def test_spectrum():
# Test utils.spectrum()
# Parseval's theorem.
for length in (2**10, 3**7):
x = np.random.random(length)
power = utils.spectrum(x)[1]
assert_allclose(np.mean(x**2), np.sum(power))
def test_falpha():
# Tests data.falpha()
x = data.falpha(length=(2**10), mean=np.pi, var=np.e)
assert_allclose(np.mean(x), np.pi)
assert_allclose(np.std(x)**2, np.e)
for length in (2**10, 3**7):
for alpha in (1.0, 2.0, 3.0):
mean, var = 1.0 + np.random.random(2)
x = data.falpha(alpha=alpha, length=length, mean=mean, var=var)
# Estimate slope of power spectrum.
freq, power = utils.spectrum(x)
desired = np.mean(
np.diff(np.log(power[1:])) / np.diff(np.log(freq[1:])))
assert_allclose(-alpha, desired)
"""Maximum Lyapunov exponent of the Rössler oscillator. The "accepted" value is 0.0714, which is quite close to what we get. """ from nolitsa import data, lyapunov, utils import numpy as np import matplotlib.pyplot as plt sample = 0.2 x0 = [-3.2916983, -1.42162302, 0.02197593] x = data.roessler(length=3000, x0=x0, sample=sample)[1][:, 0] # Choose appropriate Theiler window. # Since Rössler is an aperiodic oscillator, the average time period is # a good choice. f, p = utils.spectrum(x) window = int(1 / f[np.argmax(p)]) # Time delay. tau = 7 # Embedding dimension. dim = [3] d = lyapunov.mle_embed(x, dim=dim, tau=tau, maxt=200, window=window)[0] t = np.arange(200) plt.title(u'Maximum Lyapunov exponent for the Rössler oscillator') plt.xlabel(r'Time $t$') plt.ylabel(r'Average divergence $\langle d_i(t) \rangle$') plt.plot(sample * t, d)