# Dynamic Time Warping¶

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

The image represents cost matrix, that is the squared Euclidean distance for each time point between both time series, which are represented at the left and at the top of the cost matrix.

The optimal path, that is the path that minimizes the total cost to go from the first time point to the last one, is represented in white on the image.

```/home/docs/checkouts/readthedocs.org/user_builds/tslearn/checkouts/stable/docs/examples/metrics/plot_dtw.py:103: UserWarning: This figure includes Axes that are not compatible with tight_layout, so results might be incorrect.
plt.tight_layout()
```

```# Author: Romain Tavenard

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

path, sim = metrics.dtw_path(s_y1, s_y2)

plt.figure(1, 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(mat, origin='lower')
ax_gram.axis("off")
ax_gram.autoscale(False)
ax_gram.plot([j for (i, j) in path], [i for (i, j) in path], "w-",
linewidth=3.)

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.tight_layout()
plt.show()
```

Total running time of the script: (0 minutes 0.191 seconds)

Gallery generated by Sphinx-Gallery