Fitting multiple data sets to single curve in least square sense

I am trying to fit multiple data sets i.e., x1,x2,x3 -> y1,y2,y3 to a single cuve f following this exapmle. However, it returns error that the second coulmn must be a vector. The ultimate goal is to fit to a curve such that the sum(abs(f(x)-y))<3. How can I do it in matlab starting with the following example

x1=(0:1:10)'; % Explanatory variable
x2=(0:1:10)'+1;
x3=(0:1:10)'+2;
x=[x1 x2 x3];
y = 5 + 3*x + 7*x.^2;
y = y + 2*randn((size(x)));% Add some noise to response variable
% Define function that will be used to fit data
% (F is a vector of fitting parameters)
f = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
F_fitted = nlinfit(x,y,f,[1 1 1]); %Error: Requires a vector second input argument.
% Display fitted coefficients
disp(['F = ',num2str(F_fitted)])
% Plot the data and fit
figure
plot(x,y,'*',x,f(F_fitted,x),'g');
legend('data','fit')

The actuall dataset is attached.

Answer 1

[Abolfazl Chaman Motlagh]

the nlinfit function take a vector as second input y, because it's cost function for optimization and regression is based on scalar output. you should fitt each model separately.

x1=(0:1:10)'; 
x2=(0:1:10)'+1;
x3=(0:1:10)'+2;
x=[x1 x2 x3];
y = 5 + 3*x + 7*x.^2;
y = y + 2*randn((size(x)));
model = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
for i=1:size(x,2)
    F_fitted(i,:) = nlinfit(x(:,i),y(:,i),model,[1 1 1]);
end
for i=1:size(x,2)
subplot(1,3,i)
plot(x(:,i),y(:,i),'*',x(:,i),model(F_fitted(i,:),x(:,i)),'g')
title(['(X_' num2str(i) ',Y_' num2str(i) ')'])
end

[Abdulllah]

Actually, this way, I get multiple curves however, I want a single optimium function to fit all the datasets.

[Abolfazl Chaman Motlagh]

well then. it means different dataset are actually more information for 1 curve. then combine them together and solve the problem.

x1=(0:1:10)';
x2=(0:1:10)'+0.2;
x3=(0:1:10)'+0.6;
x=[x1 x2 x3];
y = 5 + 3*x + 7*x.^2;
y = y + 2*randn((size(x)));
model = @(F,x) F(1) + F(2).*x + F(3).*x.^2;
F_fitted = nlinfit(x(:),y(:),model,[1 1 1]);
X_sort = sort(x(:));
Y_sort = sort(y(:));
plot(X_sort,Y_sort,'*',X_sort,model(F_fitted,X_sort),'b')

[Abdulllah]

Yes, I should do like that but my data sets are complicated and forming a vector results in a low correlation fit.

The actuall data set is attached.

[Abolfazl Chaman Motlagh]

column of your data doen't seem to belong to same curve!!! they look like have a small shift to up. is this what you want?

load('dataset.mat')
figure;
scatter(x(:,1),y(:,1),'filled');hold on
for i=2:6
    scatter(x(:,i),y(:,i),'filled')
end
legend('col1','col2','col3','col4','col5','col6','Location','northwest')

if this is okay with you, you can treat this x and y like my previous comment.

[X_sort,index] = sort(x(:));
Y_sort = y(index);

the rest is same.

[Abdulllah]

No, you have scatter plot it. I have to fit the all the coumns into single curve..e, I have to find out a single curve which can fit all the coulmns. A single curve fitting multiple datasets

[Abolfazl Chaman Motlagh]

i scatterd them to demonstrate the fact that they don't lie on same curve. the solution is in second comment.

[Abdulllah]

Thanks for this. I tried sorting and then finding the curve however it gives a very bad correlation even I tried upto 4th order.

May be I am in the wrong direction. What actually I want to do is that, I have to find out a function f such that f(x1)≈y1 , f(x2)≈y2 .... f(x6)≈y6

[Abolfazl Chaman Motlagh]

let us review our conversation, because i think there's misunderstanding between us.

-------------------------------------------------------------------------

you have two variable x and y. both have same number of elements. column are x and y are different observations. but we know that all datasets obey same relation. which means there is a function like f(x) that describe all relations. which means . j here is index of dataset (which is column) and i is index of element within a dataset (which is row). so basically:

so basically all elements of x and y are some information we can use in finding function f.

--------------------------------------------------------------------------

if this is the problem making x and y, a column vector using x(:) and y(:) and then finding best fit on all data will do the work. (my second comment)

but here comes my objection. if column of x (and y) are describing same functionality and relation. then ploting them should illustrate this fact. but as we can see in scatter plot that i put in third comment, we can see the pairs don't show same relation for any i. this means there is no functionality describing all this relations at once.

[AndresVar]

@Abdulllah like Abolfazi said, your data won't fit a single f(x). The data fits a*exp(b*x) pretty well but for different values of a.

There might be another parameter that is missing that collapses the data.

Find the different coefficient by fitting each series or find them like this: simulataneous Curve fitting - (mathworks.com)

Answer 2

[AndresVar]

x1,x2,x3 are column vectors that you combined into a matrix, so then y was evaluated to a matrix also.

nlinfit expects vectors not matrices

to fix it, combine x1 x2 x3 into 1 long column vector using ";"

x=[x1;x2;x3]

Then you should probably sort it and apply same sorting to y Sort array elements - MATLAB sort (mathworks.com)

[Abdulllah]

If I combined x1,x2,x3 to a single vector and do the same with y, Matlab finds a very bad solution. I am looking for a better options to get a single optimium function to fit all the datasets.