Vensim doesn’t have a function for the cumulative normal distribution, but it’s easy to implement via a macro. I used to use a polynomial cited in Numerical Recipes (error function, Ch. 6.2):
:MACRO: NCDF(x)
NCDF = 1-Complementary Normal CDF
~ dmnl
~ |
Complementary Normal CDF= ERFCy/2 ~ dmnl ~ |
ERFCy = IF THEN ELSE(y>=0,ans,2-ans) ~ dmnl ~ http://www.library.cornell.edu/nr/bookcpdf/c6-2.pdf |
y = x/sqrt(2) ~ dmnl ~ |
ans=t*exp(-z*z-1.26551+t*(1.00002+t*(0.374092+t*(0.0967842+ t*(-0.186288+t*(0.278868+t*(-1.1352+t*(1.48852+ t*(-0.822152+t*0.170873))))))))) ~ dmnl ~ |
t=1/(1+0.5*z) ~ dmnl ~ |
z = ABS(y) ~ dmnl ~ |
:END OF MACRO:
I recently discovered a better approximation here, from algorithm 26.2.17 in Abromowitz and Stegun, Handbook of Mathematical Functions:
:MACRO: NCDF2(x)NCDF2 = IF THEN ELSE(x >= 0,
(1 - c * exp( -x * x / 2 ) * t *
( t *( t * ( t * ( t * b5 + b4 ) + b3 ) + b2 ) + b1 )), ( c * exp( -x * x / 2 ) * t *
( t *( t * ( t * ( t * b5 + b4 ) + b3 ) + b2 ) + b1 ))
)
~ dmnl
~ From http://www.sitmo.com/doc/Calculating_the_Cumulative_Normal_Distribution
Implements algorithm 26.2.17 from Abromowitz and Stegun, Handbook of Mathematical
Functions. It has a maximum absolute error of 7.5e^-8.
http://www.math.sfu.ca/
|
c = 0.398942
~ dmnl
~ |
t = IF THEN ELSE( x >= 0, 1/(1+p*x), 1/(1-p*x))
~ dmnl
~ |
b5 = 1.33027
~ dmnl
~ |
b4 = -1.82126
~ dmnl
~ |
b3 = 1.78148
~ dmnl
~ |
b2 = -0.356564
~ dmnl
~ |
b1 = 0.319382
~ dmnl
~ |
p = 0.231642
~ dmnl
~ |
:END OF MACRO:
In advanced Vensim versions, paste the macro into the header of your model (View>As Text). Otherwise, you can implement the equations inside the macro directly in your model.
Thanks Tom!