 'maxwell' |
% | Nakagami | 'nakagami' |
% | Negative Binomial | 'negbinom' |
% | Normal | 'norm' |
% | Normal-Inverse-Gaussian (NIG) | 'nig' |
% | Pareto | 'pareto' |
% | Pareto2 | 'pareto2' |
% | Pascal | 'pascal' |
% | Planck | 'planck' |
% | Poisson | 'po' |
% | Quadratic | 'quadr' |
% | Rademacher | 'rademacher' |
% | Rayleigh | 'rayl' |
% | Rice | 'rice' |
% | Semicircle | 'semicirc' |
% | Skellam | 'skellam' |
% | Student's-t | 't' |
% | Triangular | 'tri' |
% | Truncated Normal | 'normaltrunc' |
% | Tukey-Lambda | 'tukeylambda' |
% | U-shape | 'u' |
% | Uniform (continuous) | 'uniform' |
% | Von Mises | 'vonmises' |
% | Wald | 'wald' |
% | Weibull | 'weibull' |
% | Wigner Semicircle | 'wigner' |
% | Yule | 'yule' |
% | Zeta | 'zeta' |
% | Zipf | 'zipf' |
% |___________________________________________|________________________|
% Version 2.0 - August 2007
% 1) New distributions support: Nakagami and Rician !
% 2) Small typo corrections in comments
% Version 1.8 - February 2007
% GIG distribution (thanks to Mr. Demetris Lamnisos)
% Computational exceptions in the reparameterized GIG generation were fixed
% Version 1.7 - December 2006
% GIG distribution (thanks to Dr. Junbin Gao)
% Computational exceptions in the reparameterized GIG generation were fixed
% Version 1.6 - September 2006
% Exception handling: BINOMIAL distribution - special case for n*p~=0
% Geometric distibution: additional note in help section
% Version 1.5 - December 2005
% 'true' and 'false' functions were replased by ones and zeros to support Matlab releases
% below 6.5
% Version 1.4 - September 2005 -
% Bugs fix:
% 1) GAMMA distribution (thanks to Earl Lawrence):
% special case for a<1
% 2) GIG distribution (thanks to Panagiotis Braimakis):
% typo in help
% code adjustment to overcome possible computational overflows
% 3) CHI SQUARE distribution
% typo in help
% Version 1.3 - July 2005 -
% Bug fix:
% Typo in GIG distribution generation:
% should be 'out' instead of 'x' in lines 1852 and 1858
% Version 1.2 - May 2005 -
% Bugs fix:
% 1) Poisson distribution did not work for lambda < 21.4. Typo ( ti instead of t )
% 2) GIG distribution: support to chi=0 or psi=0 cases
% 3) Beta distribution: column sampleSize
% 4) Cauchy distribution: typo in example
% 5) Chi distribution: typo in example
% 6) Non-central F distribution: number of input parameters
% 7) INVERSE GAUSSIAN (IG) distribution: typo in example
%
% Version 1.1 - April 2005 - Bug fix: Generation from binomial distribution using only 'binomial'
% usage string was changed to 'binom' ( 'binomial' works too ).
% Version 1.0 - March 2005 - Initial version
% Alex Bar Guy & Alexander Podgaetsky
% alex@wavion.co.il
% These programs are distributed in the hope that they will be useful,
% but WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.
% Any comments and suggestions please send to:
% alex@wavion.co.il
% Reference links:
% 1) http://mathworld.wolfram.com/topics/StatisticalDistributions.html
% 2) http://en.wikipedia.org/wiki/Category:Probability_distributions
% 3) http://www.brighton-webs.co.uk/index.asp
% 4) http://www.jstatsoft.org/v11/i03/v11i03.pdf
% 5) http://www.quantlet.com/mdstat/scripts/csa/html/node236.html
funcName = mfilename;
if nargin == 0
help(funcName);
return;
elseif nargin == 1
runMode = 'distribHelp';
elseif nargin == 2
runMode = 'genRun';
sampleSize = [1 1];
else
runMode = 'genRun';
sampleSize = [varargin{1:end}];
end
distribNameInner = lower( distribName( ~isspace( distribName ) ) );
if strcmp(runMode, 'distribHelp')
fid = fopen( [ funcName '.m' ], 'r' );
printHelpFlag = 0;
while 1
tline = fgetl( fid );
if ~ischar( tline )
fprintf( '\n Unknown distribution name ''%s''.\n', distribName );
break;
end
if ~isempty( strfind( tline, [ 'END ', distribNameInner,' HELP' ] ) )
printHelpFlag = 0;
break;
end
if printHelpFlag
startPosition = strfind( tline, ' % ' ) + 3;
printLine = tline( startPosition : end );
if ~strcmp( funcName, 'randraw' )
indxs = strfind( printLine, 'randraw' );
while ~isempty( indxs )
headLine = printLine( 1:indxs(1)-1 );
tailLine = printLine( indxs(1)+7:end );
printLine = [ headLine, funcName, tailLine ];
indxs = strfind( printLine, 'randraw' );
end
end
pause(0.02);
fprintf( '\n%s', printLine );
end
if ~isempty( strfind( tline, [ 'START ', distribNameInner,' HELP' ] ) )
printHelpFlag = 1;
end
end
fprintf( '\n\n' );
fclose( fid );
if nargout > 0
varargout{1} = [];
end
return;
end
if length(sampleSize) == 1
sampleSize = [ sampleSize, 1 ];
end
if strcmp(runMode, 'genRun')
runExample = 0;
plotFlag = 0;
dbclear if warning;
out = [];
if prod(sampleSize) > 0
switch lower( distribNameInner )
case {'alpha'}
% START alpha HELP
% THE ALPHA DISTRIBUTION
%
% pdf(y) = b*normpdf(a-b./y) ./ (y.^2*normcdf(a)); y>0; a>0; b>0;
% cdf(y) = normcdf(a-b./y)/normcdf(a); y>0; a>0; b>0;
% where normpdf(x) = 1/sqrt(2*pi) * exp(-1/2*x.^2); is the standard normal PDF
% normcdf(x) = 0.5*(1+erf(y/sqrt(2))); is the standard normal CDF
%
% PARAMETERS:
% a - shape parameter (a>0)
% b - shape parameter (b>0)
%
% SUPPORT:
% y, y>0
%
% CLASS:
% Continuous skewed distributions
%
% USAGE:
% randraw('alpha', [], sampleSize) - generate sampleSize number
% of variates from Alpha distribution with shape parameters a and b;
% randraw('alpha') - help for Alpha distribution;
%
% EXAMPLES:
% 1. y = randraw('alpha', [1 2], [1 1e5]);
% 2. y = randraw('alpha', [2 3], 1, 1e5);
% 3. y = randraw('alpha', [10 50], 1e5 );
% 4. y = randraw('alpha', [20.5 30.5], [1e5 1] );
% 5. randraw('alpha');
% END alpha HELP
% References:
% 1. doc erf
checkParamsNum(funcName, 'Alpha', 'alpha', distribParams, [2]);
a = distribParams(1);
b = distribParams(2);
validateParam(funcName, 'Alpha', 'alpha', '[a, b]', 'a', a, {'> 0'});
validateParam(funcName, 'Alpha', 'alpha', '[a, b]', 'b', b, {'> 0'});
out = b ./ ( a - norminv(normcdf(a)*rand(sampleSize)) );
case {'anglit'}
% START anglit HELP
% THE ANGLIT DISTRIBUTION
%
% Standard form of anglit distribution:
% pdf(y) = sin(2*y+pi/2); -pi/4 <= y <= pi/4;
% cdf(y) = sin(y+pi/4).^2; -pi/4 <= y <= pi/4;
%
% Mean = Median = Mode = 0;
% Variance = (pi/4)^2 - 0.5;
%
% General form of anglit distribution:
% pdf(y) = sin(pi/2*(y-t)/s+pi/2); t-s <= y <= t+s; s>0
% cdf(y) = sin(pi/4*(y-t)/s+pi/4).^2; t-s <= y <= t+s; s>0
%
% Mean = Median = Mode = t;
% Variance = ???????;
%
% PARAMETERS:
% t - location
% s -scale; s>0
%
% SUPPORT:
% y, -pi/4 <= y <= pi.4 - standard Anglit distribution
% or
% y, t-s <= y <= t+s - generalized Anglit distribution
%
% CLASS:
% Continuous distributions
%
% USAGE:
% randraw('anglit', [], sampleSize) - generate sampleSize number
% of variates from standard Anglit distribution;
% randraw('anglit', [t, s], sampleSize) - generate sampleSize number
% of variates from generalized Anglit distribution
% with location parameter 't' and scale parameter 's';
% randraw('anglit') - help for Anglit distribution;
%
% EXAMPLES:
% 1. y = randraw('anglit', [], [1 1e5]);
% 2. y = randraw('anglit', [], 1, 1e5);
% 3. y = randraw('anglit', [], 1e5 );
% 4. y = randraw('anglit', [10 3], [1e5 1] );
% 5. randraw('anglit');
%
% END anglit HELP
checkParamsNum(funcName, 'Anglit', 'anglit', distribParams, [0, 2]);
if numel(distribParams)==2
t = distribParams(1);
s = distribParams(2);
validateParam(funcName, 'Anglit', 'anglit', '[t, s]', 's', s, {'> 0'});
else
t = 0;
s = pi/4;
end
out = t + s * (4/pi*asin(sqrt(rand(sampleSize)))-1);
case {'arcsin'}
% START arcsin HELP
% THE ARC-SINE DISTRIBUTION
%
% pdf(y) = 1 ./ (pi*sqrt(y.*(1-y))); 0<y<1;
% cdf(y) = 2*asin(sqrt(y))/pi; 0<y<1;
%
% Mean = 0.5;
% Variance = 0.125;
%
% PARAMETERS:
% None
%
% SUPPORT:
% y, 0<y<1
%
% CLASS:
% Continuous symmetric distributions
% NOTES:
% The arc-sine distribution is a special case of the beta distribution
% with both parameters equal to 1/2. The generalized arc-sine distribution
% is the special case of the beta distribution where the two parameters sum
% to 1 but are not necessarily equal to 1/2.
%
% USAGE:
% randraw('arcsin', [], sampleSize) - generate sampleSize number
% of variates from the Arc-sine distribution;
% randraw('arcsin') - help for the Arc-sine distribution;
%
% EXAMPLES:
% 1. y = randraw('arcsin', [], [1 1e5]);
% 2. y = randraw('arcsin', [], 1, 1e5);
% 3. y = randraw('arcsin', [], 1e5 );
% 4. randraw('arcsin');
% SEE ALSO:
% U distribution
% END arcsin HELP
checkParamsNum(funcName, 'Arcsin', 'arcsin', distribParams, 0);
out = sin( rand(sampleSize)*pi/2 ).^2;
case {'bernoulli', 'bern'}
% START bernoulli HELP START bern HELP
% THE BERNOULLI DISTRIBUTION
%
% pdf(y) = p.^y .* (1-p).^(1-y);
% cdf(y) = (y==0)*(1-p) + (y==1)*1;
%
% PARAMETERS:
% p is a probability of success; (0<p<1)
%
% SUPPORT:
% y = [0 1];
%
% CLASS:
% Discrete distributions
%
% USAGE:
% randraw('bern', p, sampleSize) - generate sampleSize number
% of variates from the Bernoulli distribution with probability of success p
% randraw('bern') - help for the Bernoulli distribution;
%
% EXAMPLES:
% 1. y = randraw('bern', 0.5, [1 1e5]);
% 2. y = randraw('bern', 0.1, 1, 1e5);
% 3. y = randraw('bern', 0.9, 1e5 );
% 4. randraw('bern');
% END bernoulli HELP END bern HELP
checkParamsNum(funcName, 'Bernoulli', 'bernoulli', distribParams, 1);
validateParam(funcName, 'Bernoulli', 'bernoulli', 'p', 'p', distribParams(1), {'>=0','<=1'});
out = double( rand( sampleSize ) < distribParams );
case {'beta', 'powerfunction', 'powerfunc'}
% START beta HELP
% THE BETA DISTRIBUTION
%
% ( sometimes: Power Function distribution )
%
% Standard form of the Beta distribution:
% pdf(y) = y.^(a-1).*(1-y).^(b-1) / beta(a, b);
% cdf(y) = betainc(y,a,b), if (y>=0 & y<=1); 0, if x<0; 1, if x>1
%
% Mean = a/(a+b);
% Variance = (a*b)/((a+b)^2*(a+b+1));
%
% General form of the Beta distribution:
% pdf(y) = (y-m).^(a-1).*(n-y).^(b-1) / (beta(a, b)*(n-m)^(a+b-1));
% cdf(y) = betainc((y-m)/(n-m),a,b), if (y>=m & y<=n); 0, if x<m; 1, if x>n
%
% Mean = (n*a + m*b)/(a+b);
% Variance = (a*b)*(n-m)^2/((a+b)^2*(a+b+1));
%
% PARAMETERS:
% a>0 - shape parameter
% b>0 - shape parameter
% m - location
% n -scale (upper bound); n>=m
%
% SUPPORT:
% y, 0<=y<=1 - standard beta distribution
% or
% y, m<=y<=n - generalized beta distribution
%
% CLASS:
% Continuous skewed distributions
%
% USAGE:
% randraw('beta', [a, b], sampleSize) - generate sampleSize number
% of variates from standard beta distribution with shape parameters
% 'a' and 'b'
% randraw('beta', [m, n, a, b], sampleSize) - generate sampleSize number
% of variates from generalized beta distribution on the interval [m, n]
% with shape parameters 'a' and 'b';
% randraw('beta') - help for the Beta distribution;
% EXAMPLES:
% 1. y = randraw('beta', [0.2 0.9], [1 1e5]);
% 2. y = randraw('beta', [0.6 3.2], 1, 1e5);
% 3. y = randraw('beta', [-10 20 3.1 6.2], 1e5 );
% 4. y = randraw('beta', [3 4 5.3 0.7], [1e5 1] );
% 5. randraw('beta');
% END beta HELP
% Refernce:
% Dagpunar, John.
% Principles of Random Variate Generation.
% Oxford University Press, 1988.
%
% max_ab < 0.5 Joehnk's algorithm
% 1 < min_ab Cheng's algortihm BB
% min_ab <= 1 <= max_ab Atkinson's switching algorithm
% 0.5<= max_ab < 1 Atkinson's switching algorithm
checkParamsNum(funcName, 'Beta', 'beta', distribParams, [2, 4]);
if numel(distribParams) == 2
a = distribParams(1);
b = distribParams(2);
m = 0;
n = 1;
validateParam(funcName, 'Beta', 'beta', '[a, b]', 'a', a, {'> 0'});
validateParam(funcName, 'Beta', 'beta', '[a, b]', 'b', b, {'> 0'});
else
m = distribParams(1);
n = distribParams(2);
a = distribParams(3);
b = distribParams(4);
validateParam(funcName, 'Beta', 'beta', '[m, n, a, b]', 'n-m', n-m, {'>= 0'});
validateParam(funcName, 'Beta', 'beta', '[m, n, a, b]', 'a', a, {'> 0'});
validateParam(funcName, 'Beta', 'beta', '[m, n, a, b]', 'b', b, {'> 0'});
end
sampleSizeIn = sampleSize;
sampleSize = [ 1, prod( sampleSizeIn ) ];
max_ab = max( a, b );
min_ab = min( a, b );
if max_ab < 0.5
% Use log(u1^a) and log(u2^b), rather than a and b, to avoid
% underflow for very small a or b.
loga = log(rand( sampleSize ))/a;
logb = log(rand( sampleSize ))/b;
logsum = (loga>logb).*(loga + log(1+ exp(logb-loga))) + ...
(loga<=logb).*(logb + log(1+ exp(loga-logb)));
out = exp(loga - logsum);
indxs = find( logsum > 0);
while ~isempty( indxs )
indxsSize = size( indxs );
loga = log(rand( indxsSize ))/a;
logb = log(rand( indxsSize ))/b;
logsum = (loga>logb).*(loga + log(1+ exp(logb-loga))) + ...
(loga<=logb).*(logb + log(1+ exp(loga-logb)));
l = (logsum <= 0);
out( indxs( l ) ) = exp(loga(l) - logsum(l));
indxs = indxs( ~l );
end
elseif min_ab > 1
% Algorithm BB
sum_ab = a + b;
lambda = sqrt((sum_ab-2)/(2*a*b-sum_ab));
c = min_ab+1/lambda;
u1 = rand( sampleSize );
u2 = rand( sampleSize );
v = lambda*log(u1./(1-u1));
z = u1.*u1.*u2;
clear('u1'); clear('u2');
w = min_ab*exp(v);
r = c*v-1.38629436112;
clear('v');
s = min_ab+r-w;
if a == min_ab
out = w./(max_ab+w);
else
out = max_ab./(max_ab+w);
end
t = log(z);
indxs = find( (s+2.609438 < 5*z) & (r+sum_ab*log(sum_ab./(max_ab+w)) < t) );
clear('v');
clear('z');
clear('w');
clear('r');
while ~isempty( indxs )
indxsSize = size( indxs );
u1 = rand( indxsSize );
u2 = rand( indxsSize );
v = lambda*log(u1./(1-u1));
z = u1.*u1.*u2;
clear('u1'); clear('u2');
w = min_ab*exp(v);
r = c*v-1.38629436112;
clear('v');
s = min_ab+r-w;
t = log(z);
l = (s+2.609438 >= 5*z) | (r+sum_ab*log(sum_ab./(max_ab+w)) >= t);
if a == min_ab
out( indxs( l ) ) = w(l)./(max_ab+w(l));
else
out( indxs( l ) ) = max_ab./(max_ab+w(l));
end
indxs = indxs( ~l );
end
elseif min_ab < 1 & max_ab > 1
% Atkinson's switching method
t = (1-min_ab)/(1+max_ab - min_ab);
r = max_ab*t/(max_ab*t + min_ab*(1-t)^max_ab);
u1 = rand( sampleSize );
w = zeros( sampleSize );
l = u1 < r;
w( l ) = t*(u1( l )/r).^(1/min_ab);
l = ~l;
w( l ) = 1- (1-t)*((1-u1(l))/(1-r)).^(1/max_ab);
if a == min_ab
out = w;
else
out = 1 - w;
end
u2 = rand( sampleSize );
indxs1 = find(u1 < r);
indxs2 = find(u1 >= r);
clear('u1');
indxs = [ indxs1( log(u2(indxs1)) >= (max_ab-1)*log(1-w(indxs1)) ), ...
indxs2( log(u2(indxs2)) >= (min_ab-1)*log(w(indxs2)/t) ) ];
clear('u1');
clear('u2');
while ~isempty( indxs )
indxsSize = size( indxs );
u1 = rand( indxsSize );
w = zeros( indxsSize );
l = u1 < r;
w( l ) = t*(u1( l )/r).^(1/min_ab);
l = ~l;
w( l ) = 1- (1-t)*((1-u1(l))/(1-r)).^(1/max_ab);
u2 = rand( indxsSize );
indxs1 = find(u1 < r);
indxs2 = find(u1 >= r);
clear('u1');
l = logical( zeros( indxsSize ) );
l( [ indxs1( log(u2(indxs1)) < (max_ab-1)*log(1-w(indxs1)) ), ...
indxs2( log(u2(indxs2)) < (min_ab-1)*log(w(indxs2)/t) ) ] ) = 1;
clear('u1');
clear('u2');
if a == min_ab
out( indxs(l) ) = w(l);
else
out( indxs(l) ) = 1 - w(l);
end
indxs = indxs( ~l );
end
else
% Atkinson's Algorithm
if min_ab == 1
t = 0.5;
r = 0.5;
else
t = 1/(1+sqrt(max_ab*(1-max_ab)/(min_ab*(1-min_ab))));
r = max_ab*t / (max_ab*t + min_ab*(1-t));
end
u1 = rand( sampleSize );
out = zeros( sampleSize );
w = zeros( sampleSize );
l1 = u1 < r;
w(l1) = t*(u1(l1)/r).^(1/min_ab);
l2 = u1 >= r;
w(l2) = 1 - (1-t)*((1-u1(l2))/(1-r)).^(1/max_ab);
if a == min_ab
out = w;
else
out = 1 - w;
end
u2 = rand( sampleSize );
indxs1 = find(l1);
indxs2 = find(l2);
indxs = [ indxs1( log(u2(l1)) >= (max_ab -1)*log((1-w(l1))/(1-t)) ), ...
indxs2( log(u2(l2)) >= (min_ab -1) * log(w(l2)/t) ) ];
clear('u2');
while ~isempty( indxs )
indxsSize = size( indxs );
u1 = rand( indxsSize );
w = zeros( indxsSize );
l1 = u1 < r;
w(l1) = t*(u1(l1)/r).^(1/min_ab);
l2 = u1 >= r;
w(l2) = 1 - (1-t)*((1-u1(l2))/(1-r)).^(1/max_ab);
u2 = rand( indxsSize );
indxs1 = find(l1);
indxs2 = find(l2);
clear('u1');
l = logical( zeros( indxsSize ) );
l( [ indxs1(log(u2(l1)) < (max_ab -1)*log((1-w(l1))/(1-t))), ...
indxs2(log(u2(l2))< (min_ab -1) * log(w(l2)/t)) ] ) = 1;
if a == min_ab
out(indxs(l)) = w(l);
else
out(indxs(l)) = 1 - w(l);
end
indxs = indxs( ~l );
end
end
out = m + (n-m) * out;
reshape( out, sampleSizeIn );
case {'bessel'}
% START bessel HELP
% THE BESSEL DISTRIBUTION
%
% Bessel distribution arises in the theory of stochastic processes.
% Bessel(nu,a) is a discrete distribution on the non-negative integers with
% parameters nu > -1 and a > 0.
%
% pdf(y) = (a/2).^(2*y+nu) ./ (besseli(nu,a).*factorial(y).*gamma(y+nu+1));
%
% PARAMETERS:
% nu > -1, a > 0
% SUPPORT:
% y = 0, 1, 2, 3, ...
% CLASS:
% Discrete distributions
%
% USAGE:
% randraw('bessel', [nu, a], sampleSize) - generate sampleSize number
% of variates from the Bessel distribution with parameters
% 'nu' and 'a'
% randraw('bessel') - help for the Bessel distribution;
% EXAMPLES:
% 1. y = randraw('bessel', [2 0.9], [1 1e5]);
% 2. y = randraw('bessel', [0.6 3.2], 1, 1e5);
% 3. y = randraw('bessel', [-0.2 8.1], 1e5 );
% 4. y = randraw('bessel', [4 5.3], [1e5 1] );
% 5. randraw('bessel');
% END bessel HELP
% Method:
%
% We implemented Condensed Table-Lookup method suggested in
% George Marsaglia, "Fast Generation Of Discrete Random Variables,"
% Journal of Statistical Software, July 2004, Volume 11, Issue 4
%
% Reference:
% L. Devroye, "Simulating Bessel random variables,"
% Statistics and Probability Letters, vol. 57, pp. 249-257, 2002.
%
checkParamsNum(funcName, 'Bessel', 'bessel', distribParams, [2]);
nu = distribParams(1);
a = distribParams(2);
validateParam(funcName, 'Bessel', 'bessel', '[nu, a]', 'nu', nu, {'> -1'});
validateParam(funcName, 'Bessel', 'bessel', '[nu, a]', 'a', a, {'> 0'});
% mu = 0.5*a*besseli(nu+1,a)/besseli(nu,a);
% chi2 = mu + 0.25*a^2*besseli(nu+1,a)/besseli(nu,a)*...
% (besseli(nu+2,a)/besseli(nu+1,a)-besseli(nu+1,a)/besseli(nu,a));
besseliNuA = besseli(nu, a);
proceed = 1;
if ~isfinite( besseliNuA )
warnStr{1} = [upper(funcName), ' - Bessel Variates Generation: '];
warnStr{2} = ['besseli(', num2str(nu), ', ' num2str(a), ') returns Inf.'];
warnStr{3} = ['Unable to proceed, return zeros ...'];
warning('%s\n %s\n %s',warnStr{1},warnStr{2},warnStr{3});
%warning([upper(funcName), ' - Bessel Variates Generation: besseli(', num2str(nu), ', ' num2str(a), ') returns Inf. Unable to proceed, return zeros ...']);
out = zeros( sampleSize );
proceed = 0;
end
if besseliNuA == 0
warnStr{1} = [upper(funcName), ' - Bessel Variates Generation: '];
warnStr{2} = ['besseli(', num2str(nu), ', ' num2str(a), ') returns 0.'];
warnStr{3} = ['Unable to proceed, return zeros ...'];
warning('%s\n %s\n %s',warnStr{1},warnStr{2},warnStr{3});
%warning([upper(funcName), '- Bessel Variates Generation: besseli(', num2str(nu), ', ' num2str(a), ') returns 0. Unable to proceed, return zeros ...']);
out = zeros( sampleSize );
proceed = 0;
end
if proceed
p0 = exp( nu*log(a/2) - gammaln(nu+1) ) / besseliNuA;
if p0 >= 5e-10
t = p0;
aa = (a/2)^2;
nu1 = nu+1;
i = 1;
while t*2147483648 > 1
t = t * aa/((i)*(i+nu));
i = i + 1;
end
sizeP = i-1;
offset = 0;
P = round( 2^30*p0*cumprod([1, aa./((1:sizeP-1).*((1:sizeP-1)+nu))] ) );
else % if p0 >= 5e-10
m = floor(0.5*(sqrt(a^2+nu^2)-nu));
pm = exp( (2*m+nu)*log(a/2) - log(besseliNuA) - ...
gammaln(m+1) - gammaln(m+nu+1) );
aa = (a/2)^2;
t = pm;
i = m + 1;
while t * 2147483648 > 1
t = t * aa/((i)*(i+nu));
i = i + 1;
end
last = i-2;
t = pm;
j = -1;
for i = m-1:-1:0
t = t * (i+1)*(i+1+nu)/aa;
if t*2147483648 < 1
j=i;
break;
end
end
offset = j+1;
sizeP = last-offset+1;
P = zeros(1, sizeP);
P(m-offset+1:last-offset+1) = ...
round( 2^30*pm*cumprod([1, aa./(((m+1):last).*(((m+1):last)+nu))] ) );
P(m-offset:-1:1) = ...
round( 2^30*pm*cumprod((m:-1:(offset+1)).*((m:-1:(offset+1))+nu)/aa) );
end % if p0 >= 5e-10, else ...
out = randFrom5Tbls( P, offset, sampleSize);
end % if proceed
case {'binom', 'binomial'}
% START binom HELP START binomial HELP
% THE BINOMIAL DISTRIBUTION
%
% pdf(y) = nchoosek(n,y)*p^y*(1-p)^(n-y) = ...
% exp( gammaln(n+1) - gammaln(n-y+1) - gammaln(y+1) + ...
% y*log(p) + (n-y)*log(1-p) ); 0<p<1, n>1
%
% Mean = n*p;
% Variance = n*p*(1-p);
% Mode = floor( (n+1)*p );
%
% PARAMETERS:
% p - probability of success in a single trial; (0<p<1)
% n - total number of trials; (n= 1, 2, 3, 4, ...)
%
% SUPPORT:
% y - number of success, y = 0, 1, 2, 3 ...
%
% CLASS:
% Discrete distributions
%
% NOTES:
% Constructive definition:
% We consider a random experiment with n independent trials; in each trial
% a certain random event A can occur (the urn model with replacement is
% a special case of such an experiment). Let
% p = probability of A in a single trial;
% n = total number of trials;
% y = number of successes (= number of trials where A occurs).
%
% USAGE:
% randraw('binom', [n, p], sampleSize) - generate sampleSize number
% of variates from the Binomial distribution with total number of trials
% 'n' and probability of success in a single trial 'p'
% randraw('binom') - help for the Binomial distribution;
%
% EXAMPLES:
% 1. y = randraw('binom', [10 0.9], [1 1e5]);
% 2. y = randraw('binom', [100 0.15], 1, 1e5);
% 3. y = randraw('binom', [5 0.5], 1e5 );
% 4. y = randraw('binom', [1000 0.02], [1e5 1] );
% 5. randraw('binom');
% END binom HELP END binomial HELP
% Method:
%
% We implemented Condensed Table-Lookup method suggested in
% George Marsaglia, "Fast Generation Of Discrete Random Variables,"
% Journal of Statistical Software, July 2004, Volume 11, Issue 4
checkParamsNum(funcName, 'Binomial', 'binomial', distribParams, [2]);
n = distribParams(1);
p = distribParams(2);
validateParam(funcName, 'Binomial', 'binomial', '[n, p]', 'n', n, {'> 0','==integer'});
validateParam(funcName, 'Binomial', 'binomial', '[n, p]', 'p', p, {'> 0','< 1'});
if n*p > 1e3 & n > 1e3
out = round( n*p + sqrt(n*p*(1-p))*randn( sampleSize ) );
elseif p<1e-4 & n*p > 1 & n*p < 100
out = feval(funcName,'poisson',n*p, sampleSize);
else
% if n large and p near 1, generate j=Binom(n,1-p), return n-j
switchFlag = 0;
if n>100 & p>0.99
p = 1-p;
switchFlag = 1;
end
mode = floor( (n+1)*p );
q = 1 - p;
h = p/q;
pmode = exp( gammaln(n+1) - gammaln(n-mode+1) - gammaln(mode+1) + ...
mode*log(p) + (n-mode)*log(1-p) );
if( 1-pmode < 5e-10 )
% "Fast Generation of Discrete Random Variables", p.3 citation:
% "For ... discrete variates with an infinite number of probabilities, we select only
% those for which, for a sample of size 2^31(10^9.33), the expected number of occurences exceeds
% 0.5. The other probabilities are assumed zero. For those unusual situations where occurrences
% with probability less than 5 ? 10^?10 must be accounted for, special tail-handling procedures
% should be used."
out = mode*ones( sampleSize );
else
i = mode + 1;
t = pmode;
while t*2147483648 > 1
t = t * h*(n-i+1)/i;
i = i + 1;
end
last = i - 2;
t = pmode;
j = -1;
for i=mode-1:-1:0
t = t * (i+1)/h/(n-i);
if t*2147483648 < 1
j=i;
break;
end
end
offset=j+1;
sizeP = last-offset+1;
P = zeros(1, sizeP);
P(mode-offset+1:last-offset+1) = ...
round( 2^30*pmode*cumprod([1, h*(n-(mode+1:last)+1)./(mode+1:last)] ) );
P(mode-offset:-1:1) = ...
round( 2^30*pmode*cumprod( (mode:-1:offset+1)./(h*(n-(mode-1:-1:offset)))) );
out = randFrom5Tbls( P, offset, sampleSize);
end
end
if switchFlag
out = n - out;
end
case {'bradford'}
% START bradford HELP
% THE BRADFORD DISTRIBUTION
%
% pdf(y) = b ./ ( log(1+b)*(1+b*y) ); 0<y<1
% cdf(y) = log(1+b*x) ./ log(1+b);
%
% Mean = (b - log(1+b)) / (b*log(1+b));
% Variance = (b*(log(1+b)-2) + 2*log(1+b)) / (2*b*(log(1+b))^2);
%
% PARAMETERS:
% b>-1,b~=0 - shape parameter;
%
% SUPPORT:
% 0<y<1
%
% CLASS:
% Continuous skewed distributions
%
% USAGE:
% randraw('bradford', [b], sampleSize) - generate sampleSize number
% of variates from the Bradford distribution with parameter b;
% randraw('bradford') - help for the Bradford distribution;
%
% EXAMPLES:
% 1. y = randraw('bradford', 1, [1 1e5]);
% 2. y = randraw('bradford', 2.2, 1, 1e5);
% 3. y = randraw('bradford', -0.4, 1e5 );
% 4. y = randraw('bradford', 10, [1e5 1] );
% 5. randraw('bradford');
% END bradford HELP
checkParamsNum(funcName, 'Bradford', 'bradford', distribParams, [1]);
b = distribParams(1);
validateParam(funcName, 'Bradford', 'bradford', 'b', 'b', b, {'> -1','~=0'});
out = ((1+b).^rand( sampleSize ) - 1)/b;
case {'burr', 'fisk'}
% START burr HELP START fisk HELP
% THE BURR DISTRIBUTION
% pdf(y) = c*d * y.^(-c-1) .* (1+y.^-c).^(-d-1); y>0
% cdf(y) = (1 + y.^-c).^(-d);
%
% Mean = gamma(1-1/c)*gamma(1/c+d)/gamma(d);
% Variance = COEF / gamma(d)^2;
% where COEF = gamma(d)*gamma(1-2/c)*gamma(2/c+d) - gamma(1-1/c)^2*gamma(1/c+d)^2;
%
% PARAMETERS:
% c - shape parameter (c>0)
% d - shape parameter (d>0)
%
% SUPPORT:
% y>0
%
% NOTES:
% The Burr distribution with d = 1, is often called the Fisk or
% LogLogistic distribution
% The Burr distribution is a generalization of the Fisk distribution
%
% CLASS:
% Continuous skewed distributions