_distance.py

__author__ = ["chrisholder", "TonyBagnall"]


from typing import Any, Callable, Union

import numpy as np

from sktime.distances._ddtw import _DdtwDistance
from sktime.distances._dtw import _DtwDistance
from sktime.distances._edr import _EdrDistance
from sktime.distances._erp import _ErpDistance
from sktime.distances._euclidean import _EuclideanDistance
from sktime.distances._lcss import _LcssDistance
from sktime.distances._msm import _MsmDistance
from sktime.distances._resolve_metric import (
    _resolve_dist_instance,
    _resolve_metric_to_factory,
)
from sktime.distances._squared import _SquaredDistance
from sktime.distances._twe import _TweDistance
from sktime.distances._wddtw import _WddtwDistance
from sktime.distances._wdtw import _WdtwDistance
from sktime.distances.base import (
    AlignmentPathReturn,
    DistanceAlignmentPathCallable,
    DistanceCallable,
    MetricInfo,
    NumbaDistance,
)


def erp_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    g: float = 0.0,
    **kwargs: Any,
) -> float:
    """Compute the Edit distance for real penalty (ERP) distance between two series.

    ERP, first proposed in [1]_, attempts align time series
    by better considering how indexes are carried forward through the cost matrix.
    Usually in the dtw cost matrix, if an alignment can't be found the previous value
    is carried forward. Erp instead proposes the idea of gaps or sequences of points
    that have no matches. These gaps are then punished based on their distance from 'g'.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    g: float, defaults = 0.
        The reference value to penalise gaps.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        ERP distance between x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 3 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If g is not a float.
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import erp_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> erp_distance(x_1d, y_1d)
    16.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> erp_distance(x_2d, y_2d)
    45.254833995939045

    References
    ----------
    .. [1] Lei Chen and Raymond Ng. 2004. On the marriage of Lp-norms and edit distance.
    In Proceedings of the Thirtieth international conference on Very large data bases
     - Volume 30 (VLDB '04). VLDB Endowment, 792–803.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="erp", **format_kwargs)


def edr_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    epsilon: float = None,
    **kwargs: Any,
) -> float:
    """Compute the Edit distance for real sequences (EDR) between two series.

    EDR computes the minimum number of elements (as a percentage) that must be removed
    from x and y so that the sum of the distance between the remaining signal elements
    lies within the tolerance (epsilon). EDR was originally proposed in [1]_.

    The value returned will be between 0 and 1 per time series. The value will
    represent as a percentage of elements that must be removed for the time series to
    be an exact match.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d array), defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    epsilon : float, defaults = None
        Matching threshold to determine if two subsequences are considered close
        enough to be considered 'common'. If not specified as per the original paper
        epsilon is set to a quarter of the maximum standard deviation.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Edr distance between the x and y. The value will be between 0.0 and 1.0
        where 0.0 is an exact match between time series (i.e. they are the same) and
        1.0 where there are no matching subsequences.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 3 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import edr_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> edr_distance(x_1d, y_1d)
    1.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> edr_distance(x_2d, y_2d)
    1.0

    References
    ----------
    .. [1] Lei Chen, M. Tamer Özsu, and Vincent Oria. 2005. Robust and fast similarity
    search for moving object trajectories. In Proceedings of the 2005 ACM SIGMOD
    international conference on Management of data (SIGMOD '05). Association for
    Computing Machinery, New York, NY, USA, 491–502.
    DOI:https://doi.org/10.1145/1066157.1066213
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "epsilon": epsilon,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="edr", **format_kwargs)


def lcss_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    epsilon: float = 1.0,
    **kwargs: Any,
) -> float:
    """Compute the longest common subsequence (LCSS) score between two time series.

    LCSS attempts to find the longest common sequence between two time series and
    returns a value that is the percentage that longest common sequence assumes.
    Originally present in [1]_, LCSS is computed by matching indexes that are
    similar up until a defined threshold (epsilon).

    The value returned will be between 0.0 and 1.0, where 0.0 means the two time series
    are exactly the same and 1.0 means they are complete opposites.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    epsilon : float, defaults = 1.
        Matching threshold to determine if two subsequences are considered close
        enough to be considered 'common'.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Lcss distance between x and y. The value returned will be between 0.0 and 1.0,
        where 0.0 means the two time series are exactly the same and 1.0 means they
        are complete opposites.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] M. Vlachos, D. Gunopoulos, and G. Kollios. 2002. "Discovering
        Similar Multidimensional Trajectories", In Proceedings of the
        18th International Conference on Data Engineering (ICDE '02).
        IEEE Computer Society, USA, 673.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "epsilon": epsilon,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="lcss", **format_kwargs)


def wddtw_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    compute_derivative=None,
    g: float = 0.0,
    **kwargs: Any,
) -> float:
    r"""Compute the weighted derivative dynamic time warping (WDDTW) distance.

    WDDTW was first proposed in [1]_ as an extension of DDTW. By adding a weight
    to the derivative it means the alignment isn't only considering the shape of the
    time series, but also the phase.

    Formally the derivative is calculated as:

    .. math::
        D_{x}[q] = \frac{{}(q_{i} - q_{i-1} + ((q_{i+1} - q_{i-1}/2)}{2}

    Therefore a weighted derivative can be calculated using D (the derivative) as:

    .. math::
        d_{w}(x_{i}, y_{j}) = ||w_{|i-j|}(D_{x_{i}} - D_{y_{j}})||

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    compute_derivative: Callable[[np.ndarray], np.ndarray],
                            defaults = average slope difference
        Callable that computes the derivative. If none is provided the average of the
        slope between two points used.
    g: float, defaults = 0.
        Constant that controls the curvature (slope) of the function; that is, g
        controls the level of penalisation for the points with larger phase
        difference.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Wddtw distance between x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If the compute derivative callable is not no_python compiled.
        If the value of g is not a float
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import wddtw_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> wddtw_distance(x_1d, y_1d) # doctest: +SKIP
    0.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> wddtw_distance(x_2d, y_2d) # doctest: +SKIP
    0.0

    References
    ----------
    .. [1] Young-Seon Jeong, Myong K. Jeong, Olufemi A. Omitaomu, Weighted dynamic time
    warping for time series classification, Pattern Recognition, Volume 44, Issue 9,
    2011, Pages 2231-2240, ISSN 0031-3203, https://doi.org/10.1016/j.patcog.2010.09.022.
    """
    if compute_derivative is None:
        from sktime.distances._ddtw_numba import average_of_slope

        compute_derivative = average_of_slope

    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "compute_derivative": compute_derivative,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="wddtw", **format_kwargs)


def wdtw_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    g: float = 0.05,
    **kwargs: Any,
) -> float:
    """Compute the weighted dynamic time warping (WDTW) distance between time series.

    First proposed in [1]_, WDTW adds a  adds a multiplicative weight penalty based on
    the warping distance. This means that time series with lower phase difference have
    a smaller weight imposed (i.e less penalty imposed) and time series with larger
    phase difference have a larger weight imposed (i.e. larger penalty imposed).

    Formally this can be described as:

    .. math::
        d_{w}(x_{i}, y_{j}) = ||w_{|i-j|}(x_{i} - y_{j})||

    Where d_w is the distance with a the weight applied to it for points i, j, where
    w(|i-j|) is a positive weight between the two points x_i and y_j.


    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    g: float, defaults = 0.
        Constant that controls the curvature (slope) of the function; that is, g
        controls the level of penalisation for the points with larger phase
        difference.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Wdtw distance between the x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import wdtw_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> wdtw_distance(x_1d, y_1d)
    27.975712863958133

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> wdtw_distance(x_2d, y_2d)
    243.2106560107827

    References
    ----------
    .. [1] Young-Seon Jeong, Myong K. Jeong, Olufemi A. Omitaomu, Weighted dynamic time
    warping for time series classification, Pattern Recognition, Volume 44, Issue 9,
    2011, Pages 2231-2240, ISSN 0031-3203, https://doi.org/10.1016/j.patcog.2010.09.022.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="wdtw", **format_kwargs)


def ddtw_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    compute_derivative=None,
    **kwargs: Any,
) -> float:
    r"""Compute the derivative dynamic time warping (DDTW) distance between time series.

    DDTW is an adaptation of DTW originally proposed in [1]_. DDTW attempts to
    improve on dtw by better account for the 'shape' of the time series.
    This is done by considering y axis data points as higher level features of 'shape'.
    To do this the first derivative of the sequence is taken, and then using this
    derived sequence a dtw computation is done.

    The default derivative used is:

    .. math::
        D_{x}[q] = \frac{{}(q_{i} - q_{i-1} + ((q_{i+1} - q_{i-1}/2)}{2}

    Where q is the original time series and d_q is the derived time series.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    compute_derivative: Callable[[np.ndarray], np.ndarray],
                            defaults = average slope difference
        Callable that computes the derivative. If none is provided the average of the
        slope between two points used.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Ddtw distance between the x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric or compute derivative callable is not no_python compiled.
        If the metric type cannot be determined
        If the compute derivative callable is not no_python compiled.
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import ddtw_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> ddtw_distance(x_1d, y_1d) # doctest: +SKIP
    0.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> ddtw_distance(x_2d, y_2d) # doctest: +SKIP
    0.0

    References
    ----------
    .. [1] Keogh, Eamonn & Pazzani, Michael. (2002). Derivative Dynamic Time Warping.
        First SIAM International Conference on Data Mining.
        1. 10.1137/1.9781611972719.1.
    """
    if compute_derivative is None:
        from sktime.distances._ddtw_numba import average_of_slope

        compute_derivative = average_of_slope

    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "compute_derivative": compute_derivative,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="ddtw", **format_kwargs)


def dtw_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    **kwargs: Any,
) -> float:
    r"""Compute the dynamic time warping (DTW) distance between two time series.

    Originally proposed in [1]_, DTW is an elastic
    distance measure, i.e., it is a distance computed after realigning (warping)
    two time series to best match each other via time axis distortions [2]_.

    This function computes time warping distances only for:
    * sequences, time index is ignored
    * two time series of equal length
    * the Euclidean pairwise distance

    For unequal length time series, use ``sktime.dists_kernels.DistFromAligner``
    with a time warping aligner such as ``sktime.aligners.AlignerDTW``.
    To use arbitrary pairwise distances, use ``sktime.aligners.AlignerDTWfromDist``.

    Mathematically, for two sequences
    :math:'\mathbf{a}=\{a_1,a_2,\ldots,a_m\}' and :math:'\mathbf{b}=\{b_1,b_2,\ldots,
    b_n\}',  (assumed equal length for simplicity), DTW first calculates
    the pairwise distance matrix :math:'M(
    \mathbf{a},\mathbf{b})', the :math:'m \times n',
    between series :math:'\mathbf{a}' and :math:'\mathbf{b}',
    where :math:'M_{i,j} = d(a_i, b_j)', for a chosen distance measure
    :math:`d: \mathbb{R}^h \times \mathbb{R}^h \rightarrow \mathbb{R}`.
    In this estimator, the squared Euclidean distance is used, i.e.,
    :math:`d(x, y):= (x-y)^2`. A warping path
    .. math::  P=((i_1, j_1), (i_2, j_2), \ldots, (i_s, j_s))
    is an ordered tuple of indices
    :math:`i_k \in \{1, \dots, m\}, j_k \in \{1, \dots, n\}`
    which define a traversal path of matrix :math:'M'.
    This implementation assumes for warping paths that:
    * closed paths: :math:`i_1 = j_1 = 1`; :math:`i_s = m, j_s = n`
    * monotonous paths: :math:`i_k \le i_{k+1}, j_k \le j_{k+1}` for all :math:`k`
    * strictly monotonous paths: :math:`(i_k, j_k) \neq (i_{k+1}, j_{k+1})` for all
    :math:`k`
    The DTW path between sequences is the path through :math:'M' that minimizes the total
    distance,
    over all valid paths (satisfying the above assumptions), given the sequences.
    Formally:
    The distance for a warping path :math:'P' of length :math:'s' is
    .. math::  D_P(\mathbf{a},\mathbf{b}) = \sum_{k=1}^s M_{i_k,j_k}.
    If :math:'\mathcal{P}' is the set of all possible paths, the DTW path :math:'P^*'
    is the path that has the minimum distance amongst those:
    .. math::  P^* = \argmin_{P\in \mathcal{P}} D_{P}(\mathbf{a},\mathbf{b}).
    The DTW distance between the two sequences :math:'\mathbf{a},\mathbf{b}' is
    the minimum warping path distance:
    .. math:: d_{dtw}(\mathbf{a}, \mathbf{b}) = \min_{P\in \mathcal{P}} D_{P}(\mathbf{a},\mathbf{b}) =
    D_{P^*}(\mathbf{a},\mathbf{b}).
    The optimal warping path $P^*$ can be found exactly through dynamic programming.
    This can be a time consuming operation, and it is common to put a
    restriction on the amount of warping allowed. This is implemented through
    the ``bounding_matrix`` structure, that restricts allowable warpings by a mask.
    Common bounding strategies include the Sakoe-Chiba band [3]_ and the Itakura
    parallelogram [4_]. The Sakoe-Chiba band creates a warping path window that has
    the same width along the diagonal of :math:'M'. The Itakura paralleogram allows
    for less warping at the start or end of the sequence than in the middle.

    If the function is called with multivariate time series, note that
    the matrix :math:'M' is computed with the multivariate squared Euclidean distance,
    :math:`d(x, y):= (x-y)^2` = \sum_{i=1}^h (x_i - y_i)^2`
    This is sometimes called the "dependent" version of DTW, DTW_D, see [5]_.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Dtw distance between x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import dtw_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> dtw_distance(x_1d, y_1d)
    58.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> dtw_distance(x_2d, y_2d)
    512.0

    References
    ----------
    .. [1] H. Sakoe, S. Chiba, "Dynamic programming algorithm optimization for
           spoken word recognition," IEEE Transactions on Acoustics, Speech and
           Signal Processing, vol. 26(1), pp. 43--49, 1978.
    .. [2] Ratanamahatana C and Keogh E.: Three myths about dynamic time warping data
    mining Proceedings of 5th SIAM International Conference on Data Mining, 2005
    .. [3] Sakoe H. and Chiba S.: Dynamic programming algorithm optimization for
    spoken word recognition. IEEE Transactions on Acoustics, Speech, and Signal
    Processing 26(1):43–49, 1978.
    .. [4] Itakura F: Minimum prediction residual principle applied to speech
    recognition. IEEE Transactions on Acoustics, Speech, and Signal Processing 23(
    1):67–72, 1975.
    .. [5] Shokoohi-Yekta M et al.: Generalizing DTW to the multi-dimensional case
    requires an adaptive approach. Data Mining and Knowledge Discovery, 31, 1–31 (2017).
    """  # noqa: E501
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="dtw", **format_kwargs)


def msm_distance(
    x: np.ndarray,
    y: np.ndarray,
    c: float = 1.0,
    window: float = None,
    itakura_max_slope: float = None,
    bounding_matrix: np.ndarray = None,
    **kwargs: dict,
) -> float:
    """Compute the move-split-merge distance.

    This metric uses as building blocks three fundamental operations: Move, Split,
    and Merge. A Move operation changes the value of a single element, a Split
    operation converts a single element into two consecutive elements, and a Merge
    operation merges two consecutive elements into one. Each operation has an
    associated cost, and the MSM distance between two time series is defined to be
    the cost of the cheapest sequence of operations that transforms the first time
    series into the second one.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    c: float, default = 1.0
        Cost for split or merge operation.
    window: Float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Must be between 0 and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Must be between 0 and 1.
    bounding_matrix: np.ndarray (2d array of shape (m1,m2)), defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should
        be infinity.
    kwargs: any
        extra kwargs.

    Returns
    -------
    float
        Msm distance between x and y.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
    References
    ----------
    .. [1]A.  Stefan,  V.  Athitsos,  and  G.  Das.   The  Move-Split-Merge  metric
    for time  series. IEEE  Transactions  on  Knowledge  and  Data  Engineering,
    25(6):1425–1438, 2013.
    """
    format_kwargs = {
        "c": c,
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="msm", **format_kwargs)


def twe_distance(
    x: np.ndarray,
    y: np.ndarray,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    lmbda: float = 1.0,
    nu: float = 0.001,
    p: int = 2,
    **kwargs: Any,
) -> float:
    """Time Warp Edit (TWE) distance between two time series.

    The Time Warp Edit (TWE) distance is a distance measure for discrete time series
    matching with time 'elasticity'. In comparison to other distance measures, (e.g.
    DTW (Dynamic Time Warping) or LCS (Longest Common Subsequence Problem)), TWE is a
    metric. Its computational time complexity is O(n^2), but can be drastically reduced
    in some specific situation by using a corridor to reduce the search space. Its
    memory space complexity can be reduced to O(n). It was first proposed in [1].

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    lmbda: float, defaults = 1.0
        A constant penalty that punishes the editing efforts. Must be >= 1.0.
    nu: float, defaults = 0.001
        A non-negative constant which characterizes the stiffness of the elastic
        twe measure. Must be > 0.
    p: int, defaults = 2
        Order of the p-norm for local cost.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    float
        Dtw distance between x and y.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import twe_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> twe_distance(x_1d, y_1d)
    28.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> twe_distance(x_2d, y_2d)
    78.37353236814714

    References
    ----------
    .. [1] Marteau, P.; F. (2009). "Time Warp Edit Distance with Stiffness Adjustment
    for Time Series Matching". IEEE Transactions on Pattern Analysis and Machine
    Intelligence. 31 (2): 306–318.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "lmbda": lmbda,
        "nu": nu,
        "p": p,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance(x, y, metric="twe", **format_kwargs)


def squared_distance(x: np.ndarray, y: np.ndarray, **kwargs: Any) -> float:
    r"""Compute the squared distance between two time series.

    The squared distance between two time series is defined as:

    .. math::
        sd(x, y) = \sum_{i=1}^{n} (x_i - y_i)^2

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    kwargs: Any
        Extra kwargs. For squared there are none however, this is kept for
        consistency.

    Returns
    -------
    float
        Squared distance between x and y.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import squared_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> squared_distance(x_1d, y_1d)
    64.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> squared_distance(x_2d, y_2d)
    512.0
    """
    return distance(x, y, metric="squared", **kwargs)


def euclidean_distance(x: np.ndarray, y: np.ndarray, **kwargs: Any) -> float:
    r"""Compute the Euclidean distance between two time series.

    Euclidean distance is supported for 1d, 2d and 3d arrays.

    The Euclidean distance between two time series of length m is the square root of
    the squared distance and is defined as:

    .. math::
        ed(x, y) = \sqrt{\sum_{i=1}^{n} (x_i - y_i)^2}

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    kwargs: Any
        Extra kwargs. For euclidean there are none however, this is kept for
        consistency.

    Returns
    -------
    float
        Euclidean distance between x and y.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import euclidean_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> euclidean_distance(x_1d, y_1d)
    8.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> euclidean_distance(x_2d, y_2d)
    22.627416997969522
    """
    return distance(x, y, metric="euclidean", **kwargs)


def dtw_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    **kwargs: Any,
) -> AlignmentPathReturn:
    r"""Compute the dynamic time warping (DTW) alignment path.

    Originally proposed in [1]_ DTW computes the distance between two time series by
    considering their alignments during the calculation. This is done by measuring
    the pointwise distance (normally using Euclidean) between all elements of the two
    time series and then using dynamic programming to find the warping path
    that minimises the total pointwise distance between realigned series.

    Mathematically dtw can be defined as:

    .. math::
        dtw(x, y) = \sqrt{\sum_{(i, j) \in \pi} \|x_{i} - y_{j}\|^2}

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the dtw alignment path.
    float
        Dtw distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] H. Sakoe, S. Chiba, "Dynamic programming algorithm optimization for
           spoken word recognition," IEEE Transactions on Acoustics, Speech and
           Signal Processing, vol. 26(1), pp. 43--49, 1978.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="dtw", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def wdtw_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    g: float = 0.05,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Compute the weighted dynamic time warping (wdtw) alignment path.

    First proposed in [1]_, WDTW adds a  adds a multiplicative weight penalty based on
    the warping distance. This means that time series with lower phase difference have
    a smaller weight imposed (i.e less penalty imposed) and time series with larger
    phase difference have a larger weight imposed (i.e. larger penalty imposed).

    Formally this can be described as:

    .. math::
        d_{w}(x_{i}, y_{j}) = ||w_{|i-j|}(x_{i} - y_{j})||

    Where d_w is the distance with a the weight applied to it for points i, j, where
    w(|i-j|) is a positive weight between the two points x_i and y_j.


    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    g: float, defaults = 0.
        Constant that controls the curvature (slope) of the function; that is, g
        controls the level of penalisation for the points with larger phase
        difference.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the wdtw alignment path.
    float
        Wdtw distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Young-Seon Jeong, Myong K. Jeong, Olufemi A. Omitaomu, Weighted dynamic time
    warping for time series classification, Pattern Recognition, Volume 44, Issue 9,
    2011, Pages 2231-2240, ISSN 0031-3203, https://doi.org/10.1016/j.patcog.2010.09.022.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="wdtw", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def ddtw_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: np.ndarray = None,
    compute_derivative=None,
    **kwargs: Any,
) -> AlignmentPathReturn:
    r"""Compute the derivative dynamic time warping (DDTW) alignment path.

    DDTW is an adaptation of DTW originally proposed in [1]_. DDTW attempts to
    improve on dtw by better account for the 'shape' of the time series.
    This is done by considering y axis data points as higher level features of 'shape'.
    To do this the first derivative of the sequence is taken, and then using this
    derived sequence a dtw computation is done.

    The default derivative used is:

    .. math::
        D_{x}[q] = \frac{{}(q_{i} - q_{i-1} + ((q_{i+1} - q_{i-1}/2)}{2}

    Where q is the original time series and d_q is the derived time series.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    compute_derivative: Callable[[np.ndarray], np.ndarray],
                            defaults = average slope difference
        Callable that computes the derivative. If none is provided the average of the
        slope between two points used.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the ddtw alignment path.
    float
        Ddtw distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric or compute derivative callable is not no_python compiled.
        If the metric type cannot be determined
        If the compute derivative callable is not no_python compiled.
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Keogh, Eamonn & Pazzani, Michael. (2002). Derivative Dynamic Time Warping.
        First SIAM International Conference on Data Mining.
        1. 10.1137/1.9781611972719.1.
    """
    if compute_derivative is None:
        from sktime.distances._ddtw_numba import average_of_slope

        compute_derivative = average_of_slope

    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "compute_derivative": compute_derivative,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="ddtw", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def wddtw_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    compute_derivative=None,
    g: float = 0.0,
    **kwargs: Any,
) -> AlignmentPathReturn:
    r"""Compute the weighted derivative dynamic time warping (WDDTW) alignment path.

    WDDTW was first proposed in [1]_ as an extension of DDTW. By adding a weight
    to the derivative it means the alignment isn't only considering the shape of the
    time series, but also the phase.

    Formally the derivative is calculated as:

    .. math::
        D_{x}[q] = \frac{{}(q_{i} - q_{i-1} + ((q_{i+1} - q_{i-1}/2)}{2}

    Therefore a weighted derivative can be calculated using D (the derivative) as:

    .. math::
        d_{w}(x_{i}, y_{j}) = ||w_{|i-j|}(D_{x_{i}} - D_{y_{j}})||

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    compute_derivative: Callable[[np.ndarray], np.ndarray],
                            defaults = average slope difference
        Callable that computes the derivative. If none is provided the average of the
        slope between two points used.
    g: float, defaults = 0.
        Constant that controls the curvature (slope) of the function; that is, g
        controls the level of penalisation for the points with larger phase
        difference.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the wddtw alignment path.
    float
        Wddtw distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If the compute derivative callable is not no_python compiled.
        If the value of g is not a float
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Young-Seon Jeong, Myong K. Jeong, Olufemi A. Omitaomu, Weighted dynamic time
    warping for time series classification, Pattern Recognition, Volume 44, Issue 9,
    2011, Pages 2231-2240, ISSN 0031-3203, https://doi.org/10.1016/j.patcog.2010.09.022.
    """
    if compute_derivative is None:
        from sktime.distances._ddtw_numba import average_of_slope

        compute_derivative = average_of_slope

    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "compute_derivative": compute_derivative,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="wddtw", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def edr_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    epsilon: float = None,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Compute the Edit distance for real sequences (EDR) alignment path.

    EDR computes the minimum number of elements (as a percentage) that must be removed
    from x and y so that the sum of the distance between the remaining signal elements
    lies within the tolerance (epsilon). EDR was originally proposed in [1]_.

    The value returned will be between 0 and 1 per time series. The value will
    represent as a percentage of elements that must be removed for the time series to
    be an exact match.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d array), defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    epsilon : float, defaults = None
        Matching threshold to determine if two subsequences are considered close
        enough to be considered 'common'. If not specified as per the original paper
        epsilon is set to a quarter of the maximum standard deviation.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the edr alignment path.
    float
        Edr distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.


    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 3 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Lei Chen, M. Tamer Özsu, and Vincent Oria. 2005. Robust and fast similarity
    search for moving object trajectories. In Proceedings of the 2005 ACM SIGMOD
    international conference on Management of data (SIGMOD '05). Association for
    Computing Machinery, New York, NY, USA, 491–502.
    DOI:https://doi.org/10.1145/1066157.1066213
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "epsilon": epsilon,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="edr", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def erp_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    g: float = 0.0,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Compute the Edit distance for real penalty (ERP) alignment path.

    ERP, first proposed in [1]_, attempts align time series
    by better considering how indexes are carried forward through the cost matrix.
    Usually in the dtw cost matrix, if an alignment can't be found the previous value
    is carried forward. Erp instead proposes the idea of gaps or sequences of points
    that have no matches. These gaps are then punished based on their distance from 'g'.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    g: float, defaults = 0.
        The reference value to penalise gaps.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the erp alignment path.
    float
        Erp distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 3 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If g is not a float.
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Lei Chen and Raymond Ng. 2004. On the marriage of Lp-norms and edit distance.
    In Proceedings of the Thirtieth international conference on Very large data bases
     - Volume 30 (VLDB '04). VLDB Endowment, 792–803.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "g": g,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="erp", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def lcss_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: Union[float, None] = None,
    itakura_max_slope: Union[float, None] = None,
    bounding_matrix: Union[np.ndarray, None] = None,
    epsilon: float = 1.0,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Compute the longest common subsequence (LCSS) alignment path.

    LCSS attempts to find the longest common sequence between two time series and
    returns a value that is the percentage that longest common sequence assumes.
    Originally present in [1]_, LCSS is computed by matching indexes that are
    similar up until a defined threshold (epsilon).

    The value returned will be between 0.0 and 1.0, where 0.0 means the two time series
    are exactly the same and 1.0 means they are complete opposites.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    epsilon : float, defaults = 1.
        Matching threshold to determine if two subsequences are considered close
        enough to be considered 'common'.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the lcss alignment path.
    float
        Lcss distance between x and y. The value returned will be between 0.0 and 1.0,
        where 0.0 means the two time series are exactly the same and 1.0 means they
        are complete opposites.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] M. Vlachos, D. Gunopoulos, and G. Kollios. 2002. "Discovering
        Similar Multidimensional Trajectories", In Proceedings of the
        18th International Conference on Data Engineering (ICDE '02).
        IEEE Computer Society, USA, 673.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "epsilon": epsilon,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="lcss", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def msm_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    c: float = 1.0,
    window: float = None,
    itakura_max_slope: float = None,
    bounding_matrix: np.ndarray = None,
    **kwargs: dict,
) -> AlignmentPathReturn:
    """Compute the move-split-merge alignment path.

    This metric uses as building blocks three fundamental operations: Move, Split,
    and Merge. A Move operation changes the value of a single element, a Split
    operation converts a single element into two consecutive elements, and a Merge
    operation merges two consecutive elements into one. Each operation has an
    associated cost, and the MSM distance between two time series is defined to be
    the cost of the cheapest sequence of operations that transforms the first time
    series into the second one.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    c: float, default = 1.0
        Cost for split or merge operation.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Must be between 0 and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Must be between 0 and 1.
    bounding_matrix: np.ndarray (2d array of shape (m1,m2)), defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should
        be infinity.
    kwargs: any
        extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the msm alignment path.
    float
        Msm distance between x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
    References
    ----------
    .. [1]A.  Stefan,  V.  Athitsos,  and  G.  Das.   The  Move-Split-Merge  metric
    for time  series. IEEE  Transactions  on  Knowledge  and  Data  Engineering,
    25(6):1425–1438, 2013.
    """
    format_kwargs = {
        "c": c,
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="msm", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def twe_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    return_cost_matrix: bool = False,
    window: float = None,
    itakura_max_slope: float = None,
    bounding_matrix: np.ndarray = None,
    lmbda: float = 1.0,
    nu: float = 0.001,
    p: int = 2,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Time Warp Edit (TWE) distance between two time series.

    The Time Warp Edit (TWE) distance is a distance measure for discrete time series
    matching with time 'elasticity'. In comparison to other distance measures, (e.g.
    DTW (Dynamic Time Warping) or LCS (Longest Common Subsequence Problem)), TWE is a
    metric. Its computational time complexity is O(n^2), but can be drastically reduced
    in some specific situation by using a corridor to reduce the search space. Its
    memory space complexity can be reduced to O(n). It was first proposed in [1].

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    window: float, defaults = None
        Float that is the radius of the sakoe chiba window (if using Sakoe-Chiba
        lower bounding). Value must be between 0. and 1.
    itakura_max_slope: float, defaults = None
        Gradient of the slope for itakura parallelogram (if using Itakura
        Parallelogram lower bounding). Value must be between 0. and 1.
    bounding_matrix: np.ndarray (2d of size mxn where m is len(x) and n is len(y)),
                                    defaults = None
        Custom bounding matrix to use. If defined then other lower_bounding params
        are ignored. The matrix should be structure so that indexes considered in
        bound should be the value 0. and indexes outside the bounding matrix should be
        infinity.
    lmbda: float, defaults = 1.0
        A constant penalty that punishes the editing efforts. Must be >= 1.0.
    nu: float, defaults = 0.001
        A non-negative constant which characterizes the stiffness of the elastic
        twe measure. Must be > 0.
    p: int, defaults = 2
        Order of the p-norm for local cost.
    kwargs: Any
        Extra kwargs.

    Returns
    -------
    list[tuple]
        List of tuples containing the twe alignment path.
    float
        Twe distance between x and y. The value returned will be between 0.0 and 1.0,
        where 0.0 means the two time series are exactly the same and 1.0 means they
        are complete opposites.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.

    Raises
    ------
    ValueError
        If the sakoe_chiba_window_radius is not a float.
        If the itakura_max_slope is not a float.
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined
        If both window and itakura_max_slope are set

    References
    ----------
    .. [1] Marteau, P.; F. (2009). "Time Warp Edit Distance with Stiffness Adjustment
    for Time Series Matching". IEEE Transactions on Pattern Analysis and Machine
    Intelligence. 31 (2): 306–318.
    """
    format_kwargs = {
        "window": window,
        "itakura_max_slope": itakura_max_slope,
        "bounding_matrix": bounding_matrix,
        "lmbda": lmbda,
        "nu": nu,
        "p": p,
    }
    format_kwargs = {**format_kwargs, **kwargs}

    return distance_alignment_path(
        x, y, metric="twe", return_cost_matrix=return_cost_matrix, **format_kwargs
    )


def distance(
    x: np.ndarray,
    y: np.ndarray,
    metric: Union[
        str,
        Callable[
            [np.ndarray, np.ndarray, dict], Callable[[np.ndarray, np.ndarray], float]
        ],
        Callable[[np.ndarray, np.ndarray], float],
        NumbaDistance,
    ],
    **kwargs: Any,
) -> float:
    """Compute the distance between two time series.

    First the distance metric is 'resolved'. This means the metric that is passed
    is resolved to a callable. The callable is then called with x and y and the
    value is then returned.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    metric: str or Callable
        The distance metric to use.
        If a string is given, the value must be one of the following strings:
        'euclidean', 'squared', 'dtw', 'ddtw', 'wdtw', 'wddtw', 'lcss', 'edr', 'erp',
        'msm'

        If callable then it has to be a distance factory or numba distance callable.
        If you want to pass custom kwargs to the distance at runtime, use a distance
        factory as it constructs the distance using the kwargs before distance
        computation.
        A distance callable takes the form (must be no_python compiled):
        Callable[[np.ndarray, np.ndarray], float]

        A distance factory takes the form (must return a no_python callable):
        Callable[[np.ndarray, np.ndarray, bool, dict], Callable[[np.ndarray,
        np.ndarray], float]].
    kwargs: Any
        Arguments for metric. Refer to each metrics documentation for a list of
        possible arguments.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> distance(x_1d, y_1d, metric='dtw')
    58.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> distance(x_2d, y_2d, metric='dtw')
    512.0

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> distance(x_2d, y_2d, metric='dtw', window=0.5)
    512.0

    Returns
    -------
    float
        Distance between the x and y.
    """
    from sktime.distances._numba_utils import to_numba_timeseries

    _x = to_numba_timeseries(x)
    _y = to_numba_timeseries(y)

    _metric_callable = _resolve_metric_to_factory(
        metric, _x, _y, _METRIC_INFOS, **kwargs
    )

    return _metric_callable(_x, _y)


def distance_factory(
    x: np.ndarray = None,
    y: np.ndarray = None,
    metric: Union[
        str,
        Callable[
            [np.ndarray, np.ndarray, dict], Callable[[np.ndarray, np.ndarray], float]
        ],
        Callable[[np.ndarray, np.ndarray], float],
        NumbaDistance,
    ] = "euclidean",
    **kwargs: Any,
) -> DistanceCallable:
    """Create a no_python distance callable.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array), defaults = None
        First time series.
    y: np.ndarray (1d or 2d array), defaults = None
        Second time series.
    metric: str or Callable, defaults  = 'euclidean'
        The distance metric to use.
        If a string is given, the value must be one of the following strings:
        'euclidean', 'squared', 'dtw', 'ddtw', 'wdtw', 'wddtw', 'lcss', 'edr', 'erp'

        If callable then it has to be a distance factory or numba distance callable.
        If you want to pass custom kwargs to the distance at runtime, use a distance
        factory as it constructs the distance using the kwargs before distance
        computation.
        A distance callable takes the form (must be no_python compiled):
        Callable[[np.ndarray, np.ndarray], float]

        A distance factory takes the form (must return a no_python callable):
        Callable[[np.ndarray, np.ndarray, bool, dict], Callable[[np.ndarray,
        np.ndarray], float]].
    kwargs: Any
        Arguments for metric. Refer to each metrics documentation for a list of
        possible arguments.

    Returns
    -------
    Callable[[np.ndarray, np.ndarray], float]]
        No_python compiled distance callable.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.
    """
    from sktime.distances._numba_utils import _numba_to_timeseries, to_numba_timeseries
    from sktime.utils.numba.njit import njit

    if x is None:
        x = np.zeros((1, 10))
    if y is None:
        y = np.zeros((1, 10))
    _x = to_numba_timeseries(x)
    _y = to_numba_timeseries(y)

    callable = _resolve_metric_to_factory(metric, _x, _y, _METRIC_INFOS, **kwargs)

    @njit(cache=True)
    def dist_callable(x: np.ndarray, y: np.ndarray):
        _x = _numba_to_timeseries(x)
        _y = _numba_to_timeseries(y)
        return callable(_x, _y)

    return dist_callable


def pairwise_distance(
    x: np.ndarray,
    y: np.ndarray = None,
    metric: Union[
        str,
        Callable[
            [np.ndarray, np.ndarray, dict], Callable[[np.ndarray, np.ndarray], float]
        ],
        Callable[[np.ndarray, np.ndarray], float],
        NumbaDistance,
    ] = "euclidean",
    **kwargs: Any,
) -> np.ndarray:
    """Compute the pairwise distance matrix between two time series.

    First the distance metric is 'resolved'. This means the metric that is passed
    is resolved to a callable. The callable is then called with x and y and the
    value is then returned. Then for each combination of x and y, the distance between
    the values are computed resulting in a 2d pairwise matrix.

    Parameters
    ----------
    x: np.ndarray (1d, 2d or 3d array)
        First time series.
    y: np.ndarray (1d, 2d or 3d array), defaults = None
        Second time series. If not specified then y is set to the value of x.
    metric: str or Callable, defaults = 'euclidean'
        The distance metric to use.
        If a string is given, the value must be one of the following strings:
            'euclidean', 'squared', 'dtw', 'ddtw', 'wdtw', 'wddtw',
            'lcss', 'edr', 'erp', 'msm'
        If callable then it has to be a distance factory or numba distance callable.
        If you want to pass custom kwargs to the distance at runtime, use a distance
        factory as it constructs the distance using the kwargs before distance
        computation.
        A distance callable takes the form (must be no_python compiled):
        Callable[[np.ndarray, np.ndarray], float]

        A distance factory takes the form (must return a no_python callable):
        Callable[[np.ndarray, np.ndarray, bool, dict], Callable[[np.ndarray,
        np.ndarray], float]].
    kwargs: Any
        Extra arguments for metric. Refer to each metric documentation for a list of
        possible arguments.

    Returns
    -------
    np.ndarray (2d of size mxn where m is len(x) and n is len(y)).
        Pairwise distance matrix between the two time series.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 3 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Examples
    --------
    >>> import numpy as np
    >>> from sktime.distances import pairwise_distance
    >>> x_1d = np.array([1, 2, 3, 4])  # 1d array
    >>> y_1d = np.array([5, 6, 7, 8])  # 1d array
    >>> pairwise_distance(x_1d, y_1d, metric='dtw')
    array([[16., 25., 36., 49.],
           [ 9., 16., 25., 36.],
           [ 4.,  9., 16., 25.],
           [ 1.,  4.,  9., 16.]])

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> pairwise_distance(x_2d, y_2d, metric='dtw')
    array([[256., 576.],
           [ 58., 256.]])

    >>> x_3d = np.array([[[1], [2], [3], [4]], [[5], [6], [7], [8]]])  # 3d array
    >>> y_3d = np.array([[[9], [10], [11], [12]], [[13], [14], [15], [16]]])  # 3d array
    >>> pairwise_distance(x_3d, y_3d, metric='dtw')
    array([[256., 576.],
           [ 64., 256.]])

    >>> x_2d = np.array([[1, 2, 3, 4], [5, 6, 7, 8]])  # 2d array
    >>> y_2d = np.array([[9, 10, 11, 12], [13, 14, 15, 16]])  # 2d array
    >>> pairwise_distance(x_2d, y_2d, metric='dtw', window=0.5)
    array([[256., 576.],
           [ 58., 256.]])
    """
    from sktime.distances._numba_utils import (
        _compute_pairwise_distance,
        _make_3d_series,
    )

    _x = _make_3d_series(x)
    if y is None:
        y = x
    _y = _make_3d_series(y)
    symmetric = np.array_equal(_x, _y)
    _metric_callable = _resolve_metric_to_factory(
        metric, _x[0], _y[0], _METRIC_INFOS, **kwargs
    )
    return _compute_pairwise_distance(_x, _y, symmetric, _metric_callable)


def distance_alignment_path(
    x: np.ndarray,
    y: np.ndarray,
    metric: Union[
        str,
        Callable[
            [np.ndarray, np.ndarray, dict], Callable[[np.ndarray, np.ndarray], float]
        ],
        Callable[[np.ndarray, np.ndarray], float],
        NumbaDistance,
    ],
    return_cost_matrix: bool = False,
    **kwargs: Any,
) -> AlignmentPathReturn:
    """Compute the alignment path and distance between two time series.

    First the distance metric is 'resolved'. This means the metric that is passed
    is resolved to a callable. The callable is then called with x and y and the
    value is then returned.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    metric: str or Callable
        The distance metric to use.
        If a string is given, the value must be one of the following strings:
        'euclidean', 'squared', 'dtw', 'ddtw', 'wdtw', 'wddtw', 'lcss', 'edr', 'erp',
        'msm'

        If callable then it has to be a distance factory or numba distance callable.
        If you want to pass custom kwargs to the distance at runtime, use a distance
        factory as it constructs the distance using the kwargs before distance
        computation.
        A distance callable takes the form (must be no_python compiled):
        Callable[[np.ndarray, np.ndarray], float]

        A distance factory takes the form (must return a no_python callable):
        Callable[[np.ndarray, np.ndarray, bool, dict], Callable[[np.ndarray,
        np.ndarray], float]].
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    kwargs: Any
        Arguments for metric. Refer to each metrics documentation for a list of
        possible arguments.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Returns
    -------
    list[tuple]
        List of tuples containing the alignment path for the distance.
    float
        Distance between the x and y.
    np.ndarray (of shape (len(x), len(y)).
        Optional return only given if return_cost_matrix = True.
        Cost matrix used to compute the distance.
    """
    from sktime.distances._numba_utils import to_numba_timeseries

    _x = to_numba_timeseries(x)
    _y = to_numba_timeseries(y)

    _dist_instance = _resolve_dist_instance(metric, _x, _y, _METRIC_INFOS, **kwargs)

    return _dist_instance.distance_alignment_path(
        _x, _y, return_cost_matrix=return_cost_matrix, **kwargs
    )


def distance_alignment_path_factory(
    x: np.ndarray,
    y: np.ndarray,
    metric: Union[
        str,
        Callable[
            [np.ndarray, np.ndarray, dict], Callable[[np.ndarray, np.ndarray], float]
        ],
        Callable[[np.ndarray, np.ndarray], float],
        NumbaDistance,
    ],
    return_cost_matrix: bool = False,
    **kwargs: Any,
) -> DistanceAlignmentPathCallable:
    """Produce a distance alignment path factory numba callable.

    First the distance metric is 'resolved'. This means the metric that is passed
    is resolved to callable. The callable is then called with x and y and the
    value is then returned.

    Parameters
    ----------
    x: np.ndarray (1d or 2d array)
        First time series.
    y: np.ndarray (1d or 2d array)
        Second time series.
    metric: str or Callable
        The distance metric to use.
        If a string is given, the value must be one of the following strings:
        'euclidean', 'squared', 'dtw', 'ddtw', 'wdtw', 'wddtw', 'lcss', 'edr', 'erp',
        'msm'

        If callable then it has to be a distance factory or numba distance callable.
        If you want to pass custom kwargs to the distance at runtime, use a distance
        factory as it constructs the distance using the kwargs before distance
        computation.
        A distance callable takes the form (must be no_python compiled):
        Callable[[np.ndarray, np.ndarray], float]

        A distance factory takes the form (must return a no_python callable):
        Callable[[np.ndarray, np.ndarray, bool, dict], Callable[[np.ndarray,
        np.ndarray], float]].
    return_cost_matrix: bool, defaults = False
        Boolean that when true will also return the cost matrix.
    kwargs: Any
        Arguments for metric. Refer to each metrics documentation for a list of
        possible arguments.

    Raises
    ------
    ValueError
        If the value of x or y provided is not a numpy array.
        If the value of x or y has more than 2 dimensions.
        If a metric string provided, and is not a defined valid string.
        If a metric object (instance of class) is provided and doesn't inherit from
        NumbaDistance.
        If a resolved metric is not no_python compiled.
        If the metric type cannot be determined.

    Returns
    -------
    Callable[[np.ndarray, np.ndarray], Union[np.ndarray, np.ndarray]]
        Callable for the distance path.
    """
    from sktime.distances._numba_utils import _numba_to_timeseries, to_numba_timeseries
    from sktime.utils.numba.njit import njit

    if x is None:
        x = np.zeros((1, 10))
    if y is None:
        y = np.zeros((1, 10))
    _x = to_numba_timeseries(x)
    _y = to_numba_timeseries(y)

    dist_instance = _resolve_dist_instance(metric, _x, _y, _METRIC_INFOS, **kwargs)
    callable = dist_instance.distance_alignment_path_factory(
        _x, _y, return_cost_matrix, **kwargs
    )

    @njit(cache=True)
    def dist_callable(x: np.ndarray, y: np.ndarray):
        _x = _numba_to_timeseries(x)
        _y = _numba_to_timeseries(y)
        return callable(_x, _y)

    return dist_callable


_METRIC_INFOS = [
    MetricInfo(
        canonical_name="euclidean",
        aka={"euclidean", "ed", "euclid", "pythagorean"},
        dist_func=euclidean_distance,
        dist_instance=_EuclideanDistance(),
    ),
    MetricInfo(
        canonical_name="erp",
        aka={"erp", "edit distance with real penalty"},
        dist_func=erp_distance,
        dist_instance=_ErpDistance(),
        dist_alignment_path_func=erp_alignment_path,
    ),
    MetricInfo(
        canonical_name="edr",
        aka={"edr", "edit distance for real sequences"},
        dist_func=edr_distance,
        dist_instance=_EdrDistance(),
        dist_alignment_path_func=edr_alignment_path,
    ),
    MetricInfo(
        canonical_name="lcss",
        aka={"lcss", "longest common subsequence"},
        dist_func=lcss_distance,
        dist_instance=_LcssDistance(),
        dist_alignment_path_func=lcss_alignment_path,
    ),
    MetricInfo(
        canonical_name="squared",
        aka={"squared"},
        dist_func=squared_distance,
        dist_instance=_SquaredDistance(),
    ),
    MetricInfo(
        canonical_name="dtw",
        aka={"dtw", "dynamic time warping"},
        dist_func=dtw_distance,
        dist_instance=_DtwDistance(),
        dist_alignment_path_func=dtw_alignment_path,
    ),
    MetricInfo(
        canonical_name="ddtw",
        aka={"ddtw", "derivative dynamic time warping"},
        dist_func=ddtw_distance,
        dist_instance=_DdtwDistance(),
        dist_alignment_path_func=ddtw_alignment_path,
    ),
    MetricInfo(
        canonical_name="wdtw",
        aka={"wdtw", "weighted dynamic time warping"},
        dist_func=wdtw_distance,
        dist_instance=_WdtwDistance(),
        dist_alignment_path_func=wdtw_alignment_path,
    ),
    MetricInfo(
        canonical_name="wddtw",
        aka={"wddtw", "weighted derivative dynamic time warping"},
        dist_func=wddtw_distance,
        dist_instance=_WddtwDistance(),
        dist_alignment_path_func=wddtw_alignment_path,
    ),
    MetricInfo(
        canonical_name="msm",
        aka={"msm", "move-split-merge"},
        dist_func=msm_distance,
        dist_instance=_MsmDistance(),
        dist_alignment_path_func=msm_alignment_path,
    ),
    MetricInfo(
        canonical_name="twe",
        aka={"twe", "time warped edit"},
        dist_func=twe_distance,
        dist_instance=_TweDistance(),
        dist_alignment_path_func=twe_alignment_path,
    ),
]

_METRICS = {info.canonical_name: info for info in _METRIC_INFOS}
_METRIC_ALIAS = {alias: info for info in _METRIC_INFOS for alias in info.aka}
_METRIC_CALLABLES = {info.canonical_name: info.dist_func for info in _METRIC_INFOS}
_METRICS_NAMES = list(_METRICS.keys())

ALL_DISTANCES = (
    ddtw_distance,
    dtw_distance,
    edr_distance,
    erp_distance,
    euclidean_distance,
    lcss_distance,
    msm_distance,
    squared_distance,
    wddtw_distance,
    wdtw_distance,
    twe_distance,
)