Global curve fitting for polynomial function

Hello all,

I have two sets of data (xdata1, ydata1, and xdata2, ydata2) and I would like to do a global curve fit for these. The function is:

y = y0 + ax + bx^2 + cx^3

where y0 is a shared parameter and a, b, and c can vary for each data set.

I am quite new to MATLAB so I am wondering if anyone have a clue of how to do this?

Thank you in advance for any answers.

/Christian

Answer 1

[dbmn]

I suggest you read the help for the polyfit function here

p = polyfit(xdata1,ydata1,3);

[dbmn]

I just read, that the y0 should be shared. For forcing the curve trough a specific point y_offset @x==0 you can use the following

params = [x'.^3, x'.^2, x']\(y-y_offset)';
c = params(1); b = params(2); a = params(3);

You might need of not need the ' depending on the shape of your x and y vectors.

[edit: credit to John d'Errico for the solution with the Solution with the mldivide operator]

[Torsten]

y_offset is also an unknown parameter.

Best wishes

Torsten.

[Christian Sögaard]

Thank you for your answers. However, Torsten is right. y0 is not known.

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.

Answer 2

[Torsten]

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.

[Torsten]

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]

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.

Answer 3

[Christian Sögaard]

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)