/usr/share/openturns/validation/ValidFORMKernelMixture.txt is in openturns-validation 1.5-7build2.
This file is owned by root:root, with mode 0o644.
The actual contents of the file can be viewed below.
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> with(Statistics):
> #Digits:=30:
> phi:=proc(x)
> 1/sqrt(2*Pi)*exp(-x^2/2)
> end:
> PHI:=proc(x)
> int(phi(t),t=-infinity..x)
> end:
> g:=x1+x2;
g := x1 + x2
> CDF_x1:=1/2*CDF(Normal(mu1,sigma1),x1)+1/2*CDF(Normal(mu2,sigma2),x1);
> CDF_x2:=CDF(Normal(mux2,sigmax2),x2);
1/2 1/2
2 (x1 - mu1) 2 (x1 - mu2)
CDF_x1 := 1/2 + 1/4 erf(---------------) + 1/4 erf(---------------)
2 sigma1 2 sigma2
1/2
2 (x2 - mux2)
CDF_x2 := 1/2 + 1/2 erf(----------------)
2 sigmax2
> u1:=subs(s=CDF_x1,Quantile(Normal(0,1),s));
> u2:=subs(t=CDF_x2,Quantile(Normal(0,1),t));
> X1:=subs(s=solve(CDF_x1=U1,x1),CDF(Normal(0,1),s));
> X2:=subs(t=solve(CDF_x2=U2,x2),CDF(Normal(0,1),t));
1/2
2 (x1 - mu1)
u1 := RootOf(-erf(_Z) + 1/2 erf(---------------)
2 sigma1
1/2
2 (x1 - mu2) 1/2
+ 1/2 erf(---------------)) 2
2 sigma2
1/2
2 (x2 - mux2) 1/2
u2 := RootOf(-erf(_Z) + erf(----------------)) 2
2 sigmax2
X1 := 1/2 + 1/2 erf(1/2 RootOf(
1/2 1/2
2 (_Z - mu1) 2 (_Z - mu2) 1/2
2 + erf(---------------) + erf(---------------) - 4 U1) 2 )
2 sigma1 2 sigma2
1/2
2 mux2
X2 := 1/2 + 1/2 erf(--------- + RootOf(erf(_Z) + 1 - 2 U2) sigmax2)
2
> G:=subs(x1=X1,x2=X2,g):
> L:=1/2*(U1^2+U2^2)+lambda*G:
> sys:={diff(L,U1),diff(L,U2),diff(L,lambda)}:
> ValNum:={mu1=-1.0,sigma1=0.25,mu2=2.0,sigma2=2.0,mux2=2.5,sigmax2=3.0}
> :
> evalf(subs(ValNum,sys)):
> vars:={U1,U2,lambda}:
> sol:=fsolve(subs(ValNum,sys),vars);
/ /
2 1/2 / | 1/2 |
sol := fsolve({U1 + 2 lambda exp(-1/2 %1 ) 2 / |Pi |
/ | |
\ \
2 1/2
4.000000000 exp(-8.000000000 (%1 + 1.0) ) 2
----------------------------------------------
1/2
Pi
2 1/2\\
0.5000000000 exp(-0.1250000000 (%1 - 2.0) ) 2 ||
+ ------------------------------------------------||, U2 +
1/2 ||
Pi //
3.0 lambda
1/2 2
exp(-(1.250000000 2 + 3.0 RootOf(erf(_Z) + 1 - 2 U2)) )
/ 2
/ exp(-RootOf(erf(_Z) + 1 - 2 U2) ), 1
/
1/2
+ 1/2 erf(1/2 %1 2 )
1/2
+ 1/2 erf(1.250000000 2 + 3.0 RootOf(erf(_Z) + 1 - 2 U2))
}, {lambda, U2, U1})
1/2
%1 := RootOf(2 + erf(2.000000000 2 (_Z + 1.0))
1/2
+ erf(0.2500000000 2 (_Z - 2.0)) - 4 U1)
>
> PointConception:=subs(sol,[u_E,u_F,u_L]);
> beta:=subs(sol,sqrt(u_L^2+u_F^2+u_E^2));
> FacteursImportance:=subs(sol,[F_E=(u_E/beta)^2,F_F=(u_F/beta)^2,F_L=(u
> _L/beta)^2]);
> Pdef_FORM:=PHI(-beta);
> with(plots):
> p1:=implicitplot3d(subs(ValNum,g),u_E=-4..0,u_F=0..4,u_L=0..4,style=PA
> TCHNOGRID,scaling=constrained,axes=normal,grid=[31,31,31],lightmodel='
> light1'):
> p2:=pointplot3d(PointConception,symbol=circle,symbolsize=15,style=POIN
> T,color=red,scaling=constrained):
> p3:=spacecurve([PointConception[1]*t,PointConception[2]*t,PointConcept
> ion[3]*t],t=0..1,color=blue,scaling=constrained):
> display3d({p1,p2,p3});
Warning, computation interrupted
> expr:=subs(ValNum,solve(g,u_F));
> A:=infinity;
> Pde:=Int(Int(int(phi(u_E)*phi(u_F)*phi(u_L),u_F=expr..infinity),u_E=-A
> ..A),u_L=-A..A);
/ 1/2 1/2 3
| 2 u_L 2 u_L
expr := -1/50 |-240 erf(--------) + 480 u_E - 1/125 erf(--------)
\ 2 2
1/2 2 \ / 1/2 3
2 u_L | / | 2 u_L
- 12/5 erf(--------) + 2080| / |-1/125000 erf(--------)
2 / / \ 2
1/2 2 1/2 \
2 u_L 2 u_L |
- 3/1250 erf(--------) - 6/25 erf(--------) - 8|
2 2 /
A := infinity
infinity infinity
/ / 2 2 / /
| | u_E u_L | |
Pde := | | -1/4 exp(- ---- - ----) |erf(10 |
| | 2 2 \ \
/ /
-infinity -infinity
1/2 1/2 3
2 u_L 2 u_L
-30000 erf(--------) + 60000 u_E - erf(--------)
2 2
1/2 2 \ / 1/2 3
2 u_L | 1/2 / | 2 u_L
- 300 erf(--------) + 260000| 2 / |erf(--------)
2 / / \ 2
1/2 2 1/2 \ \
2 u_L 2 u_L | |
+ 300 erf(--------) + 30000 erf(--------) + 1000000|) - 1|/
2 2 / /
Pi du_E du_L
> for i from 1 to 10 do
> evalf[i](Pde);
> od;
0.0006
0.00060
0.000604
0.0006038
0.00060378
0.000603785
0.0006037850
0.00060378500
0.000603785000
0.0006037850000
>
|