Function fitting on a set of data points

Hello,

I am trying to fit a function on a set of data point using some parameters to be optimized and the function lsqcurvefit.

The function is essentially:

y = a+b*cos(x)+c*cos(2*x)+d*sin(x)-e*sin(2*x);

So I first defined the parameters in terms on one variable, as following:

x(1) = a

x(2) = b

x(3) = c

x(4) = d

x(5) = e

And here goes the code:

DATA = load('data_f.txt');

t = DATA(:,1);
y = DATA(:,2);

plot(t,y,'ro')

F = @(x,xdata) +x(1) - x(2)*cos(xdata*(pi/180)) + x(3)*cos(2*xdata*(pi/180)) - x(4)*sin(xdata*(pi/180)) - x(5)*sin(2*xdata*(pi/180));

x0 = [1 1 1 1 1];

[x,resnorm,~,exitflag,output] = lsqcurvefit(F,x0,t,y)

hold on 
plot(t,F(t,y))
hold off

Problem is that the curve looks horrible but the parameters look fine compared to the fitting done in other programs. Is there something I am not seeing?

Thank you very much in advance!

Stay safe,

Alex

Answer 1

[John D'Errico]

What did you do wrong? It looks like Jon may be correct, in that you did not use the newly estimated set of parameters from the fit.

xy = [0.0	6.08
15	5.3
30	4.28
45	2.89
60	1.31
75	-0.16
90	-0.47
105	0.23
120	1.68
135	3.55
150	5.32
165	6.51
180	6.86
195	6.29
210	4.92
225	3.04
240	1.19
255	0
270	-0.09
285	0.75
300	2.21
315	3.89
330	5.29
345	6.09
360	6.08];
x = xy(:,1);
y = xy(:,2);

FIRST, PLOT YOUR DATA. LOOK AT IT. Does this simple trig model make sense?

plot(x,y,'ro')

And, yes, the result does seem to be a very simple sinusoidally varying curve.

However, there is no need to use lsqcurvefit to fit this, since it is a linear model in the parameters. Using a nonlinear estimation tool to fit a linear model is like using a Mack truck to carry a pea to Boston. A bit of over kill. But, yes, you could use lsqcurvefit. I won't bother to do so here.

A = [ones(size(x)),sind(x),sind(2*x),cosd(x),cosd(2*x)];
coefs = A\y

coefs = 5×1

3.1957 -0.1826 -0.4175 -0.2026 3.3489

plot(x,y,'ro')
hold on
fun = @(u,coefs) coefs(1) + coefs(2)*sind(u) + coefs(3)*sind(2*u) + coefs(4)*cosd(u) + coefs(5)*cosd(2*u);
plot(x,fun(x,coefs),'b-')
hold off

While there is a little lack of fit, it is not terrible. Look at the residual errors.

plot(x,y - fun(x,coefs),'o-')

Lack of fit is usually seen as patterning in the residuals. Here we see exactly that. So your model is probably not the correct model for this data, merely a good approximation to that data. By way of comparison, how much of the information in your original signal has been explained by the model?

[var(y),var(y - fun(x,coefs))]

ans = 1×2

6.1724 0.0445

As a percentage of the total variance...

100*(1 - var(y - fun(x,coefs))/var(y))

ans = 99.2797

So around 99% of the total variance in y has been explained by your model. So pretty good, but not perfect.

[Jon]

I didn't look that closely at the overall problem the OP was trying to solve, just at the specific error. Great point that this can be solved as a linear regression problem, as it is linear in the parameters, even though the function that is fit is not a linear equation. I think this is often overlooked.

[Alessandro Ruda]

Thank you guys for your time! I didn't consider this way of solving it

[John D'Errico]

Yes. I think that too often people overlook that point, that a "linear" regression refers to the parameters. If the model is linear in the parameters, then it is a linear regression problem. Instead, people look at what they see as a NONLINEAR model, and accept that this must be a nonlinear regression. Not the end of the world, but this is, as I am known to say, is like using a Mack truck to carry a pea to Boston. (Don't ask where I learned the phrase.) You don't want to use a nonlinear estimation tool to solve a linear problem if you can avoid it, as that carries its own set of issues.

Answer 2

[Jon]

It seems like you are not using the parameters you just solved for when evaluating your function to plot it.

Maybe you meant

plot(t,F(x,t))

[Jon]

or if you want to compare the data points to the fitted curve

numFit = 1000; % for example
tfit = linspace(t(1),t(end),numFit)
plot(t,y,'o',tfit,F(x,tfit))