Data fitting by non linear discrete equation.

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Ankur Pal 2020-11-24

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此问题的直接链接

https://ww2.mathworks.cn/matlabcentral/answers/660513-data-fitting-by-non-linear-discrete-equation

评论： Ankur Pal 2020-11-24

    xdata=[5.0	4.2	4.3	4.3	4.3	4.1	3.7 3.6	3.9	5.9	6.7	5.4	5.1	5.5	6.2	6.7	6.9	6.2	6.1	6.3	5.9	5.5	5.5	5.5	5.5];
    ydata=[2.0	2.5	1.9	2.1	2.0	2.1	2.7	4.0	3.8	3.9	3.6	3.2	3.2	3.1	3.2	3.7	3.7	3.2	3.2	3.5	3.5	3.2	3.1	3.3	3.1];

I want to fit the following equations to this data: I want to estimate all the parameter. (alpha and beta lie between 0 and 1, a lies between 0 and alpha, b lies between 0 and beta and c can be any value from 0 to infinity.) Thanks in advance.

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Rik 2020-11-24

What did you try so far?

Ankur Pal 2020-11-24

编辑：Ankur Pal 2020-11-24

I have tried lsqcurvefit but could not understand how to frame these equations. Also how to limit the parameters.

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Answer 1

Rik 2020-11-24

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The code below uses fminsearch, which means you don't need the optimization toolbox. The downside is that it is sensitive to a local minimum, depending on your initial parameter estimate.

xdata=[5.0	4.2	4.3	4.3	4.3	4.1	3.7 3.6	3.9	5.9	6.7	5.4	5.1	5.5	6.2	6.7	6.9	6.2	6.1	6.3	5.9	5.5	5.5	5.5	5.5];
ydata=[2.0	2.5	1.9	2.1	2.0	2.1	2.7	4.0	3.8	3.9	3.6	3.2	3.2	3.1	3.2	3.7	3.7	3.2	3.2	3.5	3.5	3.2	3.1	3.3	3.1];
initial_guess=0.5*ones(1,5);
fitted_params=fminsearch(@(fit_vals) costfun(fit_vals,xdata,ydata),initial_guess);
[alpha,beta,a,b,c]=deal(fitted_params(1),fitted_params(2),fitted_params(3),fitted_params(4),fitted_params(5));
x0=xdata(1);y0=ydata(1);elems=numel(xdata);
[x_fitted,y_fitted]=f_g(fitted_params,x0,y0,elems);
disp([x_fitted;y_fitted])
  Columns 1 through 17

    5.0000    3.7777    2.9288    2.3394    1.9301    1.6459    1.4485    1.3114    1.2163    1.1502    1.1043    1.0724    1.0503    1.0349    1.0241    1.0161    1.0095
    2.0000    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9821    0.9820    0.9816    0.9799    0.9767

  Columns 18 through 25

    1.0039    0.9990    0.9946    0.9906    0.9871    0.9835    0.9810    0.9765
    0.9734    0.9701    0.9670    0.9639    0.9615    0.9581    0.9577    0.9489
function cost=costfun(fit_vals,xdata,ydata)
x0=xdata(1);y0=ydata(1);elems=numel(xdata);
[x,y]=f_g(fit_vals,x0,y0,elems);
cost= sum((x-xdata).^2) + sum((y-ydata).^2);
[alpha,beta,a,b,c]=deal(fit_vals(1),fit_vals(2),fit_vals(3),fit_vals(4),fit_vals(5));
if ( alpha<0 || alpha>1 ) || ...
        ( a<0 || a>alpha ) || ...
        ( beta<0 || beta>1 ) || ...
        ( b<0 || b>beta ) || ...
        c<0
    cost=inf;
end
end
function [x,y]=f_g(params,x0,y0,elems)
[alpha,beta,a,b,c]=deal(params(1),params(2),params(3),params(4),params(5));
x=zeros(1,elems);x(1)=x0;
y=zeros(1,elems);y(1)=y0;
for n=2:elems
    x(n)=(1-alpha)*x(n-1) + a/(1+exp(-c*(x(n-1)-y(n-1))));
    y(n)=(1-beta )*y(n-1) + b/(1+exp(-c*(x(n-1)-y(n-1))));
end
end