# 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 accros DTW implementations is to use a window that indicates the maximal shift that is allowed. 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 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.

### 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) .

## DTW and keep all warping paths¶

If, next to the distance, you also want the full matrix to see all possible warping paths, also alled 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 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 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 (z-normalization)¶

If you are interested in comparing only the shape, and not the absolute differences and offset, you need to z-normalize the data first. 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])
```

## Multi-dimensionsal DTW¶

Compare two multi-dimensional sequences.

Assumes the first dimension of the data structure to be the sequence item index (or time series index).

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

```from dtaidistance import dtw_ndim

series1 = np.array([[0, 0],  # first 2-dim point at t=0
[0, 1],  # second 2-dim 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])
```