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nic_spm_Gpdf

PURPOSE ^

Probability Density function (PDF) of Gamma distribution

SYNOPSIS ^

function f = nic_spm_Gpdf(x,h,l)

DESCRIPTION ^

 Probability Density function (PDF) of Gamma distribution
 FORMAT f = spm_Gpdf(g,h,l)

 x - Gamma-variate   (Gamma has range [0,Inf) )
 h - Shape parameter (h>0)
 l - Scale parameter (l>0)
 f - PDF of Gamma-distribution with shape & scale parameters h & l
_______________________________________________________________________

 spm_Gpdf implements the Probability Density function of the Gamma
 distribution.

 Definition:
-----------------------------------------------------------------------
 The PDF of the Gamma distribution with shape parameter h and scale l
 is defined for h>0 & l>0 and for x in [0,Inf) by: (See Evans et al.,
 Ch18, but note that this referenic_e uses the alternative
 parameterisation of the Gamma with scale parameter c=1/l)

           l^h * x^(h-1) exp(-lx)
    f(x) = ---------------------
                gamma(h)

 Variate relationships: (Evans et al., Ch18 & Ch8)
-----------------------------------------------------------------------
 For natural (strictly +ve integer) shape h this is an Erlang distribution.

 The Standard Gamma distribution has a single parameter, the shape h.
 The scale taken as l=1.

 The Chi-squared distribution with v degrees of freedom is equivalent
 to the Gamma distribution with scale parameter 1/2 and shape parameter v/2.

 Algorithm:
-----------------------------------------------------------------------
 Direct computation using logs to avoid roundoff errors.

 Referenic_es:
-----------------------------------------------------------------------
 Evans M, Hastings N, Peacock B (1993)
       "Statistical Distributions"
        2nd Ed. Wiley, New York

 Abramowitz M, Stegun IA, (1964)
       "Handbook of Mathematical functions"
        US Government Printing Office

 Press WH, Teukolsky SA, Vetterling AT, Flannery BP (1992)
       "Numerical Recipes in C"
        Cambridge
_______________________________________________________________________
 @(#)spm_Gpdf.m    2.2 Andrew Holmes 99/04/26

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

DOWNLOAD ^

nic_spm_Gpdf.m

SOURCE CODE ^

0001 function f = nic_spm_Gpdf(x,h,l)
0002 % Probability Density function (PDF) of Gamma distribution
0003 % FORMAT f = spm_Gpdf(g,h,l)
0004 %
0005 % x - Gamma-variate   (Gamma has range [0,Inf) )
0006 % h - Shape parameter (h>0)
0007 % l - Scale parameter (l>0)
0008 % f - PDF of Gamma-distribution with shape & scale parameters h & l
0009 %_______________________________________________________________________
0010 %
0011 % spm_Gpdf implements the Probability Density function of the Gamma
0012 % distribution.
0013 %
0014 % Definition:
0015 %-----------------------------------------------------------------------
0016 % The PDF of the Gamma distribution with shape parameter h and scale l
0017 % is defined for h>0 & l>0 and for x in [0,Inf) by: (See Evans et al.,
0018 % Ch18, but note that this referenic_e uses the alternative
0019 % parameterisation of the Gamma with scale parameter c=1/l)
0020 %
0021 %           l^h * x^(h-1) exp(-lx)
0022 %    f(x) = ---------------------
0023 %                gamma(h)
0024 %
0025 % Variate relationships: (Evans et al., Ch18 & Ch8)
0026 %-----------------------------------------------------------------------
0027 % For natural (strictly +ve integer) shape h this is an Erlang distribution.
0028 %
0029 % The Standard Gamma distribution has a single parameter, the shape h.
0030 % The scale taken as l=1.
0031 %
0032 % The Chi-squared distribution with v degrees of freedom is equivalent
0033 % to the Gamma distribution with scale parameter 1/2 and shape parameter v/2.
0034 %
0035 % Algorithm:
0036 %-----------------------------------------------------------------------
0037 % Direct computation using logs to avoid roundoff errors.
0038 %
0039 % Referenic_es:
0040 %-----------------------------------------------------------------------
0041 % Evans M, Hastings N, Peacock B (1993)
0042 %       "Statistical Distributions"
0043 %        2nd Ed. Wiley, New York
0044 %
0045 % Abramowitz M, Stegun IA, (1964)
0046 %       "Handbook of Mathematical functions"
0047 %        US Government Printing Office
0048 %
0049 % Press WH, Teukolsky SA, Vetterling AT, Flannery BP (1992)
0050 %       "Numerical Recipes in C"
0051 %        Cambridge
0052 %_______________________________________________________________________
0053 % @(#)spm_Gpdf.m    2.2 Andrew Holmes 99/04/26
0054 
0055 %-Format arguments, note & check sizes
0056 %-----------------------------------------------------------------------
0057 if nargin<3, error('Insufficient arguments'), end
0058 
0059 ad = [ndims(x);ndims(h);ndims(l)];
0060 rd = max(ad);
0061 as = [    [size(x),ones(1,rd-ad(1))];...
0062     [size(h),ones(1,rd-ad(2))];...
0063     [size(l),ones(1,rd-ad(3))]     ];
0064 rs = max(as);
0065 xa = prod(as,2)>1;
0066 if sum(xa)>1 & any(any(diff(as(xa,:)),1))
0067     error('non-scalar args must match in size'), end
0068 
0069 %-Computation
0070 %-----------------------------------------------------------------------
0071 %-Initialise result to zeros
0072 f = zeros(rs);
0073 
0074 %-Only defined for strictly positive h & l. Return NaN if undefined.
0075 md = ( ones(size(x))  &  h>0  &  l>0 );
0076 if any(~md(:)), f(~md) = NaN;
0077     warning('Returning NaN for out of range arguments'), end
0078 
0079 %-Degenerate cases at x==0: h<1 => f=Inf; h==1 => f=l; h>1 => f=0
0080 ml = ( md  &  x==0  &  h<1 );
0081 f(ml) = Inf;
0082 ml = ( md  &  x==0  &  h==1 ); if xa(3), mll=ml; else mll=1; end
0083 f(ml) = l(mll);
0084 
0085 %-Compute where defined and x>0
0086 Q  = find( md  &  x>0 );
0087 if isempty(Q), return, end
0088 if xa(1), Qx=Q; else Qx=1; end
0089 if xa(2), Qh=Q; else Qh=1; end
0090 if xa(3), Ql=Q; else Ql=1; end
0091 
0092 %-Compute
0093 f(Q) = exp( (h(Qh)-1).*log(x(Qx)) +h(Qh).*log(l(Ql)) - l(Ql).*x(Qx)...
0094         -gammaln(h(Qh)) );

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