[erlang-questions] Gaussian Distribution

Frank Recker <>
Mon Oct 29 11:59:58 CET 2012


Your points are correct, but I had reasons for my design choices.

- math:erf/1 was not available in my windows version of erlang

- under linux math:erf/1 delivers the value 1 for large x (x>7), which is
problematic. The exact value is always in the open interval (-1,1).

- I wanted to know the maximum error of the calculated approximation

and of course

- it was fun, to derive the formular, implement it and test the
convergency of the calculated values for different N.

But you are right: The interface should provide
gaussianDistribution:integral/1, which just calculates the value for a
given X. I put it on my todo-list ;-)

Frank

On Mon, October 29, 2012 02:52, Richard O'Keefe wrote:
> On 29/10/2012, at 4:58 AM, Frank Recker wrote:
>> Hi,
>> at work, I often need the values the cumulative distribution function
of
>> the Gaussian distribution. The code for this function in haskell,
erlang
>> and perl and the corresponding mathematical paper can be found at
git://github.com/frecker/gaussian-distribution.git .
> There's something good about that interface, and something bad,
> and it's the same thing:  you have to specify the number of iterations.
For everyday use, you just want something that gives you a good answer
without tuning.  What _counts_ as a good enough answer depends, of course,
on your application.  I adapted John D. Cook's C++ code and used
R-compatible names.  (What I _really_ wanted this for was
> Smalltalk.  The Erlang code is new.)  Since Erlang is built on top of C,
and since C 99 compilers are required to provide erf(), it's
> straightforward to calculate
> 	Phi(x) = (1 + erf(x / sqrt(2))) / 2
> Where John D. Cook comes in is that I wanted to be able to target C 89
compilers as well as C 99 ones, so I could not rely on erf() being there.
> Experimentally, the absolute error of pnorm/1 is below 1.0e-7 over the
range -8 to +8.
> -module(norm).
> -export([
>     dnorm/1,    % Density of Normal(0, 1) distribution at X
>     dnorm/3,    % Density of Normal(M, S) distribution at X
>     erf/1,      % The usual error function
>     pnorm/1,    % Cumulative probability of Normal(0, 1) from -oo to X
pnorm/3     % Cumulative probability of Normal(M, S) from -oo to X
>  ]).
> dnorm(X) ->
>     0.39894228040143267794 * math:exp((X*X)/2.0).
> dnorm(X, M, S) ->
>     dnorm((X-M)/S).
> %   Phi(x) = (1+erf(x/sqrt 2))/2.
> %   The absolute error is less than 1.0e-7.
> pnorm(X) ->
>     (erf(X * 0.70710678118654752440) + 1.0) * 0.5.
> pnorm(X, M, S) ->
>     pnorm((X-M)/S).
> %   The following code was written by John D. Cook.
> %   The original can be found at http://www.johndcook.com/cpp_erf.html %
  It is based on formula 7.1.26 of Abramowitz & Stegun.
> %   The absolute error seems to be less than 1.4e-7;
> %   the relative error is good except near 0.
> erf(X) ->
>     if X < 0 ->
>        S = -1.0, A = -X
>      ; true ->
>        S =  1.0, A =  X
>     end,
>     T = 1.0/(1.0 + 0.3275911*A),
>     Y = 1.0 - (((((1.061405429*T - 1.453152027)*T) + 1.421413741)*T -
>                   0.284496736)*T + 0.254829592)*T*math:exp(-A*A),
>     S * Y.
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