MATLAB: How to see if data fit a specified curve

curve fittingmodel

I have expression data, some are correlated, but some follow a pattern of y = 1/x (only the positive values of x and y).
How do I rapidly tell if the data fit y = 1/x (I have a few thousand pairs of data to try, hence not wanting to plot each one individually). Is there an easy way of doing this with curve fitting functions?

Best Answer

There are several ways to classify your data into the three groups: 1) random, 2) y=x, and 3) y=1/x. I tend to favor low-level approaches whenever possible. The demo below creates a dataset that fits your description and then classifies each data point according to those three categories. It does so by computing the error between each coordinate and the two functions y=x and y=1/x. If the error is above a set (subjective) threshold for both functions, the coordinate is classified as random. Otherwise, it is classified according to the function with the smallest error. If your data contain a considerable amount of noise, first trying increasing the errThreshold. If that doesn't help, we may need to use a more sophistocated algorithm. 
% Create x-data
% the +.2 is to avoid values near 0 which cause problems with plotting 1/x
x = rand(1,1000)*5 + .2;  
% Create 3 groups of y-data
%1) y = x
%2) y = 1/x
%3) random
randXIdx = randperm(numel(x)); 
sections = floor(numel(x).*[.33, .66]); 
y = rand(size(x))*5 + .2;                                     % random y values
y(randXIdx(1:sections(1)-1)) = x(randXIdx(1:sections(1)-1));  %y = x
y(randXIdx(sections(1):sections(2))) = 1./x(randXIdx(sections(1):sections(2)));  %y = 1/x
% Look at data
clf()
plot(x,y, 'o')
% Compute the error between the two function y=x and y=1/x.
err1 = abs(y-x); 
err2 = abs(y - 1./x); 
% Choose a threshold.  Plotting the error may be helpful. 
% Error greater than threshold will be assigned to random class
%     clf()
%     plot(err1,'ro')
%     hold on
%     plot(err2, 'bx')
errThreshold = 0.05;    % my subjective judgement
% Classify the coordinates based on minimum error.
% If the error for both is beyond threshold, classify as random. 
group = zeros(size(y)); 
group(err1 >= errThreshold & err2 >= errThreshold) = 1; % group 1 is random group
group(err1 < errThreshold) = 2; % group 2 is y=x group
group(err2 < errThreshold) = 3; % group 3 is y=1/x groups (and y=x if the point belongs to both groups)
% Check that all points are assigned to a group
if any(group==0)
    error('Point not assigned to group.')
end
% Plot results
clf()
plot(x,y, 'ko')
hold on
plot(x(group==1),y(group==1), 'r.', 'DisplayName', 'rand')
plot(x(group==2),y(group==2), 'b.', 'DisplayName', 'y=x')
plot(x(group==3),y(group==3), 'g.', 'DisplayName', 'y=1/x')
legend()
Results of classification:
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