Fit curve when corresponding to one x value we have multiple y value?

I have a data set where corresponding to one x value multiple y values are available. I tring to a*x^b function to these data using curve fitting app.

I an getting error 'Data sizes are incompatible'.

Do'nt know to arrange these data.

Help needed.

x =  [ 0.0500 0.5000 5.0000 15.0000 25.0000];
y = [ 1.0000 0.9686 0.9893 0.9585 0.9484;...
      1.0000 0.8911 0.8813 0.8420 0.8375;...
      1.0000 0.7802 0.7747 0.7802 0.8022;...
      1.0000 0.9819 0.9392 0.9038 0.8923;...
      1.0000 0.9590 0.8974 0.8840 0.7308;...
      1.0000 0.8803 0.8872 0.6493 0.7214;...
      1.0000 0.9551 0.8923 0.8163 0.7953;...
      1.0000 0.9591 0.9084 0.8098 0.6922;...
      1.0000 0.8306 0.7610 0.6142 0.6299;...
      1.0000 0.8993 0.8583 0.8115 0.7535;...
      1.0000 0.9072 0.8780 0.7501 0.6728;...
      1.0000 0.9348 0.8529 0.6733 0.5870;...
      1.0000 0.8333 0.8167 0.7667 0.7000;...
      1.0000 0.9048 0.8493 0.7405 0.6214;...
      1.0000 0.8892 0.7967 0.6878 0.5504];

Answer 1

[Image Analyst]

Usually what I do in that situation is to add a very tiny, insignificant bit a noise to the x values. It will be so small that it won't affect the fit but it will prevent you from having duplicate x values:

x =  [ 0.0500 0.5000 5.0000 15.0000 25.0000];
y = [ 1.0000 0.9686 0.9893 0.9585 0.9484;...
	1.0000 0.8911 0.8813 0.8420 0.8375;...
	1.0000 0.7802 0.7747 0.7802 0.8022;...
	1.0000 0.9819 0.9392 0.9038 0.8923;...
	1.0000 0.9590 0.8974 0.8840 0.7308;...
	1.0000 0.8803 0.8872 0.6493 0.7214;...
	1.0000 0.9551 0.8923 0.8163 0.7953;...
	1.0000 0.9591 0.9084 0.8098 0.6922;...
	1.0000 0.8306 0.7610 0.6142 0.6299;...
	1.0000 0.8993 0.8583 0.8115 0.7535;...
	1.0000 0.9072 0.8780 0.7501 0.6728;...
	1.0000 0.9348 0.8529 0.6733 0.5870;...
	1.0000 0.8333 0.8167 0.7667 0.7000;...
	1.0000 0.9048 0.8493 0.7405 0.6214;...
	1.0000 0.8892 0.7967 0.6878 0.5504];
[rows, columns] = size(y)
xm = repmat(x, [rows, 1])
% Make into vectors.
xv = xm(:); 
yv = y(:);
plot(xv, yv, '+', 'LineWidth', 3, 'MarkerSize', 12);
% Add a tiny bit of noise to each x value
xv = xv + 0.000001 * rand(length(xv), 1);
% Fit to a cubic
coefficients = polyfit(xv, yv, 3);
% Get new values
yFit = polyval(coefficients, xv);
hold on
plot(xv, yFit, 'r-', 'LineWidth', 3);
grid on;
xlabel('x', 'FontSize', 20);
ylabel('y', 'FontSize', 20);
legend('data', 'fit');

Use fitnlm since you want a power law. I'm attaching some examples of fitnlm().

Answer 2

[Star Strider]

If you want to regress all of them to the same set of parameters, this works:

xm = repmat(x, size(y,1), 1);
fcn = @(p,x) p(1) .* x.^p(2);
cf = @(b) norm(y - fcn(b,xm));
B = fminsearch(cf, [1;1]);
RL = fcn(B,xm);

figure
plot(x, y)
hold on
plot(x, RL, '--', 'LineWidth',2)
hold off
grid
xlabel('x')
ylabel('y')

producing:

Use a loop and loop through the rows of ‘y’ to fit them individually.

.