[erlang-questions] Gaussian Distribution
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 ;-)
On Mon, October 29, 2012 02:52, Richard O'Keefe wrote:
> On 29/10/2012, at 4:58 AM, Frank Recker wrote:
>> at work, I often need the values the cumulative distribution function
>> the Gaussian distribution. The code for this function in haskell,
>> and perl and the corresponding mathematical paper can be found at
> 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.
> 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) ->
> % 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) ->
> % 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
> 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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