.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples/plot_viterbi.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:`here <sphx_glr_download_auto_examples_plot_viterbi.py>` to download the full example code .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_plot_viterbi.py: ================ Viterbi decoding ================ This notebook demonstrates how to use Viterbi decoding to impose temporal smoothing on frame-wise state predictions. Our working example will be the problem of silence/non-silence detection. .. GENERATED FROM PYTHON SOURCE LINES 12-25 .. code-block:: default # Code source: Brian McFee # License: ISC ################## # Standard imports from __future__ import print_function import numpy as np import matplotlib.pyplot as plt import librosa import librosa.display .. GENERATED FROM PYTHON SOURCE LINES 26-27 Load an example signal .. GENERATED FROM PYTHON SOURCE LINES 27-42 .. code-block:: default y, sr = librosa.load('audio/sir_duke_slow.mp3') # And compute the spectrogram magnitude and phase S_full, phase = librosa.magphase(librosa.stft(y)) ################### # Plot the spectrum plt.figure(figsize=(12, 4)) librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max), y_axis='log', x_axis='time', sr=sr) plt.colorbar() plt.tight_layout() .. image-sg:: /auto_examples/images/sphx_glr_plot_viterbi_001.png :alt: plot viterbi :srcset: /auto_examples/images/sphx_glr_plot_viterbi_001.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/core/audio.py:161: UserWarning: PySoundFile failed. Trying audioread instead. warnings.warn('PySoundFile failed. Trying audioread instead.') /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'basey' parameter of __init__() has been renamed 'base' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linthreshy' parameter of __init__() has been renamed 'linthresh' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linscaley' parameter of __init__() has been renamed 'linscale' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) .. GENERATED FROM PYTHON SOURCE LINES 43-46 As you can see, there are periods of silence and non-silence throughout this recording. .. GENERATED FROM PYTHON SOURCE LINES 46-65 .. code-block:: default # As a first step, we can plot the root-mean-square (RMS) curve rms = librosa.feature.rms(y=y)[0] times = librosa.frames_to_time(np.arange(len(rms))) plt.figure(figsize=(12, 4)) plt.plot(times, rms) plt.axhline(0.02, color='r', alpha=0.5) plt.xlabel('Time') plt.ylabel('RMS') plt.axis('tight') plt.tight_layout() # The red line at 0.02 indicates a reasonable threshold for silence detection. # However, the RMS curve occasionally dips below the threshold momentarily, # and we would prefer the detector to not count these brief dips as silence. # This is where the Viterbi algorithm comes in handy! .. image-sg:: /auto_examples/images/sphx_glr_plot_viterbi_002.png :alt: plot viterbi :srcset: /auto_examples/images/sphx_glr_plot_viterbi_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 66-76 As a first step, we will convert the raw RMS score into a likelihood (probability) by logistic mapping :math:`P[V=1 | x] = \frac{\exp(x - \tau)}{1 + \exp(x - \tau)}` where :math:`x` denotes the RMS value and :math:`\tau=0.02` is our threshold. The variable :math:`V` indicates whether the signal is non-silent (1) or silent (0). We'll normalize the RMS by its standard deviation to expand the range of the probability vector .. GENERATED FROM PYTHON SOURCE LINES 76-90 .. code-block:: default r_normalized = (rms - 0.02) / np.std(rms) p = np.exp(r_normalized) / (1 + np.exp(r_normalized)) # We can plot the probability curve over time: plt.figure(figsize=(12, 4)) plt.plot(times, p, label='P[V=1|x]') plt.axhline(0.5, color='r', alpha=0.5, label='Descision threshold') plt.xlabel('Time') plt.axis('tight') plt.legend() plt.tight_layout() .. image-sg:: /auto_examples/images/sphx_glr_plot_viterbi_003.png :alt: plot viterbi :srcset: /auto_examples/images/sphx_glr_plot_viterbi_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 91-94 which looks much like the first plot, but with the decision threshold shifted to 0.5. A simple silence detector would classify each frame independently of its neighbors, which would result in the following plot: .. GENERATED FROM PYTHON SOURCE LINES 94-108 .. code-block:: default plt.figure(figsize=(12, 6)) ax = plt.subplot(2,1,1) librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max), y_axis='log', x_axis='time', sr=sr) plt.subplot(2,1,2, sharex=ax) plt.step(times, p>=0.5, label='Non-silent') plt.xlabel('Time') plt.axis('tight') plt.ylim([0, 1.05]) plt.legend() plt.tight_layout() .. image-sg:: /auto_examples/images/sphx_glr_plot_viterbi_004.png :alt: plot viterbi :srcset: /auto_examples/images/sphx_glr_plot_viterbi_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'basey' parameter of __init__() has been renamed 'base' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linthreshy' parameter of __init__() has been renamed 'linthresh' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linscaley' parameter of __init__() has been renamed 'linscale' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) .. GENERATED FROM PYTHON SOURCE LINES 109-117 We can do better using the Viterbi algorithm. We'll use state 0 to indicate silent, and 1 to indicate non-silent. We'll assume that a silent frame is equally likely to be followed by silence or non-silence, but that non-silence is slightly more likely to be followed by non-silence. This is accomplished by building a self-loop transition matrix, where `transition[i, j]` is the probability of moving from state `i` to state `j` in the next frame. .. GENERATED FROM PYTHON SOURCE LINES 117-121 .. code-block:: default transition = librosa.sequence.transition_loop(2, [0.5, 0.6]) print(transition) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [[0.5 0.5] [0.4 0.6]] .. GENERATED FROM PYTHON SOURCE LINES 122-124 Our `p` variable only indicates the probability of non-silence, so we need to also compute the probability of silence as its complement. .. GENERATED FROM PYTHON SOURCE LINES 124-128 .. code-block:: default full_p = np.vstack([1 - p, p]) print(full_p) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none [[0.666662 0.66666806 0.66667175 0.6666764 0.6666662 0.6666547 0.66665447 0.6666441 0.6666499 0.6666609 0.6666493 0.6666572 0.6666585 0.65281963 0.5593039 0.50396335 0.4687572 0.44503105 0.44209725 0.44649702 0.45015687 0.45296526 0.47192842 0.50088567 0.533761 0.57154465 0.60663986 0.63306737 0.6560469 0.66306615 0.6656199 0.66632414 0.66658187 0.6666868 0.6666943 0.6666857 0.6666565 0.66665673 0.66667044 0.6666879 0.666713 0.6667024 0.6666863 0.66668093 0.66668844 0.6667025 0.6666882 0.66659105 0.666566 0.66656137 0.66658425 0.6666702 0.6666781 0.6666808 0.66664445 0.6666428 0.66665864 0.6666496 0.66597325 0.6332568 0.565287 0.51315165 0.46493763 0.4289661 0.41747576 0.43100828 0.4557107 0.44279826 0.40919942 0.36801213 0.33117193 0.32734305 0.33589602 0.34995198 0.36854565 0.38243645 0.3977514 0.40770727 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GENERATED FROM PYTHON SOURCE LINES 129-132 Now, we're ready to decode! We'll use `viterbi_discriminative` here, since the inputs are state likelihoods conditional on data (in our case, data is rms). .. GENERATED FROM PYTHON SOURCE LINES 132-151 .. code-block:: default states = librosa.sequence.viterbi_discriminative(full_p, transition) # sphinx_gallery_thumbnail_number = 5 plt.figure(figsize=(12, 6)) ax = plt.subplot(2,1,1) librosa.display.specshow(librosa.amplitude_to_db(S_full, ref=np.max), y_axis='log', x_axis='time', sr=sr) plt.xlabel('') ax.tick_params(labelbottom=False) plt.subplot(2, 1, 2, sharex=ax) plt.step(times, p>=0.5, label='Frame-wise') plt.step(times, states, linestyle='--', color='orange', label='Viterbi') plt.xlabel('Time') plt.axis('tight') plt.ylim([0, 1.05]) plt.legend() .. image-sg:: /auto_examples/images/sphx_glr_plot_viterbi_005.png :alt: plot viterbi :srcset: /auto_examples/images/sphx_glr_plot_viterbi_005.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'basey' parameter of __init__() has been renamed 'base' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linthreshy' parameter of __init__() has been renamed 'linthresh' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) /tmp/tmpfl4ra6qp/b0064fe7dbe8048b1d4148e61a568b6fe3fca91b/librosa/display.py:862: MatplotlibDeprecationWarning: The 'linscaley' parameter of __init__() has been renamed 'linscale' since Matplotlib 3.3; support for the old name will be dropped two minor releases later. scaler(mode, **kwargs) .. GENERATED FROM PYTHON SOURCE LINES 152-157 Note how the Viterbi output has fewer state changes than the frame-wise predictor, and it is less sensitive to momentary dips in energy. This is controlled directly by the transition matrix. A higher self-transition probability means that the decoder is less likely to change states. .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 2.488 seconds) .. _sphx_glr_download_auto_examples_plot_viterbi.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_viterbi.py <plot_viterbi.py>` .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_viterbi.ipynb <plot_viterbi.ipynb>` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_