def test_line(self):
# Test dimension.afn() by embedding a line.
# Particle moving uniformly in 1-D.
a, b = np.random.random(2)
t = np.arange(100)
x = a + b * t
dim = np.arange(1, 10 + 2)
window = 10
# Chebyshev distances between near-neighbors remain bounded.
# This gives "cleaner" results when embedding known objects like
# a line. For a line, E = 1.0 for all dimensions as expected,
# whereas it is (d + 1) / d (for cityblock) and sqrt(d + 1) /
# sqrt(d) for Euclidean. In both cases, E -> 1.0 at large d,
# but E = 1.0 is definitely preferred.
for metric in ('chebyshev', 'cityblock', 'euclidean'):
Es_des = (window + 1) * b
if metric == 'chebyshev':
E_des = 1.0
elif metric == 'cityblock':
E_des = (dim + 1) / dim
elif metric == 'euclidean':
E_des = np.sqrt((dim + 1) / dim)
E, Es = dimension.afn(x, dim=dim, metric=metric)
assert_allclose(E_des, E)
assert_allclose(Es_des, Es)
def test_noise(self):
# Test dimension.afn() using uncorrelated random numbers.
x = np.random.random(1000)
dim = np.arange(1, 5 + 2)
window = 10
metric = 'chebyshev'
E, Es = dimension.afn(x, dim=dim, metric=metric, window=window)
_, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1]
# The standard deviation of E2 should be ~ 0 for uncorrelated
# random numbers [Ramdani et al., Physica D 223, 229 (2006)].
# Additionally, the mean of E2 should be ~ 1.0.
assert_allclose(np.std(E2), 0, atol=0.1)
assert_allclose(np.mean(E2), 1, atol=0.1)
#!/usr/bin/env python # -*- coding: utf-8 -*- """AFN for time series from the Henon map. E1 saturates near an embedding dimension of 2. E2 != 1 at many values of d. Thus the series is definitely deterministic. The plot matches Fig. 1 of Cao (1997) pretty well. """ from nolitsa import data, dimension import matplotlib.pyplot as plt import numpy as np # Generate data. x = data.henon()[:, 0] # AFN algorithm. dim = np.arange(1, 10 + 2) E, Es = dimension.afn(x, tau=1, dim=dim, window=5) E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1] plt.title(r'AFN for time series from the Henon map') plt.xlabel(r'Embedding dimension $d$') plt.ylabel(r'$E_1(d)$ and $E_2(d)$') plt.plot(dim[:-1], E1, 'bo-', label=r'$E_1(d)$') plt.plot(dim[:-1], E2, 'go-', label=r'$E_2(d)$') plt.legend() plt.show()
The minimum embedding dimension comes out to be 5-7 (depending on the initial condition) with both E1 and E2 curves giving very strong hints of determinism. According to Grassberger & Procaccia (1983) the correlation dimension of the Mackey-Glass system with a delay of 23 is ~ 2.5. Thus, the results are definitely comparable. """ import numpy as np import matplotlib.pyplot as plt from nolitsa import data, dimension x = data.mackey_glass(tau=23.0, sample=0.46, n=1000) # Since we're resampling the time series using a sampling step of # 0.46, the time delay required is 23.0/0.46 = 50. tau = 50 dim = np.arange(1, 16 + 2) # AFN algorithm. E, Es = dimension.afn(x, tau=tau, dim=dim, window=100) E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1] plt.title(r'AFN for time series from the Mackey-Glass system') plt.xlabel(r'Embedding dimension $d$') plt.ylabel(r'$E_1(d)$ and $E_2(d)$') plt.plot(dim[:-1], E1, 'bo-', label=r'$E_1(d)$') plt.plot(dim[:-1], E2, 'go-', label=r'$E_2(d)$') plt.legend() plt.show()
#!/usr/bin/env python # -*- coding: utf-8 -*- """AFN for time series from the Rössler oscillator. E1 saturates near an embedding dimension of 4. E2 != 1 at many values of d. Thus, the series is definitely deterministic. """ from nolitsa import data, dimension import matplotlib.pyplot as plt import numpy as np # Generate data. x = data.roessler()[1][:, 0] # AFN algorithm. dim = np.arange(1, 10 + 2) E, Es = dimension.afn(x, tau=14, dim=dim, window=45, metric='cityblock') E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1] plt.title(r'AFN for time series from the Rössler oscillator') plt.xlabel(r'Embedding dimension $d$') plt.ylabel(r'$E_1(d)$ and $E_2(d)$') plt.plot(dim[:-1], E1, 'bo-', label=r'$E_1(d)$') plt.plot(dim[:-1], E2, 'go-', label=r'$E_2(d)$') plt.legend() plt.show()
import matplotlib.pyplot as plt import numpy as np # Generate data. x = data.roessler(length=5000)[1][:, 0] # Convert to 8-bit. x = np.int8(utils.rescale(x, (-127, 127))) # Add uniform noise of two different noise levels. y1 = x + (-0.001 + 0.002 * np.random.random(len(x))) y2 = x + (-0.5 + 1.0 * np.random.random(len(x))) # AFN algorithm. dim = np.arange(1, 10 + 2) F, Fs = dimension.afn(y1, tau=14, dim=dim, window=40) F1, F2 = F[1:] / F[:-1], Fs[1:] / Fs[:-1] E, Es = dimension.afn(y2, tau=14, dim=dim, window=40) E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1] plt.figure(1) plt.title(r'AFN after corrupting with uniform noise in $[-0.001, 0.001]$') plt.xlabel(r'Embedding dimension $d$') plt.ylabel(r'$E_1(d)$ and $E_2(d)$') plt.plot(dim[:-1], F1, 'bo-', label=r'$E_1(d)$') plt.plot(dim[:-1], F2, 'go-', label=r'$E_2(d)$') plt.legend() plt.figure(2) plt.title(r'AFN after corrupting with uniform noise in $[-0.5, 0.5]$')
np.random.seed(999)
eta = np.random.normal(size=(N), loc=0, scale=1.0)
a = 0.99
x[0] = eta[0]
for i in range(1, N):
x[i] = a * x[i - 1] + eta[i]
x = utils.rescale(x)
# Calculate the autocorrelation time.
tau = np.argmax(delay.acorr(x) < 1.0 / np.e)
# AFN without any minimum temporal separation.
dim = np.arange(1, 10 + 2)
F, Fs = dimension.afn(x, tau=tau, dim=dim, window=0)
F1, F2 = F[1:] / F[:-1], Fs[1:] / Fs[:-1]
# AFN with a minimum temporal separation (equal to the autocorrelation
# time) between near-neighbors.
dim = np.arange(1, 10 + 2)
E, Es = dimension.afn(x, tau=tau, dim=dim, window=tau)
E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1]
plt.figure(1)
plt.title(r'AR(1) process with $a = 0.99$')
plt.xlabel(r'i')
plt.ylabel(r'$x_i$')
plt.plot(x)
plt.figure(2)
#!/usr/bin/env python # -*- coding: utf-8 -*- """AFN for time series from the Lorenz attractor. E1 saturates near an embedding dimension of 3. E2 != 1 at many values of d. Thus the series is definitely deterministic. The plot matches Fig. 3 of Cao (1997) rather nicely. """ from nolitsa import data, dimension import matplotlib.pyplot as plt import numpy as np # Generate data. x = data.lorenz()[1][:, 0] # AFN algorithm. dim = np.arange(1, 10 + 2) E, Es = dimension.afn(x, tau=5, dim=dim, window=20) E1, E2 = E[1:] / E[:-1], Es[1:] / Es[:-1] plt.title(r'AFN for time series from the Lorenz attractor') plt.xlabel(r'Embedding dimension $d$') plt.ylabel(r'$E_1(d)$ and $E_2(d)$') plt.plot(dim[:-1], E1, 'bo-', label=r'$E_1(d)$') plt.plot(dim[:-1], E2, 'go-', label=r'$E_2(d)$') plt.legend() plt.show()