# 37dynamic time warping

There are beautiful explanations our there of how the Dynamic Time Warping algorithm works. An excellent source is this tutorial.

I also used as sources for this lecture the following pages:

import stuff
import numpy as np
import matplotlib.pyplot as plt
import matplotlib as mpl
import pandas as pd
from pandas.plotting import register_matplotlib_converters
register_matplotlib_converters()  # datetime converter for a matplotlib
import seaborn as sns
sns.set(style="ticks", font_scale=1.5)
import matplotlib.gridspec as gridspec
define functions for calculating DTW
def fill_dtw_cost_matrix(abs_diff):
N1, N2 = abs_diff.shape
cost_matrix = abs_diff.copy()  # make a copy
# prepend row full of nan
cost_matrix = np.vstack([np.full(N1, np.nan), cost_matrix])
# prepend col full of nan
cost_matrix = np.hstack([np.full((N2+1,1), np.nan), cost_matrix])
# make origin zero
cost_matrix[0, 0] = 0
for i in range(1, N1+1):
for j in range(1, N2+1):
south     = cost_matrix[i-1, j  ]
west      = cost_matrix[i  , j-1]
southwest = cost_matrix[i-1, j-1]
# print(([south, west, southwest]))
# print(np.nanmin([south, west, southwest]))
cost_matrix[i, j] = cost_matrix[i, j] + np.nanmin([south, west, southwest])
return cost_matrix

def find_min_path(m):
N1, N2 = m.shape
path=[[N1-1, N2-1]]
i,j = path[-1][0], path[-1][1]
while True:
south = m[i-1, j]
west = m[i, j-1]
southwest = m[i-1, j-1]
if southwest <= min(south, west):
i = i-1
j = j-1
elif south <= west:
i = i-1
else:
j = j-1
path.append([i,j])
if (i==1) & (j==1):
break
return path
useful functions for plotting
def plot_series(s1, s2):
fig, ax = plt.subplots()

ax.plot(s1, color="tab:blue", lw=3)
ax.plot(s2, color="tab:orange", lw=3)

i = len(s1)//2
for j in range(i-2,i+2):
ax.plot([i, j], [s1[i], s2[j]], color="black", alpha=0.3, ls="--")
ax.text((i+j)/2, s2[j] + (s1[i]-s2[j])/2, f"{np.abs(s1[i]-s2[j]):.2f}" )

ax.text(-0.5, s1[0], "P[i]", color="tab:blue")
ax.text(-0.5, s2[0], "Q[i]", color="tab:orange")

ax.set(xticks=np.arange(0,len(s1)),
xlim=[-1, len(s1)],
title=fr"absolute vertical distance between $P[{i}]$ and $Q[i]$");

def plot_abs_diff(s1, s2, c):
fig = plt.figure(1, figsize=(8, 8))
gs = gridspec.GridSpec(2, 2, width_ratios=[1,len(s2)], height_ratios=[len(s1),1])
gs.update(left=0.10, right=0.90,top=0.90, bottom=0.10,
hspace=0.05, wspace=0.05
)

ax_left = plt.subplot(gs[0, 0])
ax_bottom = plt.subplot(gs[1, 1])
ax_matrix = plt.subplot(gs[0, 1])

text_dict = {'ha':'center', 'va':'center', 'color':"orangered",
'fontsize':24, 'weight':'bold'}

s1_reshaped = s1.reshape(len(s1), 1)
s2_reshaped = s2.reshape(1, len(s2))

# Plot the matrix
ax_matrix.imshow(c, cmap='Blues', origin="lower")
ax_matrix.set(xticks=[],
yticks=[],
title="absolute difference: "+r"|P[i]-Q[j]|")
# https://stackoverflow.com/a/33829001
for (j,i),label in np.ndenumerate(c):
ax_matrix.text(i, j, f"{label:.2f}", **text_dict)

# Plot the array
ax_left.imshow(s1_reshaped, cmap='Greys', origin="lower")
ax_bottom.imshow(s2_reshaped, cmap='Greys')

ax_left.set(xticks=[],
yticks=np.arange(len(s1)),
xlabel="P[i]"
)
ax_left.xaxis.set_label_position("top")

ax_bottom.set(yticks=[],
xticks=np.arange(len(s2)),
)
ax_bottom.yaxis.set_label_position("right")
ax_left.tick_params(axis='y', which='both', left=False)
ax_bottom.tick_params(axis='x', which='both', bottom=False)

for (j,i),label in np.ndenumerate(s1_reshaped):
ax_left.text(i, j, f"{label:.2f}", **text_dict)

for (j,i),label in np.ndenumerate(s2_reshaped):
ax_bottom.text(i, j, f"{label:.2f}", **text_dict)

# fig.savefig("abs_diff_3.png")

def plot_cost_and_path(m, p):
fig, ax = plt.subplots(figsize=(10,10))

m_plot = ax.imshow(m, origin="lower")

text_dict = {'ha':'center', 'va':'center', 'color':"deepskyblue",
'fontsize':24, 'weight':'bold'}

# https://stackoverflow.com/a/33829001
for (j,i),label in np.ndenumerate(m):
if np.isnan(label):
label = r"$\times$"
else: label = f"{label:.2f}"
ax.text(i,j,label, **text_dict, zorder=10)

ax.plot(p[:,1], p[:,0], color=[0.6]*3)
ax.set(title="path of smallest total distance")

def plot_warped_series(s1, s2, p):
fig, ax = plt.subplots()

Q_offset = 4
ax.plot(s1, color="tab:blue", lw=3)
ax.plot(s2-Q_offset, color="tab:orange", lw=3)

p1 = p - 1

for (i,j) in p1:
ax.plot([i, j], [s1[i], s2[j]-Q_offset], color="black", alpha=0.3, ls="--")

ax.text(-1, s1[0], "P[i]", color="tab:blue")
ax.text(-1, s2[0]-Q_offset, "Q[i]", color="tab:orange")

ax.set(xticks=np.arange(0,len(s1)),
xlim=[-1, len(s1)],
# yticks=np.arange(-2,2),
title=r"absolute vertical distance between $P[1]$ and $Q[i]$");

def normalize(series):
return (series-series.mean()) / series.std()

def impose_causality(m):
N1, N2 = m.shape
lower_right = np.tril_indices(n=N1, m=N2, k=-4)
cost_matrix[lower_right] = np.nan
a few useful time series
s1_a = np.array([1,5,4,8,4,7,6,5,3,3])
s2_a = np.array([1,2,3,5,5,2,6,4,5,5]) - 4
c_a = np.abs(np.subtract.outer(s1_a, s2_a))
##Call DTW function
m_a = fill_dtw_cost_matrix(c_a)
p_a = np.array(find_min_path(m_a))

s1_b = np.array([1, 3, 4, 1, 2])-2
s2_b = np.array([-1, 0, 2, 3, 0])-1
# s1_b = normalize(s1_b)
# s2_b = normalize(s2_b)
c_b = np.abs(np.subtract.outer(s1_b, s2_b))
##Call DTW function
m_b = fill_dtw_cost_matrix(c_b)
p_b = np.array(find_min_path(m_b))

s1_c = np.array([1, 3, 4, 1, 2, 2, 2, 4, 3, 3])
s2_c = np.array([-1, 0, 2, 3, 0, 2, 3, 1, 1, 0])
s1_c = normalize(s1_c)
s2_c = normalize(s2_c)
c_c = np.abs(np.subtract.outer(s1_c, s2_c))
##Call DTW function
m_c = fill_dtw_cost_matrix(c_c)
p_c = np.array(find_min_path(m_c))

Let’s plot two time series, P and Q.

Plot P and Q
plot_series(s1_b, s2_b)
Let’s build a matrix for the absolute differences between P and Q
plot_abs_diff(s1_b, s2_b, c_b)
Let’s build the cost matrix and the best path
plot_cost_and_path(m_b, p_b)
plot warped time series
plot_warped_series(s1_b, s2_b, p_b)

Let’s see another example.

Plot P and Q
plot_series(s1_c, s2_c)

These are exactly the same series from before, with additional data points.

Let’s build the cost matrix and the best path
plot_cost_and_path(m_c, p_c)

Now the path meanders both above and below the diagonal, meaning that the second time series sometimes lags behind (bottom) and sometimes leads ahead (top) the first time series,

plot warped time series
plot_warped_series(s1_c, s2_c, p_c)

## 37.1 causality

Show the code
s1_c = np.array([1, 3, 4, 1, 2, 2, 2, 4, 3, 3])
s2_c = np.array([-1, 0, 2, 3, 0, 2, 3, 1, 1, 0])
s1_c = normalize(s1_c)
s2_c = normalize(s2_c)
c_c = np.abs(np.subtract.outer(s1_c, s2_c))
##Call DTW function
m_c = fill_dtw_cost_matrix(c_c)

N1, N2 = m_c.shape
noncausal = np.tril_indices(n=N1,m=N2, k=-1)
m_c[noncausal] = np.nan

# p_c = np.array(find_min_path(m_c))
IndexError: index -12 is out of bounds for axis 1 with size 11
Show the code
m_c
array([[ 0.        ,         nan,         nan,         nan,         nan,
nan,         nan,         nan,         nan,         nan,
nan],
[        nan,  0.15153451,  0.76923077,  2.92538857,  5.85077714,
6.46847341,  8.62463121, 11.55001978, 12.93694681, 14.32387384,
14.94157011],
[        nan,         nan,  1.48563839,  0.97358843,  1.94717685,
3.28128073,  3.48563839,  4.45922681,  5.02409993,  5.58897304,
6.92307692],
[        nan,         nan,         nan,  1.74513084,  0.97590007,
3.28590403,  4.05282315,  3.48795004,  5.02872322,  6.56487311,
7.898977  ],
[        nan,         nan,         nan,         nan,  3.90128864,
1.59359634,  3.74975414,  6.41333861,  4.87487707,  6.2618041 ,
6.87950036],
[        nan,         nan,         nan,         nan,         nan,
1.95180015,  2.77385406,  4.72334256,  5.13436952,  5.28590403,
5.64410784],
[        nan,         nan,         nan,         nan,         nan,
nan,  3.13205787,  4.72334256,  5.13436952,  5.54539648,
5.64410784],
[        nan,         nan,         nan,         nan,         nan,
nan,         nan,  5.08154637,  5.13436952,  5.54539648,
5.90360029],
[        nan,         nan,         nan,         nan,         nan,
nan,         nan,         nan,  4.98283502,  6.5236082 ,
7.85540044],
[        nan,         nan,         nan,         nan,         nan,
nan,         nan,         nan,         nan,  4.57180806,
5.90591194],
[        nan,         nan,         nan,         nan,         nan,
nan,         nan,         nan,         nan,         nan,
5.90591194]])
Show the code
plot_abs_diff(s1_b, s2_b, c_b)
Show the code
##Fill DTW Matrix
def fill_dtw_cost_matrix_causal(abs_diff):
N1, N2 = abs_diff.shape
cost_matrix = abs_diff.copy()  # make a copy
# prepend row full of nan
cost_matrix = np.vstack([np.full(N1, np.nan), cost_matrix])
# prepend col full of nan
cost_matrix = np.hstack([np.full((N2+1,1), np.nan), cost_matrix])
# make origin zero
cost_matrix[0, 0] = 0

for i in range(1, N1+1):
for j in range(1, N2+1):
south     = cost_matrix[i-1, j  ]
west      = cost_matrix[i  , j-1]
southwest = cost_matrix[i-1, j-1]
# print(([south, west, southwest]))
# print(np.nanmin([south, west, southwest]))
cost_matrix[i, j] = cost_matrix[i, j] + np.nanmin([south, west, southwest])
return cost_matrix
Show the code
##Call DTW function
m_a_causal = fill_dtw_cost_matrix_causal(c_a)
p_a_causal = np.array(find_min_path(m_a))
/var/folders/kv/9cqw3y_s6c75xmgqm9n0t5d40000gn/T/ipykernel_29281/239877799.py:21: RuntimeWarning: All-NaN axis encountered
cost_matrix[i, j] = cost_matrix[i, j] + np.nanmin([south, west, southwest])
Show the code
fig, ax = plt.subplots(figsize=(10,10))

m = ax.imshow(m_a_causal, origin="lower")
# plt.colorbar(m, ax=ax, shrink=1.0)

text_dict = {'ha':'center', 'va':'center', 'color':"deepskyblue",
'fontsize':24, 'weight':'bold'}

# https://stackoverflow.com/a/33829001
for (j,i),label in np.ndenumerate(m_a_causal):
if np.isnan(label):
label = r"$\times$"
else: label = f"{label:.0f}"
ax.text(i,j,label, **text_dict, zorder=10)

ax.plot(p_a_causal[:,1], p_a_causal[:,0], color=[0.6]*3)

ax.set(title="path of smallest total distance")
Show the code
fig, ax = plt.subplots()

ax.plot(s1_a, color="tab:blue", lw=3)
ax.plot(s2_a, color="tab:orange", lw=3)

p1 = p_a_causal - 1

for (i,j) in p1:
ax.plot([i, j], [s1_a[i], s2_a[j]], color="black", alpha=0.3, ls="--")

ax.text(0.5, 5.7, "P[i]", color="tab:blue")
ax.text(0.5, 0.0, "Q[i]", color="tab:orange")

ax.set(xticks=np.arange(0,3),
yticks=np.arange(-2,7),
title=r"absolute vertical distance between $P[1]$ and $Q[i]$");
Show the code
a = 1.0*np.arange(16).reshape(4, 4)
# iu1 = np.triu_indices(4)
iu1 = np.tril_indices(n=4,m=5, k=-1)
a[iu1] = np.nan
a
array([[ 0.,  1.,  2.,  3.],
[nan,  5.,  6.,  7.],
[nan, nan, 10., 11.],
[nan, nan, nan, 15.]])