# Dynamic Time Warping (DTW)

```
from dtaidistance import dtw
from dtaidistance import dtw_visualisation as dtwvis
import numpy as np
s1 = np.array([0., 0, 1, 2, 1, 0, 1, 0, 0, 2, 1, 0, 0])
s2 = np.array([0., 1, 2, 3, 1, 0, 0, 0, 2, 1, 0, 0, 0])
path = dtw.warping_path(s1, s2)
dtwvis.plot_warping(s1, s2, path, filename="warp.png")
```

## DTW Distance Measure Between Two Time Series

Only the distance measure based on two sequences of numbers:

```
from dtaidistance import dtw
s1 = [0, 0, 1, 2, 1, 0, 1, 0, 0]
s2 = [0, 1, 2, 0, 0, 0, 0, 0, 0]
distance = dtw.distance(s1, s2)
print(distance)
```

The fastest version (30-300 times) uses c directly but requires an array
as input (with the double type), and (optionally) also prunes computations
by setting `max_dist`

to the Euclidean upper bound:

```
from dtaidistance import dtw
import array
s1 = array.array('d',[0, 0, 1, 2, 1, 0, 1, 0, 0])
s2 = array.array('d',[0, 1, 2, 0, 0, 0, 0, 0, 0])
d = dtw.distance_fast(s1, s2, use_pruning=True)
```

Or you can use a numpy array (with dtype double or float):

```
from dtaidistance import dtw
import numpy as np
s1 = np.array([0, 0, 1, 2, 1, 0, 1, 0, 0], dtype=np.double)
s2 = np.array([0.0, 1, 2, 0, 0, 0, 0, 0, 0], dtype=np.double)
d = dtw.distance_fast(s1, s2, use_pruning=True)
```

Check the `__doc__`

for information about the available arguments:

```
print(dtw.distance.__doc__)
```

### DTW Complexity and Early-Stopping

The `distance`

function has linear space complexity but quadratic
time complexity. To reduce the time complexity, a number of options
are available. The most used approach across DTW implementations is
to use a window that indicates the maximal shift that is allowed (also
known as a Sakoe-Chiba band).
This reduces the complexity to the product of window size and
largest sequence length:

`window`

: Only allow for shifts up to this amount away from the two diagonals.

A number of other options are foreseen to early stop some or all paths the dynamic programming algorithm is exploring:

`max_dist`

: Avoid computing partial paths that will be larger than this value. If no solution is found that is smaller or equal to this value, then return infinity.`use_pruning`

: A good way of pruning partial paths is to set`max_dist`

to the Euclidean upper bound. If this option is set to true, this is done automatically.`max_step`

: Do not allow steps larger than this value, replace them with infinity.`max_length_diff`

: Return infinity if difference in length of two sequences is larger than this value.

### DTW Tuning

A number of options are foreseen to tune how the cost is computed:

`penalty`

: Penalty to add if compression or expansion is applied (on top of the distance).`psi`

: Up to`psi`

number of start and end points of a sequence can be ignored if this would lead to a lower distance. This is also called psi-relaxation (for cyclical sequences) [2].

## DTW and keep all warping paths

If, next to the distance, you also want the full matrix to see all possible warping paths (also called the accumulated cost matrix):

```
from dtaidistance import dtw
s1 = [0, 0, 1, 2, 1, 0, 1, 0, 0]
s2 = [0, 1, 2, 0, 0, 0, 0, 0, 0]
distance, paths = dtw.warping_paths(s1, s2)
print(distance)
print(paths)
```

The matrix with all warping paths (or accumulated cost matrix) can be visualised as follows:

```
from dtaidistance import dtw
from dtaidistance import dtw_visualisation as dtwvis
import random
import numpy as np
x = np.arange(0, 20, .5)
s1 = np.sin(x)
s2 = np.sin(x - 1)
random.seed(1)
for idx in range(len(s2)):
if random.random() < 0.05:
s2[idx] += (random.random() - 0.5) / 2
d, paths = dtw.warping_paths(s1, s2, window=25, psi=2)
best_path = dtw.best_path(paths)
dtwvis.plot_warpingpaths(s1, s2, paths, best_path)
```

Notice the `psi`

parameter that relaxes the matching at the beginning
and end. In this example this results in a perfect match even though the
sine waves are slightly shifted.

## DTW between multiple Time series

To compute the DTW distance measures between all sequences in a list of
sequences, use the method `dtw.distance_matrix`

. You can speed up the
computation by using the `dtw.distance_matrix_fast`

method that tries
to run all algorithms in C. Also parallelization can be activated using
the `parallel`

argument.

The `distance_matrix`

and `distance_matrix_fast`

methods expect a
list of lists/arrays:

```
from dtaidistance import dtw
import numpy as np
timeseries = [
np.array([0, 0, 1, 2, 1, 0, 1, 0, 0], dtype=np.double),
np.array([0.0, 1, 2, 0, 0, 0, 0, 0, 0, 0, 0]),
np.array([0.0, 0, 1, 2, 1, 0, 0, 0])]
ds = dtw.distance_matrix_fast(timeseries)
```

or a matrix (in case all time series have the same length):

```
from dtaidistance import dtw
import numpy as np
timeseries = np.array([
[0.0, 0, 1, 2, 1, 0, 1, 0, 0],
[0.0, 1, 2, 0, 0, 0, 0, 0, 0],
[0.0, 0, 1, 2, 1, 0, 0, 0, 0]])
ds = dtw.distance_matrix_fast(timeseries)
```

The result is stored in a matrix representation. Since only the upper
triangular matrix is required, this representation uses more memory then necessary.
This behaviour can be deactivated by setting the argument `compact`

to
true. The method will then return a 1-dimensional array with all results.
This array represents the concatenation of all upper triangular rows.

## DTW between multiple time series, limited to block

You can instruct the computation to only fill part of the distance measures matrix. For example to distribute the computations over multiple computing nodes, or to only compare source time series to target time series.

```
from dtaidistance import dtw
import numpy as np
timeseries = np.array([
[0., 0, 1, 2, 1, 0, 1, 0, 0],
[0., 1, 2, 0, 0, 0, 0, 0, 0],
[1., 2, 0, 0, 0, 0, 0, 1, 1],
[0., 0, 1, 2, 1, 0, 1, 0, 0],
[0., 1, 2, 0, 0, 0, 0, 0, 0],
[1., 2, 0, 0, 0, 0, 0, 1, 1]])
ds = dtw.distance_matrix_fast(timeseries, block=((1, 4), (3, 5)))
```

The output in this case will be:

```
# 0 1 2 3 4 5
[[ inf inf inf inf inf inf] # 0
[ inf inf inf 1.4142 0.0000 inf] # 1
[ inf inf inf 2.2360 1.7320 inf] # 2
[ inf inf inf inf 1.4142 inf] # 3
[ inf inf inf inf inf inf] # 4
[ inf inf inf inf inf inf]] # 5
```

Especially for blocks the matrix representation uses a lot of unnecesary
memory. This can be avoided by setting the `compact`

argument to true:

```
from dtaidistance import dtw
import numpy as np
timeseries = np.array([
[0., 0, 1, 2, 1, 0, 1, 0, 0],
[0., 1, 2, 0, 0, 0, 0, 0, 0],
[1., 2, 0, 0, 0, 0, 0, 1, 1],
[0., 0, 1, 2, 1, 0, 1, 0, 0],
[0., 1, 2, 0, 0, 0, 0, 0, 0],
[1., 2, 0, 0, 0, 0, 0, 1, 1]])
ds = dtw.distance_matrix_fast(timeseries, block=((1, 4), (3, 5)), compact=True)
```

The result will now be:

```
[1.4142 0.0000 2.2360 1.7320 1.4142]
```

## DTW based on shape

If you are interested in comparing only the shape, and not the absolute differences and offset, you need to transform the data first.

**z-normalization**

Z-normalize is the most popular transformation. This can be achieved
using the SciPy `zscore`

function:

```
import numpy as np
a = np.array([0.1, 0.3, 0.2, 0.1])
from scipy import stats
az = stats.zscore(a)
# az = array([-0.90453403, 1.50755672, 0.30151134, -0.90453403])
```

**Differencing**

Z-normalization has the disadvantage that constant baselines are not necessarily at the same level. The causes a small error but it accumulates over a long distance. To avoid this, use differencing (see the clustering K-means documentation for a visual example).

```
series = dtaidistance.preprocessing.differencing(series, smooth=0.1)
```

## Multi-dimensionsal DTW

To compare two multivariate sequences, a multivariate time series with n_timesteps and at each timestep a vector with n_values is stored in a two dimensional array of size (n_timesteps,n_values). The first dimension of the data structure is the sequence item index (i.e., time series index, time step) and the second dimension is the index of the value in the vector.

For example, two 2-dimensional multivariate series with five timesteps:

```
from dtaidistance import dtw_ndim
series1 = np.array([[0, 0], # first point at t=0
[0, 1], # second point at t=1
[2, 1],
[0, 1],
[0, 0]], dtype=np.double)
series2 = np.array([[0, 0],
[2, 1],
[0, 1],
[0, .5],
[0, 0]], dtype=np.double)
d = dtw_ndim.distance(series1, series2)
```

This method returns the dependent DTW (DTW_D) distance between two n-dimensional sequences. If you want to compute the independent DTW (DTW_I) distance, use the 1-dimensional version:

```
dtw_i = 0
for dim in range(ndim):
dtw_i += dtw.distance(s1[:,dim], s2[:,dim])
```

To compute a distance matrix between multivariate time series, the same data structures are for univariate DTW are supported. The only difference is that when all data is stored in a Numpy array, this is now a 3-dimensional array with as size (n_series, n_timesteps, n_values).