I am trying to fit dataset with a function, but with little success, as illustrated below. The blue line is the data I wish to fit, and the red line is my attempt.
Below is the function I used:
$$ g(x) = -A\sin\left(f \frac{\pi}{2}\cos(x)\right)^2 + D $$
where $A$, $f$, and $D$ are parameters. In Python,
-A*np.sin(f*(np.pi/2)*np.cos(x))**2+D
Any help would be greatly appreciated!
On
You have $n$ data points $(x_i,y_i)$ and you want to fit based on the model $$y= -A\sin\left(f \,\frac{\pi}{2}\,\cos(x)\right)^2 + D$$ or (notations are not clear) $$y= -A\sin^2\left(f \,\frac{\pi}{2}\,\cos(x)\right)+ D$$ but this does not matter.
The model is nonlinear because of $f$. So, give it a value and define $$t_i=\sin^2\left(f \,\frac{\pi}{2}\,\cos(x_i)\right)$$ Now, the model is linear and for any $f$ you have $A(f)$ and $D(f)$ (even explicitly).
For this value of $f$ compute the sum of squares of residuals $SSQ(f)$ and change the value of $f$ until you see a minimum.
At this point, you then have good estimates of the three parameters and you can start a nonlinear regression.
On
Of course the shape of the experimental curve seems sinusoidal in the range of the higher values of y. But in the range of the smaller values of y the shape looks more as arcs of exponential than sinusoidal. That is why I tried an exponential function of a sinusoidal function (Instead of a sinusoidal function of a sinusoidal function).
A simplified function (slightly less accurate) is :
Half the frequency of your function (replace
xwithx / 2), and then square the entire result. The resulting function should be proportional to a rough approximation of the function you are trying to approximate, so you can just scale it down until you are happy with the result. Eyeballing it from your image, I would guess that an added factor of $0.2$ would do the trick ($10^2\div20=5$, $1\div5=0.2$). (Bear in mind that you would be multiplying your entire function by $0.5$, not dividing by $0.5$.)