def soft_dtw(ts1, ts2, gamma=1.):
r"""Compute Soft-DTW metric between two time series.
Soft-DTW was originally presented in [1]_ and is
discussed in more details in our
:ref:`user-guide page on DTW and its variants<dtw>`.
Soft-DTW is computed as:
.. math::
\text{soft-DTW}_{\gamma}(X, Y) =
\min_{\pi}{}^\gamma \sum_{(i, j) \in \pi} \|X_i, Y_j\|^2
where :math:`\min^\gamma` is the soft-min operator of parameter
:math:`\gamma`.
In the limit case :math:`\gamma = 0`, :math:`\min^\gamma` reduces to a
hard-min operator and soft-DTW is defined as the square of the DTW
similarity measure.
Parameters
----------
ts1
A time series
ts2
Another time series
gamma : float (default 1.)
Gamma paraneter for Soft-DTW
Returns
-------
float
Similarity
Examples
--------
>>> soft_dtw([1, 2, 2, 3],
... [1., 2., 3., 4.],
... gamma=1.) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
-0.89...
>>> soft_dtw([1, 2, 3, 3],
... [1., 2., 2.1, 3.2],
... gamma=0.01) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
0.089...
See Also
--------
cdist_soft_dtw : Cross similarity matrix between time series datasets
References
----------
.. [1] M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for
Time-Series," ICML 2017.
"""
if gamma == 0.:
return dtw(ts1, ts2)**2
return SoftDTW(SquaredEuclidean(ts1[:ts_size(ts1)], ts2[:ts_size(ts2)]),
gamma=gamma).compute()
def _fit_one_init(self, X, x_squared_norms, rs):
n_ts, _, d = X.shape
sz = min([ts_size(ts) for ts in X])
if hasattr(self.init, '__array__'):
self.cluster_centers_ = self.init.copy()
elif self.init == "k-means++":
self.cluster_centers_ = _k_init(X[:, :sz, :].reshape(
(n_ts, -1)), self.n_clusters, x_squared_norms, rs).reshape(
(-1, sz, d))
elif self.init == "random":
indices = rs.choice(X.shape[0], self.n_clusters)
self.cluster_centers_ = X[indices].copy()
else:
raise ValueError("Value %r for parameter 'init' is invalid" %
self.init)
self.cluster_centers_ = _check_full_length(self.cluster_centers_)
old_inertia = numpy.inf
for it in range(self.max_iter):
self._assign(X)
if self.verbose:
print("%.3f" % self.inertia_, end=" --> ")
self._update_centroids(X)
if numpy.abs(old_inertia - self.inertia_) < self.tol:
break
old_inertia = self.inertia_
if self.verbose:
print("")
return self
def _mm_update_barycenter(X, diag_sum_v_k, list_w_k):
"""Update barycenters using the formula from Algorithm 2 in [1]_.
Parameters
----------
X : numpy.array of shape (n, sz, d)
Time-series to be averaged
diag_sum_v_k : numpy.array of shape (barycenter_size, )
sum of weighted :math:`V^{(k)}` diagonals (as a vector)
list_w_k : list of numpy.array of shape (barycenter_size, sz_k)
list of weighted :math:`W^{(k)}` matrices
Returns
-------
numpy.array of shape (barycenter_size, d)
Updated barycenter
References
----------
.. [1] D. Schultz and B. Jain. Nonsmooth Analysis and Subgradient Methods
for Averaging in Dynamic Time Warping Spaces.
Pattern Recognition, 74, 340-358.
"""
d = X.shape[2]
barycenter_size = diag_sum_v_k.shape[0]
sum_w_x = numpy.zeros((barycenter_size, d))
for k, (w_k, x_k) in enumerate(zip(list_w_k, X)):
sum_w_x += w_k.dot(x_k[:ts_size(x_k)])
barycenter = numpy.diag(1. / diag_sum_v_k).dot(sum_w_x)
return barycenter
def _check_full_length(centroids):
"""Check that provided centroids are full-length (ie. not padded with nans).
If some centroids are found to be padded with nans, the last value is
repeated until the end.
Examples
--------
>>> centroids = to_time_series_dataset([[1, 2, 3], [1, 2, 3, 4, 5]])
>>> _check_full_length(centroids)
array([[[ 1.],
[ 2.],
[ 3.],
[ 3.],
[ 3.]],
<BLANKLINE>
[[ 1.],
[ 2.],
[ 3.],
[ 4.],
[ 5.]]])
"""
centroids_ = numpy.empty(centroids.shape)
n, max_sz = centroids.shape[:2]
for i in range(n):
sz = ts_size(centroids[i])
centroids_[i, :sz] = centroids[i, :sz]
if sz < max_sz:
centroids_[i, sz:] = centroids[i, sz - 1]
return centroids_
def fit_transform(self, X, **kwargs):
"""Fit to data, then transform it.
Parameters
----------
X : array-like
Time series dataset to be resampled.
Returns
-------
numpy.ndarray
Resampled time series dataset.
"""
X_ = to_time_series_dataset(X)
n_ts, sz, d = X_.shape
equal_size = check_equal_size(X_)
X_out = numpy.empty((n_ts, self.sz_, d))
for i in range(X_.shape[0]):
xnew = numpy.linspace(0, 1, self.sz_)
if not equal_size:
sz = ts_size(X_[i])
for di in range(d):
f = interp1d(numpy.linspace(0, 1, sz),
X_[i, :sz, di],
kind="slinear")
X_out[i, :, di] = f(xnew)
return X_out
def _check_series_length(self, X):
"""Ensures that time series in X matches the following requirements:
- their length is greater than the size of the longest shapelet
- (at predict time) their length is lower than the maximum allowed
length, as set by self.max_size
"""
sizes = numpy.array([ts_size(Xi) for Xi in X])
self._min_sz_fit = sizes.min()
if self.n_shapelets_per_size is not None:
max_sz_shp = max(self.n_shapelets_per_size.keys())
if max_sz_shp > self._min_sz_fit:
raise ValueError("Sizes in X do not match maximum "
"shapelet size: there is at least one "
"series in X that is shorter than one of the "
"shapelets. Shortest time series is of "
"length {} and longest shapelet is of length "
"{}".format(self._min_sz_fit, max_sz_shp))
if hasattr(self, 'model_') or self.max_size is not None:
# Model is already fitted
max_sz_X = sizes.max()
if hasattr(self, 'model_'):
max_size = self._X_fit_dims[1]
else:
max_size = self.max_size
if max_size < max_sz_X:
raise ValueError("Sizes in X do not match maximum allowed "
"size as set by max_size. "
"Longest time series is of "
"length {} and max_size is "
"{}".format(max_sz_X, max_size))
def soft_dtw(ts1, ts2, gamma=1.):
r"""Compute Soft-DTW metric between two time series.
Soft-DTW was originally presented in [1]_.
Parameters
----------
ts1
A time series
ts2
Another time series
gamma : float (default 1.)
Gamma paraneter for Soft-DTW
Returns
-------
float
Similarity
Examples
--------
>>> soft_dtw([1, 2, 2, 3],
... [1., 2., 3., 4.],
... gamma=1.) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
-0.89...
>>> soft_dtw([1, 2, 3, 3],
... [1., 2., 2.1, 3.2],
... gamma=0.01) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
0.089...
See Also
--------
cdist_soft_dtw : Cross similarity matrix between time series datasets
References
----------
.. [1] M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for
Time-Series," ICML 2017.
"""
if gamma == 0.:
return dtw(ts1, ts2)
return SoftDTW(SquaredEuclidean(ts1[:ts_size(ts1)], ts2[:ts_size(ts2)]),
gamma=gamma).compute()
def _transform(self, X, y=None):
n_ts, sz, d = X.shape
X_transformed = numpy.empty((n_ts, self.n_segments, d))
for i_ts in range(n_ts):
sz_segment = ts_size(X[i_ts]) // self.n_segments
for i_seg in range(self.n_segments):
start = i_seg * sz_segment
end = start + sz_segment
segment = X[i_ts, start:end, :]
X_transformed[i_ts, i_seg, :] = segment.mean(axis=0)
return X_transformed
def sigma_gak(dataset, n_samples=100, random_state=None):
r"""Compute sigma value to be used for GAK.
This method was originally presented in [1]_.
Parameters
----------
dataset
A dataset of time series
n_samples : int (default: 100)
Number of samples on which median distance should be estimated
random_state : integer or numpy.RandomState or None (default: None)
The generator used to draw the samples. If an integer is given, it
fixes the seed. Defaults to the global numpy random number generator.
Returns
-------
float
Suggested bandwidth (:math:`\\sigma`) for the Global Alignment kernel
Examples
--------
>>> dataset = [[1, 2, 2, 3], [1., 2., 3., 4.]]
>>> sigma_gak(dataset=dataset,
... n_samples=200,
... random_state=0) # doctest: +ELLIPSIS
2.0...
See Also
--------
gak : Compute Global Alignment kernel
cdist_gak : Compute cross-similarity matrix using Global Alignment kernel
References
----------
.. [1] M. Cuturi, "Fast global alignment kernels," ICML 2011.
"""
random_state = check_random_state(random_state)
dataset = to_time_series_dataset(dataset)
n_ts, sz, d = dataset.shape
if not check_equal_size(dataset):
sz = numpy.min([ts_size(ts) for ts in dataset])
if n_ts * sz < n_samples:
replace = True
else:
replace = False
sample_indices = random_state.choice(n_ts * sz,
size=n_samples,
replace=replace)
dists = pdist(dataset[:, :sz, :].reshape((-1, d))[sample_indices],
metric="euclidean")
return numpy.median(dists) * numpy.sqrt(sz)
def _kmeans_init_shapelets(X, n_shapelets, shp_len, n_draw=10000):
n_ts, sz, d = X.shape
indices_ts = numpy.random.choice(n_ts, size=n_draw, replace=True)
indices_time = numpy.array(
[numpy.random.choice(ts_size(ts) - shp_len + 1, size=1)[0]
for ts in X[indices_ts]]
)
subseries = numpy.zeros((n_draw, shp_len, d))
for i in range(n_draw):
subseries[i] = X[indices_ts[i],
indices_time[i]:indices_time[i] + shp_len]
return TimeSeriesKMeans(n_clusters=n_shapelets,
metric="euclidean",
verbose=False).fit(subseries).cluster_centers_
def VisualizeShapelets(self):
'''
visualize all of shapelets learned by shapelet classifier
'''
plt.figure()
for i, sz in enumerate(self.shapelet_sizes.keys()):
plt.subplot(len(self.shapelet_sizes), 1, i + 1)
plt.title("%d shapelets of size %d" %
(self.shapelet_sizes[sz], sz))
for shapelet in self.shapelet_clf.shapelets_:
if ts_size(shapelet) == sz:
plt.plot(shapelet.ravel())
plt.xlim([0, max(self.shapelet_sizes.keys())])
plt.tight_layout()
plt.show()
def _check_full_length(centroids):
"""Check that provided centroids are full-length (ie. not padded with
nans).
If some centroids are found to be padded with nans, the last value is
repeated until the end.
"""
centroids_ = numpy.empty(centroids.shape)
n, max_sz = centroids.shape[:2]
for i in range(n):
sz = ts_size(centroids[i])
centroids_[i, :sz] = centroids[i, :sz]
if sz < max_sz:
centroids_[i, sz:] = centroids[i, sz - 1]
return centroids_
def _subgradient_update_barycenter(X, list_diag_v_k, list_w_k, weights_sum,
barycenter, eta):
"""Update barycenters using the formula from Algorithm 1 in [1]_.
Parameters
----------
X : numpy.array of shape (n, sz, d)
Time-series to be averaged
list_diag_v_k : list of numpy.array of shape (barycenter_size, )
list of weighted :math:`V^{(k)}` diagonals (as vectors)
list_w_k : list of numpy.array of shape (barycenter_size, sz_k)
list of weighted :math:`W^{(k)}` matrices
weights_sum : float
sum of weights applied to matrices :math:`V^{(k)}` and :math:`W^{(k)}`
barycenter : numpy.array of shape (barycenter_size, d)
Barycenter as computed at the previous iteration of the algorithm
eta : float
Step-size for the subgradient descent algorithm
Returns
-------
numpy.array of shape (barycenter_size, d)
Updated barycenter
References
----------
.. [1] D. Schultz and B. Jain. Nonsmooth Analysis and Subgradient Methods
for Averaging in Dynamic Time Warping Spaces.
Pattern Recognition, 74, 340-358.
"""
d = X.shape[2]
barycenter_size = barycenter.shape[0]
delta_bar = numpy.zeros((barycenter_size, d))
for k, (v_k, w_k, x_k) in enumerate(zip(list_diag_v_k, list_w_k, X)):
delta_bar += v_k.reshape((-1, 1)) * barycenter
delta_bar -= w_k.dot(x_k[:ts_size(x_k)])
barycenter -= (2. * eta / weights_sum) * delta_bar
return barycenter
def _fit_one_init(self, X, x_squared_norms, rs):
n_ts, _, d = X.shape
sz = min([ts_size(ts) for ts in X])
self.cluster_centers_ = _k_init(X[:, :sz, :].reshape((n_ts, -1)),
self.n_clusters, x_squared_norms, rs).reshape((-1, sz, d))
old_inertia = numpy.inf
for it in range(self.max_iter):
self._assign(X)
if self.verbose:
print("%.3f" % self.inertia_, end=" --> ")
self._update_centroids(X)
if numpy.abs(old_inertia - self.inertia_) < self.tol:
break
old_inertia = self.inertia_
if self.verbose:
print("")
return self
def cdist_soft_dtw(dataset1, dataset2=None, gamma=1.):
r"""Compute cross-similarity matrix using Soft-DTW metric.
Soft-DTW was originally presented in [1]_ and is
discussed in more details in our
:ref:`user-guide page on DTW and its variants<dtw>`.
Soft-DTW is computed as:
.. math::
\text{soft-DTW}_{\gamma}(X, Y) =
\min_{\pi}{}^\gamma \sum_{(i, j) \in \pi} \|X_i, Y_j\|^2
where :math:`\min^\gamma` is the soft-min operator of parameter
:math:`\gamma`.
In the limit case :math:`\gamma = 0`, :math:`\min^\gamma` reduces to a
hard-min operator and soft-DTW is defined as the square of the DTW
similarity measure.
Parameters
----------
dataset1
A dataset of time series
dataset2
Another dataset of time series
gamma : float (default 1.)
Gamma paraneter for Soft-DTW
Returns
-------
numpy.ndarray
Cross-similarity matrix
Examples
--------
>>> cdist_soft_dtw([[1, 2, 2, 3], [1., 2., 3., 4.]], gamma=.01)
array([[-0.01098612, 1. ],
[ 1. , 0. ]])
>>> cdist_soft_dtw([[1, 2, 2, 3], [1., 2., 3., 4.]],
... [[1, 2, 2, 3], [1., 2., 3., 4.]], gamma=.01)
array([[-0.01098612, 1. ],
[ 1. , 0. ]])
See Also
--------
soft_dtw : Compute Soft-DTW
cdist_soft_dtw_normalized : Cross similarity matrix between time series
datasets using a normalized version of Soft-DTW
References
----------
.. [1] M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for
Time-Series," ICML 2017.
"""
dataset1 = to_time_series_dataset(dataset1, dtype=numpy.float64)
self_similarity = False
if dataset2 is None:
dataset2 = dataset1
self_similarity = True
else:
dataset2 = to_time_series_dataset(dataset2, dtype=numpy.float64)
dists = numpy.empty((dataset1.shape[0], dataset2.shape[0]))
equal_size_ds1 = check_equal_size(dataset1)
equal_size_ds2 = check_equal_size(dataset2)
for i, ts1 in enumerate(dataset1):
if equal_size_ds1:
ts1_short = ts1
else:
ts1_short = ts1[:ts_size(ts1)]
for j, ts2 in enumerate(dataset2):
if equal_size_ds2:
ts2_short = ts2
else:
ts2_short = ts2[:ts_size(ts2)]
if self_similarity and j < i:
dists[i, j] = dists[j, i]
else:
dists[i, j] = soft_dtw(ts1_short, ts2_short, gamma=gamma)
return dists
def cdist_soft_dtw(dataset1, dataset2=None, gamma=1.):
"""Compute cross-similarity matrix using Soft-DTW metric.
Soft-DTW was originally presented in [1]_.
Parameters
----------
dataset1
A dataset of time series
dataset2
Another dataset of time series
gamma : float (default 1.)
Gamma paraneter for Soft-DTW
Returns
-------
numpy.ndarray
Cross-similarity matrix
Examples
--------
>>> cdist_soft_dtw([[1, 2, 2, 3], [1., 2., 3., 4.]], gamma=.01) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
array([[-0.01..., 1. ],
[ 1. , 0. ]])
>>> cdist_soft_dtw([[1, 2, 2, 3], [1., 2., 3., 4.]], [[1, 2, 2, 3], [1., 2., 3., 4.]], gamma=.01) # doctest: +NORMALIZE_WHITESPACE +ELLIPSIS
array([[-0.01..., 1. ],
[ 1. , 0. ]])
See Also
--------
soft_dtw : Compute Soft-DTW
cdist_soft_dtw_normalized : Cross similarity matrix between time series
datasets using a normalized version of Soft-DTW
References
----------
.. [1] M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for Time-Series," ICML 2017.
"""
dataset1 = to_time_series_dataset(dataset1, dtype=numpy.float64)
self_similarity = False
if dataset2 is None:
dataset2 = dataset1
self_similarity = True
else:
dataset2 = to_time_series_dataset(dataset2, dtype=numpy.float64)
dists = numpy.empty((dataset1.shape[0], dataset2.shape[0]))
equal_size_ds1 = check_equal_size(dataset1)
equal_size_ds2 = check_equal_size(dataset2)
for i, ts1 in enumerate(dataset1):
if equal_size_ds1:
ts1_short = ts1
else:
ts1_short = ts1[:ts_size(ts1)]
for j, ts2 in enumerate(dataset2):
if equal_size_ds2:
ts2_short = ts2
else:
ts2_short = ts2[:ts_size(ts2)]
if self_similarity and j < i:
dists[i, j] = dists[j, i]
else:
dists[i, j] = soft_dtw(ts1_short, ts2_short, gamma=gamma)
return dists
def test_kmeans():
n, sz, d = 15, 10, 3
rng = np.random.RandomState(0)
time_series = rng.randn(n, sz, d)
km = TimeSeriesKMeans(n_clusters=3, metric="euclidean", max_iter=5,
verbose=False, random_state=rng).fit(time_series)
dists = cdist(time_series.reshape((n, -1)),
km.cluster_centers_.reshape((3, -1)))
np.testing.assert_allclose(km.labels_, dists.argmin(axis=1))
np.testing.assert_allclose(km.labels_, km.predict(time_series))
km_dba = TimeSeriesKMeans(n_clusters=3,
metric="dtw",
max_iter=5,
verbose=False,
random_state=rng).fit(time_series)
dists = cdist_dtw(time_series, km_dba.cluster_centers_)
np.testing.assert_allclose(km_dba.labels_, dists.argmin(axis=1))
np.testing.assert_allclose(km_dba.labels_, km_dba.predict(time_series))
km_sdtw = TimeSeriesKMeans(n_clusters=3,
metric="softdtw",
max_iter=5,
verbose=False,
random_state=rng).fit(time_series)
dists = cdist_soft_dtw(time_series, km_sdtw.cluster_centers_)
np.testing.assert_allclose(km_sdtw.labels_, dists.argmin(axis=1))
np.testing.assert_allclose(km_sdtw.labels_, km_sdtw.predict(time_series))
km_nofit = TimeSeriesKMeans(n_clusters=101,
verbose=False,
random_state=rng).fit(time_series)
assert(km_nofit._X_fit is None)
X_bis = to_time_series_dataset([[1, 2, 3, 4],
[1, 2, 3],
[2, 5, 6, 7, 8, 9]])
TimeSeriesKMeans(n_clusters=2, verbose=False, max_iter=5,
metric="softdtw", random_state=0).fit(X_bis)
TimeSeriesKMeans(n_clusters=2, verbose=False, max_iter=5,
metric="dtw", random_state=0,
init="random").fit(X_bis)
TimeSeriesKMeans(n_clusters=2, verbose=False, max_iter=5,
metric="dtw", random_state=0,
init="k-means++").fit(X_bis)
TimeSeriesKMeans(n_clusters=2, verbose=False, max_iter=5,
metric="dtw", init=X_bis[:2]).fit(X_bis)
# Barycenter size (nb of timestamps)
# Case 1. kmeans++ / random init
n, sz, d = 15, 10, 1
n_clusters = 3
time_series = rng.randn(n, sz, d)
sizes_all_same_series = [sz] * n_clusters
km_euc = TimeSeriesKMeans(n_clusters=3,
metric="euclidean",
max_iter=5,
verbose=False,
init="k-means++",
random_state=rng).fit(time_series)
np.testing.assert_equal(sizes_all_same_series,
[ts_size(b) for b in km_euc.cluster_centers_])
km_dba = TimeSeriesKMeans(n_clusters=3,
metric="dtw",
max_iter=5,
verbose=False,
init="random",
random_state=rng).fit(time_series)
np.testing.assert_equal(sizes_all_same_series,
[ts_size(b) for b in km_dba.cluster_centers_])
# Case 2. forced init
barys = to_time_series_dataset([[1., 2., 3.],
[1., 2., 2., 3., 4.],
[3., 2., 1.]])
sizes_all_same_bary = [barys.shape[1]] * n_clusters
# If Euclidean is used, barycenters size should be that of the input series
km_euc = TimeSeriesKMeans(n_clusters=3,
metric="euclidean",
max_iter=5,
verbose=False,
init=barys,
random_state=rng)
np.testing.assert_raises(ValueError, km_euc.fit, time_series)
km_dba = TimeSeriesKMeans(n_clusters=3,
metric="dtw",
max_iter=5,
verbose=False,
init=barys,
random_state=rng).fit(time_series)
np.testing.assert_equal(sizes_all_same_bary,
[ts_size(b) for b in km_dba.cluster_centers_])
km_sdtw = TimeSeriesKMeans(n_clusters=3,
metric="softdtw",
max_iter=5,
verbose=False,
init=barys,
random_state=rng).fit(time_series)
np.testing.assert_equal(sizes_all_same_bary,
[ts_size(b) for b in km_sdtw.cluster_centers_])
# A simple dataset, can we extract the correct number of clusters?
time_series = to_time_series_dataset([[1, 2, 3],
[7, 8, 9, 11],
[.1, .2, 2.],
[1, 1, 1, 9],
[10, 20, 30, 1000]])
preds = TimeSeriesKMeans(n_clusters=3, metric="dtw", max_iter=5,
random_state=rng).fit_predict(time_series)
np.testing.assert_equal(set(preds), set(range(3)))
preds = TimeSeriesKMeans(n_clusters=4, metric="dtw", max_iter=5,
random_state=rng).fit_predict(time_series)
np.testing.assert_equal(set(preds), set(range(4)))
# Define the model using parameters provided by the authors (except that we use
# fewer iterations here)
shp_clf = ShapeletModel(n_shapelets_per_size=shapelet_sizes,
optimizer=Adagrad(lr=.1),
weight_regularizer=.01,
max_iter=200,
verbose=0)
shp_clf.fit(X_train, y_train)
predicted_labels = shp_clf.predict(X_test)
print("Correct classification rate:", accuracy_score(y_test, predicted_labels))
plt.figure()
for i, sz in enumerate(shapelet_sizes.keys()):
plt.subplot(len(shapelet_sizes), 1, i + 1)
plt.title("%d shapelets of size %d" % (shapelet_sizes[sz], sz))
for shp in shp_clf.shapelets_:
if ts_size(shp) == sz:
plt.plot(shp.ravel())
plt.xlim([0, max(shapelet_sizes.keys()) - 1])
plt.tight_layout()
plt.show()
# The loss history is accessible via the `model` attribute that is a keras
# model
plt.figure()
plt.plot(numpy.arange(1, 201), shp_clf.model_.history.history["loss"])
plt.title("Evolution of cross-entropy loss during training")
plt.xlabel("Epochs")
plt.show()
def soft_dtw_alignment(ts1, ts2, gamma=1.):
r"""Compute Soft-DTW metric between two time series and return both the
similarity measure and the alignment matrix.
Soft-DTW was originally presented in [1]_ and is
discussed in more details in our
:ref:`user-guide page on DTW and its variants<dtw>`.
Soft-DTW is computed as:
.. math::
\text{soft-DTW}_{\gamma}(X, Y) =
\min_{\pi}{}^\gamma \sum_{(i, j) \in \pi} \|X_i, Y_j\|^2
where :math:`\min^\gamma` is the soft-min operator of parameter
:math:`\gamma`.
In the limit case :math:`\gamma = 0`, :math:`\min^\gamma` reduces to a
hard-min operator and soft-DTW is defined as the square of the DTW
similarity measure.
Parameters
----------
ts1
A time series
ts2
Another time series
gamma : float (default 1.)
Gamma paraneter for Soft-DTW
Returns
-------
numpy.ndarray
Soft-alignment matrix
float
Similarity
Examples
--------
>>> a, dist = soft_dtw_alignment([1, 2, 2, 3],
... [1., 2., 3., 4.],
... gamma=1.) # doctest: +ELLIPSIS
>>> dist
-0.89...
>>> a # doctest: +ELLIPSIS +NORMALIZE_WHITESPACE
array([[1.00...e+00, 1.88...e-01, 2.83...e-04, 4.19...e-11],
[3.40...e-01, 8.17...e-01, 8.87...e-02, 3.94...e-05],
[5.05...e-02, 7.09...e-01, 5.30...e-01, 6.98...e-03],
[1.37...e-04, 1.31...e-01, 7.30...e-01, 1.00...e+00]])
See Also
--------
soft_dtw : Returns soft-DTW score alone
References
----------
.. [1] M. Cuturi, M. Blondel "Soft-DTW: a Differentiable Loss Function for
Time-Series," ICML 2017.
"""
if gamma == 0.:
path, dist = dtw_path(ts1, ts2)
dist_sq = dist**2
a = numpy.zeros((ts_size(ts1), ts_size(ts2)))
for i, j in path:
a[i, j] = 1.
else:
sdtw = SoftDTW(SquaredEuclidean(ts1[:ts_size(ts1)],
ts2[:ts_size(ts2)]),
gamma=gamma)
dist_sq = sdtw.compute()
a = sdtw.grad()
return a, dist_sq