# Source code for mars.tensor.reduction.var

```#!/usr/bin/env python
# -*- coding: utf-8 -*-
# Copyright 1999-2021 Alibaba Group Holding Ltd.
#
# Licensed under the Apache License, Version 2.0 (the "License");
# you may not use this file except in compliance with the License.
# You may obtain a copy of the License at
#
#
# Unless required by applicable law or agreed to in writing, software
# distributed under the License is distributed on an "AS IS" BASIS,
# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
# See the License for the specific language governing permissions and
# limitations under the License.

import numpy as np
from math import factorial

from ... import opcodes as OperandDef
from ...serialization.serializables import Int32Field
from ..datasource import tensor as astensor
from ..array_utils import as_same_device, device, get_array_module
from .core import TensorReduction, TensorReductionMixin, numel

def reduce_var_square(var_square, avg_diff, count, op, axis, sum_func):
moment = op.moment
dtype = op.dtype
kw = dict(axis=axis, dtype=dtype, keepdims=bool(op.keepdims))

reduced_var_square = var_square[..., moment - 2].sum(**kw) + sum_func(
count * avg_diff**moment, **kw
)
for i in range(1, moment - 1):
coeff = factorial(moment) / float(factorial(i) * factorial(moment - i))
reduced_var_square += coeff * sum_func(
var_square[..., moment - i - 2] * avg_diff**moment, **kw
)
return reduced_var_square

class TensorMoment(TensorReduction, TensorReductionMixin):
_op_type_ = OperandDef.MOMENT

_moment = Int32Field("moment", default=2)
_ddof = Int32Field("ddof")

def __init__(
self,
axis=None,
keepdims=None,
moment=None,
ddof=None,
combine_size=None,
stage=None,
**kw
):
stage = self._rewrite_stage(stage)
if moment is not None:
kw["_moment"] = moment
super().__init__(
_axis=axis,
_keepdims=keepdims,
_ddof=ddof,
_combine_size=combine_size,
stage=stage,
**kw
)

@property
def moment(self):
return getattr(self, "_moment", 2)

@property
def ddof(self):
return self._ddof

@classmethod
def execute_agg(cls, ctx, op):
axis = cls.get_axis(op.axis)
dtype = op.dtype

(_data, _count, _var_square), device_id, xp = as_same_device(
ctx[op.inputs[0].key], device=op.device, ret_extra=True
)

with device(device_id):
chunk_count = xp.sum(_count, axis=axis, dtype=np.int64, keepdims=True)
chunk_sum = xp.sum(_data, axis=axis, dtype=dtype, keepdims=True)
avg = xp.true_divide(chunk_sum, chunk_count, dtype=dtype)
avg_diff = xp.true_divide(_data, _count, dtype=dtype) - avg
var_square = reduce_var_square(
_var_square, avg_diff, _count, op, axis, xp.sum
)

ctx[op.outputs[0].key] = xp.true_divide(
var_square,
xp.sum(chunk_count, axis=axis, dtype=dtype, keepdims=bool(op.keepdims))
- op.ddof,
dtype=dtype,
)

@classmethod
def execute_map(cls, ctx, op):
(in_chunk,), device_id, xp = as_same_device(
[ctx[c.key] for c in op.inputs], device=op.device, ret_extra=True
)

axis = cls.get_axis(op.axis)
moment = op.moment
dtype = op.dtype
empty = get_array_module(in_chunk, nosparse=True).empty

with device(device_id):
chunk_count = numel(
in_chunk, axis=axis, dtype=np.int64, keepdims=bool(op.keepdims)
)
chunk_sum = xp.sum(
in_chunk, axis=axis, dtype=dtype, keepdims=bool(op.keepdims)
)
avg = xp.true_divide(chunk_sum, chunk_count)
var_square = empty(chunk_count.shape + (moment - 1,), dtype=dtype)
for i in range(2, moment + 1):
var_square[..., i - 2] = xp.sum(
(in_chunk - avg) ** i,
axis=axis,
dtype=dtype,
keepdims=bool(op.keepdims),
)
ctx[op.outputs[0].key] = (chunk_sum, chunk_count, var_square)

@classmethod
def execute_combine(cls, ctx, op):
axis = cls.get_axis(op.axis)
moment = op.moment
dtype = op.dtype

(_data, _count, _var_square), device_id, xp = as_same_device(
ctx[op.inputs[0].key], device=op.device, ret_extra=True
)
empty = get_array_module(_data, nosparse=True).empty

with device(device_id):
chunk_count = xp.sum(
_count, axis=axis, dtype=np.int64, keepdims=bool(op.keepdims)
)
chunk_sum = xp.sum(
_data, axis=axis, dtype=dtype, keepdims=bool(op.keepdims)
)
avg = xp.true_divide(chunk_sum, chunk_count, dtype=dtype)
avg_diff = xp.true_divide(_data, _count, dtype=dtype) - avg
var_square = empty(chunk_count.shape + (moment - 1,), dtype=dtype)

for m in range(2, moment + 1):
var_square[..., m - 2] = reduce_var_square(
_var_square, avg_diff, _count, op, axis, xp.sum
)

ctx[op.outputs[0].key] = (chunk_sum, chunk_count, var_square)

class TensorVar(TensorReduction, TensorReductionMixin):
_op_type_ = OperandDef.VAR

_ddof = Int32Field("ddof")

def __new__(cls, *args, **kwargs):
if kwargs.get("stage") is not None:
return TensorMoment(*args, **kwargs)
return super().__new__(cls)

def __init__(self, axis=None, keepdims=None, ddof=0, combine_size=None, **kw):
super().__init__(
_axis=axis, _keepdims=keepdims, _ddof=ddof, _combine_size=combine_size, **kw
)

@property
def ddof(self):
return self._ddof

def _get_op_kw(self):
kw = dict()
kw["ddof"] = self.ddof
return kw

@classmethod
def execute(cls, ctx, op):
axis = cls.get_axis(op.axis)
(in_chunk,), device_id, xp = as_same_device(
[ctx[c.key] for c in op.inputs], device=op.device, ret_extra=True
)

with device(device_id):
ctx[op.outputs[0].key] = xp.var(
in_chunk,
axis=axis,
dtype=op.dtype,
ddof=op.ddof,
keepdims=bool(op.keepdims),
)

[docs]def var(a, axis=None, dtype=None, out=None, ddof=0, keepdims=None, combine_size=None):
"""
Compute the variance along the specified axis.

Returns the variance of the tensor elements, a measure of the spread of a
distribution.  The variance is computed for the flattened tensor by
default, otherwise over the specified axis.

Parameters
----------
a : array_like
Tensor containing numbers whose variance is desired.  If `a` is not a
tensor, a conversion is attempted.
axis : None or int or tuple of ints, optional
Axis or axes along which the variance is computed.  The default is to
compute the variance of the flattened array.

If this is a tuple of ints, a variance is performed over multiple axes,
instead of a single axis or all the axes as before.
dtype : data-type, optional
Type to use in computing the variance.  For arrays of integer type
the default is `float32`; for tensors of float types it is the same as
the tensor type.
out : Tensor, optional
Alternate output array in which to place the result.  It must have
the same shape as the expected output, but the type is cast if
necessary.
ddof : int, optional
"Delta Degrees of Freedom": the divisor used in the calculation is
``N - ddof``, where ``N`` represents the number of elements. By
default `ddof` is zero.
keepdims : 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 tensor.

If the default value is passed, then `keepdims` will not be
passed through to the `var` method of sub-classes of
`Tensor`, however any non-default value will be.  If the
sub-classes `sum` method does not implement `keepdims` any
exceptions will be raised.
combine_size: int, optional
The number of chunks to combine.

Returns
-------
variance : Tensor, see dtype parameter above
If ``out=None``, returns a new tensor containing the variance;
otherwise, a reference to the output tensor is returned.

--------
std , mean, nanmean, nanstd, nanvar

Notes
-----
The variance is the average of the squared deviations from the mean,
i.e.,  ``var = mean(abs(x - x.mean())**2)``.

The mean is normally calculated as ``x.sum() / N``, where ``N = len(x)``.
If, however, `ddof` is specified, the divisor ``N - ddof`` is used
instead.  In standard statistical practice, ``ddof=1`` provides an
unbiased estimator of the variance of a hypothetical infinite population.
``ddof=0`` provides a maximum likelihood estimate of the variance for
normally distributed variables.

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

For floating-point input, the variance is computed using the same
precision the input has.  Depending on the input data, this can cause
the results to be inaccurate, especially for `float32` (see example
below).  Specifying a higher-accuracy accumulator using the ``dtype``
keyword can alleviate this issue.

Examples
--------
>>> import mars.tensor as mt

>>> a = mt.array([[1, 2], [3, 4]])
>>> mt.var(a).execute()
1.25
>>> mt.var(a, axis=0).execute()
array([ 1.,  1.])
>>> mt.var(a, axis=1).execute()
array([ 0.25,  0.25])

In single precision, var() can be inaccurate:

>>> a = mt.zeros((2, 512*512), dtype=mt.float32)
>>> a[0, :] = 1.0
>>> a[1, :] = 0.1
>>> mt.var(a).execute()
0.20250003

Computing the variance in float64 is more accurate:

>>> mt.var(a, dtype=mt.float64).execute()
0.20249999932944759
>>> ((1-0.55)**2 + (0.1-0.55)**2)/2
0.2025

"""
a = astensor(a)
if dtype is None:
dtype = np.var(np.ones((1,), dtype=a.dtype)).dtype
op = TensorVar(
axis=axis, dtype=dtype, keepdims=keepdims, ddof=ddof, combine_size=combine_size
)
return op(a, out=out)
```