Nov 2009

Some notes about the Distribution Functions toolbox.

1. Problems in Scilab

There is a massive loss of digits in the inverse cumulative beta distribution
function.
R,      options(digits=20), qbeta(0.5, 1, 2),                0.2928932188134525        > Correct
Scilab, format(25),         p = 0.5; cdfbet("XY",1,2,p,1-p), 0.29289321750687375       > Inaccurate
(5.2.beta-1)
ScicosLab                                                    0.2928932175068738041901  > Inaccurate
Scilab (4.1.2)                                               0.2928932175068737486789  > Inaccurate
Directly from DCDFLIB, C++ version from Burkardt,            0.29289321881345187       > Correct
Octave, format long, betainv ( 0.5 , 1 , 2 )                 0.292893218813453         > Correct
Matlab,                                                      0.292893218813453         > Correct
There is only 7 digits accurate in Scilab. 

Is that sure that Scilab is inaccurate ? Yes.
p = cdfbet ( "PQ", x , y , 1, 2)
-->(p-0.5)/0.5
 ans  =
  - 3.696D-09

Why is Scilab inaccurate ?

R uses the file qbeta.c, which does not use the DCDFLIB.

Octave implements a Newton method based on betacdf and betapdf.

In Scilab, the module is in modules/statistics/src/dcdflib/cdfbet.f

2. Problems in Stixbox

There is no hypergeometric function in Scilab.
The Stixbox / dhypg functions returns the pdf for the 
hypergeometric function.
The problem is that it is inaccurate:

p = dhypg ( 200 , 515 , 500 , 1030 )
p = 0.0  // Expected result = 1.65570E-ˆ’10

3. Issues with the Poisson pdf

Scilab does not offer the Poisson pdf.
We could naively think that it is easy to program 
such a law. The naive code is :

function p = dfpoisson1 ( k , lambda )
  p = exp ( - lambda ) * lambda^k / factorial ( k )
endfunction

The not-so-naive code is 

function p = dfpoisson2 ( k , lambda )
  p = exp ( - lambda )
  for i = 1 : k
    p = p * lambda / i
  end
endfunction

// Exact is 2.23568e-14
f = dfpoisson1 ( 314 ,   200 ), mprintf("%e\n", f ) // Nan
f = dfpoisson2 ( 314 ,   200 ), mprintf("%e\n", f ) // 2.235682e-014

Conclusion: it is not-so-easy to compute such a simple distribution
function.

4. Implementations of distribution functions

The f77 original version of the dcdflib:
http://www.netlib.org/random/

A C++ port of the DCDFLIB, by John Burkardt:
https://people.sc.fsu.edu/~burkardt/cpp_src/dcdflib/dcdflib.html

A project aiming at using the /safe flag of Visual compiler :
http://www.codewiz51.com/wiki/DCDFLIB00.ashx

5. Issues with the gamma distribution function

The scale parameter is inverted with respect to Matlab !
In R, the scale parameter is the same as in Scilab.

Matlab    format long; gamcdf(5,1,2)              0.91791500137610       correct
Octave                                            0.917915001376101      correct
Scilab    format(20); cdfgam("PQ",5,1,1/2)        0.91791500137610116    correct
R         options(digits=20); pgamma(5,1,2)       0.9999546000702375     wrong

Matlab    format long; gamcdf(5,2,1)              0.95957231800549       correct
Scilab    format(20); cdfgam("PQ",5,2,1)          0.95957231800548715    correct
R         options(digits=20); pgamma(5,2,1)       0.9595723180054872     correct

Matlab    format long; gamcdf(5,2,2)              0.71270250481635       correct
Scilab    format(20); cdfgam("PQ",5,2,2)          0.71270250481635422    correct
R         options(digits=20); pgamma(5,2,1/2)     0.7127025048163542     correct

// Caution : the scale factor is to be inverted !
x = linspace(0,20,100);
a = 1*ones(1,100);
b = 1/2*ones(1,100);
y = cdfgam("PQ",x,a,b);
plot(x,y)

======================================

About the S parameter of the cdfnbn function
[P,Q]=cdfnbn("PQ",S,Xn,Pr,Ompr)
P,Q (Q=1-P)
  The cumulation from 0 to S of the negative binomial distribution.
S
  The upper limit of cumulation of the binomial distribution. 
  There are F or fewer failures before the XNth success.
  In Matlab: X
Xn
  The number of successes. 
  In Matlab: R
Pr
  The probability of success in each binomial trial. 
  In Matlab: P
Ompr
  1-Pr
  In Matlab: does not exist
  
In Matlab :
X = 5
R = 3
P = 0.3
Y = nbincdf(X,R,P) //     0.4482

In Scilab:
S = 5
Xn = 3
Pr = 0.3
Ompr = 0.7
[P,Q]=cdfnbn("PQ",S,Xn,Pr,Ompr)

======================================

In Matlab :
X = 1.2
NU1 = 5
NU2 = 20
DELTA = 2
ncfcdf(X,NU1,NU2,DELTA) // 0.4650

In Scilab
F = 1.2
Dfn = 5
Dfd = 20
Pnonc = 2
[P,Q]=distfun_cdffnc("PQ",F,Dfn,Dfd,Pnonc)

