Soft Dynamic Time Warping

This example illustrates Soft Dynamic Time Warping (DTW) computation between time series and plots the optimal soft alignment matrices [1].

[1]

M. Cuturi, M. Blondel “Soft-DTW: a Differentiable Loss Function for Time-Series,” ICML 2017.

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# Author: Romain Tavenard
# License: BSD 3 clause
# sphinx_gallery_thumbnail_number = 3

import numpy
from scipy.spatial.distance import cdist
import matplotlib.pyplot as plt

from tslearn import metrics

numpy.random.seed(0)

s_x = numpy.array(
    [-0.790, -0.765, -0.734, -0.700, -0.668, -0.639, -0.612, -0.587, -0.564,
     -0.544, -0.529, -0.518, -0.509, -0.502, -0.494, -0.488, -0.482, -0.475,
     -0.472, -0.470, -0.465, -0.464, -0.461, -0.458, -0.459, -0.460, -0.459,
     -0.458, -0.448, -0.431, -0.408, -0.375, -0.333, -0.277, -0.196, -0.090,
     0.047, 0.220, 0.426, 0.671, 0.962, 1.300, 1.683, 2.096, 2.510, 2.895,
     3.219, 3.463, 3.621, 3.700, 3.713, 3.677, 3.606, 3.510, 3.400, 3.280,
     3.158, 3.038, 2.919, 2.801, 2.676, 2.538, 2.382, 2.206, 2.016, 1.821,
     1.627, 1.439, 1.260, 1.085, 0.917, 0.758, 0.608, 0.476, 0.361, 0.259,
     0.173, 0.096, 0.027, -0.032, -0.087, -0.137, -0.179, -0.221, -0.260,
     -0.293, -0.328, -0.359, -0.385, -0.413, -0.437, -0.458, -0.480, -0.498,
     -0.512, -0.526, -0.536, -0.544, -0.552, -0.556, -0.561, -0.565, -0.568,
     -0.570, -0.570, -0.566, -0.560, -0.549, -0.532, -0.510, -0.480, -0.443,
     -0.402, -0.357, -0.308, -0.256, -0.200, -0.139, -0.073, -0.003, 0.066,
     0.131, 0.186, 0.229, 0.259, 0.276, 0.280, 0.272, 0.256, 0.234, 0.209,
     0.186, 0.162, 0.139, 0.112, 0.081, 0.046, 0.008, -0.032, -0.071, -0.110,
     -0.147, -0.180, -0.210, -0.235, -0.256, -0.275, -0.292, -0.307, -0.320,
     -0.332, -0.344, -0.355, -0.363, -0.367, -0.364, -0.351, -0.330, -0.299,
     -0.260, -0.217, -0.172, -0.128, -0.091, -0.060, -0.036, -0.022, -0.016,
     -0.020, -0.037, -0.065, -0.104, -0.151, -0.201, -0.253, -0.302, -0.347,
     -0.388, -0.426, -0.460, -0.491, -0.517, -0.539, -0.558, -0.575, -0.588,
     -0.600, -0.606, -0.607, -0.604, -0.598, -0.589, -0.577, -0.558, -0.531,
     -0.496, -0.454, -0.410, -0.364, -0.318, -0.276, -0.237, -0.203, -0.176,
     -0.157, -0.145, -0.142, -0.145, -0.154, -0.168, -0.185, -0.206, -0.230,
     -0.256, -0.286, -0.318, -0.351, -0.383, -0.414, -0.442, -0.467, -0.489,
     -0.508, -0.523, -0.535, -0.544, -0.552, -0.557, -0.560, -0.560, -0.557,
     -0.551, -0.542, -0.531, -0.519, -0.507, -0.494, -0.484, -0.476, -0.469,
     -0.463, -0.456, -0.449, -0.442, -0.435, -0.431, -0.429, -0.430, -0.435,
     -0.442, -0.452, -0.465, -0.479, -0.493, -0.506, -0.517, -0.526, -0.535,
     -0.548, -0.567, -0.592, -0.622, -0.655, -0.690, -0.728, -0.764, -0.795,
     -0.815, -0.823, -0.821])

s_y1 = numpy.concatenate((s_x, s_x))[::2].reshape((-1, 1))
s_y2 = numpy.concatenate((s_x, s_x[::-1]))[::2].reshape((-1, 1))
sz = s_y1.shape[0]

for gamma in [0., .1, 1.]:
    alignment, sim = metrics.soft_dtw_alignment(s_y1, s_y2, gamma=gamma)

    plt.figure(figsize=(8, 8))

    # definitions for the axes
    left, bottom = 0.01, 0.1
    w_ts = h_ts = 0.2
    left_h = left + w_ts + 0.02
    width = height = 0.65
    bottom_h = bottom + height + 0.02

    rect_s_y = [left, bottom, w_ts, height]
    rect_gram = [left_h, bottom, width, height]
    rect_s_x = [left_h, bottom_h, width, h_ts]

    ax_gram = plt.axes(rect_gram)
    ax_s_x = plt.axes(rect_s_x)
    ax_s_y = plt.axes(rect_s_y)

    mat = cdist(s_y1, s_y2)

    ax_gram.imshow(alignment, origin='lower')
    ax_gram.axis("off")
    ax_gram.autoscale(False)
    plt.suptitle("$\\gamma={:.1f}$".format(gamma), fontsize=24)

    ax_s_x.plot(numpy.arange(sz), s_y2, "b-", linewidth=3.)
    ax_s_x.axis("off")
    ax_s_x.set_xlim((0, sz - 1))

    ax_s_y.plot(- s_y1, numpy.arange(sz), "b-", linewidth=3.)
    ax_s_y.axis("off")
    ax_s_y.set_ylim((0, sz - 1))

    plt.show()