Finding Linearity

Hi everyone ,

In the attached image you can see a curve initially a linear response at some point it is becoming nonlinear . Is there any simple procedure to find out that point at which it behavior become nonlinear ??

http://img801.imageshack.us/img801/9980/unbenanntuhu.png

Regards

Answer 1

[Grzegorz Knor]

My idea is as follows: fit line to x & y data, next fit to x(1:end-1) & y(1:end-1), next x(1:end-2) & etc. Then check to which point the data behave in linear way.

For example:

% generate example data
x = linspace(0,1,100);
y = smooth(interp1([0 .4 1],[0 .9 1],x),20);
x = x(:);
y = y(:);
plot(x,y)
% fit coefficients 
hold on
nrm = zeros(length(x)-1,1);
coef = zeros(length(x)-1,2);
for k = length(x):-1:2
    coef(k-1,:) = ([x(1:k) ones(k,1)]\y(1:k))';
    nrm(k-1) = norm(x(1:k)*coef(k-1,1)+coef(k-1,2)-y(1:k));
end
% plot nrm vector
figure
plot(nrm)
% define critical value
crit = 5e-2;
% find the value that satisfies the given condition
idx = find(nrm<crit,1,'last');
% plot result
figure(1)
plot(x(1:idx),x(1:idx)*coef(idx,1)+coef(idx,2),'r')

[Syed Umair Uddin]

Could you please mathematically elaborate it a bit more ?

Answer 2

[bym]

perhaps a simple check of the derivative

find(abs(diff(y)) > tol)  % where tol is a suitable tolerance

[edit]

maybe this will give you some ideas to build upon:

im = imread('http://img801.imageshack.us/img801/9980/unbenanntuhu.png');
imshow(im)
gimg = rgb2gray(im); % convert to gray
bw = gimg<50;        % convert to BW
figure
bwt = bwmorph(bw,'thin','Inf'); % thin pixels to line
imshow(~bwt)
[r,c] = find(bwt);
figure
ddc = diff(diff(r)); %2nd derivative  suggests 5 is a threshold
plot(ddc)

[Syed Umair Uddin]

for example in the above case what value for the tolerance ?

[bym]

well, since you didn't provide any scaling it is difficult to say. Do you have the data points that generated this curve? Or, are you asking how to determine non-linearity from the image itself?

[Syed Umair Uddin]

Here it is

http://img26.imageshack.us/img26/4121/81400288.jpg

off course it is not an image . I am generating the graph from the measured data at any x point the behavior is changing into non linearity I am interested to find out that value . Thanks alot in advance .

Answer 3

[bym]

Here is one way of accomplishing your goal (with the caveat that I have not tested this on different curves. I used Jiro's FEX submission to convert your image into data points called Data001

plot(Data001(:,1),Data001(:,2))
hold
d = diff(Data001(:,2));
rd = flipud(d);
r = find(d-rd==0);  % this line will need tweaking
plot(Data001(r+1,1),Data001(r+1,2),'ro')

I am sure this is not very robust to a great variety of curves, but might serve your purposes

Data001 =

    1.2538    1.4481
    1.2594    1.4528
    1.2675    1.4609
    1.2762    1.4704
    1.2805    1.4744
    1.2886    1.4832
    1.2999    1.4967
    1.3117    1.5096
    1.3179    1.5177
    1.3279    1.5292
    1.3385    1.5420
    1.3479    1.5528
    1.3585    1.5636
    1.3684    1.5737
    1.3790    1.5832
    1.3871    1.5899
    1.3977    1.5987
    1.4132    1.6081
    1.4263    1.6149
    1.4412    1.6209
    1.4536    1.6242
    1.4660    1.6269
    1.4796    1.6282
    1.4932    1.6295
    1.5044    1.6302
    1.5211    1.6315
    1.5335    1.6314
    1.5490    1.6327
    1.5638    1.6327
    1.5756    1.6333
    1.5892    1.6339
    1.5991    1.6339

[Syed Umair Uddin]

Hi thanks .. Could you please explain how exactly we will get the data points which are linear for example

d = diff(Data001(:,2)); % differentiating Y data points

rd = flipud(d); % Inverting d

r = find(d-rd==0);

Thanks in advance