.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/plot_hmm_sampling_and_decoding.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_hmm_sampling_and_decoding.py: Sampling from and decoding an HMM --------------------------------- This script shows how to sample points from a Hidden Markov Model (HMM): we use a 4-state model with specified mean and covariance. The plot shows the sequence of observations generated with the transitions between them. We can see that, as specified by our transition matrix, there are no transition between component 1 and 3. Then, we decode our model to recover the input parameters. .. GENERATED FROM PYTHON SOURCE LINES 14-63 .. code-block:: default import numpy as np import matplotlib.pyplot as plt from hmmlearn import hmm # Prepare parameters for a 4-components HMM # Initial population probability startprob = np.array([0.6, 0.3, 0.1, 0.0]) # The transition matrix, note that there are no transitions possible # between component 1 and 3 transmat = np.array([[0.7, 0.2, 0.0, 0.1], [0.3, 0.5, 0.2, 0.0], [0.0, 0.3, 0.5, 0.2], [0.2, 0.0, 0.2, 0.6]]) # The means of each component means = np.array([[0.0, 0.0], [0.0, 11.0], [9.0, 10.0], [11.0, -1.0]]) # The covariance of each component covars = .5 * np.tile(np.identity(2), (4, 1, 1)) # Build an HMM instance and set parameters gen_model = hmm.GaussianHMM(n_components=4, covariance_type="full") # Instead of fitting it from the data, we directly set the estimated # parameters, the means and covariance of the components gen_model.startprob_ = startprob gen_model.transmat_ = transmat gen_model.means_ = means gen_model.covars_ = covars # Generate samples X, Z = gen_model.sample(500) # Plot the sampled data fig, ax = plt.subplots() ax.plot(X[:, 0], X[:, 1], ".-", label="observations", ms=6, mfc="orange", alpha=0.7) # Indicate the component numbers for i, m in enumerate(means): ax.text(m[0], m[1], 'Component %i' % (i + 1), size=17, horizontalalignment='center', bbox=dict(alpha=.7, facecolor='w')) ax.legend(loc='best') fig.show() .. image-sg:: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_001.png :alt: plot hmm sampling and decoding :srcset: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 64-65 Now, let's ensure we can recover our parameters. .. GENERATED FROM PYTHON SOURCE LINES 65-90 .. code-block:: default scores = list() models = list() for n_components in (3, 4, 5): for idx in range(10): # define our hidden Markov model model = hmm.GaussianHMM(n_components=n_components, covariance_type='full', random_state=idx) model.fit(X[:X.shape[0] // 2]) # 50/50 train/validate models.append(model) scores.append(model.score(X[X.shape[0] // 2:])) print(f'Converged: {model.monitor_.converged}' f'\tScore: {scores[-1]}') # get the best model model = models[np.argmax(scores)] n_states = model.n_components print(f'The best model had a score of {max(scores)} and {n_states} ' 'states') # use the Viterbi algorithm to predict the most likely sequence of states # given the model states = model.predict(X) .. rst-class:: sphx-glr-script-out .. code-block:: none Converged: True Score: -1576.1753905467317 Converged: True Score: -1179.2235368637366 Converged: True Score: -1250.4232852250461 Converged: True Score: -1250.423285225043 Converged: True Score: -1225.1550378569789 Converged: True Score: -1250.4232852250436 Converged: True Score: -1250.4232852250425 Converged: True Score: -1250.4232852250402 Converged: True Score: -1250.423285225039 Converged: True Score: -1250.4232852250395 Converged: True Score: -1135.0399742332147 Converged: True Score: -1078.5651921877454 Converged: True Score: -1209.7488003377325 Converged: True Score: -1343.9580402942222 Converged: True Score: -1123.7381638734348 Converged: True Score: -1223.7070036303828 Converged: True Score: -1197.5094287450333 Converged: True Score: -1121.7247334015203 Converged: True Score: -1117.5244481532716 Converged: True Score: -1117.9925570287362 Converged: True Score: -1264.910201511487 Converged: True Score: -1015.3354311749436 Converged: True Score: -1139.4481688328324 Converged: True Score: -1253.126595190962 Converged: True Score: -1164.0728223476915 Converged: True Score: -1149.9952377598836 Converged: True Score: -1220.4604840715806 Converged: True Score: -1022.4576591442977 Converged: True Score: -1052.616926381614 Converged: True Score: -1128.1136948555557 The best model had a score of -1015.3354311749436 and 5 states .. GENERATED FROM PYTHON SOURCE LINES 91-97 Let's plot our states compared to those generated and our transition matrix to get a sense of our model. We can see that the recovered states follow the same path as the generated states, just with the identities of the states transposed (i.e. instead of following a square as in the first figure, the nodes are switch around but this does not change the basic pattern). The same is true for the transition matrix. .. GENERATED FROM PYTHON SOURCE LINES 97-118 .. code-block:: default # plot model states over time fig, ax = plt.subplots() ax.plot(Z, states) ax.set_title('States compared to generated') ax.set_xlabel('Generated State') ax.set_ylabel('Recovered State') fig.show() # plot the transition matrix fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(8, 5)) ax1.imshow(gen_model.transmat_, aspect='auto', cmap='spring') ax1.set_title('Generated Transition Matrix') ax2.imshow(model.transmat_, aspect='auto', cmap='spring') ax2.set_title('Recovered Transition Matrix') for ax in (ax1, ax2): ax.set_xlabel('State To') ax.set_ylabel('State From') fig.tight_layout() fig.show() .. rst-class:: sphx-glr-horizontal * .. image-sg:: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_002.png :alt: States compared to generated :srcset: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_002.png :class: sphx-glr-multi-img * .. image-sg:: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_003.png :alt: Generated Transition Matrix, Recovered Transition Matrix :srcset: /auto_examples/images/sphx_glr_plot_hmm_sampling_and_decoding_003.png :class: sphx-glr-multi-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 1.837 seconds) .. _sphx_glr_download_auto_examples_plot_hmm_sampling_and_decoding.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_hmm_sampling_and_decoding.py ` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_hmm_sampling_and_decoding.ipynb ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_