.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/plot_poisson_hmm.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_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. .. GENERATED FROM PYTHON SOURCE LINES 15-42 .. code-block:: Python 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() .. image-sg:: /auto_examples/images/sphx_glr_plot_poisson_hmm_001.png :alt: plot poisson hmm :srcset: /auto_examples/images/sphx_glr_plot_poisson_hmm_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 43-44 Now, fit a Poisson Hidden Markov Model to the data. .. GENERATED FROM PYTHON SOURCE LINES 44-67 .. code-block:: Python 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]) .. rst-class:: sphx-glr-script-out .. code-block:: none Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -391.9189281654951 Converged: True Score: -341.89397049875447 Converged: True Score: -341.88244772721396 Converged: True Score: -342.14454823785394 Converged: True Score: -341.89296748597104 Converged: True Score: -341.8855538199331 Converged: True Score: -342.28762276127594 Converged: True Score: -342.5369292103581 Converged: True Score: -341.88750207762195 Converged: True Score: -341.87893633799735 Converged: True Score: -342.9703881743685 Converged: True Score: -343.0429169039876 Converged: True Score: -342.0845203955183 Converged: True Score: -342.6892743201989 Converged: True Score: -328.5977427805315 Converged: True Score: -341.6613001106409 Converged: True Score: -329.0639942771591 Converged: True Score: -329.5050253557605 Converged: True Score: -338.90786993240823 Converged: True Score: -328.7562782189667 Converged: True Score: -337.65157449610496 Converged: True Score: -328.140456803197 Converged: True Score: -340.59777383065614 Converged: True Score: -330.4670538297964 Converged: True Score: -341.5633429271246 Converged: True Score: -328.02126544237626 Converged: True Score: -340.86474747756444 Converged: True Score: -339.55978555721845 Converged: True Score: -328.69847130568786 Converged: True Score: -328.87604629227917 Converged: True Score: -340.1318877583548 The best model had a score of -328.02126544237626 and 4 components .. GENERATED FROM PYTHON SOURCE LINES 68-72 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. .. GENERATED FROM PYTHON SOURCE LINES 72-80 .. code-block:: Python # 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') .. image-sg:: /auto_examples/images/sphx_glr_plot_poisson_hmm_002.png :alt: States compared to generated :srcset: /auto_examples/images/sphx_glr_plot_poisson_hmm_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(0.5, 23.52222222222222, 'State') .. GENERATED FROM PYTHON SOURCE LINES 81-85 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. .. GENERATED FROM PYTHON SOURCE LINES 85-92 .. code-block:: Python 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') .. image-sg:: /auto_examples/images/sphx_glr_plot_poisson_hmm_003.png :alt: Transition Matrix :srcset: /auto_examples/images/sphx_glr_plot_poisson_hmm_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none Text(31.222222222222214, 0.5, 'State From') .. GENERATED FROM PYTHON SOURCE LINES 93-96 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. .. GENERATED FROM PYTHON SOURCE LINES 96-112 .. code-block:: Python # 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() .. image-sg:: /auto_examples/images/sphx_glr_plot_poisson_hmm_004.png :alt: Histogram of Earthquakes with Fitted Poisson States :srcset: /auto_examples/images/sphx_glr_plot_poisson_hmm_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 0.891 seconds) .. _sphx_glr_download_auto_examples_plot_poisson_hmm.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_poisson_hmm.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_poisson_hmm.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_