hurst

Hurst exponent evaluation and R/S-analysis

hurst is a small Python module for analysing random walks and evaluating the Hurst exponent (H).

H = 0.5 — Brownian motion,
0.5 < H < 1.0 — persistent behavior,
0 < H < 0.5 — anti-persistent behavior.

Installation

Install hurst module with
pip install hurst
or
pip install -e git+https://github.com/Mottl/hurst#egg=hurst

Usage

import numpy as np
import matplotlib.pyplot as plt
from hurst import compute_Hc, random_walk

# Use random_walk() function or generate a random walk series manually:
# series = random_walk(99999, cumprod=True)
np.random.seed(42)
random_changes = 1. + np.random.randn(99999) / 1000.
series = np.cumprod(random_changes)  # create a random walk from random changes

# Evaluate Hurst equation
H, c, data = compute_Hc(series, kind='price', simplified=True)

# Plot
f, ax = plt.subplots()
ax.plot(data[0], c*data[0]**H, color="deepskyblue")
ax.scatter(data[0], data[1], color="purple")
ax.set_xscale('log')
ax.set_yscale('log')
ax.set_xlabel('Time interval')
ax.set_ylabel('R/S ratio')
ax.grid(True)
plt.show()

print("H={:.4f}, c={:.4f}".format(H,c))

H=0.4964, c=1.4877

Kinds of series

The kind parameter of the compute_Hc function can have the following values:
'change': a series is just random values (i.e. np.random.randn(...))
'random_walk': a series is a cumulative sum of changes (i.e. np.cumsum(np.random.randn(...)))
'price': a series is a cumulative product of changes (i.e. np.cumprod(1+epsilon*np.random.randn(...))

Brownian motion, persistent and antipersistent random walks

You can generate random walks with random_walk() function as following:

Brownian

brownian = random_walk(99999, proba=0.5)

Persistent

persistent = random_walk(99999, proba=0.7)

Antipersistent

antipersistent = random_walk(99999, proba=0.3)