I have some data (x and y coordinates) coming from a study and I have to plot them and to find the best curve that fits data. My curves are:

• polynomial up to 6th degree;
• power law; and
• exponential.

I am able to find the best fit for polynomial with

while(i < 6):
    coefs, val = poly.polyfit(x, y, i, full=True)

and I take the degree that minimizes val.

When I have to fit a power law (the most probable in my study), I do not know how to do it correctly. This is what I have done. I have applied the log function to all x and y and I have tried to fit it with a linear polynomial. If the error (val) is lower than the others polynomial tried before, I have chosen the power law function. Am I correct?

Now how can I reconstruct my power law starting from the line y = mx + q in order to draw it with the original points? I need also to display the function found.

I have tried with:

def power_law(x, m, q):
    return q * (x**m)

using

x_new = np.linspace(x[0], x[-1], num=len(x)*10)
y1 = power_law(x_new, coefs[0], coefs[1])
popt, pcov = curve_fit(power_law, x_new, y1)

but it seems not to work well.