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"""
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",
"convolve",
"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
[docs]
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
)
[docs]
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
)
[docs]
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 _convolve_impl(a, v, mode, method, rdtype):
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[::-1], l_pad, r_pad, rdtype)
elif method == "fft":
r = _convolve_fft(a, v, l_pad, r_pad, rdtype)
else:
raise ValueError(
f"Unknown method: {method}. Supported methods: auto, direct, fft"
)
return r
[docs]
def convolve(a, v, mode="full", method="auto"):
r"""
Returns the discrete, linear convolution of two one-dimensional sequences.
The convolution operator is often seen in signal processing, where it
models the effect of a linear time-invariant system on a signal [1]_. In
probability theory, the sum of two independent random variables is
distributed according to the convolution of their individual
distributions.
If `v` is longer than `a`, the arrays are swapped before computation.
For full documentation refer to :obj:`numpy.convolve`.
Parameters
----------
a : {dpnp.ndarray, usm_ndarray}
First input array.
v : {dpnp.ndarray, usm_ndarray}
Second input array.
mode : {'full', 'valid', 'same'}, optional
- 'full': This returns the convolution
at each point of overlap, with an output shape of (N+M-1,). At
the end-points of the convolution, the signals do not overlap
completely, and boundary effects may be seen.
- 'same': Mode 'same' returns output of length ``max(M, N)``. Boundary
effects are still visible.
- 'valid': Mode 'valid' returns output of length
``max(M, N) - min(M, N) + 1``. The convolution product is only given
for points where the signals overlap completely. Values outside
the signal boundary have no effect.
Default: ``'full'``.
method : {'auto', 'direct', 'fft'}, optional
- 'direct': The convolution 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, linear convolution of `a` and `v`.
See Also
--------
:obj:`dpnp.correlate` : Cross-correlation of two 1-dimensional sequences.
Notes
-----
The discrete convolution operation is defined as
.. math:: (a * v)_n = \sum_{m = -\infty}^{\infty} a_m v_{n - m}
It can be shown that a convolution :math:`x(t) * y(t)` in time/space
is equivalent to the multiplication :math:`X(f) Y(f)` in the Fourier
domain, after appropriate padding (padding is necessary to prevent
circular convolution). Since multiplication is more efficient (faster)
than convolution, the function implements two approaches - direct and fft
which are regulated by the keyword `method`.
References
----------
.. [1] Wikipedia, "Convolution",
https://en.wikipedia.org/wiki/Convolution
Examples
--------
Note how the convolution operator flips the second array
before "sliding" the two across one another:
>>> import dpnp as np
>>> a = np.array([1, 2, 3], dtype=np.float32)
>>> v = np.array([0, 1, 0.5], dtype=np.float32)
>>> np.convolve(a, v)
array([0. , 1. , 2.5, 4. , 1.5], dtype=float32)
Only return the middle values of the convolution.
Contains boundary effects, where zeros are taken
into account:
>>> np.convolve(a, v, 'same')
array([1. , 2.5, 4. ], dtype=float32)
The two arrays are of the same length, so there
is only one position where they completely overlap:
>>> np.convolve(a, v, 'valid')
array([2.5], dtype=float32)
"""
a, v = dpnp.atleast_1d(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}"
)
device = a.sycl_device
rdtype = result_type_for_device([a.dtype, v.dtype], device)
if v.size > a.size:
a, v = v, a
r = _convolve_impl(a, v, mode, method, rdtype)
return dpnp.asarray(r, dtype=rdtype, order="C")
[docs]
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]
[docs]
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
r = _convolve_impl(a, v[::-1], mode, method, rdtype)
if revert:
r = r[::-1]
return dpnp.asarray(r, dtype=rdtype, order="C")
[docs]
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,
)
[docs]
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,
)
[docs]
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")
[docs]
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,
)
[docs]
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
[docs]
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
[docs]
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