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Re: nonLinearFit and nonLinearRegress
*To*: mathgroup at smc.vnet.net
*Subject*: [mg55346] Re: nonLinearFit and nonLinearRegress
*From*: "Carl K. Woll" <carlw at u.washington.edu>
*Date*: Sat, 19 Mar 2005 04:47:03 -0500 (EST)
*Organization*: University of Washington
*References*: <d1ecpc$etg$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
John,
See below for my comments.
"John Hawkin" <john.hawkin at gmail.com> wrote in message
news:d1ecpc$etg$1 at smc.vnet.net...
> I'm trying to do a nonLinear fit on some data, fitting a parameter
> which appears in the limits of the integral that I'm trying to fit.
> When I do a nonLinearFit, I get about a page of errors, but it does
> work. It takes about 10 minutes to do it for 20 points with a
> reasonable initial estimate. However when I use the nonLinearRegress
> command, even with the best fit parameter from nonLinearFit given as
> the initial guess, it takes hours to work. Is it possible there is a
> way to speed this up, and is this normal?
>
> The commands that I'm using to integrate this function are:
>
> h[B]:=2/(B pi) NIntegrate[x Sin[x] Exp[-(x/B)^3/2], {x, 0, inf},
> Method-Oscillatory];
> holts[c_, d_, Q_]:= NIntegrate[h[B], {B, c/Q, d/Q}];
>
There are a number of syntax errors in your definition of h[B]. I will
assume in the following that you meant to define h[B_] as follows:
h[B_] := 2/(B Pi) NIntegrate[x Sin[x] Exp[-(x/B)^(3/2)],{x,0,Infinity},
Method->Oscillatory]
If the above is indeed the proper definition of h[B_], then you should be
able to turn your holts function into a single integral instead of a double
integral. First, let x=B t in the integral for h, to obtain
h[B_] := 2B/Pi NIntegrate[t Sin[B t] Exp[-t^(3/2)], {t,0,Infinity}]
Then change the order of integration in the holts integral to obtain
holts[c_,d_,Q_]:=2/Pi NIntegrate[t Exp[-t^(3/2) <stuff>, {t,0,Infinity}]
where
<stuff> = Integral[B Sin[B t],{B,c/Q,d/Q}]
Evaluating the revised holts function is practically instantaneous as
compared to your definition, and produces the same answer.
> where holts is the function I am fitting (the h[B] integral is known
> as the Holtsmark integral). The fitting command I'm using is:
>
> NonlinearFit[fitData, holts[c,d,Q], {c,d}, {Q, estimateQ}];
>
> where fitData contains sets of size 3, with the values of c and d as
> the first two values, and the data point that I'm fitting the integral
> to as the 3rd value. If anyone has any ideas on how I can perform the
> nonlinear regression fast enough so that I can do it many times (maybe
> 50), I would greatly appreciate it. Thanks very much,
>
> John Hawkin
>
Returning to your versions of h and holts, the first thing you should do is
to make sure the code for holts and h only executes when the arguments are
numerical.
h[B_?NumericQ] := 2/(B Pi) NIntegrate[ etc. ]
holts[c_?NumericQ,d_?NumericQ,Q_?NumericQ] := NIntegrate[ etc. ]
Next, in order to use NonlinearFit or NonlinearRegress with functions which
Mathematica is unable to differentiate, I like to teach Mathematica these
derivatives. So, using your definition of holts, it's easy to see that the
derivative of holts with respect to c, d and Q are
holtsc[c_?NumericQ,d_?NumericQ,Q_?NumericQ] := -h[c/Q]/Q
holtsd[c_?NumericQ,d_?NumericQ,Q_?NumericQ] := h[d/Q]/Q
holtsQ[c_?NumericQ,d_?NumericQ,Q_?NumericQ] := c h[c/Q]/Q^2 - d h[d/Q]/Q^2
Then, we teach these derivatives to Mathematica as follows:
Derivative[1,0,0][holts] = holtsc
Derivative[0,1,0][holts] = holtsd
Derivative[0,0,1][holts] = holtsQ
Then, use NonlinearFit or NonlinearRegress in the usual way. With your
definitions of h and holts, this takes a long time and produces lots of
error messages associated with oscillating functions and recursion. Using
the version of holts that I proposed above, using NonlinearFit or
NonlinearRegress is an order of magnitude faster, but still produces a bunch
of error messages.
Carl Woll
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