Repository: aeon-toolkit/aeon (python)

Issue:
[ENH] wrap/implement Global Alignment DTW
### Describe the feature or idea you want to propose

there is an interesting paper here, that describes an extension of DTW 
https://arxiv.org/pdf/2002.03848.pdf
described here
https://rtavenar.github.io/hdr/parts/01/dtw/dtw_gi.html


### Describe your proposed solution

It might already be in tslearn, in which case we can maybe wrap it, although might be better to port it in for numba reasons? 


### Describe alternatives you've considered, if relevant

_No response_

### Additional context

_No response_

Pull request description:
[ENH] Implement DTW with Global alignment
Fixes #444 

Added DTW with global invariance in aeon

Changes
- Implentation of DTW with GI added
- Uses Numba for JIT compilation

TODO:
- Add Tests

### Comparision of dtw_gi implentations:
first implementation is implented in aeon
second implementation taken from https://rtavenar.github.io/hdr/parts/01/dtw/dtw_gi.html

The cost of 2nd is square root of the 1st cost($18.95^2$ = 359.20) because `tslearn` uses the `Euclidean_distance` but `aeon` uses `squared Euclidean_distance` 
![image](https://github.com/user-attachments/assets/0c0255ef-9f09-4238-b00e-50c0756ae476)


##### For all contributions
- [ ] I've added myself to the [list of contributors](https://github.com/aeon-toolkit/aeon/blob/main/.all-contributorsrc). Alternatively, you can use the [@all-contributors](https://allcontributors.org/docs/en/bot/usage) bot to do this for you after the PR has been merged.
- [x] The PR title starts with either [ENH], [MNT], [DOC], [BUG], [REF], [DEP] or [GOV] indicating whether the PR topic is related to enhancement, maintenance, documentation, bugs, refactoring, deprecation or governance.

##### For new estimators and functions
- [ ] I've added the estimator/function to the online [API documentation](https://www.aeon-toolkit.org/en/latest/api_reference.html).
- [ ] (OPTIONAL) I've added myself as a `__maintainer__` at the top of relevant files and want to be contacted regarding its maintenance. Unmaintained files may be removed. This is for the full file, and you should not add yourself if you are just making minor changes or do not want to help maintain its contents.

##### For developers with write access
- [ ] (OPTIONAL) I've updated aeon's [CODEOWNERS](https://github.com/aeon-toolkit/aeon/blob/main/CODEOWNERS) to receive notifications about future changes to these files.

Patch:
diff --git a/aeon/distances/__init__.py b/aeon/distances/__init__.py
index e1d3205e..d6ff3f77 100644
--- a/aeon/distances/__init__.py
+++ b/aeon/distances/__init__.py
@@ -18,6 +18,10 @@ __all__ = [
     "dtw_pairwise_distance",
     "dtw_cost_matrix",
     "dtw_alignment_path",
+    "dtw_gi_distance",
+    "dtw_gi_pairwise_distance",
+    "dtw_gi_cost_matrix",
+    "dtw_gi_alignment_path",
     "ddtw_distance",
     "ddtw_pairwise_distance",
     "ddtw_alignment_path",
@@ -111,6 +115,10 @@ from aeon.distances.elastic import (
     dtw_alignment_path,
     dtw_cost_matrix,
     dtw_distance,
+    dtw_gi_alignment_path,
+    dtw_gi_cost_matrix,
+    dtw_gi_distance,
+    dtw_gi_pairwise_distance,
     dtw_pairwise_distance,
     edr_alignment_path,
     edr_cost_matrix,
diff --git a/aeon/distances/_distance.py b/aeon/distances/_distance.py
index 1263e11c..33f91414 100644
--- a/aeon/distances/_distance.py
+++ b/aeon/distances/_distance.py
@@ -24,6 +24,10 @@ from aeon.distances.elastic import (
     dtw_alignment_path,
     dtw_cost_matrix,
     dtw_distance,
+    dtw_gi_alignment_path,
+    dtw_gi_cost_matrix,
+    dtw_gi_distance,
+    dtw_gi_pairwise_distance,
     dtw_pairwise_distance,
     edr_alignment_path,
     edr_cost_matrix,
@@ -447,6 +451,7 @@ def get_distance_function(method: Union[str, DistanceFunction]) -> DistanceFunct
     method          Distance Function
     =============== ========================================
     'dtw'           distances.dtw_distance
+    'dtw_gi'        distances.dtw_gi_distance
     'shape_dtw'     distances.shape_dtw_distance
     'ddtw'          distances.ddtw_distance
     'wdtw'          distances.wdtw_distance
@@ -728,6 +733,16 @@ DISTANCES = [
         "symmetric": True,
         "unequal_support": True,
     },
+    {
+        "name": "dtw_gi",
+        "distance": dtw_gi_distance,
+        "pairwise_distance": dtw_gi_pairwise_distance,
+        "cost_matrix": dtw_gi_cost_matrix,
+        "alignment_path": dtw_gi_alignment_path,
+        "type": DistanceType.ELASTIC,
+        "symmetric": False,
+        "unequal_support": True,
+    },
     {
         "name": "ddtw",
         "distance": ddtw_distance,
diff --git a/aeon/distances/elastic/__init__.py b/aeon/distances/elastic/__init__.py
index 0c386245..8e5d1aa9 100644
--- a/aeon/distances/elastic/__init__.py
+++ b/aeon/distances/elastic/__init__.py
@@ -10,6 +10,10 @@ __all__ = [
     "dtw_pairwise_distance",
     "dtw_cost_matrix",
     "dtw_alignment_path",
+    "dtw_gi_distance",
+    "dtw_gi_pairwise_distance",
+    "dtw_gi_cost_matrix",
+    "dtw_gi_alignment_path",
     "ddtw_distance",
     "ddtw_pairwise_distance",
     "ddtw_alignment_path",
@@ -71,6 +75,12 @@ from aeon.distances.elastic._dtw import (
     dtw_distance,
     dtw_pairwise_distance,
 )
+from aeon.distances.elastic._dtw_gi import (
+    dtw_gi_alignment_path,
+    dtw_gi_cost_matrix,
+    dtw_gi_distance,
+    dtw_gi_pairwise_distance,
+)
 from aeon.distances.elastic._edr import (
     edr_alignment_path,
     edr_cost_matrix,
diff --git a/aeon/distances/elastic/_dtw_gi.py b/aeon/distances/elastic/_dtw_gi.py
new file mode 100644
index 00000000..0026c377
--- /dev/null
+++ b/aeon/distances/elastic/_dtw_gi.py
@@ -0,0 +1,551 @@
+r"""Dynamic time warping with Global Invariances (DTW-GI) between two time series."""
+
+__maintainer__ = []
+
+from typing import Optional, Union
+
+import numpy as np
+from numba import njit
+from numba.typed import List as NumbaList
+
+from aeon.distances.elastic._dtw import dtw_alignment_path, dtw_cost_matrix
+from aeon.utils.conversion._convert_collection import _convert_collection_to_numba_list
+from aeon.utils.validation.collection import _is_numpy_list_multivariate
+
+
+@njit(cache=True, fastmath=True)
+def _path2mat(
+    path: list[tuple[int, int]],
+    x_timepoints: int,
+    y_timepoints: int,
+):
+    r"""Convert a warping alignment path to a binary warping matrix."""
+    w = np.zeros((x_timepoints, y_timepoints))
+    for i, j in path:
+        w[i, j] = 1
+    return w
+
+
+@njit(cache=True, fastmath=True)
+def _dtw_gi(
+    x: np.ndarray,
+    y: np.ndarray,
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+):
+    r"""
+    Compute Dynamic Time Warping with Global Invariance between the two time series.
+
+    Parameters
+    ----------
+    x : np.ndarray
+        First time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    y : np.ndarray
+        Second time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    window : float or None, default=None
+        The window to use for the bounding matrix. If None, no bounding matrix
+        is used. window is a percentage deviation, so if ``window = 0.1`` then
+        10% of the series length is the max warping allowed.
+        is used.
+    itakura_max_slope : float, default=None
+        Maximum slope as a proportion of the number of time points used to create
+        Itakura parallelogram on the bounding matrix. Must be between 0. and 1.
+    init_p : array-like of shape (x_channels, y_channels), default=None
+        Initial linear transformation. If None, the identity matrix is used.
+    max_iter : int, default=20
+        Maximum number of iterations for the iterative optimization.
+    use_bias : bool, default=False
+        If True, the feature space map is affine (with a bias term).
+
+    Returns
+    -------
+    - w_pi: binary warping matrix of shape (n0, n1)
+    - p: the final linear (Stiefel) matrix of shape (x_channels, y_channels)
+    - cost: final DTW cost considering global invariances
+
+    If use_bias is True, also returns:
+      - bias
+
+    """
+    if x.ndim == 1 and y.ndim == 1:
+        x_ = x.reshape((1, x.shape[0]))
+        y_ = y.reshape((1, y.shape[0]))
+    elif x.ndim == 2 and y.ndim == 2:
+        x_ = x
+        y_ = y
+    else:
+        raise ValueError("x and y must be 1D or 2D")
+
+    x_channels, x_timepoints = x_.shape
+    y_channels, y_timepoints = y_.shape
+
+    x_m = np.sum(x_, axis=1) / x_.shape[1]
+    x_m = x_m.reshape((-1, 1))
+    y_m = np.sum(y_, axis=1) / y_.shape[1]
+    y_m = y_m.reshape((-1, 1))
+
+    w_pi = np.zeros((x_timepoints, y_timepoints))
+    if init_p is None:
+        p = np.eye(x_channels, y_channels, dtype=np.float64)
+    else:
+        p = init_p
+
+    y_ = y_.astype(np.float64)
+    x_ = x_.astype(np.float64)
+
+    bias = np.zeros((x_channels, 1))
+
+    for _ in range(max_iter):
+        w_pi_old = w_pi.copy()
+        y_transformed = p.dot(y_) + bias
+
+        path, cost = dtw_alignment_path(x_, y_transformed, window, itakura_max_slope)
+        w_pi = _path2mat(path, x_timepoints, y_timepoints)
+
+        if np.allclose(w_pi, w_pi_old):
+            break
+
+        if use_bias:
+            m = (x_ - x_m).dot(w_pi).dot((y_ - y_m).T)
+        else:
+            m = x_.dot(w_pi).dot(y_.T)
+
+        u, sigma, vt = np.linalg.svd(m, full_matrices=False)
+        p = u.dot(vt)
+        if use_bias:
+            bias = x_m - p.dot(y_m)
+
+    y_trans = p.dot(y_) + bias
+    path, cost = dtw_alignment_path(x_, y_trans, window, itakura_max_slope)
+
+    if use_bias:
+        return w_pi, p, bias, cost, x_, y_trans
+    else:
+        dummy_bias = np.zeros((x_channels, 1), dtype=np.float64)
+        return w_pi, p, dummy_bias, cost, x_, y_trans
+
+
+@njit(cache=True, fastmath=True)
+def dtw_gi_distance(
+    x: np.ndarray,
+    y: np.ndarray,
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> float:
+    r"""Compute the DTW_GI distance between two time series x and y.
+
+    The DTW_gi distance between 2 timeseries x and y is the similarity
+    measure that estimates both temporal alignment and does feature space
+    transformation between time series simultaneously. This means that the
+    time series do not have to lie in the same ambient space.
+    A good background into DTW with global invariances can be found in [1]_.
+    This implementation is inspired by [2]_.
+
+    For example, if we have two time series x and y of lengths n and m
+    respectively, and we assume that the time series do not lie in the
+    same ambient space. Lets assume that features of x lie in :math:`\mathbb{R}^p`
+    and features of y lie in :math:`\mathbb{R}^q`. To compare the two time series,
+    we need to find an optimum mapping from the feature space of y to the feature space
+    where features of x lie. So think of it as optimizing on a family of functions F
+    that map features from y onto the feature space in which features of x
+    lie. (This is just one way to do it, the mapping can be
+    in the opposite direction as well. But this function assumes the former way).
+
+    More formally, we define Dynamic Time Warping with Global Invariances (DTW-GI)
+    as the solution of the following joint optimization problem:
+
+    :math:`\text{DTW-GI}(\mathbf{x}, \mathbf{y}) =
+    \min_{f \in \mathcal{F}, \pi \in \mathcal{A}(\mathbf{x}, \mathbf{y})}
+    \sqrt{\sum_{(i,j) \in \pi} d(x_i, f(y_j))^2},`
+
+    This similarity measure estimates temporal alignment
+    with feature space transformation between time series
+    allowing the alignment of time series that do not exist in the
+    same ambient space.
+
+    Note: The optimal warping path :math:`P^*` for a given cost matrix can be found
+    exactly through a dynamic programming formulation. 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 supplies a mask for
+    allowable warpings. The most common bounding strategies include the
+    Sakoe-Chiba band [3]_. The width of the allowed warping is controlled through the
+     ``window`` parameter which sets the maximum proportion of warping allowed.
+
+    Parameters
+    ----------
+    x : np.ndarray
+        First time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    y : np.ndarray
+        Second time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    window : float, default=None
+        The window to use for the bounding matrix. If None, no bounding matrix
+        is used. window is a percentage deviation, so if ``window = 0.1``,
+        10% of the series length is the max warping allowed.
+        is used.
+    itakura_max_slope : float, default=None
+        Maximum slope as a proportion of the number of time points used to create
+        Itakura parallelogram on the bounding matrix. Must be between 0. and 1.
+    init_p : array-like of shape (x_channels, y_channels), default=None
+        Initial linear transformation. If None, the identity matrix is used.
+    max_iter : int, default=20
+        Maximum number of iterations for the iterative optimization.
+    use_bias : bool, default=False
+        If True, the feature space map is affine (with a bias term).
+
+    Returns
+    -------
+    float
+        DTW_GI distance between x and y, minimum value 0.
+
+    Raises
+    ------
+    ValueError
+        If x and y are not 1D or 2D arrays.
+
+    References
+    ----------
+    .. [1] T. Vayer, R. Tavenard, L. Chapel, N. Courty, R. Flamary, and Y. Soullard,
+    “Time Series Alignment with Global Invariances,” arXiv.org, 2020.
+    https://arxiv.org/abs/2002.03848
+
+    .. [2] Romain Tavenard, “DTW with Global Invariances,” Github.io, Dec. 17, 2020.
+    https://rtavenar.github.io/hdr/parts/01/dtw/dtw_gi.html
+
+    .. [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.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from aeon.distances import dtw_gi_distance
+    >>> x = np.array([1, 2, 3, 4, 5, 6, 7, 8, 9, 10])
+    >>> y = np.array([11, 12, 13, 14, 15, 16, 17, 18, 19, 20])
+    >>> dtw_gi_distance(x, y) # 1D series
+    768.0
+    >>> x = np.array([[1, 2, 3, 4, 5], [6, 7, 8, 9, 10], [0, 1, 0, 2, 0]])
+    >>> y = np.array([[11, 12, 13, 14],[7, 8, 9, 20],[1, 3, 4, 5]] )
+    >>> round(dtw_gi_distance(x, y), 1) # 2D series with 3 channels, unequal length
+    359.2
+    """
+    if x.ndim == 1 and y.ndim == 1:
+        _x = x.reshape((1, x.shape[0]))
+        _y = y.reshape((1, y.shape[0]))
+        return _dtw_gi(_x, _y, window, itakura_max_slope, init_p, max_iter, use_bias)[3]
+    if x.ndim == 2 and y.ndim == 2:
+        return _dtw_gi(x, y, window, itakura_max_slope, init_p, max_iter, use_bias)[3]
+    raise ValueError("x and y must be 1D or 2D")
+
+
+@njit(cache=True, fastmath=True)
+def dtw_gi_cost_matrix(
+    x: np.ndarray,
+    y: np.ndarray,
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> np.ndarray:
+    r"""Compute the DTW_GI cost matrix between two time series.
+
+    The cost matrix is the pairwise Euclidean distance between all points
+    :math:`M_{i,j}=(x_i-y_{\text{trans},j})^2`. Where `y_trans` is the time
+    series we get by finding the optimal mapping from feature space of y
+    to feature space where features of x lie. It is used in the DTW GI
+    path calculations.
+
+    Parameters
+    ----------
+    x : np.ndarray
+        First time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    y : np.ndarray
+        Second time series, either univariate, shape ``(n_timepoints,)``, or
+        multivariate, shape ``(n_channels, n_timepoints)``.
+    window : float, default=None
+        The window to use for the bounding matrix. If None, no bounding matrix
+        is used. window is a percentage deviation, so if ``window = 0.1``,
+        10% of the series length is the max warping allowed.
+        is used.
+    itakura_max_slope : float, default=None
+        Maximum slope as a proportion of the number of time points used to create
+        Itakura parallelogram on the bounding matrix. Must be between 0. and 1.
+    init_p : array-like of shape (x_channels, y_channels), default=None
+        Initial linear transformation. If None, the identity matrix is used.
+    max_iter : int, default=20
+        Maximum number of iterations for the iterative optimization.
+    use_bias : bool, default=False
+        If True, the feature space map is affine (with a bias term).
+
+    Returns
+    -------
+    np.ndarray (n_timepoints, m_timepoints)
+        dtw gi cost matrix between x and y.
+
+    Raises
+    ------
+    ValueError
+        If x and y are not 1D or 2D arrays.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from aeon.distances import dtw_gi_cost_matrix
+    >>> x = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]])
+    >>> y = np.array([[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]])
+    >>> dtw_gi_cost_matrix(x, y)
+    array([[  0.,   1.,   5.,  14.,  30.,  55.,  91., 140., 204., 285.],
+           [  1.,   0.,   1.,   5.,  14.,  30.,  55.,  91., 140., 204.],
+           [  5.,   1.,   0.,   1.,   5.,  14.,  30.,  55.,  91., 140.],
+           [ 14.,   5.,   1.,   0.,   1.,   5.,  14.,  30.,  55.,  91.],
+           [ 30.,  14.,   5.,   1.,   0.,   1.,   5.,  14.,  30.,  55.],
+           [ 55.,  30.,  14.,   5.,   1.,   0.,   1.,   5.,  14.,  30.],
+           [ 91.,  55.,  30.,  14.,   5.,   1.,   0.,   1.,   5.,  14.],
+           [140.,  91.,  55.,  30.,  14.,   5.,   1.,   0.,   1.,   5.],
+           [204., 140.,  91.,  55.,  30.,  14.,   5.,   1.,   0.,   1.],
+           [285., 204., 140.,  91.,  55.,  30.,  14.,   5.,   1.,   0.]])
+    """
+    _, _, _, _, xnew, y_trans = _dtw_gi(
+        x, y, window, itakura_max_slope, init_p, max_iter, use_bias
+    )
+
+    return dtw_cost_matrix(xnew, y_trans, window, itakura_max_slope)
+
+
+def dtw_gi_pairwise_distance(
+    X: Union[np.ndarray, list[np.ndarray]],
+    y: Optional[Union[np.ndarray, list[np.ndarray]]] = None,
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    unequal_length: bool = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> np.ndarray:
+    r"""Compute the DTW_GI pairwise distance between a set of time series.
+
+    By default, this takes a collection of :math:`n` time series :math:`X` and returns a
+    matrix
+    :math:`D` where :math:`D_{i,j}` is the DTW_GI distance between the :math:`i^{th}`
+    and the :math:`j^{th}` series in :math:`X`. If :math:`X` is 2 dimensional,
+    it is assumed to be a collection of univariate series with shape ``(n_cases,
+    n_timepoints)``. If it is 3 dimensional, it is assumed to be shape ``(n_cases,
+    n_channels, n_timepoints)``.
+
+    This function has an optional argument, :math:`y`, to allow calculation of the
+    distance matrix between :math:`X` and one or more series stored in :math:`y`. If
+    :math:`y` is 1 dimensional, we assume it is a single univariate series and the
+    distance matrix returned is shape ``(n_cases,1)``. If it is 2D, we assume it
+    is a collection of univariate series with shape ``(m_cases, m_timepoints)``
+    and the distance ``(n_cases,m_cases)``. If it is 3 dimensional,
+    it is assumed to be shape ``(m_cases, m_channels, m_timepoints)``.
+
+    Parameters
+    ----------
+    X : np.ndarray or List of np.ndarray
+        A collection of time series instances  of shape ``(n_cases, n_timepoints)``
+        or ``(n_cases, n_channels, n_timepoints)``.
+    y : np.ndarray or List of np.ndarray or None, default=None
+        A single series or a collection of time series of shape ``(m_timepoints,)`` or
+        ``(m_cases, m_timepoints)`` or ``(m_cases, m_channels, m_timepoints)``.
+        If None, then the dtw gi pairwise distance between the instances of X is
+        calculated.
+    window : float or None, default=None
+        The window to use for the bounding matrix. If None, no bounding matrix
+        is used.
+    itakura_max_slope : float, default=None
+        Maximum slope as a proportion of the number of time points used to create
+        Itakura parallelogram on the bounding matrix. Must be between 0. and 1.
+    init_p : array-like of shape (x_channels, y_channels), default=None
+        Initial linear transformation. If None, the identity matrix is used.
+    max_iter : int, default=20
+        Maximum number of iterations for the iterative optimization.
+    use_bias : bool, default=False
+        If True, the feature space map is affine (with a bias term).
+
+    Returns
+    -------
+    np.ndarray
+        DTW_GI pairwise matrix between the instances of X of shape
+        ``(n_cases, n_cases)`` or between X and y of shape ``(n_cases,
+        n_cases)``.
+
+    Raises
+    ------
+    ValueError
+        If X is not 2D or 3D array and if y is not 1D, 2D or 3D arrays when passing y.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from aeon.distances import dtw_gi_pairwise_distance
+    >>> # Distance between each time series in a collection of time series
+    >>> X = np.array([[[1, 2, 3]],[[4, 5, 6]], [[7, 8, 9]]])
+    >>> dtw_gi_pairwise_distance(X)
+    array([[  0.,  26., 108.],
+           [ 26.,   0.,  26.],
+           [108.,  26.,   0.]])
+
+    >>> # Distance between two collections of time series
+    >>> X = np.array([[[1, 2, 3]],[[4, 5, 6]], [[7, 8, 9]]])
+    >>> y = np.array([[[11, 12, 13]],[[14, 15, 16]], [[17, 18, 19]]])
+    >>> dtw_gi_pairwise_distance(X, y)
+    array([[300., 507., 768.],
+           [147., 300., 507.],
+           [ 48., 147., 300.]])
+
+    >>> X = np.array([[[1, 2, 3]],[[4, 5, 6]], [[7, 8, 9]]])
+    >>> y_univariate = np.array([11, 12, 13])
+    >>> dtw_gi_pairwise_distance(X, y_univariate)
+    array([[300.],
+           [147.],
+           [ 48.]])
+
+    >>> # Distance between each TS in a collection of unequal-length time series
+    >>> X = [np.array([1, 2, 3]), np.array([4, 5, 6, 7]), np.array([8, 9, 10, 11, 12])]
+    >>> dtw_gi_pairwise_distance(X)
+    array([[  0.,  42., 292.],
+           [ 42.,   0.,  83.],
+           [292.,  83.,   0.]])
+    """
+    multivariate_conversion = _is_numpy_list_multivariate(X, y)
+    _X, unequal_length = _convert_collection_to_numba_list(
+        X, "X", multivariate_conversion
+    )
+
+    if y is None:
+        # To self
+        return _dtw_gi_pairwise_distance(
+            _X, window, itakura_max_slope, unequal_length, init_p, max_iter, use_bias
+        )
+    _y, unequal_length = _convert_collection_to_numba_list(
+        y, "y", multivariate_conversion
+    )
+    return _dtw_gi_from_multiple_to_multiple_distance(
+        _X, _y, window, itakura_max_slope, unequal_length, init_p, max_iter, use_bias
+    )
+
+
+@njit(cache=True, fastmath=True)
+def _dtw_gi_from_multiple_to_multiple_distance(
+    x: NumbaList[np.ndarray],
+    y: NumbaList[np.ndarray],
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    unequal_length: bool = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> np.ndarray:
+    n_cases = len(x)
+    m_cases = len(y)
+    distances = np.zeros((n_cases, m_cases))
+
+    for i in range(n_cases):
+        for j in range(m_cases):
+            x1, y1 = x[i], y[j]
+            distances[i, j] = dtw_gi_distance(
+                x1, y1, window, itakura_max_slope, init_p, max_iter, use_bias
+            )
+    return distances
+
+
+@njit(cache=True, fastmath=True)
+def _dtw_gi_pairwise_distance(
+    X: NumbaList[np.ndarray],
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    unequal_length: bool = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> np.ndarray:
+    n_cases = len(X)
+    distances = np.zeros((n_cases, n_cases))
+
+    for i in range(n_cases):
+        for j in range(i + 1, n_cases):
+            x1, x2 = X[i], X[j]
+            distances[i, j] = dtw_gi_distance(
+                x1, x2, window, itakura_max_slope, init_p, max_iter, use_bias
+            )
+            distances[j, i] = distances[i, j]
+
+    return distances
+
+
+@njit(cache=True, fastmath=True)
+def dtw_gi_alignment_path(
+    x: np.ndarray,
+    y: np.ndarray,
+    window: Optional[float] = None,
+    itakura_max_slope: Optional[float] = None,
+    init_p: np.ndarray = None,
+    max_iter: int = 20,
+    use_bias: bool = False,
+) -> tuple[list[tuple[int, int]], float]:
+    """Compute the DTW_GI alignment path between two time series.
+
+    Parameters
+    ----------
+    x : np.ndarray
+        First time series, shape ``(n_channels, n_timepoints)`` or ``(n_timepoints,)``.
+    y : np.ndarray
+        Second time series, shape ``(m_channels, m_timepoints)`` or ``(m_timepoints,)``.
+    window : float, default=None
+        The window to use for the bounding matrix. If None, no bounding matrix
+        is used.
+    itakura_max_slope : float, default=None
+        Maximum slope as a proportion of the number of time points used to create
+        Itakura parallelogram on the bounding matrix. Must be between 0. and 1.
+    init_p : array-like of shape (x_channels, y_channels), default=None
+        Initial linear transformation. If None, the identity matrix is used.
+    max_iter : int, default=20
+        Maximum number of iterations for the iterative optimization.
+    use_bias : bool, default=False
+        If True, the feature space map is affine (with a bias term).
+
+    Returns
+    -------
+    List[Tuple[int, int]]
+        The alignment path between the two time series where each element is a tuple
+        of the index in x and the index in y that have the best alignment according
+        to the cost matrix.
+    float
+        The DTW_GI distance betweeen the two time series.
+
+    Raises
+    ------
+    ValueError
+        If x and y are not 1D or 2D arrays.
+
+    Examples
+    --------
+    >>> import numpy as np
+    >>> from aeon.distances import dtw_gi_alignment_path
+    >>> x = np.array([[1, 2, 3, 6]])
+    >>> y = np.array([[1, 2, 3, 4]])
+    >>> dtw_gi_alignment_path(x, y)
+    ([(0, 0), (1, 1), (2, 2), (3, 3)], 4.0)
+    """
+    w_pi, _, _, cost, _, _ = _dtw_gi(
+        x, y, window, itakura_max_slope, init_p, max_iter, use_bias
+    )
+    min_alignment_path = []
+    for i in range(len(w_pi)):
+        for j in range(len(w_pi[0])):
+            if w_pi[i, j] == 1:
+                min_alignment_path.append((i, j))
+
+    return min_alignment_path, cost


Test patch:
diff --git a/aeon/distances/elastic/tests/test_distance_correctness.py b/aeon/distances/elastic/tests/test_distance_correctness.py
index f33ca088..2a0c203b 100644
--- a/aeon/distances/elastic/tests/test_distance_correctness.py
+++ b/aeon/distances/elastic/tests/test_distance_correctness.py
@@ -10,6 +10,7 @@ from aeon.datasets import load_basic_motions, load_unit_test
 from aeon.distances import (
     ddtw_distance,
     dtw_distance,
+    dtw_gi_distance,
     edr_distance,
     erp_distance,
     euclidean_distance,
@@ -23,6 +24,7 @@ from aeon.distances import (
 
 distances = [
     "dtw",
+    "dtw_gi",
     "wdtw",
     "lcss",
     "msm",
@@ -36,6 +38,7 @@ distances = [
 
 distance_parameters = {
     "dtw": [0.0, 0.1, 1.0],  # window
+    "dtw_gi": [0.0, 0.1, 1.0],  # window
     "wdtw": [0.0, 0.1, 1.0],  # parameter g
     "wddtw": [0.0, 0.1, 1.0],  # parameter g
     "erp": [0.0, 0.1, 1.0],  # window
@@ -63,6 +66,7 @@ basic_motions_distances = {
     "euclidean": 27.51835240,
     "squared": 757.25971908652,
     "dtw": [757.259719, 330.834497, 330.834497],
+    "dtw_gi": [259.5333502342899, 310.10738471013804, 310.10738471013804],
     "wdtw": [165.41724, 3.308425, 0],
     "msm": [70.014828, 89.814828, 268.014828],
     "erp": [169.3715, 102.0979, 102.097904],
@@ -90,6 +94,8 @@ def test_multivariate_correctness():
     for j in range(0, 3):
         d = dtw_distance(case1, case2, window=distance_parameters["dtw"][j])
         assert_almost_equal(d, basic_motions_distances["dtw"][j], 4)
+        d = dtw_gi_distance(case1, case2, window=distance_parameters["dtw_gi"][j])
+        assert_almost_equal(d, basic_motions_distances["dtw_gi"][j], 4)
         d = wdtw_distance(case1, case2, g=distance_parameters["wdtw"][j])
         assert_almost_equal(d, basic_motions_distances["wdtw"][j], 4)
         d = lcss_distance(case1, case2, epsilon=distance_parameters["lcss"][j] / 50.0)
diff --git a/aeon/distances/tests/test_distances.py b/aeon/distances/tests/test_distances.py
index 3633c9ca..4efe396a 100644
--- a/aeon/distances/tests/test_distances.py
+++ b/aeon/distances/tests/test_distances.py
@@ -57,6 +57,7 @@ def _validate_distance_result(
     assert isinstance(dist_result_to_self, float)
 
     # If unequal length swap where x and y are to ensure it works both ways around
+
     if symmetric and original_x.shape[-1] != original_y.shape[-1] and check_xy_permuted:
         _validate_distance_result(
             original_y,
diff --git a/aeon/testing/expected_results/expected_distance_results.py b/aeon/testing/expected_results/expected_distance_results.py
index 8e7c6873..7126c5c6 100644
--- a/aeon/testing/expected_results/expected_distance_results.py
+++ b/aeon/testing/expected_results/expected_distance_results.py
@@ -41,6 +41,13 @@ _expected_distance_results = {
         4.0997661869195205,
         25.0,
     ],
+    "dtw_gi": [
+        0.344520787316184,
+        0.344520787316184,
+        0.2998607605839068,
+        5.893210968537887,
+        25.0,
+    ],
     "ddtw": [
         0.2963709096971962,
         0.2963709096971962,