FindFit[] Blues
- To: mathgroup at smc.vnet.net
- Subject: [mg118139] FindFit[] Blues
- From: "Ernest Jackson" <ernestrjackson2 at verizon.net>
- Date: Thu, 14 Apr 2011 04:48:37 -0400 (EDT)
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