Hurst exponent

The Hurst exponent is a measure of the long-term memory of time series data. It can be used to determine whether a time series is trending, mean-reverting, or random. The Hurst exponent can take values in the range [0, 1]:

• If \(H < 0.5\), the time series is mean-reverting (anti-persistent).

• If \(H = 0.5\), the time series is a random walk (Brownian motion).

• If \(H > 0.5\), the time series is trending (persistent).

The Hurst exponent can be calculated using the hurst_exponent method from the DiscreteDynamicalSystem class. The method takes the initial conditions and the number of iterations as input and returns the Hurst exponent for each time series. The following code snippet shows how to calculate the Hurst exponent for the standard map with three different parameter values.

from pynamicalsys import DiscreteDynamicalSystem as dds
import numpy as np

# Create the discrete dynamical system object
ds = dds(model="standard map")

# Create the random initial conditions
num_ic = 250
x_range = (0, 1)  # x range for initial conditions
y_range = (0, 1)  # y range for initial conditions
np.random.seed(0)  # Set the seed for reproducibility
x_ic = np.random.uniform(x_range[0], x_range[1], num_ic)
y_ic = np.random.uniform(y_range[0], y_range[1], num_ic)
u = np.column_stack((x_ic, y_ic))  # Initial conditions array with shape (num_ic, d)

# Parameter values for the standard map
k_values = [0.9, 1.5, 3.9]

# Calculate the Hurst exponent for each parameter value
H = [
     ds.hurst_exponent(u[i], total_time, parameters=k[j])
     for i in range(num_ic)
     for j in range(len(k))
 ]
H = np.array(H).reshape(num_ic, len(k), 2)

# We also calculate the trajectories for each initial condition
trajectories = [
     ds.trajectory(u, total_time, parameters=k[i])
     for i in range(len(k))
 ]
trajectories_reshaped = []
for trajectory in trajectories:
   trajectory_reshaped = trajectory.reshape(num_ic, total_time, 2)
   trajectories_reshaped.append(trajectory_reshaped)

We can visualize the Hurst exponent for the \(x\) time series of the standard map with three different parameter values. The following code snippet shows how to create a plot of the Hurst exponent for each initial condition.

from pynamicalsys import PlotStyler
import matplotlib.pyplot as plt

# Set the plot style
ps = PlotStyler(fontsize=24)
ps.apply_style()

# Create the figure and axes
fig, ax = plt.subplots(1, 3, figsize=(15, 5), sharey=True, sharex=True)
[ps.set_tick_padding(ax[i], pad_x = 8) for i in range(3)]

# Plot the Hurst exponent for each initial condition
hms = [0, 0, 0]
for j in range(len(k)):
   for i in range(num_ic):
      hm = ax[j].scatter(trajectories_reshaped[j][i, :, 0],
                           trajectories_reshaped[j][i, :, 1],
                           c=H[i, j, 0] * np.ones(total_time),
                           s=0.05,
                           edgecolor='none',
                           cmap="nipy_spectral",
                           vmin=0,
                           vmax=H[:, j, 0].max())
      hms[j] = hm

# Add colorbars and labels
[plt.colorbar(hms[i], ax=ax[i], label=rf"Hurst exponent with $k = {k[i]:.1f}$", location="top", aspect=40, pad=0.01) for i in range(len(k))]
ax[0].set_xticks([0, 0.2, 0.4, 0.6, 0.8, 1])
ax[0].set_xlim(0, 1)
ax[0].set_ylim(0, 1)
ax[0].set_ylabel("$y$")
[ax[i].set_xlabel("$x$") for i in range(len(k))]

plt.tight_layout(pad=0.05)
plt.show()

Hurst exponent of the \(x\) time series for the standard map with three different parameter values.