examples/plot_poisson_hmm.py

"""
Using a Hidden Markov Model with Poisson Emissions to Understand Earthquakes
----------------------------------------------------------------------------

Let's look at data of magnitude 7+ earthquakes between 1900-2006 in the
world collected by the US Geological Survey as described in this textbook:
Zucchini & MacDonald, "Hidden Markov Models for Time Series"
(https://ayorho.files.wordpress.com/2011/05/chapter1.pdf). The goal is to
see if we can separate out different tectonic processes that cause
earthquakes based on their frequency of occurance. The idea is that each
tectonic boundary may cause earthquakes with a particular distribution
of waiting times depending on how active it is. This might tell help us
predict future earthquake danger, espeically on a geological time scale.
"""

import numpy as np
import matplotlib.pyplot as plt

from scipy.stats import poisson
from hmmlearn import hmm

# earthquake data from http://earthquake.usgs.gov/
earthquakes = np.array([
    13, 14, 8, 10, 16, 26, 32, 27, 18, 32, 36, 24, 22, 23, 22, 18,
    25, 21, 21, 14, 8, 11, 14, 23, 18, 17, 19, 20, 22, 19, 13, 26,
    13, 14, 22, 24, 21, 22, 26, 21, 23, 24, 27, 41, 31, 27, 35, 26,
    28, 36, 39, 21, 17, 22, 17, 19, 15, 34, 10, 15, 22, 18, 15, 20,
    15, 22, 19, 16, 30, 27, 29, 23, 20, 16, 21, 21, 25, 16, 18, 15,
    18, 14, 10, 15, 8, 15, 6, 11, 8, 7, 18, 16, 13, 12, 13, 20,
    15, 16, 12, 18, 15, 16, 13, 15, 16, 11, 11])


# Plot the sampled data
fig, ax = plt.subplots()
ax.plot(earthquakes, ".-", ms=6, mfc="orange", alpha=0.7)
ax.set_xticks(range(0, earthquakes.size, 10))
ax.set_xticklabels(range(1906, 2007, 10))
ax.set_xlabel('Year')
ax.set_ylabel('Count')
fig.show()

# %%
# Now, fit a Poisson Hidden Markov Model to the data.

scores = list()
models = list()
for n_components in range(1, 5):
    for idx in range(10):  # ten different random starting states
        # define our hidden Markov model
        model = hmm.PoissonHMM(n_components=n_components, random_state=idx,
                               n_iter=10)
        model.fit(earthquakes[:, None])
        models.append(model)
        scores.append(model.score(earthquakes[:, None]))
        print(f'Converged: {model.monitor_.converged}\t\t'
              f'Score: {scores[-1]}')

# get the best model
model = models[np.argmax(scores)]
print(f'The best model had a score of {max(scores)} and '
      f'{model.n_components} components')

# use the Viterbi algorithm to predict the most likely sequence of states
# given the model
states = model.predict(earthquakes[:, None])

# %%
# Let's plot the waiting times from our most likely series of states of
# earthquake activity with the earthquake data. As we can see, the
# model with the maximum likelihood had different states which may reflect
# times of varying earthquake danger.

# plot model states over time
fig, ax = plt.subplots()
ax.plot(model.lambdas_[states], ".-", ms=6, mfc="orange")
ax.plot(earthquakes)
ax.set_title('States compared to generated')
ax.set_xlabel('State')

# %%
# Fortunately, 2006 ended with a period of relative tectonic stability, and,
# if we look at our transition matrix, we can see that the off-diagonal terms
# are small, meaning that the state transitions are rare and it's unlikely that
# there will be high earthquake danger in the near future.

fig, ax = plt.subplots()
ax.imshow(model.transmat_, aspect='auto', cmap='spring')
ax.set_title('Transition Matrix')
ax.set_xlabel('State To')
ax.set_ylabel('State From')

# %%
# Finally, let's look at the distribution of earthquakes compared to our
# waiting time parameter values. We can see that our model fits the
# distribution fairly well, replicating results from the reference.

# get probabilities for each state given the data, take the average
# to find the proportion of time in that state
prop_per_state = model.predict_proba(earthquakes[:, None]).mean(axis=0)

# earthquake counts to plot
bins = sorted(np.unique(earthquakes))

fig, ax = plt.subplots()
ax.hist(earthquakes, bins=bins, density=True)
ax.plot(bins, poisson.pmf(bins, model.lambdas_).T @ prop_per_state)
ax.set_title('Histogram of Earthquakes with Fitted Poisson States')
ax.set_xlabel('Number of Earthquakes')
ax.set_ylabel('Proportion')

plt.show()