lppls 0.1.15

A Python module for fitting the LPPLS model to data.

Log Periodic Power Law Singularity (LPPLS) Model

lppls is a Python module for fitting the LPPLS model to data.

Overview

The LPPLS model provides a flexible framework to detect bubbles and predict regime changes of a financial asset. A bubble is defined as a faster-than-exponential increase in asset price, that reflects positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations. It models a bubble price as a power law with a finite-time singularity decorated by oscillations with a frequency increasing with time.

Here is the model:

where:

• expected log price at the date of the termination of the bubble
• critical time (date of termination of the bubble and transition in a new regime)
• expected log price at the peak when the end of the bubble is reached at
• amplitude of the power law acceleration
• amplitude of the log-periodic oscillations
• degree of the super exponential growth
• scaling ratio of the temporal hierarchy of oscillations
• time scale of the oscillations

The model has three components representing a bubble. The first, , handles the hyperbolic power law. For when the price growth becomes unsustainable, and at the growth rate becomes infinite. The second term, , controls the amplitude of the oscillations. It drops to zero at the critical time . The third term, , models the frequency of the osciallations. They become infinite at .

Important links

• Official source code repo: https://github.com/Boulder-Investment-Technologies/lppls
• Download releases: https://pypi.org/project/lppls/
• Issue tracker: https://github.com/Boulder-Investment-Technologies/lppls/issues

Installation

Dependencies

lppls requires:

• Pandas (>= 0.25.0)
• Python (>= 3.6)
• NumPy (>= 1.17.0)
• SciPy (>= 1.3.0)
• Matplotlib (>= 3.1.1)

User installation

pip install -U lppls

Example Use

from lppls import lppls
import numpy as np
import pandas as pd
%matplotlib inline

# read example dataset into df 
data = pd.read_csv('data/sp500.csv', index_col='Date')

# convert index col to evenly spaced numbers over a specified interval
time = np.linspace(0, len(data)-1, len(data))

# create list of observation data, in this case, 
# daily adjusted close prices of the S&P 500
price = [p for p in data['Adj Close']]

# create Mx2 matrix (expected format for LPPLS observations)
observations = np.array([time, price])

# create reasonable bounds for critical time initialization
first = time[0]
last = time[-1]
pct_delta = (last - first) * 0.20
tc_init_min = last - pct_delta
tc_init_max = last + pct_delta

# set random initialization limits for non-linear params
init_limits = [
    (tc_init_min, tc_init_max), # tc : Critical Time 
    (0.1, 0.9),                 # m : 0.1 ≤ m ≤ 0.9
    (6, 13),                    # ω : 6 ≤ ω ≤ 13
]

# set the max number for searches to perform before giving-up
# the literature suggests 25
MAX_SEARCHES = 25

# instantiate a new LPPLS model with the S&P 500 dataset
lppls_model = lppls.LPPLS(use_ln=True, observations=observations)

# fit the model to the data and get back the params
tc, m, w, a, b, c = lppls_model.fit(observations, MAX_SEARCHES, init_limits, minimizer='Nelder-Mead')

# visualize the fit
lppls_model.plot_fit(tc, m, w, observations)

# should give a plot like the following...

References

• Filimonov, V. and Sornette, D. A Stable and Robust Calibration Scheme of the Log-Periodic Power Law Model. Physica A: Statistical Mechanics and its Applications. 2013
• Sornette, D. Why Stock Markets Crash: Critical Events in Complex Financial Systems. Princeton University Press. 2002.