# Source code for stumpy.stump

```
# STUMPY
# Copyright 2019 TD Ameritrade. Released under the terms of the 3-Clause BSD license.
# STUMPY is a trademark of TD Ameritrade IP Company, Inc. All rights reserved.
import numba
import numpy as np
from numba import njit, prange
from . import config, core
from .aamp import aamp
@njit(
# "(f8[:], f8[:], i8, f8[:], f8[:], f8[:], f8[:], f8[:], f8[:], f8[:], f8[:],"
# "b1[:], b1[:], b1[:], b1[:], i8[:], i8, i8, i8, f8[:, :, :], f8[:, :],"
# "f8[:, :], i8[:, :, :], i8[:, :], i8[:, :], b1)",
fastmath=True,
)
def _compute_diagonal(
T_A,
T_B,
m,
μ_Q,
M_T,
σ_Q_inverse,
Σ_T_inverse,
cov_a,
cov_b,
cov_c,
cov_d,
T_A_subseq_isfinite,
T_B_subseq_isfinite,
T_A_subseq_isconstant,
T_B_subseq_isconstant,
diags,
diags_start_idx,
diags_stop_idx,
thread_idx,
ρ,
ρL,
ρR,
I,
IL,
IR,
ignore_trivial,
):
"""
Compute (Numba JIT-compiled) and update the (top-k) Pearson correlation (ρ),
ρL, ρR, I, IL, and IR sequentially along individual diagonals using a single
thread and avoiding race conditions.
Parameters
----------
T_A : numpy.ndarray
The time series or sequence for which to compute the matrix profile
T_B : numpy.ndarray
The time series or sequence that will be used to annotate `T_A`. For every
subsequence in `T_A`, its nearest neighbor in `T_B` will be recorded.
m : int
Window size
μ_Q : numpy.ndarray
Mean of the query sequence, `Q`, relative to the current sliding window
M_T : numpy.ndarray
Sliding mean of time series, `T`
σ_Q_inverse : numpy.ndarray
Inverse standard deviation of the query sequence, `Q`, relative to the current
sliding window
Σ_T_inverse : numpy.ndarray
Inverse sliding standard deviation of time series, `T`
cov_a : numpy.ndarray
The first covariance term relating T_A[i + g + m - 1] and M_T_m_1[i + g]
cov_b : numpy.ndarray
The second covariance term relating T_B[i + m - 1] and μ_Q_m_1[i]
cov_c : numpy.ndarray
The third covariance term relating T_A[i + g - 1] and M_T_m_1[i + g]
cov_d : numpy.ndarray
The fourth covariance term relating T_B[i - 1] and μ_Q_m_1[i]
T_A_subseq_isfinite : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_A` contains a
`np.nan`/`np.inf` value (False)
T_B_subseq_isfinite : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_B` contains a
`np.nan`/`np.inf` value (False)
T_A_subseq_isconstant : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_A` is constant (True)
T_B_subseq_isconstant : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_B` is constant (True)
diags : numpy.ndarray
The diagonal indices
diags_start_idx : int
The starting (inclusive) diagonal index
diags_stop_idx : int
The stopping (exclusive) diagonal index
thread_idx : int
The thread index
ρ : numpy.ndarray
The (top-k) Pearson correlations, sorted in ascending order per row
ρL : numpy.ndarray
The top-1 left Pearson correlations
ρR : numpy.ndarray
The top-1 right Pearson correlations
I : numpy.ndarray
The (top-k) matrix profile indices
IL : numpy.ndarray
The top-1 left matrix profile indices
IR : numpy.ndarray
The top-1 right matrix profile indices
ignore_trivial : bool
Set to `True` if this is a self-join. Otherwise, for AB-join, set this to
`False`. Default is `True`.
Returns
-------
None
Notes
-----
`DOI: 10.1007/s10115-017-1138-x \
<https://www.cs.ucr.edu/~eamonn/ten_quadrillion.pdf>`__
See Section 4.5
The above reference outlines a general approach for traversing the distance
matrix in a diagonal fashion rather than in a row-wise fashion.
`DOI: 10.1145/3357223.3362721 \
<https://www.cs.ucr.edu/~eamonn/public/GPU_Matrix_profile_VLDB_30DraftOnly.pdf>`__
See Section 3.1 and Section 3.3
The above reference outlines the use of the Pearson correlation via Welford's
centered sum-of-products along each diagonal of the distance matrix in place of the
sliding window dot product found in the original STOMP method.
"""
n_A = T_A.shape[0]
n_B = T_B.shape[0]
m_inverse = 1.0 / m
constant = (m - 1) * m_inverse * m_inverse # (m - 1)/(m * m)
uint64_m = np.uint64(m)
for diag_idx in range(diags_start_idx, diags_stop_idx):
g = diags[diag_idx]
if g >= 0:
iter_range = range(0, min(n_A - m + 1, n_B - m + 1 - g))
else:
iter_range = range(-g, min(n_A - m + 1, n_B - m + 1 - g))
for i in iter_range:
uint64_i = np.uint64(i)
uint64_j = np.uint64(i + g)
if uint64_i == 0 or uint64_j == 0:
cov = (
np.dot(
(T_B[uint64_j : uint64_j + uint64_m] - M_T[uint64_j]),
(T_A[uint64_i : uint64_i + uint64_m] - μ_Q[uint64_i]),
)
* m_inverse
)
else:
# The next lines are equivalent and left for reference
# cov = cov + constant * (
# (T_B[i + g + m - 1] - M_T_m_1[i + g])
# * (T_A[i + m - 1] - μ_Q_m_1[i])
# - (T_B[i + g - 1] - M_T_m_1[i + g]) * (T_A[i - 1] - μ_Q_m_1[i])
# )
cov = cov + constant * (
cov_a[uint64_j] * cov_b[uint64_i]
- cov_c[uint64_j] * cov_d[uint64_i]
)
if T_B_subseq_isfinite[uint64_j] and T_A_subseq_isfinite[uint64_i]:
# Neither subsequence contains NaNs
if T_B_subseq_isconstant[uint64_j] and T_A_subseq_isconstant[uint64_i]:
pearson = 1.0
elif T_B_subseq_isconstant[uint64_j] or T_A_subseq_isconstant[uint64_i]:
pearson = 0.5
else:
pearson = cov * Σ_T_inverse[uint64_j] * σ_Q_inverse[uint64_i]
pearson = min(1.0, pearson)
# `ρ[thread_idx, i, :]` is sorted in ascending order and MUST be updated
# when the newly-calculated `pearson` value becomes greater than the
# first (i.e. smallest) element in this array. Note that a higher
# pearson value corresponds to a lower distance.
if pearson > ρ[thread_idx, uint64_i, 0]:
idx = np.searchsorted(ρ[thread_idx, uint64_i], pearson)
core._shift_insert_at_index(
ρ[thread_idx, uint64_i], idx, pearson, shift="left"
)
core._shift_insert_at_index(
I[thread_idx, uint64_i], idx, uint64_j, shift="left"
)
if ignore_trivial: # self-joins only
if pearson > ρ[thread_idx, uint64_j, 0]:
idx = np.searchsorted(ρ[thread_idx, uint64_j], pearson)
core._shift_insert_at_index(
ρ[thread_idx, uint64_j], idx, pearson, shift="left"
)
core._shift_insert_at_index(
I[thread_idx, uint64_j], idx, uint64_i, shift="left"
)
if uint64_i < uint64_j:
# left pearson correlation and left matrix profile index
if pearson > ρL[thread_idx, uint64_j]:
ρL[thread_idx, uint64_j] = pearson
IL[thread_idx, uint64_j] = uint64_i
# right pearson correlation and right matrix profile index
if pearson > ρR[thread_idx, uint64_i]:
ρR[thread_idx, uint64_i] = pearson
IR[thread_idx, uint64_i] = uint64_j
return
@njit(
# "(f8[:], f8[:], i8, f8[:], f8[:], f8[:], f8[:], f8[:], f8[:], b1[:], b1[:],"
# "b1[:], b1[:], i8[:], b1, i8)",
parallel=True,
fastmath=True,
)
def _stump(
T_A,
T_B,
m,
μ_Q,
M_T,
σ_Q_inverse,
Σ_T_inverse,
μ_Q_m_1,
M_T_m_1,
T_A_subseq_isfinite,
T_B_subseq_isfinite,
T_A_subseq_isconstant,
T_B_subseq_isconstant,
diags,
ignore_trivial,
k,
):
"""
A Numba JIT-compiled version of STOMPopt with Pearson correlations for parallel
computation of the (top-k) matrix profile, the (top-k) matrix profile indices,
the top-1 left matrix profile and its matrix profile index, and the top-1 right
matrix profile and its matrix profile index.
Parameters
----------
T_A : numpy.ndarray
The time series or sequence for which to compute the matrix profile
T_B : numpy.ndarray
The time series or sequence that will be used to annotate `T_A`. For every
subsequence in `T_A`, its nearest neighbor in `T_B` will be recorded.
m : int
Window size
μ_Q : numpy.ndarray
Mean of the query sequence, `Q`, relative to the current sliding window
M_T : numpy.ndarray
Sliding mean of time series, `T`
σ_Q_inverse : numpy.ndarray
Inverse standard deviation of the query sequence, `Q`, relative to the current
sliding window
Σ_T_inverse : numpy.ndarray
Inverse sliding standard deviation of time series, `T`
μ_Q_m_1 : numpy.ndarray
Mean of the query sequence, `Q`, relative to the current sliding window and
using a window size of `m-1`
M_T_m_1 : numpy.ndarray
Sliding mean of time series, `T`, using a window size of `m-1`
T_A_subseq_isfinite : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_A` contains a
`np.nan`/`np.inf` value (False)
T_B_subseq_isfinite : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_B` contains a
`np.nan`/`np.inf` value (False)
T_A_subseq_isconstant : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_A` is constant (True)
T_B_subseq_isconstant : numpy.ndarray
A boolean array that indicates whether a subsequence in `T_B` is constant (True)
diags : numpy.ndarray
The diagonal indices
ignore_trivial : bool
Set to `True` if this is a self-join. Otherwise, for AB-join, set this to
`False`. Default is `True`.
k : int
The number of top `k` smallest distances used to construct the matrix profile.
Note that this will increase the total computational time and memory usage
when k > 1.
Returns
-------
out1 : numpy.ndarray
The (top-k) matrix profile
out2 : numpy.ndarray
The (top-1) left matrix profile
out3 : numpy.ndarray
The (top-1) right matrix profile
out4 : numpy.ndarray
The (top-k) matrix profile indices
out5 : numpy.ndarray
The (top-1) left matrix profile indices
out6 : numpy.ndarray
The (top-1) right matrix profile indices
Notes
-----
`DOI: 10.1007/s10115-017-1138-x \
<https://www.cs.ucr.edu/~eamonn/ten_quadrillion.pdf>`__
See Section 4.5
The above reference outlines a general approach for traversing the distance
matrix in a diagonal fashion rather than in a row-wise fashion.
`DOI: 10.1145/3357223.3362721 \
<https://www.cs.ucr.edu/~eamonn/public/GPU_Matrix_profile_VLDB_30DraftOnly.pdf>`__
See Section 3.1 and Section 3.3
The above reference outlines the use of the Pearson correlation via Welford's
centered sum-of-products along each diagonal of the distance matrix in place of the
sliding window dot product found in the original STOMP method.
`DOI: 10.1109/ICDM.2016.0085 \
<https://www.cs.ucr.edu/~eamonn/STOMP_GPU_final_submission_camera_ready.pdf>`__
See Table II
Timeseries, T_A, will be annotated with the distance location
(or index) of all its subsequences in another times series, T_B.
Return: For every subsequence, Q, in T_A, you will get a distance
and index for the closest subsequence in T_B. Thus, the array
returned will have length T_A.shape[0]-m+1. Additionally, the
left and right matrix profiles are also returned.
Note: Unlike in the Table II where T_A.shape is expected to be equal
to T_B.shape, this implementation is generalized so that the shapes of
T_A and T_B can be different. In the case where T_A.shape == T_B.shape,
then our algorithm reduces down to the same algorithm found in Table II.
Additionally, unlike STAMP where the exclusion zone is m/2, the default
exclusion zone for STOMP is m/4 (See Definition 3 and Figure 3).
For self-joins, set `ignore_trivial = True` in order to avoid the
trivial match.
Note that left and right matrix profiles are only available for self-joins.
"""
n_A = T_A.shape[0]
n_B = T_B.shape[0]
l = n_A - m + 1
n_threads = numba.config.NUMBA_NUM_THREADS
ρ = np.full((n_threads, l, k), np.NINF, dtype=np.float64)
I = np.full((n_threads, l, k), -1, dtype=np.int64)
ρL = np.full((n_threads, l), np.NINF, dtype=np.float64)
IL = np.full((n_threads, l), -1, dtype=np.int64)
ρR = np.full((n_threads, l), np.NINF, dtype=np.float64)
IR = np.full((n_threads, l), -1, dtype=np.int64)
ndist_counts = core._count_diagonal_ndist(diags, m, n_A, n_B)
diags_ranges = core._get_array_ranges(ndist_counts, n_threads, False)
cov_a = T_B[m - 1 :] - M_T_m_1[:-1]
cov_b = T_A[m - 1 :] - μ_Q_m_1[:-1]
# The next lines are equivalent and left for reference
# cov_c = np.roll(T_A, 1)
# cov_ = cov_c[:M_T_m_1.shape[0]] - M_T_m_1[:]
cov_c = np.empty(M_T_m_1.shape[0], dtype=np.float64)
cov_c[1:] = T_B[: M_T_m_1.shape[0] - 1]
cov_c[0] = T_B[-1]
cov_c[:] = cov_c - M_T_m_1
# The next lines are equivalent and left for reference
# cov_d = np.roll(T_B, 1)
# cov_d = cov_d[:μ_Q_m_1.shape[0]] - μ_Q_m_1[:]
cov_d = np.empty(μ_Q_m_1.shape[0], dtype=np.float64)
cov_d[1:] = T_A[: μ_Q_m_1.shape[0] - 1]
cov_d[0] = T_A[-1]
cov_d[:] = cov_d - μ_Q_m_1
for thread_idx in prange(n_threads):
# Compute and update pearson correlations and matrix profile indices
# within a single thread and avoiding race conditions
_compute_diagonal(
T_A,
T_B,
m,
μ_Q,
M_T,
σ_Q_inverse,
Σ_T_inverse,
cov_a,
cov_b,
cov_c,
cov_d,
T_A_subseq_isfinite,
T_B_subseq_isfinite,
T_A_subseq_isconstant,
T_B_subseq_isconstant,
diags,
diags_ranges[thread_idx, 0],
diags_ranges[thread_idx, 1],
thread_idx,
ρ,
ρL,
ρR,
I,
IL,
IR,
ignore_trivial,
)
# Reduction of results from all threads
for thread_idx in range(1, n_threads):
# update top-k arrays
core._merge_topk_ρI(ρ[0], ρ[thread_idx], I[0], I[thread_idx])
# update left matrix profile and matrix profile indices
mask = ρL[0] < ρL[thread_idx]
ρL[0][mask] = ρL[thread_idx][mask]
IL[0][mask] = IL[thread_idx][mask]
# update right matrix profile and matrix profile indices
mask = ρR[0] < ρR[thread_idx]
ρR[0][mask] = ρR[thread_idx][mask]
IR[0][mask] = IR[thread_idx][mask]
# Reverse top-k rho (and its associated I) to be in descending order and
# then convert from Pearson correlations to Euclidean distances (ascending order)
p_norm = np.abs(2 * m * (1 - ρ[0, :, ::-1]))
I = I[0, :, ::-1]
p_norm_L = np.abs(2 * m * (1 - ρL[0, :]))
p_norm_R = np.abs(2 * m * (1 - ρR[0, :]))
for i in prange(p_norm.shape[0]):
for j in range(p_norm.shape[1]):
if p_norm[i, j] < config.STUMPY_P_NORM_THRESHOLD:
p_norm[i, j] = 0.0
if p_norm_L[i] < config.STUMPY_P_NORM_THRESHOLD:
p_norm_L[i] = 0.0
if p_norm_R[i] < config.STUMPY_P_NORM_THRESHOLD:
p_norm_R[i] = 0.0
return (
np.sqrt(p_norm),
np.sqrt(p_norm_L),
np.sqrt(p_norm_R),
I,
IL[0],
IR[0],
)
[docs]
@core.non_normalized(
aamp,
exclude=["normalize", "p", "T_A_subseq_isconstant", "T_B_subseq_isconstant"],
)
def stump(
T_A,
m,
T_B=None,
ignore_trivial=True,
normalize=True,
p=2.0,
k=1,
T_A_subseq_isconstant=None,
T_B_subseq_isconstant=None,
):
"""
Compute the z-normalized matrix profile
This is a convenience wrapper around the Numba JIT-compiled parallelized
`_stump` function which computes the (top-k) matrix profile according to
STOMPopt with Pearson correlations.
Parameters
----------
T_A : numpy.ndarray
The time series or sequence for which to compute the matrix profile
m : int
Window size
T_B : numpy.ndarray, default None
The time series or sequence that will be used to annotate T_A. For every
subsequence in T_A, its nearest neighbor in T_B will be recorded. Default is
`None` which corresponds to a self-join.
ignore_trivial : bool, default True
Set to `True` if this is a self-join. Otherwise, for AB-join, set this
to `False`. Default is `True`.
normalize : bool, default True
When set to `True`, this z-normalizes subsequences prior to computing distances.
Otherwise, this function gets re-routed to its complementary non-normalized
equivalent set in the `@core.non_normalized` function decorator.
p : float, default 2.0
The p-norm to apply for computing the Minkowski distance. Minkowski distance is
typically used with `p` being 1 or 2, which correspond to the Manhattan distance
and the Euclidean distance, respectively. This parameter is ignored when
`normalize == True`.
k : int, default 1
The number of top `k` smallest distances used to construct the matrix profile.
Note that this will increase the total computational time and memory usage
when k > 1. If you have access to a GPU device, then you may be able to
leverage `gpu_stump` for better performance and scalability.
T_A_subseq_isconstant : numpy.ndarray or function, default None
A boolean array that indicates whether a subsequence in `T_A` is constant
(True). Alternatively, a custom, user-defined function that returns a
boolean array that indicates whether a subsequence in `T_A` is constant
(True). The function must only take two arguments, `a`, a 1-D array,
and `w`, the window size, while additional arguments may be specified
by currying the user-defined function using `functools.partial`. Any
subsequence with at least one np.nan/np.inf will automatically have its
corresponding value set to False in this boolean array.
T_B_subseq_isconstant : numpy.ndarray or function, default None
A boolean array that indicates whether a subsequence in `T_B` is constant
(True). Alternatively, a custom, user-defined function that returns a
boolean array that indicates whether a subsequence in `T_B` is constant
(True). The function must only take two arguments, `a`, a 1-D array,
and `w`, the window size, while additional arguments may be specified
by currying the user-defined function using `functools.partial`. Any
subsequence with at least one np.nan/np.inf will automatically have its
corresponding value set to False in this boolean array.
Returns
-------
out : numpy.ndarray
When k = 1 (default), the first column consists of the matrix profile,
the second column consists of the matrix profile indices, the third column
consists of the left matrix profile indices, and the fourth column consists
of the right matrix profile indices. However, when k > 1, the output array
will contain exactly 2 * k + 2 columns. The first k columns (i.e., out[:, :k])
consists of the top-k matrix profile, the next set of k columns
(i.e., out[:, k:2k]) consists of the corresponding top-k matrix profile
indices, and the last two columns (i.e., out[:, 2k] and out[:, 2k+1] or,
equivalently, out[:, -2] and out[:, -1]) correspond to the top-1 left
matrix profile indices and the top-1 right matrix profile indices, respectively.
See Also
--------
stumpy.stumped : Compute the z-normalized matrix profile with a distributed dask
cluster
stumpy.gpu_stump : Compute the z-normalized matrix profile with one or more GPU
devices
stumpy.scrump : Compute an approximate z-normalized matrix profile
Notes
-----
`DOI: 10.1007/s10115-017-1138-x \
<https://www.cs.ucr.edu/~eamonn/ten_quadrillion.pdf>`__
See Section 4.5
The above reference outlines a general approach for traversing the distance
matrix in a diagonal fashion rather than in a row-wise fashion.
`DOI: 10.1145/3357223.3362721 \
<https://www.cs.ucr.edu/~eamonn/public/GPU_Matrix_profile_VLDB_30DraftOnly.pdf>`__
See Section 3.1 and Section 3.3
The above reference outlines the use of the Pearson correlation via Welford's
centered sum-of-products along each diagonal of the distance matrix in place of the
sliding window dot product found in the original STOMP method.
`DOI: 10.1109/ICDM.2016.0085 \
<https://www.cs.ucr.edu/~eamonn/STOMP_GPU_final_submission_camera_ready.pdf>`__
See Table II
Timeseries, T_A, will be annotated with the distance location
(or index) of all its subsequences in another times series, T_B.
Return: For every subsequence, Q, in T_A, you will get a distance
and index for the closest subsequence in T_B. Thus, the array
returned will have length T_A.shape[0]-m+1. Additionally, the
left and right matrix profiles are also returned.
Note: Unlike in the Table II where T_A.shape is expected to be equal
to T_B.shape, this implementation is generalized so that the shapes of
T_A and T_B can be different. In the case where T_A.shape == T_B.shape,
then our algorithm reduces down to the same algorithm found in Table II.
Additionally, unlike STAMP where the exclusion zone is m/2, the default
exclusion zone for STOMP is m/4 (See Definition 3 and Figure 3).
For self-joins, set `ignore_trivial = True` in order to avoid the
trivial match.
Note that left and right matrix profiles are only available for self-joins.
Examples
--------
>>> import stumpy
>>> import numpy as np
>>> stumpy.stump(np.array([584., -11., 23., 79., 1001., 0., -19.]), m=3)
array([[0.11633857113691416, 4, -1, 4],
[2.694073918063438, 3, -1, 3],
[3.0000926340485923, 0, 0, 4],
[2.694073918063438, 1, 1, -1],
[0.11633857113691416, 0, 0, -1]], dtype=object)
"""
if T_B is None:
ignore_trivial = True
T_B = T_A
T_B_subseq_isconstant = T_A_subseq_isconstant
(
T_A,
μ_Q,
σ_Q_inverse,
μ_Q_m_1,
T_A_subseq_isfinite,
T_A_subseq_isconstant,
) = core.preprocess_diagonal(T_A, m, T_subseq_isconstant=T_A_subseq_isconstant)
(
T_B,
M_T,
Σ_T_inverse,
M_T_m_1,
T_B_subseq_isfinite,
T_B_subseq_isconstant,
) = core.preprocess_diagonal(T_B, m, T_subseq_isconstant=T_B_subseq_isconstant)
if T_A.ndim != 1: # pragma: no cover
raise ValueError(
f"T_A is {T_A.ndim}-dimensional and must be 1-dimensional. "
"For multidimensional STUMP use `stumpy.mstump` or `stumpy.mstumped`"
)
if T_B.ndim != 1: # pragma: no cover
raise ValueError(
f"T_B is {T_B.ndim}-dimensional and must be 1-dimensional. "
"For multidimensional STUMP use `stumpy.mstump` or `stumpy.mstumped`"
)
core.check_window_size(m, max_size=min(T_A.shape[0], T_B.shape[0]))
ignore_trivial = core.check_ignore_trivial(T_A, T_B, ignore_trivial)
n_A = T_A.shape[0]
n_B = T_B.shape[0]
l = n_A - m + 1
excl_zone = int(np.ceil(m / config.STUMPY_EXCL_ZONE_DENOM))
if ignore_trivial:
diags = np.arange(excl_zone + 1, n_A - m + 1, dtype=np.int64)
else:
diags = np.arange(-(n_A - m + 1) + 1, n_B - m + 1, dtype=np.int64)
P, PL, PR, I, IL, IR = _stump(
T_A,
T_B,
m,
μ_Q,
M_T,
σ_Q_inverse,
Σ_T_inverse,
μ_Q_m_1,
M_T_m_1,
T_A_subseq_isfinite,
T_B_subseq_isfinite,
T_A_subseq_isconstant,
T_B_subseq_isconstant,
diags,
ignore_trivial,
k,
)
out = np.empty((l, 2 * k + 2), dtype=object) # last two columns are to
# store left and right matrix profile indices
out[:, :k] = P
out[:, k:] = np.column_stack((I, IL, IR))
core._check_P(out[:, 0])
return out
```