Python least squares fit on data

Question

I am currently working on a scientific paper for my university and got some data on which I would like to do a regression. The data looks like this:

Both, P (red) and w(blue) seem to follow a sin-function.

My functions to fit the data look like this:

def test_P(x, P0, P1, P2, P3):
    return P0 * np.sin(x * P1 + P2) + P3


def test_w(x, w0, w1, w2, w3):
    return w0 * np.sin(x * w1 + w2) + w3

Given the time-array time , w and p , I did the following:

paramp, paramp_covariance = optimize.curve_fit(test_P, time, P, maxfev=20000)
paramw, paramw_covariance = optimize.curve_fit(test_w, time, w, maxfev=20000)

Which results in:

You can see that the deflection w has been fit very nicely with: R^2 w = 0.9997 . Although the Force P didn't get fit at all.

I tried to reduce the number of parameters for P so no shifting along t or w itself is possible:

def test_P(x, P0, P1):
     return P0 * np.sin(x * P1)

This actually gets fit much better:

Although you can see that it's still not really a perfect fit as test_P(x, P0, P1, P2, P3) could theoretically do.

I am not sure how the data is fitted but due to its non-linearity, I assume that its simply the solution it wants to converge to because of local minima. If I could give some initial starting values for P0, P1, P2, P3 , I could solve this problem.

I am very happy if someone could help me.

Appendix

def test_P(x, P0, P1):
    return P0 * np.sin(x * P1)


def test_w(x, w0, w1, w2, w3):
    return w0 * np.sin(x * w1 + w2) + w3



# time, j, tau,  w, P = compute()



time = np.fromstring("0.00000000e+00 2.80568971e-06 5.61137943e-06 8.41706914e-06 "
                     "1.12227589e-05 1.40284486e-05 1.68341383e-05 1.96398280e-05 "
                     "2.24455177e-05 2.52512074e-05 2.80568971e-05 3.08625868e-05 "
                     "3.36682766e-05 3.64739663e-05 3.92796560e-05 4.20853457e-05 "
                     "4.48910354e-05 4.76967251e-05 5.05024148e-05 5.33081045e-05 "
                     "5.61137943e-05 5.89194840e-05 6.17251737e-05 6.45308634e-05 "
                     "6.73365531e-05 7.01422428e-05", sep=' ')

j = 26


w = np.fromstring("0.00000000e+00 5.38570360e-06 6.91685941e-06 1.85449532e-05 "
                     "3.74039599e-05 6.40181749e-05 9.84056769e-05 1.40161109e-04 "
                     "1.88501856e-04 2.42324540e-04 3.00295181e-04 3.60927587e-04 "
                     "4.22660154e-04 4.83951704e-04 5.43352668e-04 5.99555945e-04 "
                     "6.51467980e-04 6.98222382e-04 7.39199688e-04 7.74056091e-04 "
                     "8.02681759e-04 8.25178050e-04 8.41902951e-04 8.53367116e-04 "
                     "8.60248942e-04 8.63521680e-04", sep=' ')

P = np.fromstring("0. 7.28709546 20.71085451 37.0721402 55.07986215 "
                     "73.54180405 91.39806157 107.70934459 121.67898126 132.68066578 "
                     "140.27838808 144.23755455 144.52399949 141.28824859 134.84108157 "
                     "125.62238298 114.1621182 101.04496874 86.87495208 72.24302972 "
                     "57.7072657 43.77853371 30.9118352 19.52425605 10.03199405 "
                     "2.97389719 ", sep=' ')



paramp, paramp_covariance = optimize.curve_fit(test_P, time, P, maxfev=100000)
paramw, paramw_covariance = optimize.curve_fit(test_w, time, w, maxfev=100000)
P_fit = np.zeros(j)
w_fit = np.zeros(j)
for i in range(0, j):
    P_fit[i] = test_P(time[i], paramp[0], paramp[1])
    w_fit[i] = test_w(time[i], paramw[0], paramw[1],paramw[2], paramw[3])
print('R^2 P:  ', r2_score(P, P_fit))
print('R^2 w:  ', r2_score(w, w_fit))

# ------------------------------------------------------------------------------
# P L O T T E N   D E R   E R G E B N I S S E

fig, ax1 = plt.subplots()
ax1.set_xlabel('time[s]')
ax1.set_ylabel('Power [kg]')
l1, = ax1.plot(time, P, 'r.', label='P')
l2, = ax1.plot(time, test_P(time, paramp[0], paramp[1]), 'r-', label='P_fit')
ax1.tick_params(axis='y', colors='r')

ax2 = ax1.twinx()
ax2.set_ylabel('w,z [cm]')
l3, = ax2.plot(time, w, 'b.', label='w')
l4, = ax2.plot(time, test_w(time, paramw[0], paramw[1],paramw[2], paramw[3]), 'b-', label='w_fit')
# ax2.plot(time,z,color='tab:cyan',label='z')
ax2.tick_params(axis='y', colors='b')

lines = [l1, l2, l3, l4]
plt.legend(lines, ["P", "P_fit", "w", "w_fit"])
fig.tight_layout()
plt.show()

Answer 1

Short answer : It's because the phase of the sine function should be bound to the interval [0,2*np.pi] . If you leave the parameter out, you obviously don't have the bounding problem. You can specify the bounds in the scipy.optimize :

paramp, paramp_covariance = optimize.curve_fit(test_P, time, P, maxfev=100000, bounds = ([-np.inf,-np.inf,0,-np.inf],[np.inf,np.inf,2*np.pi,np.inf]))

Long answer :

I can't reproduce your w fit, if I use your code I get following image:

So the problem is at least consistent for both optimize functions. If you then bound the phase of the sine , you get the same result as before.

I don't know why this fixes it, I would just think that internally the optimize function searches for an gradient over P2 internally and not finding it, searching until it reaches some internal "max steps" parameter, so it's best optimization is its init.

Anyone know the math behind this one?

Full code below. It demonstrates the solution, as well as the W that forms a line.

import numpy as np 
import matplotlib.pyplot as plt
from scipy import optimize
def test_P(x, P0, P1, P2, P3):
    return P0 * np.sin(x * P1 + P2) + P3 


def test_w(x, w0, w1, w2, w3):
    return w0 * np.sin(x * w1 + w2) + w3



# time, j, tau,  w, P = compute()



time = np.fromstring("0.00000000e+00 2.80568971e-06 5.61137943e-06 8.41706914e-06 "
                     "1.12227589e-05 1.40284486e-05 1.68341383e-05 1.96398280e-05 "
                     "2.24455177e-05 2.52512074e-05 2.80568971e-05 3.08625868e-05 "
                     "3.36682766e-05 3.64739663e-05 3.92796560e-05 4.20853457e-05 "
                     "4.48910354e-05 4.76967251e-05 5.05024148e-05 5.33081045e-05 "
                     "5.61137943e-05 5.89194840e-05 6.17251737e-05 6.45308634e-05 "
                     "6.73365531e-05 7.01422428e-05", sep=' ')

j = 26


w = np.fromstring("0.00000000e+00 5.38570360e-06 6.91685941e-06 1.85449532e-05 "
                     "3.74039599e-05 6.40181749e-05 9.84056769e-05 1.40161109e-04 "
                     "1.88501856e-04 2.42324540e-04 3.00295181e-04 3.60927587e-04 "
                     "4.22660154e-04 4.83951704e-04 5.43352668e-04 5.99555945e-04 "
                     "6.51467980e-04 6.98222382e-04 7.39199688e-04 7.74056091e-04 "
                     "8.02681759e-04 8.25178050e-04 8.41902951e-04 8.53367116e-04 "
                     "8.60248942e-04 8.63521680e-04", sep=' ')

P = np.fromstring("0. 7.28709546 20.71085451 37.0721402 55.07986215 "
                     "73.54180405 91.39806157 107.70934459 121.67898126 132.68066578 "
                     "140.27838808 144.23755455 144.52399949 141.28824859 134.84108157 "
                     "125.62238298 114.1621182 101.04496874 86.87495208 72.24302972 "
                     "57.7072657 43.77853371 30.9118352 19.52425605 10.03199405 "
                     "2.97389719 ", sep=' ')

print(P)

paramp, paramp_covariance = optimize.curve_fit(test_P, time, P, maxfev=100000, bounds = ([-np.inf,-np.inf,0,-np.inf],[np.inf,np.inf,2*np.pi,np.inf]))
print(paramp)

paramw, paramw_covariance = optimize.curve_fit(test_w, time, w, maxfev=100000)
print(paramw)
P_fit = np.zeros(j)

w_fit = np.zeros(j)
for i in range(0, j):
    P_fit[i] = test_P(time[i], paramp[0], paramp[1], paramp[2], paramp[3])
    w_fit[i] = test_w(time[i], paramw[0], paramw[1],paramw[2], paramw[3])

# ------------------------------------------------------------------------------
# P L O T T E N   D E R   E R G E B N I S S E

fig, ax1 = plt.subplots()
ax1.set_xlabel('time[s]')
ax1.set_ylabel('Power [kg]')
l1, = ax1.plot(time, P, 'r.', label='P')
l2, = ax1.plot(time, test_P(time, paramp[0], paramp[1], paramp[2], paramp[3]), 'r-', label='P_fit')
ax1.tick_params(axis='y', colors='r')

ax2 = ax1.twinx()
ax2.set_ylabel('w,z [cm]')
l3, = ax2.plot(time, w, 'b.', label='w')
l4, = ax2.plot(time, test_w(time, paramw[0], paramw[1],paramw[2], paramw[3]), 'b-', label='w_fit')
# ax2.plot(time,z,color='tab:cyan',label='z')
ax2.tick_params(axis='y', colors='b')

lines = [l1, l2, l3, l4]
plt.legend(lines, ["P", "P_fit", "w", "w_fit"])
fig.tight_layout()
plt.show()

yields: