# [erlang-questions] Gaussian Distribution

Frank Recker <>
Mon Oct 29 16:52:55 CET 2012

```Sorry, but I don't overlook the ramifications of this approach. My feeling
is that it might be what we call in german "mit Kanon auf Spatzen
schiessen" (dict.leo.org translates this to "to crack a nut with a
sledgehammer").

Frank

On Mon, October 29, 2012 14:10, Robert Virding wrote:
> Wouldn't a better solution be to have a BEAM implementation of the error
> function implemented when it doesn't occur in the system math library so
> that the math module always provides this function?
>
> Robert
>
> ----- Original Message -----
>> From: "Frank Recker" <>
>> To:
>> Sent: Monday, 29 October, 2012 3:59:58 AM
>> Subject: Re: [erlang-questions] Gaussian Distribution
>>
>> 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 .
>> > 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
>> 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.
>> > _______________________________________________
>> > erlang-questions mailing list
>> >
>> > http://erlang.org/mailman/listinfo/erlang-questions
>>
>>
>>
>>
>>
>>
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>

```