from numpy import arange, sin, pi, random, array
x = arange(0, 6e-2, 6e-2 / 30)
A, k, theta = 10, 1.0 / 3e-2, pi / 6
y_true = A * sin(2 * pi * k * x + theta)
y_meas = y_true + 2*random.randn(len(x))

def residuals(p, y, x):
    A, k, theta = p
    err = y - A * sin(2 * pi * k * x + theta)
    return err

def peval(x, p):
    return p[0] * sin(2 * pi * p[1] * x + p[2])

p0 = [8, 1 / 2.3e-2, pi / 3]
array(p0)
# array([8., 43.4783, 1.0472])

from scipy.optimize import leastsq
plsq = leastsq(residuals, p0, args=(y_meas, x))
plsq[0]
# array([10.9437, 33.3605, 0.5834])  # may vary (periodicity)

# Notice that this example has multiple equivalent minima related by
# :math:`\theta \to \theta + \pi` and :math:`A\to -A`. The optimization
# process may converge to any of these minima, depending on the
# initial guess.

A, k, theta
# (10., 33.3333, 0.5236)

import matplotlib.pyplot as plt
plt.plot(x, peval(x, plsq[0]),x,y_meas,'o',x,y_true)
plt.title('Least-squares fit to noisy data')
plt.legend(['Fit', 'Noisy', 'True'])
plt.show()