Doubt on data of a SISO system

Hello all,

How can I know if the relationship between these data is linear or nonlinear?

u=[0.158 0.105  0.158  0.158  0.158  0.158  0.105  0.158  0.105  0.105  0.158  0.158  0.158  0.158  0.211  0.211  0.158  0.105  0.158  0.211  0.158  0.105  0.105  0.105  0.158  0.158  0.158  0.158  0.158  0.158]

y=[0.760 0.759  0.757  0.759  0.763  0.763  0.763  0.767  0.770  0.771  0.770  0.770  0.766  0.758  0.758  0.758  0.757  0.757  0.756  0.753  0.750  0.749  0.749  0.754  0.756  0.756  0.756  0.756  0.755  0.755]

Where, u is the input and y is the output, and 0.158 really indicate the same location. Consider that there are no errors in the measured signals.

In fact, I want to know if the relationship between u and y is linear or nonlinear.

Does anyone have any suggestions?

Thanks.

[Information merged from answer]

I found this paper:

S.A. Billings, W.S.F. Voon (1983), Structure detection and model validity tests in the identification of nonlinear systems, IEE Proceedings, Vol. 130, No. 4, JULY 1983.

In Section 4 the authors use a high-order correlation function applied to the response signal performed in a linear identification. Then, if the result is within the confidence inteval of 95% means that the system is linear.

How can I implement the function described by Billings e Voon in matlab? If I'm not mistaken ,I will need to use a FIR model, is not it?

Can you help me with this?

Thanks

Answer 1

[Walter Roberson]

If one accepts that there are errors in the measurement of the values, then you cannot prove that the relationship is non-linear: the best you could do would be to calculate an error term.

The multiple occurrences of 0.158 in u (or is it x?): are those intended to represent the exact same location, or are they intended to indicate locations that are certainly different but are the same to within 3 decimal places?

[Walter Roberson]

If there are no errors in the measurements, then the fact that there are multiple copies of the same x but which have different y, establishes that y is not any deterministic function of x; likewise one can easily establish that x is not any deterministic function of y.

Linear systems are always bijective, but your system is not either "one-to-one" _or_ "onto".

The non-linear relationship of your data becomes quite clear if you

scatter(x,y)

[Emanuel]

This is true, but for longer periods of data, the cross- correlation function indicates a significant relationship.

How can I measure how much linearity or nonlinearity have these data ?

[Walter Roberson]

[ux, ua, ub] = unique(x);

[P,S] = polyfit(x, y, 1);

[Y, delta] = polyval(P, ux, S);

%Y are the mean y at the unique x positions, and delta are one standard deviation for each Y.

%visual representation

plot(ux, Y, 'b', ux, Y + delta, 'g--', ux, Y - delta, 'g--', x, y, 'c*')

or if you have the stats toolbox and just want the visual representation,

[ux, ua, ub] = unique(x);

xlab = cellstr(num2str(ux.'));

boxplot(y, xlab(ub));

[Walter Roberson]

In the above, if the measurements are considered to be exact, then a non-zero standard deviation for any unique x point establishes that the system is non-linear.

[Walter Roberson]

Emanual, I have not read that paper, and I have not studied system identification and I have not studied digital filters. I would have to do a fair bit of reading to figure out how to implement an algorithm such as that. I do not have the resources for doing that.

[Emanuel]

Walter, I think the CRA algorithm calculates the correlation functions of high order.

Thanks for helping me!