Re: Re: Fitting parameters of nonlinear diff equation system

• To: mathgroup at smc.vnet.net
• Subject: [mg81549] Re: [mg81303] Re: Fitting parameters of nonlinear diff equation system
• From: "Paul Thomas" <paul.pgt at gmail.com>
• Date: Wed, 26 Sep 2007 21:48:43 -0400 (EDT)
• References: <fcnl8h\$s85\$1@smc.vnet.net> <200709181002.GAA04068@smc.vnet.net>

```Hi Szabolcs,

A couple things, thanks for your  reply!

Replacing f with f[t] in fun[] does seem to return a function. I tested this
by just running fun with a given a,b,c and with the f[t] syntax it produces
an interpolating function (as NDSolve should)--it doesn't with just f as I
think it doesn't know what "f" is in that case--the output of just using "f"
is literally just "f" with no function associated.

On the /@ priority issue, I actually changed one other thing, which maybe is
a problem too.
Instead of using "/@" there I used "/."--that I think does require the
parentheses. I kept getting an error using "/@" and "/." seemed to fix it.
The error would say that essentially a non-numerical equation was resulting
from the /@ and it would write it out--it did seem to be doing the wrong
equation, but it looks like the right one with /.
Now it seems to be doing exactly what you designed it to do.

Once the parameters are fit I feed them back into "fun" and check the work
essentially of the process and plot the optimization over time. For example,
once I've done a particular fit:

g[t_]:=fun[1,2,3] {pretending 1,2,3 were our best a,b,c fits}

g[t] is now my best fit function and matches the findminimum minimum at the
particular t points I have. The curves also look generally as they should.

Do you think I have an error in this? Thanks for all your help with it, like
I said, it seems like a very powerful approach at least for the types of
things that I've been trying to do--very flexible!

Paul

On 9/25/07, Szabolcs Horv=E1t <szhorvat at gmail.com> wrote:
>
> Hi Paul,
>
> I am writing to warn you that something *might* have gone wrong with
> the calculations.  (Let's hope that it is just a misunderstanding).
>
> On 24/09/2007, Paul Thomas <paul.pgt at gmail.com> wrote:
> > Hi everyone,
> >
> > Just wanted to follow up with this. The e-mail below from Szabolcs was
> very
> > helpful and it did work, with just a minor syntax change. He suggested
> the
> > following code for fitting a system of differential equations with
> multiple
> > parameters:
> >
> > fun[a_, b_, c_] :=
> >    First[
> >      f /. NDSolve[
> >             deq /. {b1 -> a, h -> b, r -> c},
> >             {v1, v2, v3, v4, v5, v6, v7, v8, v9, v10, v11, v12, v13, f} ,
> >             {t, 0, 10} (* dummy values *)
> >           ]]
> >
> > opt[a_?NumericQ, b_?NumericQ, c_?NumericQ] :=
> >    Norm[fun[a, b, c] /@ tvals - fvals]
> >
> > Now use FindMinimum on 'opt', like this:
> >
> > FindMinimum[opt[a,b,c], {{a,1.}, {b,2.}, {c,3.}}]
> >
> > One change I had to make was in the "fun[a_,b_,c_]" equation--instead of
> > "f/.NDSolve" it should be "f[t]/.NDSolve" etc etc.
>
> While I admit that I did not test these commands thoroughly (as I did
> not have access to the data), something seems to be wrong with these
> modifications.  I meant fun[] to return a function, not a number (note
> that we later map this function to a list) ... if f is changed to
> f[t], then the return value will not be a function.
>
> > The other change I had to make was in the opt equation. I just needed to
> put
> > parentheses around the "(fun[a, b, c] /@ tvals)" term--otherwise it
> thinks
> > the "/@" Map command is being applied to tvals-fvals.
>
> Something seems to be wrong here too, as /@ has a higher precedence than -
> .
>
> Example:
>
> In[1]:=
> a={1,2,3};
> b={4,5,6};
>
> In[3]:= f/@a-b
> Out[3]= {-4+f[1],-5+f[2],-6+f[3]}
>
> > I want to thank everyone for their comments, I'm so pleased with this
> > technique now. I think it works very much like the documentation says
> > FindFit works, but it is much more flexible in terms of the data you can
> > input for the fit and also allows a whole system to be solved with data
> only
> > from a single equation. To me, those are both huge advantages. Thanks!
> >
> > Paul
>
> Szabolcs
>

```

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