Re: Global Fitting multiple equations to multiple
- To: mathgroup at smc.vnet.net
- Subject: [mg84637] Re: Global Fitting multiple equations to multiple
- From: dh <dh at metrohm.ch>
- Date: Mon, 7 Jan 2008 07:28:17 -0500 (EST)
- References: <flqc6h$jv4$1@smc.vnet.net>
Hi Michael,
you want to fit a vector valued function but FindFit can only deal with
scalar valued functions. Well, all you have to do is to help FindFit a
bit. If we consider the norm of the error we have a scalar function that
we may fit to the zero function.
Say we have data yi[tj] where yi are the vector components and tj the
discrete time. Further yic[t] is the theortical expression for the
component i. Then
err=Sum( (yic[tj]-yi[tj])^2 ) where the sum is over i and j
is the norm square of the error (you may even take another norm).
You may now fit err againste the data: {{t1,0},{t2,0}...} to get the
parameters.
hope this helps, Daniel
Michael Parker Latham wrote:
> Hi,
> I have collected a time dependent data set. At each time point there is four observables (IntDiagBo, IntDiagFr, IntCrossFrBo and IntCrossBoFr) which can be described by four equations (ibb, iff, ifb, ibf). These four equations are related and depend on six variables (ib, if, rb, rf, kbf, kfb). I would like to globally fit the four data sets to the four equations to extract the six variable. What I have done so far is below
>
> In[1]:= << "NonlinearRegression`"
>
> Setup Equations
>
> In[2]:= a11 = rb + kbf;
> a12 = -kfb;
> a21 = -kbf;
> a22 = rf + kfb;
> \[Lambda]1 = 1/2*((a11 + a22) + Sqrt[(a11 - a22)^2 + 4*kfb*kbf]);
> \[Lambda]2 = 1/2*((a11 + a22) - Sqrt[(a11 - a22)^2 + 4*kfb*kbf]);
>
> Diagonal Peaks Equations
>
> In[8]:= ibb = ib*((\[Lambda]1 - a11) Exp[-\[Lambda]2*x1] - (\[Lambda]2 - a11)
> Exp[-\[Lambda]1*x1])/(\[Lambda]1 - \[Lambda]2);
> iff = if*((\[Lambda]1 - a22) Exp[-\[Lambda]2*x1] - (\[Lambda]2 - a22)
> Exp[-\[Lambda]1*x1])/(\[Lambda]1 - \[Lambda]2);
>
> Cross Peaks Equations
>
> In[10]:= ibf = ib*(a21*Exp[-\[Lambda]1*x1] -
> a21*Exp[-\[Lambda]2*x1])/(\[Lambda]1 - \[Lambda]2);
> ifb = if*(a12*Exp[-\[Lambda]1*x1] -
> a12*Exp[-\[Lambda]2*x1])/(\[Lambda]1 - \[Lambda]2);
>
> Data for 1 set of peaks
>
> In[12]:= IntDiagBo = {{0.01938, 5111831}, {0.03673, 3996563}, {0.03673, 4073294}, {0.05409, 3057033}, {0.07144, 2487152}, {0.08879, 1967903}, {0.10615, 1588128}, {0.10615, 1500203}, {0.1235, 1222629} , {0.14086, 1046551}};
> IntDiagFr = {{0.01938 , 13757210}, {0.03673, 11098630}, {0.03673, 11044150}, {0.05409, 9100536}, {0.07144, 7166819}, {0.08879, 6045878}, {0.10615, 4976833}, {0.10615, 5150198}, {0.1235, 4106922}, {0.14086, 3583284}};
> IntCrossFrBo = {{0.01938, 418536.5}, {0.03673, 607315.1}, {0.03673, 647085.9}, {0.05409, 663520.1}, {0.07144, 653818.1}, {0.08879, 643523.8}, {0.10615, 577511.9}, {0.10615, 647762.6}, {0.1235, 626453.3}, {0.14086, 564194.7}};
> IntCrossBoFr = {{0.01938, 450925.1}, {0.03673, 623400.1}, {0.03673, 719430.4}, {0.05409, 782262.1}, {0.07144, 796733.9}, {0.08879, 730650.5}, {0.10615, 709815.9}, {0.10615, 707971.9}, {0.1235, 754398.6}, {0.14086, 645057.8}};
>
> Restrain Exchange Rates with Kd
>
> In[28]:= 0.002 == kfb/kbf;
>
> I can use FindFit to fit each data set to its respective equation, but I would like to globally, simultaneously fit the four equations to the four data sets to get "better" values for the six variables.
>
> Any help with this would be greatly appreciated. Please let me know if there is any other information that can be supplied.
> Thanks in advance,
> Mike
>
>
> Michael Latham
> Graduate Student
> Department of Chemistry and Biochemistry
> University of Colorado, Boulder
> Phone: 303-492-8085
> Fax: 303-492-2439
>