dpnp_iface_statistics.py

# -*- coding: utf-8 -*-
# *****************************************************************************
# Copyright (c) 2016-2025, Intel Corporation
# All rights reserved.
#
# Redistribution and use in source and binary forms, with or without
# modification, are permitted provided that the following conditions are met:
# - Redistributions of source code must retain the above copyright notice,
#   this list of conditions and the following disclaimer.
# - Redistributions in binary form must reproduce the above copyright notice,
#   this list of conditions and the following disclaimer in the documentation
#   and/or other materials provided with the distribution.
#
# THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS"
# AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE
# IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE
# ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT HOLDER OR CONTRIBUTORS BE
# LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR
# CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF
# SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS
# INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN
# CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE)
# ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF
# THE POSSIBILITY OF SUCH DAMAGE.
# *****************************************************************************

"""
Interface of the statistics function of the DPNP

Notes
-----
This module is a face or public interface file for the library
it contains:
 - Interface functions
 - documentation for the functions
 - The functions parameters check

"""

import math

import dpctl.tensor as dpt
import dpctl.tensor._tensor_elementwise_impl as ti
import dpctl.utils as dpu
import numpy
from dpctl.tensor._numpy_helper import normalize_axis_index

import dpnp

# pylint: disable=no-name-in-module
import dpnp.backend.extensions.statistics._statistics_impl as statistics_ext
from dpnp.dpnp_utils.dpnp_utils_common import (
    result_type_for_device,
    to_supported_dtypes,
)

from .dpnp_utils import get_usm_allocations
from .dpnp_utils.dpnp_utils_reduction import dpnp_wrap_reduction_call
from .dpnp_utils.dpnp_utils_statistics import dpnp_cov, dpnp_median

__all__ = [
    "amax",
    "amin",
    "average",
    "corrcoef",
    "correlate",
    "cov",
    "max",
    "mean",
    "median",
    "min",
    "ptp",
    "std",
    "var",
]


def _count_reduce_items(arr, axis, where=True):
    """
    Calculates the number of items used in a reduction operation
    along the specified axis or axes.

    Parameters
    ----------
    arr : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        axis or axes along which the number of items used in a reduction
        operation must be counted. If a tuple of unique integers is given,
        the items are counted over multiple axes. If ``None``, the variance
        is computed over the entire array.

        Default: ``None``.

    Returns
    -------
    out : int
        The number of items should be used in a reduction operation.

    Limitations
    -----------
    Parameters `where` is only supported with its default value.

    """
    if where is True:
        # no boolean mask given, calculate items according to axis
        if axis is None:
            axis = tuple(range(arr.ndim))
        elif not isinstance(axis, tuple):
            axis = (axis,)
        items = 1
        for ax in axis:
            items *= arr.shape[normalize_axis_index(ax, arr.ndim)]
        items = dpnp.intp(items)
    else:  # pragma: no cover
        raise NotImplementedError(
            "where keyword argument is only supported with its default value."
        )
    return items


def _divide_by_scalar(a, v):
    """
    Divide input array by a scalar.

    The division is implemented through dedicated ``ti._divide_by_scalar``
    function which has a better performance comparing to standard ``divide``
    function, because there is no need to have internal call of ``asarray``
    for the denominator `v` and so it avoids allocating extra device memory.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    v : scalar
        The scalar denominator.

    Returns
    -------
    out : dpnp.ndarray
        An array containing the result of division.

    """

    usm_a = dpnp.get_usm_ndarray(a)
    queue = usm_a.sycl_queue
    _manager = dpu.SequentialOrderManager[queue]
    dep_evs = _manager.submitted_events

    # pylint: disable=protected-access
    ht_ev, div_ev = ti._divide_by_scalar(
        src=usm_a, scalar=v, dst=usm_a, sycl_queue=queue, depends=dep_evs
    )
    _manager.add_event_pair(ht_ev, div_ev)
    return a


def amax(a, axis=None, out=None, keepdims=False, initial=None, where=True):
    """
    Return the maximum of an array or maximum along an axis.

    `amax` is an alias of :obj:`dpnp.max`.

    See Also
    --------
    :obj:`dpnp.max` : alias of this function
    :obj:`dpnp.ndarray.max` : equivalent method

    """

    return max(
        a, axis=axis, out=out, keepdims=keepdims, initial=initial, where=where
    )


def amin(a, axis=None, out=None, keepdims=False, initial=None, where=True):
    """
    Return the minimum of an array or minimum along an axis.

    `amin` is an alias of :obj:`dpnp.min`.

    See Also
    --------
    :obj:`dpnp.min` : alias of this function
    :obj:`dpnp.ndarray.min` : equivalent method

    """

    return min(
        a, axis=axis, out=out, keepdims=keepdims, initial=initial, where=where
    )


def average(a, axis=None, weights=None, returned=False, *, keepdims=False):
    """
    Compute the weighted average along the specified axis.

    For full documentation refer to :obj:`numpy.average`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        Axis or axes along which the averages must be computed. If
        a tuple of unique integers, the averages are computed over multiple
        axes. If ``None``, the average is computed over the entire array.

        Default: ``None``.
    weights : {array_like}, optional
        An array of weights associated with the values in `a`. Each value in
        `a` contributes to the average according to its associated weight.
        The weights array can either be 1-D (in which case its length must be
        the size of `a` along the given axis) or of the same shape as `a`.
        If `weights=None`, then all data in `a` are assumed to have a
        weight equal to one. The 1-D calculation is::

            avg = sum(a * weights) / sum(weights)

        The only constraint on `weights` is that `sum(weights)` must not be 0.

        Default: ``None``.
    returned : {bool}, optional
        If ``True``, the tuple (`average`, `sum_of_weights`) is returned,
        otherwise only the average is returned. If `weights=None`,
        `sum_of_weights` is equivalent to the number of elements over which
        the average is taken.

        Default: ``False``.
    keepdims : {None, bool}, optional
        If ``True``, the reduced axes (dimensions) are included in the result
        as singleton dimensions, so that the returned array remains
        compatible with the input array according to Array Broadcasting
        rules. Otherwise, if ``False``, the reduced axes are not included in
        the returned array.

        Default: ``False``.

    Returns
    -------
    out, [sum_of_weights] : dpnp.ndarray, dpnp.ndarray
        Return the average along the specified axis. When `returned` is
        ``True``, return a tuple with the average as the first element and
        the sum of the weights as the second element. `sum_of_weights` is of
        the same type as `out`. The result dtype follows a general pattern.
        If `weights` is ``None``, the result dtype will be that of `a` , or
        default floating point data type for the device where input array `a`
        is allocated. Otherwise, if `weights` is not ``None`` and `a` is
        non-integral, the result type will be the type of lowest precision
        capable of representing values of both `a` and `weights`. If `a`
        happens to be integral, the previous rules still applies but the result
        dtype will at least be default floating point data type for the device
        where input array `a` is allocated.

    See Also
    --------
    :obj:`dpnp.mean` : Compute the arithmetic mean along the specified axis.
    :obj:`dpnp.sum` : Sum of array elements over a given axis.

    Examples
    --------
    >>> import dpnp as np
    >>> data = np.arange(1, 5)
    >>> data
    array([1, 2, 3, 4])
    >>> np.average(data)
    array(2.5)
    >>> np.average(np.arange(1, 11), weights=np.arange(10, 0, -1))
    array(4.0)

    >>> data = np.arange(6).reshape((3, 2))
    >>> data
    array([[0, 1],
        [2, 3],
        [4, 5]])
    >>> np.average(data, axis=1, weights=[1./4, 3./4])
    array([0.75, 2.75, 4.75])
    >>> np.average(data, weights=[1./4, 3./4])
    TypeError: Axis must be specified when shapes of a and weights differ.

    With ``keepdims=True``, the following result has shape (3, 1).

    >>> np.average(data, axis=1, keepdims=True)
    array([[0.5],
        [2.5],
        [4.5]])

    >>> a = np.ones(5, dtype=np.float64)
    >>> w = np.ones(5, dtype=np.complex64)
    >>> avg = np.average(a, weights=w)
    >>> print(avg.dtype)
    complex128

    """

    dpnp.check_supported_arrays_type(a)
    usm_type, exec_q = get_usm_allocations([a, weights])

    if weights is None:
        avg = dpnp.mean(a, axis=axis, keepdims=keepdims)
        scl = dpnp.asanyarray(
            avg.dtype.type(a.size / avg.size),
            usm_type=usm_type,
            sycl_queue=exec_q,
        )
    else:
        if not dpnp.is_supported_array_type(weights):
            weights = dpnp.asarray(
                weights, usm_type=usm_type, sycl_queue=exec_q
            )

        a_dtype = a.dtype
        if not dpnp.issubdtype(a_dtype, dpnp.inexact):
            default_dtype = dpnp.default_float_type(a.device)
            res_dtype = dpnp.result_type(a_dtype, weights.dtype, default_dtype)
        else:
            res_dtype = dpnp.result_type(a_dtype, weights.dtype)

        # Sanity checks
        wgt_shape = weights.shape
        a_shape = a.shape
        if a_shape != wgt_shape:
            if axis is None:
                raise TypeError(
                    "Axis must be specified when shapes of input array and "
                    "weights differ."
                )
            if weights.ndim != 1:
                raise TypeError(
                    "1D weights expected when shapes of input array and "
                    "weights differ."
                )
            if wgt_shape[0] != a_shape[axis]:
                raise ValueError(
                    "Length of weights not compatible with specified axis."
                )

            # setup weights to broadcast along axis
            weights = dpnp.broadcast_to(
                weights, (a.ndim - 1) * (1,) + wgt_shape
            )
            weights = weights.swapaxes(-1, axis)

        scl = weights.sum(axis=axis, dtype=res_dtype, keepdims=keepdims)
        if dpnp.any(scl == 0.0):
            raise ZeroDivisionError("Weights sum to zero, can't be normalized")

        avg = dpnp.multiply(a, weights).sum(
            axis=axis, dtype=res_dtype, keepdims=keepdims
        )
        avg /= scl

    if returned:
        if scl.shape != avg.shape:
            scl = dpnp.broadcast_to(scl, avg.shape).copy()
        return avg, scl
    return avg


def corrcoef(x, y=None, rowvar=True, *, dtype=None):
    """
    Return Pearson product-moment correlation coefficients.

    For full documentation refer to :obj:`numpy.corrcoef`.

    Parameters
    ----------
    x : {dpnp.ndarray, usm_ndarray}
        A 1-D or 2-D array containing multiple variables and observations.
        Each row of `x` represents a variable, and each column a single
        observation of all those variables. Also see `rowvar` below.
    y : {None, dpnp.ndarray, usm_ndarray}, optional
        An additional set of variables and observations. `y` has the same
        shape as `x`.

        Default: ``None``.
    rowvar : {bool}, optional
        If `rowvar` is ``True``, then each row represents a variable,
        with observations in the columns. Otherwise, the relationship
        is transposed: each column represents a variable, while the rows
        contain observations.

        Default: ``True``.
    dtype : {None, str, dtype object}, optional
        Data-type of the result.

        Default: ``None``.

    Returns
    -------
    R : {dpnp.ndarray}
        The correlation coefficient matrix of the variables.

    See Also
    --------
    :obj:`dpnp.cov` : Covariance matrix.

    Examples
    --------
    In this example we generate two random arrays, ``xarr`` and ``yarr``, and
    compute the row-wise and column-wise Pearson correlation coefficients,
    ``R``. Since `rowvar` is true by default, we first find the row-wise
    Pearson correlation coefficients between the variables of ``xarr``.

    >>> import dpnp as np
    >>> np.random.seed(123)
    >>> xarr = np.random.rand(3, 3).astype(np.float32)
    >>> xarr
    array([[7.2858386e-17, 2.2066992e-02, 3.9520904e-01],
           [4.8012391e-01, 5.9377134e-01, 4.5147297e-01],
           [9.0728188e-01, 9.9387854e-01, 5.8399546e-01]], dtype=float32)
    >>> R1 = np.corrcoef(xarr)
    >>> R1
    array([[ 0.99999994, -0.6173796 , -0.9685411 ],
           [-0.6173796 ,  1.        ,  0.7937219 ],
           [-0.9685411 ,  0.7937219 ,  0.9999999 ]], dtype=float32)

    If we add another set of variables and observations ``yarr``, we can
    compute the row-wise Pearson correlation coefficients between the
    variables in ``xarr`` and ``yarr``.

    >>> yarr = np.random.rand(3, 3).astype(np.float32)
    >>> yarr
    array([[0.17615308, 0.65354985, 0.15716429],
           [0.09373496, 0.2123185 , 0.84086883],
           [0.9011005 , 0.45206687, 0.00225109]], dtype=float32)
    >>> R2 = np.corrcoef(xarr, yarr)
    >>> R2
    array([[ 0.99999994, -0.6173796 , -0.968541  , -0.48613155,  0.9951523 ,
            -0.8900264 ],
           [-0.6173796 ,  1.        ,  0.7937219 ,  0.9875833 , -0.53702235,
             0.19083664],
           [-0.968541  ,  0.7937219 ,  0.9999999 ,  0.6883078 , -0.9393724 ,
             0.74857277],
           [-0.48613152,  0.9875833 ,  0.6883078 ,  0.9999999 , -0.39783284,
             0.0342579 ],
           [ 0.9951523 , -0.53702235, -0.9393725 , -0.39783284,  0.99999994,
            -0.9305482 ],
           [-0.89002645,  0.19083665,  0.7485727 ,  0.0342579 , -0.9305482 ,
             1.        ]], dtype=float32)

    Finally if we use the option ``rowvar=False``, the columns are now
    being treated as the variables and we will find the column-wise Pearson
    correlation coefficients between variables in ``xarr`` and ``yarr``.

    >>> R3 = np.corrcoef(xarr, yarr, rowvar=False)
    >>> R3
    array([[ 1.        ,  0.9724453 , -0.9909503 ,  0.8104691 , -0.46436927,
            -0.1643624 ],
           [ 0.9724453 ,  1.        , -0.9949381 ,  0.6515728 , -0.6580445 ,
             0.07012729],
           [-0.99095035, -0.994938  ,  1.        , -0.72450536,  0.5790461 ,
             0.03047091],
           [ 0.8104691 ,  0.65157276, -0.72450536,  1.        ,  0.14243561,
            -0.71102554],
           [-0.4643693 , -0.6580445 ,  0.57904613,  0.1424356 ,  0.99999994,
            -0.79727215],
           [-0.1643624 ,  0.07012729,  0.03047091, -0.7110255 , -0.7972722 ,
             0.99999994]], dtype=float32)
    """

    out = dpnp.cov(x, y, rowvar, dtype=dtype)
    if out.ndim == 0:
        # scalar covariance
        # nan if incorrect value (nan, inf, 0), 1 otherwise
        return out / out

    d = dpnp.diag(out)

    stddev = dpnp.sqrt(d.real)
    out /= stddev[:, None]
    out /= stddev[None, :]

    # Clip real and imaginary parts to [-1, 1]. This does not guarantee
    # abs(a[i, j]) <= 1 for complex arrays, but is the best we can do without
    # excessive work.
    dpnp.clip(out.real, -1, 1, out=out.real)
    if dpnp.iscomplexobj(out):
        dpnp.clip(out.imag, -1, 1, out=out.imag)

    return out


def _get_padding(a_size, v_size, mode):
    assert v_size <= a_size

    if mode == "valid":
        l_pad, r_pad = 0, 0
    elif mode == "same":
        l_pad = v_size // 2
        r_pad = v_size - l_pad - 1
    elif mode == "full":
        l_pad, r_pad = v_size - 1, v_size - 1
    else:  # pragma: no cover
        raise ValueError(
            f"Unknown mode: {mode}. Only 'valid', 'same', 'full' are supported."
        )

    return l_pad, r_pad


def _choose_conv_method(a, v, rdtype):
    assert a.size >= v.size
    if rdtype == dpnp.bool:
        # to avoid accuracy issues
        return "direct"

    if v.size < 10**4 or a.size < 10**4:
        # direct method is faster for small arrays
        return "direct"

    if dpnp.issubdtype(rdtype, dpnp.integer):
        max_a = int(dpnp.max(dpnp.abs(a)))
        sum_v = int(dpnp.sum(dpnp.abs(v)))
        max_value = int(max_a * sum_v)

        default_float = dpnp.default_float_type(a.sycl_device)
        if max_value > 2 ** numpy.finfo(default_float).nmant - 1:
            # can't represent the result in the default float type
            return "direct"  # pragma: no covers

    if dpnp.issubdtype(rdtype, dpnp.number):
        return "fft"

    raise ValueError(f"Unsupported dtype: {rdtype}")  # pragma: no cover


def _run_native_sliding_dot_product1d(a, v, l_pad, r_pad, rdtype):
    queue = a.sycl_queue
    device = a.sycl_device

    supported_types = statistics_ext.sliding_dot_product1d_dtypes()
    supported_dtype = to_supported_dtypes(rdtype, supported_types, device)

    if supported_dtype is None:  # pragma: no cover
        raise ValueError(
            f"function does not support input types "
            f"({a.dtype.name}, {v.dtype.name}), "
            "and the inputs could not be coerced to any "
            f"supported types. List of supported types: "
            f"{[st.name for st in supported_types]}"
        )

    a_casted = dpnp.asarray(a, dtype=supported_dtype, order="C")
    v_casted = dpnp.asarray(v, dtype=supported_dtype, order="C")

    usm_type = dpu.get_coerced_usm_type([a_casted.usm_type, v_casted.usm_type])
    out_size = l_pad + r_pad + a_casted.size - v_casted.size + 1
    # out type is the same as input type
    out = dpnp.empty_like(a_casted, shape=out_size, usm_type=usm_type)

    a_usm = dpnp.get_usm_ndarray(a_casted)
    v_usm = dpnp.get_usm_ndarray(v_casted)
    out_usm = dpnp.get_usm_ndarray(out)

    _manager = dpu.SequentialOrderManager[queue]

    mem_ev, corr_ev = statistics_ext.sliding_dot_product1d(
        a_usm,
        v_usm,
        out_usm,
        l_pad,
        r_pad,
        depends=_manager.submitted_events,
    )
    _manager.add_event_pair(mem_ev, corr_ev)

    return out


def _convolve_fft(a, v, l_pad, r_pad, rtype):
    assert a.size >= v.size
    assert l_pad < v.size

    # +1 is needed to avoid circular convolution
    padded_size = a.size + r_pad + 1
    fft_size = 2 ** int(math.ceil(math.log2(padded_size)))

    af = dpnp.fft.fft(a, fft_size)  # pylint: disable=no-member
    vf = dpnp.fft.fft(v, fft_size)  # pylint: disable=no-member

    r = dpnp.fft.ifft(af * vf)  # pylint: disable=no-member
    if dpnp.issubdtype(rtype, dpnp.floating):
        r = r.real
    elif dpnp.issubdtype(rtype, dpnp.integer) or rtype == dpnp.bool:
        r = r.real.round()

    start = v.size - 1 - l_pad
    end = padded_size - 1

    return r[start:end]


def correlate(a, v, mode="valid", method="auto"):
    r"""
    Cross-correlation of two 1-dimensional sequences.

    This function computes the correlation as generally defined in signal
    processing texts [1]_:

    .. math:: c_k = \sum_n a_{n+k} \cdot \overline{v}_n

    with `a` and `v` sequences being zero-padded where necessary and
    :math:`\overline v` denoting complex conjugation.

    For full documentation refer to :obj:`numpy.correlate`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        First input array.
    v : {dpnp.ndarray, usm_ndarray}
        Second input array.
    mode : {"valid", "same", "full"}, optional
        Refer to the :obj:`dpnp.convolve` docstring. Note that the default
        is ``"valid"``, unlike :obj:`dpnp.convolve`, which uses ``"full"``.

        Default: ``"valid"``.
    method : {"auto", "direct", "fft"}, optional
        Specifies which method to use to calculate the correlation:

        - `"direct"` : The correlation is determined directly from sums.
        - `"fft"` : The Fourier Transform is used to perform the calculations.
          This method is faster for long sequences but can have accuracy issues.
        - `"auto"` : Automatically chooses direct or Fourier method based on
          an estimate of which is faster.

        Note: Use of the FFT convolution on input containing NAN or INF
        will lead to the entire output being NAN or INF.
        Use method='direct' when your input contains NAN or INF values.

        Default: ``"auto"``.

    Returns
    -------
    out : dpnp.ndarray
        Discrete cross-correlation of `a` and `v`.

    Notes
    -----
    The definition of correlation above is not unique and sometimes
    correlation may be defined differently. Another common definition is [1]_:

    .. math:: c'_k = \sum_n a_{n} \cdot \overline{v_{n+k}}

    which is related to :math:`c_k` by :math:`c'_k = c_{-k}`.

    References
    ----------
    .. [1] Wikipedia, "Cross-correlation",
           https://en.wikipedia.org/wiki/Cross-correlation

    See Also
    --------
    :obj:`dpnp.convolve` : Discrete, linear convolution of two one-dimensional
                        sequences.


    Examples
    --------
    >>> import dpnp as np
    >>> a = np.array([1, 2, 3], dtype=np.float32)
    >>> v = np.array([0, 1, 0.5], dtype=np.float32)
    >>> np.correlate(a, v)
    array([3.5], dtype=float32)
    >>> np.correlate(a, v, "same")
    array([2. , 3.5, 3. ], dtype=float32)
    >>> np.correlate([a, v, "full")
    array([0.5, 2. , 3.5, 3. , 0. ], dtype=float32)

    Using complex sequences:

    >>> ac = np.array([1+1j, 2, 3-1j], dtype=np.complex64)
    >>> vc = np.array([0, 1, 0.5j], dtype=np.complex64)
    >>> np.correlate(ac, vc, 'full')
    array([0.5-0.5j, 1. +0.j , 1.5-1.5j, 3. -1.j , 0. +0.j ], dtype=complex64)

    Note that you get the time reversed, complex conjugated result
    (:math:`\overline{c_{-k}}`) when the two input sequences `a` and `v` change
    places:

    >>> np.correlate(vc, ac, 'full')
    array([0. +0.j , 3. +1.j , 1.5+1.5j, 1. +0.j , 0.5+0.5j], dtype=complex64)

    """

    dpnp.check_supported_arrays_type(a, v)

    if a.size == 0 or v.size == 0:
        raise ValueError(
            f"Array arguments cannot be empty. "
            f"Received sizes: a.size={a.size}, v.size={v.size}"
        )
    if a.ndim != 1 or v.ndim != 1:
        raise ValueError(
            f"Only 1-dimensional arrays are supported. "
            f"Received shapes: a.shape={a.shape}, v.shape={v.shape}"
        )

    supported_methods = ["auto", "direct", "fft"]
    if method not in supported_methods:
        raise ValueError(
            f"Unknown method: {method}. Supported methods: {supported_methods}"
        )

    device = a.sycl_device
    rdtype = result_type_for_device([a.dtype, v.dtype], device)

    if dpnp.issubdtype(v.dtype, dpnp.complexfloating):
        v = dpnp.conj(v)

    revert = False
    if v.size > a.size:
        revert = True
        a, v = v, a

    l_pad, r_pad = _get_padding(a.size, v.size, mode)

    if method == "auto":
        method = _choose_conv_method(a, v, rdtype)

    if method == "direct":
        r = _run_native_sliding_dot_product1d(a, v, l_pad, r_pad, rdtype)
    elif method == "fft":
        r = _convolve_fft(a, v[::-1], l_pad, r_pad, rdtype)
    else:  # pragma: no cover
        raise ValueError(f"Unknown method: {method}")

    if revert:
        r = r[::-1]

    return dpnp.asarray(r, dtype=rdtype, order="C")


def cov(
    m,
    y=None,
    rowvar=True,
    bias=False,
    ddof=None,
    fweights=None,
    aweights=None,
    *,
    dtype=None,
):
    """
    Estimate a covariance matrix, given data and weights.

    For full documentation refer to :obj:`numpy.cov`.

    Parameters
    ----------
    m : {dpnp.ndarray, usm_ndarray}
        A 1-D or 2-D array containing multiple variables and observations.
        Each row of `m` represents a variable, and each column a single
        observation of all those variables. Also see `rowvar` below.
    y : {None, dpnp.ndarray, usm_ndarray}, optional
        An additional set of variables and observations. `y` has the same form
        as that of `m`.

        Default: ``None``.
    rowvar : bool, optional
        If `rowvar` is ``True``, then each row represents a variable, with
        observations in the columns. Otherwise, the relationship is transposed:
        each column represents a variable, while the rows contain observations.

        Default: ``True``.
    bias : bool, optional
        Default normalization is by ``(N - 1)``, where ``N`` is the number of
        observations given (unbiased estimate). If `bias` is ``True``, then
        normalization is by ``N``. These values can be overridden by using the
        keyword `ddof`.

        Default: ``False``.
    ddof : {None, int}, optional
        If not ``None`` the default value implied by `bias` is overridden. Note
        that ``ddof=1`` will return the unbiased estimate, even if both
        `fweights` and `aweights` are specified, and ``ddof=0`` will return the
        simple average. See the notes for the details.

        Default: ``None``.
    fweights : {None, dpnp.ndarray, usm_ndarray}, optional
        1-D array of integer frequency weights; the number of times each
        observation vector should be repeated.
        It is required that ``fweights >= 0``. However, the function will not
        raise an error when ``fweights < 0`` for performance reasons.

        Default: ``None``.
    aweights : {None, dpnp.ndarray, usm_ndarray}, optional
        1-D array of observation vector weights. These relative weights are
        typically large for observations considered "important" and smaller for
        observations considered less "important". If ``ddof=0`` the array of
        weights can be used to assign probabilities to observation vectors.
        It is required that ``aweights >= 0``. However, the function will not
        error when ``aweights < 0`` for performance reasons.

        Default: ``None``.
    dtype : {None, str, dtype object}, optional
        Data-type of the result. By default, the return data-type will have
        the default floating point data-type of the device on which the input
        arrays reside.

        Default: ``None``.

    Returns
    -------
    out : dpnp.ndarray
        The covariance matrix of the variables.

    See Also
    --------
    :obj:`dpnp.corrcoef` : Normalized covariance matrix.

    Notes
    -----
    Assume that the observations are in the columns of the observation array `m`
    and let ``f = fweights`` and ``a = aweights`` for brevity. The steps to
    compute the weighted covariance are as follows::

        >>> import dpnp as np
        >>> m = np.arange(10, dtype=np.float32)
        >>> f = np.arange(10) * 2
        >>> a = np.arange(10) ** 2.0
        >>> ddof = 1
        >>> w = f * a
        >>> v1 = np.sum(w)
        >>> v2 = np.sum(w * a)
        >>> m -= np.sum(m * w, axis=None, keepdims=True) / v1
        >>> cov = np.dot(m * w, m.T) * v1 / (v1**2 - ddof * v2)

    Note that when ``a == 1``, the normalization factor
    ``v1 / (v1**2 - ddof * v2)`` goes over to ``1 / (np.sum(f) - ddof)``
    as it should.

    Examples
    --------
    >>> import dpnp as np
    >>> x = np.array([[0, 2], [1, 1], [2, 0]]).T

    Consider two variables, :math:`x_0` and :math:`x_1`, which correlate
    perfectly, but in opposite directions:

    >>> x
    array([[0, 1, 2],
           [2, 1, 0]])

    Note how :math:`x_0` increases while :math:`x_1` decreases. The covariance
    matrix shows this clearly:

    >>> np.cov(x)
    array([[ 1., -1.],
           [-1.,  1.]])

    Note that element :math:`C_{0, 1}`, which shows the correlation between
    :math:`x_0` and :math:`x_1`, is negative.

    Further, note how `x` and `y` are combined:

    >>> x = np.array([-2.1, -1,  4.3])
    >>> y = np.array([3,  1.1,  0.12])
    >>> X = np.stack((x, y), axis=0)
    >>> np.cov(X)
    array([[11.71      , -4.286     ], # may vary
           [-4.286     ,  2.14413333]])
    >>> np.cov(x, y)
    array([[11.71      , -4.286     ], # may vary
           [-4.286     ,  2.14413333]])
    >>> np.cov(x)
    array(11.71)

    """

    arrays = [m]
    if y is not None:
        arrays.append(y)
    dpnp.check_supported_arrays_type(*arrays)

    if m.ndim > 2:
        raise ValueError("m has more than 2 dimensions")

    if y is not None:
        if y.ndim > 2:
            raise ValueError("y has more than 2 dimensions")

    if ddof is not None:
        if not isinstance(ddof, int):
            raise ValueError("ddof must be integer")
    else:
        ddof = 0 if bias else 1

    def_float = dpnp.default_float_type(m.sycl_queue)
    if dtype is None:
        dtype = dpnp.result_type(*arrays, def_float)

    if fweights is not None:
        dpnp.check_supported_arrays_type(fweights)
        if not dpnp.issubdtype(fweights.dtype, numpy.integer):
            raise TypeError("fweights must be integer")

        if fweights.ndim > 1:
            raise ValueError("cannot handle multidimensional fweights")

        fweights = dpnp.astype(fweights, def_float)

    if aweights is not None:
        dpnp.check_supported_arrays_type(aweights)
        if aweights.ndim > 1:
            raise ValueError("cannot handle multidimensional aweights")

        aweights = dpnp.astype(aweights, def_float)

    return dpnp_cov(
        m,
        y=y,
        rowvar=rowvar,
        ddof=ddof,
        dtype=dtype,
        fweights=fweights,
        aweights=aweights,
    )


def max(a, axis=None, out=None, keepdims=False, initial=None, where=True):
    """
    Return the maximum of an array or maximum along an axis.

    For full documentation refer to :obj:`numpy.max`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int or tuple of ints}, optional
        Axis or axes along which to operate. By default, flattened input is
        used. If this is a tuple of integers, the minimum is selected over
        multiple axes, instead of a single axis or all the axes as before.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. Must be of the
        same shape and buffer length as the expected output.

        Default: ``None``.
    keepdims : {None, bool}, optional
        If this is set to ``True``, the axes which are reduced are left in the
        result as dimensions with size one. With this option, the result will
        broadcast correctly against the input array.

        Default: ``False``.

    Returns
    -------
    out : dpnp.ndarray
        Maximum of `a`. If `axis` is ``None``, the result is a zero-dimensional
        array. If `axis` is an integer, the result is an array of dimension
        ``a.ndim - 1``. If `axis` is a tuple, the result is an array of
        dimension ``a.ndim - len(axis)``.

    Limitations
    -----------
    Parameters `where`, and `initial` are only supported with their default
    values. Otherwise ``NotImplementedError`` exception will be raised.

    See Also
    --------
    :obj:`dpnp.min` : Return the minimum of an array.
    :obj:`dpnp.maximum` : Element-wise maximum of two arrays, propagates NaNs.
    :obj:`dpnp.fmax` : Element-wise maximum of two arrays, ignores NaNs.
    :obj:`dpnp.amax` : The maximum value of an array along a given axis,
                       propagates NaNs.
    :obj:`dpnp.nanmax` : The maximum value of an array along a given axis,
                         ignores NaNs.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.arange(4).reshape((2, 2))
    >>> a
    array([[0, 1],
           [2, 3]])
    >>> np.max(a)
    array(3)

    >>> np.max(a, axis=0)   # Maxima along the first axis
    array([2, 3])
    >>> np.max(a, axis=1)   # Maxima along the second axis
    array([1, 3])

    >>> b = np.arange(5, dtype=float)
    >>> b[2] = np.nan
    >>> np.max(b)
    array(nan)

    """

    dpnp.check_limitations(initial=initial, where=where)
    usm_a = dpnp.get_usm_ndarray(a)

    return dpnp_wrap_reduction_call(
        usm_a,
        out,
        dpt.max,
        a.dtype,
        axis=axis,
        keepdims=keepdims,
    )


def mean(a, /, axis=None, dtype=None, out=None, keepdims=False, *, where=True):
    """
    Compute the arithmetic mean along the specified axis.

    For full documentation refer to :obj:`numpy.mean`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        Axis or axes along which the arithmetic means must be computed. If
        a tuple of unique integers, the means are computed over multiple
        axes. If ``None``, the mean is computed over the entire array.

        Default: ``None``.
    dtype : {None, str, dtype object}, optional
        Type to use in computing the mean. By default, if `a` has a
        floating-point data type, the returned array will have
        the same data type as `a`.
        If `a` has a boolean or integral data type, the returned array
        will have the default floating point data type for the device
        where input array `a` is allocated.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. It must have
        the same shape as the expected output but the type (of the calculated
        values) will be cast if necessary.

        Default: ``None``.
    keepdims : {None, bool}, optional
        If ``True``, the reduced axes (dimensions) are included in the result
        as singleton dimensions, so that the returned array remains
        compatible with the input array according to Array Broadcasting
        rules. Otherwise, if ``False``, the reduced axes are not included in
        the returned array.

        Default: ``False``.

    Returns
    -------
    out : dpnp.ndarray
        An array containing the arithmetic means along the specified axis(axes).
        If the input is a zero-size array, an array containing NaN values is
        returned.

    Limitations
    -----------
    Parameter `where` is only supported with its default value.
    Otherwise ``NotImplementedError`` exception will be raised.

    See Also
    --------
    :obj:`dpnp.average` : Weighted average.
    :obj:`dpnp.std` : Compute the standard deviation along the specified axis.
    :obj:`dpnp.var` : Compute the variance along the specified axis.
    :obj:`dpnp.nanmean` : Compute the arithmetic mean along the specified axis,
                          ignoring NaNs.
    :obj:`dpnp.nanstd` : Compute the standard deviation along
                         the specified axis, while ignoring NaNs.
    :obj:`dpnp.nanvar` : Compute the variance along the specified axis,
                         while ignoring NaNs.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.array([[1, 2], [3, 4]])
    >>> np.mean(a)
    array(2.5)
    >>> np.mean(a, axis=0)
    array([2., 3.])
    >>> np.mean(a, axis=1)
    array([1.5, 3.5])

    """

    dpnp.check_limitations(where=where)

    usm_a = dpnp.get_usm_ndarray(a)
    usm_res = dpt.mean(usm_a, axis=axis, keepdims=keepdims)
    if dtype is not None:
        usm_res = dpt.astype(usm_res, dtype)

    return dpnp.get_result_array(usm_res, out, casting="unsafe")


def median(a, axis=None, out=None, overwrite_input=False, keepdims=False):
    """
    Compute the median along the specified axis.

    For full documentation refer to :obj:`numpy.median`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple or list of ints}, optional
        Axis or axes along which the medians are computed. The default,
        ``axis=None``, will compute the median along a flattened version of
        the array. If a sequence of axes, the array is first flattened along
        the given axes, then the median is computed along the resulting
        flattened axis.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. It must have
        the same shape as the expected output but the type (of the calculated
        values) will be cast if necessary.

        Default: ``None``.
    overwrite_input : bool, optional
       If ``True``, then allow use of memory of input array `a` for
       calculations. The input array will be modified by the call to
       :obj:`dpnp.median`. This will save memory when you do not need to
       preserve the contents of the input array. Treat the input as undefined,
       but it will probably be fully or partially sorted.

       Default: ``False``.
    keepdims : {None, bool}, optional
        If ``True``, the reduced axes (dimensions) are included in the result
        as singleton dimensions, so that the returned array remains
        compatible with the input array according to Array Broadcasting
        rules. Otherwise, if ``False``, the reduced axes are not included in
        the returned array.

        Default: ``False``.

    Returns
    -------
    out : dpnp.ndarray
        A new array holding the result. If `a` has a floating-point data type,
        the returned array will have the same data type as `a`. If `a` has a
        boolean or integral data type, the returned array will have the
        default floating point data type for the device where input array `a`
        is allocated.

    See Also
    --------
    :obj:`dpnp.mean` : Compute the arithmetic mean along the specified axis.
    :obj:`dpnp.percentile` : Compute the q-th percentile of the data
                             along the specified axis.

    Notes
    -----
    Given a vector ``V`` of length ``N``, the median of ``V`` is the
    middle value of a sorted copy of ``V``, ``V_sorted`` - i.e.,
    ``V_sorted[(N-1)/2]``, when ``N`` is odd, and the average of the
    two middle values of ``V_sorted`` when ``N`` is even.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.array([[10, 7, 4], [3, 2, 1]])
    >>> a
    array([[10,  7,  4],
           [ 3,  2,  1]])
    >>> np.median(a)
    array(3.5)

    >>> np.median(a, axis=0)
    array([6.5, 4.5, 2.5])
    >>> np.median(a, axis=1)
    array([7., 2.])
    >>> np.median(a, axis=(0, 1))
    array(3.5)

    >>> m = np.median(a, axis=0)
    >>> out = np.zeros_like(m)
    >>> np.median(a, axis=0, out=m)
    array([6.5, 4.5, 2.5])
    >>> m
    array([6.5, 4.5, 2.5])

    >>> b = a.copy()
    >>> np.median(b, axis=1, overwrite_input=True)
    array([7., 2.])
    >>> assert not np.all(a==b)
    >>> b = a.copy()
    >>> np.median(b, axis=None, overwrite_input=True)
    array(3.5)
    >>> assert not np.all(a==b)

    """

    dpnp.check_supported_arrays_type(a)
    return dpnp_median(
        a, axis, out, overwrite_input, keepdims, ignore_nan=False
    )


def min(a, axis=None, out=None, keepdims=False, initial=None, where=True):
    """
    Return the minimum of an array or maximum along an axis.

    For full documentation refer to :obj:`numpy.min`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int or tuple of ints}, optional
        Axis or axes along which to operate. By default, flattened input is
        used. If this is a tuple of integers, the minimum is selected over
        multiple axes, instead of a single axis or all the axes as before.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. Must be of the
        same shape and buffer length as the expected output.

        Default: ``None``.
    keepdims : {None, bool}, optional
        If this is set to ``True``, the axes which are reduced are left in the
        result as dimensions with size one. With this option, the result will
        broadcast correctly against the input array.

        Default: ``False``.

    Returns
    -------
    out : dpnp.ndarray
        Minimum of `a`. If `axis` is ``None``, the result is a zero-dimensional
        array. If `axis` is an integer, the result is an array of dimension
        ``a.ndim - 1``. If `axis` is a tuple, the result is an array of
        dimension ``a.ndim - len(axis)``.

    Limitations
    -----------
    Parameters `where`, and `initial` are only supported with their default
    values. Otherwise ``NotImplementedError`` exception will be raised.

    See Also
    --------
    :obj:`dpnp.max` : Return the maximum of an array.
    :obj:`dpnp.minimum` : Element-wise minimum of two arrays, propagates NaNs.
    :obj:`dpnp.fmin` : Element-wise minimum of two arrays, ignores NaNs.
    :obj:`dpnp.amin` : The minimum value of an array along a given axis,
                       propagates NaNs.
    :obj:`dpnp.nanmin` : The minimum value of an array along a given axis,
                         ignores NaNs.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.arange(4).reshape((2, 2))
    >>> a
    array([[0, 1],
           [2, 3]])
    >>> np.min(a)
    array(0)

    >>> np.min(a, axis=0)   # Minima along the first axis
    array([0, 1])
    >>> np.min(a, axis=1)   # Minima along the second axis
    array([0, 2])

    >>> b = np.arange(5, dtype=float)
    >>> b[2] = np.nan
    >>> np.min(b)
    array(nan)

    """

    dpnp.check_limitations(initial=initial, where=where)
    usm_a = dpnp.get_usm_ndarray(a)

    return dpnp_wrap_reduction_call(
        usm_a,
        out,
        dpt.min,
        a.dtype,
        axis=axis,
        keepdims=keepdims,
    )


def ptp(
    a,
    /,
    axis=None,
    out=None,
    keepdims=False,
):
    """
    Range of values (maximum - minimum) along an axis.

    For full documentation refer to :obj:`numpy.ptp`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        Axis along which to find the peaks. By default, flatten the array.
        `axis` may be negative, in which case it counts from the last to the
        first axis. If this is a tuple of ints, a reduction is performed on
        multiple axes, instead of a single axis or all the axes as before.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. It must have the
        same shape and buffer length as the expected output, but the type of
        the output values will be cast if necessary.

        Default: ``None``.
    keepdims : {None, bool}, optional
        If this is set to ``True``, the axes which are reduced are left in the
        result as dimensions with size one. With this option, the result will
        broadcast correctly against the input array.

        Default: ``None``.

    Returns
    -------
    ptp : dpnp.ndarray
        The range of a given array.

    Examples
    --------
    >>> import dpnp as np
    >>> x = np.array([[4, 9, 2, 10], [6, 9, 7, 12]])
    >>> np.ptp(x, axis=1)
    array([8, 6])

    >>> np.ptp(x, axis=0)
    array([2, 0, 5, 2])

    >>> np.ptp(x)
    array(10)

    This example shows that a negative value can be returned when the input is
    an array of signed integers:

    >>> y = np.array([[1, 127],
    ...               [0, 127],
    ...               [-1, 127],
    ...               [-2, 127]], dtype="i1")
    >>> np.ptp(y, axis=1)
    array([ 126,  127, -128, -127], dtype=int8)

    """

    return dpnp.subtract(
        dpnp.max(a, axis=axis, keepdims=keepdims, out=out),
        dpnp.min(a, axis=axis, keepdims=keepdims),
        out=out,
    )


# pylint: disable=redefined-outer-name
def std(
    a,
    axis=None,
    dtype=None,
    out=None,
    ddof=0,
    keepdims=False,
    *,
    where=True,
    mean=None,
    correction=None,
):
    r"""
    Compute the standard deviation along the specified axis.

    For full documentation refer to :obj:`numpy.std`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        Axis or axes along which the standard deviations must be computed.
        If a tuple of unique integers is given, the standard deviations
        are computed over multiple axes. If ``None``, the standard deviation
        is computed over the entire array.

        Default: ``None``.
    dtype : {None, str, dtype object}, optional
        Type to use in computing the standard deviation. By default, if `a` has
        a floating-point data type, the returned array will have the same data
        type as `a`. If `a` has a boolean or integral data type, the returned
        array will have the default floating point data type for the device
        where input array `a` is allocated.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. It must have
        the same shape as the expected output but the type (of the calculated
        values) will be cast if necessary.

        Default: ``None``.
    ddof : {int, float}, optional
        Means Delta Degrees of Freedom. The divisor used in calculations is
        ``N - ddof``, where ``N`` corresponds to the total number of elements
        over which the standard deviation is calculated.

        Default: ``0.0``.
    keepdims : {None, bool}, optional
        If ``True``, the reduced axes (dimensions) are included in the result
        as singleton dimensions, so that the returned array remains compatible
        with the input array according to Array Broadcasting rules. Otherwise,
        if ``False``, the reduced axes are not included in the returned array.

        Default: ``False``.
    mean : {dpnp.ndarray, usm_ndarray}, optional
        Provide the mean to prevent its recalculation. The mean should have
        a shape as if it was calculated with ``keepdims=True``.
        The axis for the calculation of the mean should be the same as used in
        the call to this `std` function.

        Default: ``None``.

    correction : {None, int, float}, optional
        Array API compatible name for the `ddof` parameter. Only one of them
        can be provided at the same time.

        Default: ``None``.

    Returns
    -------
    out : dpnp.ndarray
        An array containing the standard deviations. If the standard deviation
        was computed over the entire array, a zero-dimensional array is
        returned.

    Limitations
    -----------
    Parameters `where` is only supported with its default value.
    Otherwise ``NotImplementedError`` exception will be raised.

    Notes
    -----
    There are several common variants of the array standard deviation
    calculation. Assuming the input `a` is a one-dimensional array and `mean`
    is either provided as an argument or computed as ``a.mean()``, DPNP
    computes the standard deviation of an array as::

        N = len(a)
        d2 = abs(a - mean)**2  # abs is for complex `a`
        var = d2.sum() / (N - ddof)  # note use of `ddof`
        std = var**0.5

    Different values of the argument `ddof` are useful in different contexts.
    The default ``ddof=0`` corresponds with the expression:

    .. math::

        \sqrt{\frac{\sum_i{|a_i - \bar{a}|^2 }}{N}}

    which is sometimes called the "population standard deviation" in the field
    of statistics because it applies the definition of standard deviation to `a`
    as if `a` were a complete population of possible observations.

    Many other libraries define the standard deviation of an array
    differently, e.g.:

    .. math::

        \sqrt{\frac{\sum_i{|a_i - \bar{a}|^2 }}{N - 1}}

    In statistics, the resulting quantity is sometimes called the "sample
    standard deviation" because if `a` is a random sample from a larger
    population, this calculation provides the square root of an unbiased
    estimate of the variance of the population. The use of :math:`N-1` in the
    denominator is often called "Bessel's correction" because it corrects for
    bias (toward lower values) in the variance estimate introduced when the
    sample mean of `a` is used in place of the true mean of the population.
    The resulting estimate of the standard deviation is still biased, but less
    than it would have been without the correction. For this quantity, use
    ``ddof=1``.

    Note that, for complex numbers, the absolute value is taken before squaring,
    so that the result is always real and non-negative.

    See Also
    --------
    :obj:`dpnp.ndarray.std` : corresponding function for ndarrays.
    :obj:`dpnp.var` : Compute the variance along the specified axis.
    :obj:`dpnp.mean` : Compute the arithmetic mean along the specified axis.
    :obj:`dpnp.nanmean` : Compute the arithmetic mean along the specified axis,
                          ignoring NaNs.
    :obj:`dpnp.nanstd` : Compute the standard deviation along
                         the specified axis, while ignoring NaNs.
    :obj:`dpnp.nanvar` : Compute the variance along the specified axis,
                         while ignoring NaNs.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.array([[1, 2], [3, 4]])
    >>> np.std(a)
    array(1.11803399)
    >>> np.std(a, axis=0)
    array([1., 1.])
    >>> np.std(a, axis=1)
    array([0.5, 0.5])

    Using the mean keyword to save computation time:

    >>> a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])
    >>> mean = np.mean(a, axis=1, keepdims=True)
    >>> np.std(a, axis=1, mean=mean)
    array([2.16506351, 1.47901995, 3.53553391])

    """

    dpnp.check_supported_arrays_type(a)
    dpnp.check_limitations(where=where)

    if correction is not None:
        if ddof != 0:
            raise ValueError(
                "ddof and correction can't be provided simultaneously."
            )
        ddof = correction

    if not isinstance(ddof, (int, float)):
        raise TypeError(
            f"An integer or float is required, but got {type(ddof)}"
        )

    if dpnp.issubdtype(a.dtype, dpnp.complexfloating) or mean is not None:
        result = dpnp.var(
            a,
            axis=axis,
            dtype=None,
            out=out,
            ddof=ddof,
            keepdims=keepdims,
            where=where,
            mean=mean,
        )
        dpnp.sqrt(result, out=result)
    else:
        usm_a = dpnp.get_usm_ndarray(a)
        usm_res = dpt.std(usm_a, axis=axis, correction=ddof, keepdims=keepdims)
        result = dpnp.get_result_array(usm_res, out)

    if dtype is not None and out is None:
        result = result.astype(dtype, casting="same_kind")
    return result


# pylint: disable=redefined-outer-name
def var(
    a,
    axis=None,
    dtype=None,
    out=None,
    ddof=0,
    keepdims=False,
    *,
    where=True,
    mean=None,
    correction=None,
):
    r"""
    Compute the variance along the specified axis.

    For full documentation refer to :obj:`numpy.var`.

    Parameters
    ----------
    a : {dpnp.ndarray, usm_ndarray}
        Input array.
    axis : {None, int, tuple of ints}, optional
        Axis or axes along which the variances must be computed. If a tuple
        of unique integers is given, the variances are computed over multiple
        axes. If ``None``, the variance is computed over the entire array.

        Default: ``None``.
    dtype : {None, str, dtype object}, optional
        Type to use in computing the variance. By default, if `a` has a
        floating-point data type, the returned array will have the same data
        type as `a`. If `a` has a boolean or integral data type, the returned
        array will have the default floating point data type for the device
        where input array `a` is allocated.

        Default: ``None``.
    out : {None, dpnp.ndarray, usm_ndarray}, optional
        Alternative output array in which to place the result. It must have
        the same shape as the expected output but the type (of the calculated
        values) will be cast if necessary.

        Default: ``None``.
    ddof : {int, float}, optional
        Means Delta Degrees of Freedom. The divisor used in calculations is
        ``N - ddof``, where ``N`` corresponds to the total number of elements
        over which the variance is calculated.

        Default: ``0.0``.
    keepdims : {None, bool}, optional
        If ``True``, the reduced axes (dimensions) are included in the result
        as singleton dimensions, so that the returned array remains compatible
        with the input array according to Array Broadcasting rules. Otherwise,
        if ``False``, the reduced axes are not included in the returned array.

        Default: ``False``.
    mean : {dpnp.ndarray, usm_ndarray}, optional
        Provide the mean to prevent its recalculation. The mean should have
        a shape as if it was calculated with ``keepdims=True``.
        The axis for the calculation of the mean should be the same as used in
        the call to this `var` function.

        Default: ``None``.

    correction : {None, int, float}, optional
        Array API compatible name for the `ddof` parameter. Only one of them
        can be provided at the same time.

        Default: ``None``.

    Returns
    -------
    out : dpnp.ndarray
        An array containing the variances. If the variance was computed over
        the entire array, a zero-dimensional array is returned.

    Limitations
    -----------
    Parameters `where` is only supported with its default value.
    Otherwise ``NotImplementedError`` exception will be raised.

    Notes
    -----
    There are several common variants of the array variance calculation.
    Assuming the input `a` is a one-dimensional array and `mean` is either
    provided as an argument or computed as ``a.mean()``, DPNP computes the
    variance of an array as::

        N = len(a)
        d2 = abs(a - mean)**2  # abs is for complex `a`
        var = d2.sum() / (N - ddof)  # note use of `ddof`

    Different values of the argument `ddof` are useful in different contexts.
    The default ``ddof=0`` corresponds with the expression:

    .. math::

        \frac{\sum_i{|a_i - \bar{a}|^2 }}{N}

    which is sometimes called the "population variance" in the field of
    statistics because it applies the definition of variance to `a` as if `a`
    were a complete population of possible observations.

    Many other libraries define the variance of an array differently, e.g.:

    .. math::

        \frac{\sum_i{|a_i - \bar{a}|^2}}{N - 1}

    In statistics, the resulting quantity is sometimes called the "sample
    variance" because if `a` is a random sample from a larger population, this
    calculation provides an unbiased estimate of the variance of the population.
    The use of :math:`N-1` in the denominator is often called "Bessel's
    correction" because it corrects for bias (toward lower values) in the
    variance estimate introduced when the sample mean of `a` is used in place
    of the true mean of the population. For this quantity, use ``ddof=1``.

    Note that, for complex numbers, the absolute value is taken before squaring,
    so that the result is always real and non-negative.

    See Also
    --------
    :obj:`dpnp.ndarray.var` : corresponding function for ndarrays.
    :obj:`dpnp.std` : Compute the standard deviation along the specified axis.
    :obj:`dpnp.mean` : Compute the arithmetic mean along the specified axis.
    :obj:`dpnp.nanmean` : Compute the arithmetic mean along the specified axis,
                          ignoring NaNs.
    :obj:`dpnp.nanstd` : Compute the standard deviation along
                         the specified axis, while ignoring NaNs.
    :obj:`dpnp.nanvar` : Compute the variance along the specified axis,
                         while ignoring NaNs.

    Examples
    --------
    >>> import dpnp as np
    >>> a = np.array([[1, 2], [3, 4]])
    >>> np.var(a)
    array(1.25)
    >>> np.var(a, axis=0)
    array([1., 1.])
    >>> np.var(a, axis=1)
    array([0.25, 0.25])

    Using the mean keyword to save computation time:

    >>> a = np.array([[14, 8, 11, 10], [7, 9, 10, 11], [10, 15, 5, 10]])
    >>> mean = np.mean(a, axis=1, keepdims=True)
    >>> np.var(a, axis=1, mean=mean)
    array([ 4.6875,  2.1875, 12.5   ])

    """

    dpnp.check_supported_arrays_type(a)
    dpnp.check_limitations(where=where)

    if correction is not None:
        if ddof != 0:
            raise ValueError(
                "ddof and correction can't be provided simultaneously."
            )
        ddof = correction

    if not isinstance(ddof, (int, float)):
        raise TypeError(
            f"An integer or float is required, but got {type(ddof)}"
        )

    if dpnp.issubdtype(a.dtype, dpnp.complexfloating) or mean is not None:
        # cast bool and integer types to default floating type
        if dtype is None and not dpnp.issubdtype(a.dtype, dpnp.inexact):
            dtype = dpnp.default_float_type(device=a.device)

        if mean is not None:
            arrmean = mean
        else:
            # Compute the mean.
            # Note that if dtype is not of inexact type
            # then `arrmean` will not be either.
            arrmean = dpnp.mean(
                a, axis=axis, dtype=dtype, keepdims=True, where=where
            )

        # Compute sum of squared deviations from mean.
        # Note that `x` may not be inexact.
        x = dpnp.subtract(a, arrmean)
        if dpnp.issubdtype(x.dtype, dpnp.complexfloating):
            x = dpnp.multiply(x, x.conj(), out=x).real
        else:
            x = dpnp.square(x, out=x)

        result = dpnp.sum(
            x,
            axis=axis,
            dtype=dtype,
            out=out,
            keepdims=keepdims,
            where=where,
        )

        # compute degrees of freedom and make sure it is not negative
        cnt = _count_reduce_items(a, axis, where)
        cnt = numpy.max(cnt - ddof, 0).astype(result.dtype, casting="same_kind")
        if not cnt:
            cnt = dpnp.nan

        # divide by degrees of freedom
        result = _divide_by_scalar(result, cnt)
    else:
        usm_a = dpnp.get_usm_ndarray(a)
        usm_res = dpt.var(usm_a, axis=axis, correction=ddof, keepdims=keepdims)
        result = dpnp.get_result_array(usm_res, out)

    if out is None and dtype is not None:
        result = result.astype(dtype, casting="same_kind")
    return result