Re: FindFit[] Blues
- To: mathgroup at smc.vnet.net
- Subject: [mg118168] Re: FindFit[] Blues
- From: Peter Pein <petsie at dordos.net>
- Date: Fri, 15 Apr 2011 03:55:48 -0400 (EDT)
- References: <io6cvt$dou$1@smc.vnet.net>
Hi Ernest,
two minor changes gave me fitted values (0.5,0.3):
1.) define errorFu as follows:
errorFu[s1_?NumericQ, s2_?NumericQ,
laterNum_?NumericQ,
n1_?NumericQ, n2_?NumericQ] :=
Total[(toLaterFu[s1, s2, laterNum] - {n1, n2})^2];
2.) Use the Option Method->NMinimize in Fit.
(this was my first attempt; maybe other methods are suitable too.
hth,
Peter
Am 14.04.2011 10:58, schrieb Ernest Jackson:
> Hi,
>
> I have been struggling with a situation where FindFit[] gives me correct answers when the model includes either
> Nest[] or Total[] but not when Nest[] is inside Total[] . I DON'T THINK THIS IS A CASE WHERE FindFit[]
> IS STUCK IN A LOCAL SOLUTION (see the simplified example code below). It appears to me that either
> I have made a dumb error which I cannot see or I do not understand some critical aspects of the problem/code.
> The problem is demonstrated with code for a two element system with simple interactions modeled in discrete
> time. Any help would be greatly appreciated.
>
> (*--------------------------------------------------------*)
> (* DEFINE FUNCTIONS TO MODEL SYSTEM *)
> (* the state of each element depends on the *)
> (* the state of the other during the prior time *)
> (* and the value of the parameters, a and b *)
> (*--------------------------------------------------------*)
> toNextFu = {a Sin[#[[2]]],b Cos[#[[1]]] }&; (* calculate next 2-value state from current state *)
> toLaterFu= Function[{s1,s2,laterNum},Flatten[Nest[toNextFu,{s1,s2},Round[laterNum]] ]]; (* calculate state laterNum of time units after current state *)
> errorFu=Function[{s1,s2, laterNum,n1,n2},Total[(toLaterFu[s1,s2, laterNum]-{n1,n2})^2]]; (* convert calculated state to an error so the function can be used inside FindFit[] *)
>
> (*----------------------------------------------------------------------------------------------------------------*)
> (* GENERATE 100 TRAINING DATA POINTS *)
> (* of {state1, state2, laterNum, nextState1, nextState2, 0} when a= 0.5 and b 0.3 *)(* 0 added at the end because I want the model's error for each training point to be 0 *)
> (*---------------------------------------------------------------------------------------------------------------*)
> pSub={a->.5,b->.3}; (* parameter values selected to produce training dataset --*)
> data=Table[Flatten[{s1=RandomReal[{0,8}],s2=RandomReal[{0,8}],laterNum=RandomInteger[{1,2}],toLaterFu[s1,s2,laterNum]/.pSub,0}],{100}];
>
> (*-----------------------------*)
> (* FIT THE DATA *)
> (*-----------------------------*)
> Clear[a,b,s1,s2,laterNum,n1,n2];
> Quiet[fit=FindFit[data,errorFu[s1,s2, laterNum,n1,n2],{a,b},{s1,s2, laterNum,n1,n2}]];
> Print["Fitted parameters: ",fit];
> Print["Correct parameters: ",pSub];
>
> (*------------------------------------------------------------------------------------------*)
> (* SHOW A SMOOTH TOTAL ERROR SURFACE AROUND THE SOLUTION *)
> (*------------------------------------------------------------------------------------------*)
> (*--fitPoint - {fit for parm-A, fit for parm-B, total error at fit} --*)
> fitPoint=Flatten[{fit[[All,2]],Total[Map[(errorFu[Drop[#,-1]/.List->Sequence]/.fit)&,data]] }];
>
> (*--targPoint - {actual parm-A, parm-B, total error with correct parms} --*)
> targPoint=Flatten[{{0.5,0.3},0}];
>
> (*-- Start and end a and b parameter values to be plotted --*)
> {startA, startB}=fit[[All,2]];
> {endA, endB}= {startA, startB} +0.4;
>
> (*-------------------------------------------------------------------*)
> (*The total error surface plot around the fitted (RED) *)
> (* and correct parameter values (BLACK) *)
> (* THIS TAKES ABOUT 17 SEC ON MY COMPUTER *)
> (*--------------------------------------------------------------------*)
> p=Plot3D[Total[ParallelMap[(errorFu[Drop[#,-1]/.List->Sequence]/.{a->valueA,b->valueB})&,data]],{valueA,startA,endA},{valueB,startB,endB},AxesLabel->Automatic];
> Show[p,Graphics3D[{{PointSize[0.02],Black,Point[targPoint]},{PointSize[0.02],Red,Point[fitPoint]}}] ]
>
> Finally, if FindFit[] is forced to start the search with a and b parameters values from -10 to 10 the same wrong solution is found suggesting further that a local solution
> is not being found. So where have I gone wrong? Again, any help will be greatly appreciated!
>
> Thanks
>
> Ernie Jackson