Global curve fitting for polynomial function

Let me clarify. A fitting needs to be done which produces a the same shared value for y0 but but still produces individual values for a, b, and c, for both data-sets. y0 should be determined so that it gives the best fit possible for both data-sets.

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Answer 2

Torsten 2016-10-10

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https://ww2.mathworks.cn/matlabcentral/answers/306574-global-curve-fitting-for-polynomial-function#answer_238380

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The objective function for your problem is given by

f(y0,a1,b1,c1,a2,b2,c2)=
sum_{i=1}^{n1}(ydata1_{i}-(y0+a1*xdata1_{i}+b1*xdata1_{i}^2+c1*xdata1_{i}^3))^2 +
sum_{i=1}^{n2}(ydata2_{i}-(y0+a2*xdata2_{i}+b2*xdata2_{i}^2+c2*xdata2_{i}^3))^2

Now take partial derivatives of f with respect to y0,a1,b1,c1,a2,b2,c2.

You'll arrive at a (7x7) linear system of equations in the unknows y0,a1,b1,c1,a2,b2,c2. This can be solved using \ (backslash).

Best wishes

Torsten.

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Torsten 2016-10-11

编辑：Torsten 2016-10-11

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Alternativly, solve the (overdetermined) linear system of equations using \ (backslash):

y0+a1*xdata1_{i}+b1*xdata1_{i}^2+c1*xdata1_{i}^3=ydata1_{i} (i=1,...,n1)
y0+a2*xdata2_{i}+b2*xdata2_{i}^2+c2*xdata2_{i}^3=ydata2_{i} (i=1,...,n2)

These are (n1+n2) equations in the unknowns y0,a1,a2,a3,b1,b2,b3.

Best wishes

Torsten.

Christian Sögaard 2016-10-14

Tank you Torsten for suggesting solutions to my problem. However, I was not able to work out how to define the objective function in matlab. Instead I worked out another way of solving my problem, which you can see below.

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Answer 3

Christian Sögaard 2016-10-14

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I have managed to solve this rather crudely myself. I have written a while loop that takes small steps for the y0 parameter and fits the other parameters with regard to this y0 value. All of the different fittings is then evaluated and the best one is chosen. Below you can see the script code for this:

%initial fitting.
n=6;
p=polyfit(x,y,n)
c1=polyval(p,x);
figure
plot(x,y,'s')
hold on
plot(x,c1)
Rsq=1-sum((y-c1).^2)/sum((y - mean(y)).^2)
n=6;
p0=polyfit(x0,y0,n)
c2=polyval(p0,x0);
figure
plot(x0,y0,'s')
hold on
plot(x0,c2)
Rsq=1-sum((y0-c2).^2)/sum((y0 - mean(y0)).^2)
%global fitting
y_g=p(end)
i=1
%step size for the shared parameter
step=0.001;
while y_g<p0(end)
%blanc
    f0=fitoptions('Method','NonlinearLeastSquares',...
    'Lower',[-2,-2],...
    'Upper',[Inf,max(x0)],...
    'StartPoint',[-1 -1 -1 -1 -1 -1]);
    ft=fittype('y_global+a*x+b*x^2+c*x^3+d*x^4+e*x^5+f*x^6','problem','y_global','options',f0);
    [curve0,gof0]=fit(x0,y0,ft,'problem',y_g)  
    koef_0(i,:)=coeffvalues(curve0)
    R2_0(i,:)=gof0.rsquare
%sample
    [curve1,gof1]=fit(x,y,ft,'problem',y_g)  
    koef1=coeffvalues(curve1)
    koef_1(i,:)=coeffvalues(curve1)
    R2_1(i,:)=gof1.rsquare
% y_g parameter save
    y_save(i,:)=y_g;
%while loop parameters    
    y_g=y_g+step;
    i=i+1;
end
%calc of best fit
R_average=(R2_0+R2_1)./2;
val=1;
tmp=abs(R_average-val);
[idx idx]=min(tmp);
closestR=R_average(idx)
closest_koef0=koef_0(idx,:)
closest_koef1=koef_1(idx,:)
closest_R2_0=R2_0(idx)
closest_R2_1=R2_1(idx)
closest_y0=y_save(idx)