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MathGroup Archive 2004

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Re: Using LevenbergMarquardt Method with a complicated function

  • To: mathgroup at smc.vnet.net
  • Subject: [mg53084] Re: Using LevenbergMarquardt Method with a complicated function
  • From: algaba at alumni.uv.es (algaba)
  • Date: Thu, 23 Dec 2004 07:58:14 -0500 (EST)
  • Sender: owner-wri-mathgroup at wolfram.com

Seems it doesn't work. I have tried some examples to understand the
proceeding and it worked, but when I wanted to implement it on the
program it fails. I have tried to add the instruction
R = Join[ResAlpha, ResDelta, ResAlphaComp, ResDeltaComp];
to specify the residual explicitly and use it in the minimization, but
it gives me the error message 
FindMinimum::fmgz: Encountered a gradient which is effectively zero.
The \
result returned may not be a minimum; it may be a maximum or a saddle
point.
It is true that mathematica now gives me some results that, in
principle, appear to be good, but in the next step I see this is not
the case: R is a null list and posterior work with data gives me
indeterminates expressions.
I think now the problem is that the residuals are not constants but
can vary (in fact, the number of residuals can also be different each
time we run the algorith), but I don't know how does it affect to
mathematica.
I have also tried to remove some summands or introduce the residuals
in other several ways, such as creating a list before the execution,
inside the function, just when implementing L-M method, etc.
So,  think I'm in the correct way, because I get some (but surely
false) results. So, what's the next step?
Thanks.

---

algaba wrote:
> Hi.
> I have defined a very long function like this:
>
> ChiSquare[Per0_?NumericQ, Ppa0_?NumericQ, Ecc0_?NumericQ] :=
>
>  (some steps and definitions here)
>
> ChiSQ = Sum[ResAlpha[[i]]^2 + ResDelta[[i]]^2, {i, 1, Length[
>             TExp]}] + Sum[ResAlphaComp[[i]]^2 +
>            ResDeltaComp[[i]]^2, {i, 1, Length[TComp]}]);
>
> which tries to find the Chi-Square of an array of data. Now, I want
to
> minimize it and I use FindMinimum, which works well. The problem
> arises when I want to use the Levenberg-Marquardt method, which
seems
> to be better for this kind of functions (As you can see, it is a sum
> of squares) But when I run Mathematica 5 it gives me the next error
> message:
>
> FindMinimum::notlm: The objective function for the method
> LevenbergMarquardt \
> must be in a least-squares form: Sum[f[i][x]^2,{i,1,n}] or Sum[w[i]
\
> f[i][x]^2,{i,1,n}] with positive w[i].
>
> I think the function accomplishes all the requirements. Why I get
this
> error? Is it maybe because of the long definition of the function?
Is
> it because Mathematica doesn't see this function as a sum of squares
> but as a sequence of steps?
> What can I do to solve this problem? I do want to use this method to
> minimize the Chi-Square. Thanks.
>

Because the definition is set up to only evaluate with numerical
values
of the arguments, Mathematica cannot do the computations necessary to
decompose the sum of squares.  To use the Levenberg-Marquardt method,
the function needs to be decomposed into a residual function r[X] such
that f[X] = r[X].r[X]/2.

The LevenbergMarquardt method has a method option that allows you to
specify the residual explicitly for cases like this where you may not
want the sum of squares evaluated symbolically.  Here is a simple
example that should give you an idea of how it works:

In[1]:=
SS[x_?NumberQ, y_?NumberQ] := (x - 1)^2 + 100 (y - x^2)^2

In[2]:=
FindMinimum[SS[x,y],{{x,1},{y,-1}}, Method->"LevenbergMarquardt"]

>From In[2]:=
FindMinimum::notlm:
The objective function for the method
LevenbergMarquardt must be in a least-squares
form: Sum[f[i][x]^2,{i,1,n}] or Sum[w[i]
f[i][x]^2,{i,1,n}] with positive w[i]. More...

Out[2]=
FindMinimum[SS[x, y], {{x, 1}, {y, -1}},

Method -> LevenbergMarquardt]

In[3]:=
R[x_?NumberQ, y_?NumberQ] := Sqrt[2] {x-1, 10 (y -x^2)}

In[4]:= FindMinimum[SS[x,y], {{x,1},{y,-1}},
Method->{"LevenbergMarquardt", "Residual"->R[x,y]}]

Out[4]=
{0., {x -> 1., y -> 1.}}

Note you could leave out the Sqrt[2] and minimize 2 SS[x,y] which
would
be slightly more efficient.


There is more explanation and examples in the Advanced Documentation
for unconstrained optimization.  Look in the Mathematica help browser
under
Advanced Documentation->Optimization->Unconstrained
Optimization->Methods for Local Minimization->Gauss Newton Methods
Rob Knapp
Wolfram Research


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