`heat.statistics`

Distributed statistical operations.

Module Contents

argmax(x: heat.core.dndarray.DNDarray, axis: int | None = None, out: heat.core.dndarray.DNDarray | None = None, **kwargs: object) → heat.core.dndarray.DNDarray

Returns an array of the indices of the maximum values along an axis. It has the same shape as x.shape with the dimension along axis removed.

Parameters:

x (DNDarray) – Input array.
axis (int, optional) – By default, the index is into the flattened array, otherwise along the specified axis.
out (DNDarray, optional.) – If provided, the result will be inserted into this array. It should be of the appropriate shape and dtype.

Examples

>>> a = ht.random.randn(3, 3)
>>> a
DNDarray([[ 1.0661,  0.7036, -2.0908],
          [-0.7534, -0.4986, -0.7751],
          [-0.4815,  1.9436,  0.6400]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.argmax(a)
DNDarray([7], dtype=ht.int64, device=cpu:0, split=None)
>>> ht.argmax(a, axis=0)
DNDarray([0, 2, 2], dtype=ht.int64, device=cpu:0, split=None)
>>> ht.argmax(a, axis=1)
DNDarray([0, 1, 1], dtype=ht.int64, device=cpu:0, split=None)

argmin(x: heat.core.dndarray.DNDarray, axis: int | None = None, out: heat.core.dndarray.DNDarray | None = None, **kwargs: object) → heat.core.dndarray.DNDarray

Returns an array of the indices of the minimum values along an axis. It has the same shape as x.shape with the dimension along axis removed.

Parameters:

x (DNDarray) – Input array.
axis (int, optional) – By default, the index is into the flattened array, otherwise along the specified axis.
out (DNDarray, optional) – Issue #100 If provided, the result will be inserted into this array. It should be of the appropriate shape and dtype.

Examples

>>> a = ht.random.randn(3, 3)
>>> a
DNDarray([[ 1.0661,  0.7036, -2.0908],
          [-0.7534, -0.4986, -0.7751],
          [-0.4815,  1.9436,  0.6400]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.argmin(a)
DNDarray([2], dtype=ht.int64, device=cpu:0, split=None)
>>> ht.argmin(a, axis=0)
DNDarray([1, 1, 0], dtype=ht.int64, device=cpu:0, split=None)
>>> ht.argmin(a, axis=1)
DNDarray([2, 2, 0], dtype=ht.int64, device=cpu:0, split=None)

average(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int, Ellipsis] | None = None, weights: heat.core.dndarray.DNDarray | None = None, returned: bool = False) → heat.core.dndarray.DNDarray | Tuple[heat.core.dndarray.DNDarray, Ellipsis]

Compute the weighted average along the specified axis.

If returned=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 average.

Parameters:

x (DNDarray) – Array containing data to be averaged.
axis (None or int or Tuple[int,...], optional) – Axis or axes along which to average x. The default, axis=None, will average over all of the elements of the input array. If axis is negative it counts from the last to the first axis. #TODO Issue #351: If axis is a tuple of ints, averaging is performed on all of the axes specified in the tuple instead of a single axis or all the axes as before.
weights (DNDarray, optional) – An array of weights associated with the values in x. Each value in x contributes to the average according to its associated weight. The weights array can either be 1D (in which case its length must be the size of x along the given axis) or of the same shape as x. If weights=None, then all data in x are assumed to have a weight equal to one, the result is equivalent to mean().
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.

Raises:

ZeroDivisionError – When all weights along axis are zero.
TypeError – When the length of 1D weights is not the same as the shape of x along axis.

Examples

>>> data = ht.arange(1,5, dtype=float)
>>> data
DNDarray([1., 2., 3., 4.], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.average(data)
DNDarray(2.5000, dtype=ht.float32, device=cpu:0, split=None)
>>> ht.average(ht.arange(1,11, dtype=float), weights=ht.arange(10,0,-1))
DNDarray([4.], dtype=ht.float64, device=cpu:0, split=None)
>>> data = ht.array([[0, 1],
                     [2, 3],
                    [4, 5]], dtype=float, split=1)
>>> weights = ht.array([1./4, 3./4])
>>> ht.average(data, axis=1, weights=weights)
DNDarray([0.7500, 2.7500, 4.7500], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.average(data, weights=weights)
Traceback (most recent call last):
    ...
TypeError: Axis must be specified when shapes of x and weights differ.

bincount(x: heat.core.dndarray.DNDarray, weights: heat.core.dndarray.DNDarray | None = None, minlength: int = 0) → heat.core.dndarray.DNDarray

Count number of occurrences of each value in array of non-negative ints. Return a non-distributed DNDarray of length max(x) + 1 if input is non-empty, else 0.

The number of bins (size 1) is one larger than the largest value in x unless x is empty, in which case the result is a tensor of size 0. If minlength is specified, the number of bins is at least minlength and if x is empty, then the result is tensor of size minlength filled with zeros. If n is the value at position i, out[n] += weights[i] if weights is specified else out[n] += 1.

Parameters:

x (DNDarray) – 1-dimensional, non-negative ints
weights (DNDarray, optional) – Weight for each value in the input tensor. Array of the same shape as x. Same split as x.
minlength (int, non-negative, optional) – Minimum number of bins

Raises:

ValueError – If x and weights don’t have the same distribution.

Examples

>>> ht.bincount(ht.arange(5))
DNDarray([1, 1, 1, 1, 1], dtype=ht.int64, device=cpu:0, split=None)
>>> ht.bincount(ht.array([0, 1, 3, 2, 1]), weights=ht.array([0, 0.5, 1, 1.5, 2]))
DNDarray([0.0000, 2.5000, 1.5000, 1.0000], dtype=ht.float32, device=cpu:0, split=None)

bucketize(input: heat.core.dndarray.DNDarray, boundaries: heat.core.dndarray.DNDarray | torch.Tensor, out_int32: bool = False, right: bool = False, out: heat.core.dndarray.DNDarray = None) → heat.core.dndarray.DNDarray

Returns the indices of the buckets to which each value in the input belongs, where the boundaries of the buckets are set by boundaries.

Parameters:

input (DNDarray) – The input array.
boundaries (DNDarray or torch.Tensor) – monotonically increasing sequence defining the bucket boundaries, 1-dimensional, not distributed
out_int32 (bool, optional) – set the dtype of the output to ht.int64 (False) or ht.int32 (True)
right (bool, optional) – indicate whether the buckets include the right (False) or left (True) boundaries, see Notes.
out (DNDarray, optional) – The output array, must be the shame shape and split as the input array.

Notes

This function uses the PyTorch’s setting for right:

right	returned index i satisfies
False	boundaries[i-1] < x <= boundaries[i]
True	boundaries[i-1] <= x < boundaries[i]

Raises:: RuntimeError – If boundaries is distributed.

See also

histc()

kurtosis(x: heat.core.dndarray.DNDarray, axis: int | None = None, unbiased: bool = True, Fischer: bool = True) → heat.core.dndarray.DNDarray

Compute the kurtosis (Fisher or Pearson) of a dataset. Kurtosis is the fourth central moment divided by the square of the variance. If Fisher’s definition is used, then 3.0 is subtracted from the result to give 0.0 for a normal distribution.

If unbiased is True (defualt) then the kurtosis is calculated using k statistics to eliminate bias coming from biased moment estimators

Parameters:

x (ht.DNDarray) – Input array
axis (NoneType or Int) – Axis along which skewness is calculated, Default is to compute over the whole array x
unbiased (Bool) – if True (default) the calculations are corrected for bias
Fischer (bool) – Whether use Fischer’s definition or not. If true 3. is subtracted from the result.

Warning

UserWarning: Dependent on the axis given and the split configuration, a UserWarning may be thrown during this function as data is transferred between processes.

max(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int, Ellipsis] | None = None, out: heat.core.dndarray.DNDarray | None = None, keepdims: bool | None = None) → heat.core.dndarray.DNDarray

Return the maximum along a given axis.

Parameters:

x (DNDarray) – Input array.
axis (None or int or Tuple[int,...], optional) – Axis or axes along which to operate. By default, flattened input is used. If this is a tuple of ints, the maximum is selected over multiple axes, instead of a single axis or all the axes as before.
out (DNDarray, optional) – Tuple of two output arrays (max, max_indices). Must be of the same shape and buffer length as the expected output. The minimum value of an output element. Must be present to allow computation on empty slice.
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 original array.

Examples

>>> a = ht.float32([
    [1, 2, 3],
    [4, 5, 6],
    [7, 8, 9],
    [10, 11, 12]
])
>>> ht.max(a)
DNDarray([12.], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.max(a, axis=0)
DNDarray([10., 11., 12.], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.max(a, axis=1)
DNDarray([ 3.,  6.,  9., 12.], dtype=ht.float32, device=cpu:0, split=None)

maximum(x1: heat.core.dndarray.DNDarray, x2: heat.core.dndarray.DNDarray, out: heat.core.dndarray.DNDarray | None = None) → heat.core.dndarray.DNDarray

Compares two DNDarrays and returns a new DNDarray containing the element-wise maxima. The DNDarrays must have the same shape, or shapes that can be broadcast to a single shape. For broadcasting semantics, see: https://pytorch.org/docs/stable/notes/broadcasting.html If one of the elements being compared is NaN, then that element is returned. TODO: Check this: If both elements are NaNs then the first is returned. The latter distinction is important for complex NaNs, which are defined as at least one of the real or imaginary parts being NaN. The net effect is that NaNs are propagated.

Parameters:

x1 (DNDarray) – The first array containing the elements to be compared.
x2 (DNDarray) – The second array containing the elements to be compared.
out (DNDarray, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned.

Examples

>>> import heat as ht
>>> a = ht.random.randn(3, 4)
>>> a
DNDarray([[ 0.2701, -0.6993,  1.2197,  0.0579],
          [ 0.6815,  0.4722, -0.3947, -0.3030],
          [ 1.0101, -1.2460, -1.3953, -0.6879]], dtype=ht.float32, device=cpu:0, split=None)
>>> b = ht.random.randn(3, 4)
>>> b
DNDarray([[ 0.9664,  0.6159, -0.8555,  0.8204],
          [-1.2200, -0.0759,  0.0437,  0.4700],
          [ 1.2271,  1.0530,  0.1095,  0.8386]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.maximum(a, b)
DNDarray([[0.9664, 0.6159, 1.2197, 0.8204],
          [0.6815, 0.4722, 0.0437, 0.4700],
          [1.2271, 1.0530, 0.1095, 0.8386]], dtype=ht.float32, device=cpu:0, split=None)
>>> c = ht.random.randn(1, 4)
>>> c
DNDarray([[-0.5363, -0.9765,  0.4099,  0.3520]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.maximum(a, c)
DNDarray([[ 0.2701, -0.6993,  1.2197,  0.3520],
          [ 0.6815,  0.4722,  0.4099,  0.3520],
          [ 1.0101, -0.9765,  0.4099,  0.3520]], dtype=ht.float32, device=cpu:0, split=None)
>>> d = ht.random.randn(3, 4, 5)
>>> ht.maximum(a, d)
ValueError: operands could not be broadcast, input shapes (3, 4) (3, 4, 5)

mean(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int, Ellipsis] | None = None) → heat.core.dndarray.DNDarray

Calculates and returns the mean of a DNDarray. If an axis is given, the mean will be taken in that direction.

Parameters:

x (DNDarray) – Values for which the mean is calculated for. The dtype of x must be a float
axis (None or int or iterable) – Axis which the mean is taken in. Default None calculates mean of all data items.

Notes

Split semantics when axis is an integer:

if axis==x.split, then mean(x).split=None
if axis>split, then mean(x).split=x.split
if axis<split, then mean(x).split=x.split-1

Examples

>>> a = ht.random.randn(1,3)
>>> a
DNDarray([[-0.1164,  1.0446, -0.4093]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.mean(a)
DNDarray(0.1730, dtype=ht.float32, device=cpu:0, split=None)
>>> a = ht.random.randn(4,4)
>>> a
DNDarray([[-1.0585,  0.7541, -1.1011,  0.5009],
          [-1.3575,  0.3344,  0.4506,  0.7379],
          [-0.4337, -0.6516, -1.3690, -0.8772],
          [ 0.6929, -1.0989, -0.9961,  0.3547]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.mean(a, 1)
DNDarray([-0.2262,  0.0413, -0.8328, -0.2619], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.mean(a, 0)
DNDarray([-0.5392, -0.1655, -0.7539,  0.1791], dtype=ht.float32, device=cpu:0, split=None)
>>> a = ht.random.randn(4,4)
>>> a
DNDarray([[-0.1441,  0.5016,  0.8907,  0.6318],
          [-1.1690, -1.2657,  1.4840, -0.1014],
          [ 0.4133,  1.4168,  1.3499,  1.0340],
          [-0.9236, -0.7535, -0.2466, -0.9703]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.mean(a, (0,1))
DNDarray(0.1342, dtype=ht.float32, device=cpu:0, split=None)

median(x: heat.core.dndarray.DNDarray, axis: int | None = None, keepdims: bool = False) → heat.core.dndarray.DNDarray

Compute the median of the data along the specified axis. Returns the median of the DNDarray elements.

Parameters:

x (DNDarray) – Input tensor
axis (int, or None, optional) – Axis along which the median is computed. Default is None, i.e., the median is computed along a flattened version of the DNDarray.
keepdims (bool, optional) – If True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result can broadcast correctly against the original array a.

min(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int, Ellipsis] | None = None, out: heat.core.dndarray.DNDarray | None = None, keepdims: bool | None = None) → heat.core.dndarray.DNDarray

Return the minimum along a given axis.

Parameters:

x (DNDarray) – Input array.
axis (None or int or Tuple[int,...]) – Axis or axes along which to operate. By default, flattened input is used. If this is a tuple of ints, the minimum is selected over multiple axes, instead of a single axis or all the axes as before.
out (Tuple[DNDarray,DNDarray], optional) – Tuple of two output arrays (min, min_indices). Must be of the same shape and buffer length as the expected output. The maximum value of an output element. Must be present to allow computation on empty slice.
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 original array.

Examples

>>> a = ht.float32([
        [1, 2, 3],
        [4, 5, 6],
        [7, 8, 9],
        [10, 11, 12]
    ])
>>> ht.min(a)
DNDarray([1.], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.min(a, axis=0)
DNDarray([1., 2., 3.], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.min(a, axis=1)
DNDarray([ 1.,  4.,  7., 10.], dtype=ht.float32, device=cpu:0, split=None)

minimum(x1: heat.core.dndarray.DNDarray, x2: heat.core.dndarray.DNDarray, out: heat.core.dndarray.DNDarray | None = None) → heat.core.dndarray.DNDarray

Compares two DNDarrays and returns a new DNDarray containing the element-wise minima. If one of the elements being compared is NaN, then that element is returned. They must have the same shape, or shapes that can be broadcast to a single shape. For broadcasting semantics, see: https://pytorch.org/docs/stable/notes/broadcasting.html TODO: Check this: If both elements are NaNs then the first is returned. The latter distinction is important for complex NaNs, which are defined as at least one of the real or imaginary parts being NaN. The net effect is that NaNs are propagated.

Parameters:

x1 (DNDarray) – The first array containing the elements to be compared.
x2 (DNDarray) – The second array containing the elements to be compared.
out (DNDarray, optional) – A location into which the result is stored. If provided, it must have a shape that the inputs broadcast to. If not provided or None, a freshly-allocated array is returned.

Examples

>>> import heat as ht
>>> a = ht.random.randn(3,4)
>>> a
DNDarray([[-0.5462,  0.0079,  1.2828,  1.4980],
          [ 0.6503, -1.1069,  1.2131,  1.4003],
          [-0.3203, -0.2318,  1.0388,  0.4439]], dtype=ht.float32, device=cpu:0, split=None)
>>> b = ht.random.randn(3,4)
>>> b
DNDarray([[ 1.8505,  2.3055, -0.2825, -1.4718],
          [-0.3684,  1.6866, -0.8570, -0.4779],
          [ 1.0532,  0.3775, -0.8669, -1.7275]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.minimum(a,b)
DNDarray([[-0.5462,  0.0079, -0.2825, -1.4718],
          [-0.3684, -1.1069, -0.8570, -0.4779],
          [-0.3203, -0.2318, -0.8669, -1.7275]], dtype=ht.float32, device=cpu:0, split=None)
>>> c = ht.random.randn(1,4)
>>> c
DNDarray([[-1.4358,  1.2914, -0.6042, -1.4009]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.minimum(a,c)
DNDarray([[-1.4358,  0.0079, -0.6042, -1.4009],
          [-1.4358, -1.1069, -0.6042, -1.4009],
          [-1.4358, -0.2318, -0.6042, -1.4009]], dtype=ht.float32, device=cpu:0, split=None)
>>> d = ht.random.randn(3,4,5)
>>> ht.minimum(a,d)
ValueError: operands could not be broadcast, input shapes (3, 4) (3, 4, 5)

Compute the q-th percentile of the data along the specified axis. Returns the q-th percentile(s) of the tensor elements.

Parameters:

x (DNDarray) – Input tensor
q (DNDarray, scalar, or list of scalars) – Percentile or sequence of percentiles to compute. Must belong to the interval [0, 100].
axis (int, or None, optional) – Axis along which the percentiles are computed. Default is None.
out (DNDarray, optional.) – Output buffer.
interpolation (str, optional) –
Interpolation method to use when the desired percentile lies between two data points \(i < j\). Can be one of:
- ‘linear’: \(i + (j - i) \cdot fraction\), where fraction is the fractional part of the index surrounded by i and j.
- ‘lower’: i.
- ‘higher’: j.
- ‘nearest’: i or j, whichever is nearest.
- ‘midpoint’: \((i + j) / 2\).
keepdims (bool, optional) – If True, the axes which are reduced are left in the result as dimensions with size one. With this option, the result can broadcast correctly against the original array x.

skew(x: heat.core.dndarray.DNDarray, axis: int = None, unbiased: bool = True) → heat.core.dndarray.DNDarray

Compute the sample skewness of a data set.

Parameters:

x (ht.DNDarray) – Input array
axis (NoneType or Int) – Axis along which skewness is calculated, Default is to compute over the whole array x
unbiased (Bool) – if True (default) the calculations are corrected for bias

Warning

UserWarning: Dependent on the axis given and the split configuration, a UserWarning may be thrown during this function as data is transferred between processes.

std(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int] | List[int] = None, ddof: int = 0, **kwargs: object) → heat.core.dndarray.DNDarray

Calculates the standard deviation of a DNDarray with the bessel correction. If an axis is given, the variance will be taken in that direction.

Parameters:

x (DNDarray) – array for which the std is calculated for. The datatype of x must be a float
axis (None or int or iterable) – Axis which the std is taken in. Default None calculates std of all data items.
ddof (int, optional) – Delta Degrees of Freedom: the denominator implicitely used in the calculation is N - ddof, where N represents the number of elements. If ddof=1, the Bessel correction will be applied. Setting ddof>1 raises a NotImplementedError.

Examples

>>> a = ht.random.randn(1, 3)
>>> a
DNDarray([[ 0.5714,  0.0048, -0.2942]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.std(a)
DNDarray(0.3590, dtype=ht.float32, device=cpu:0, split=None)
>>> a = ht.random.randn(4,4)
>>> a
DNDarray([[ 0.8488,  1.2225,  1.2498, -1.4592],
          [-0.5820, -0.3928,  0.1509, -0.0174],
          [ 0.6426, -1.8149,  0.1369,  0.0042],
          [-0.6043, -0.0523, -1.6653,  0.6631]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.std(a, 1, ddof=1)
DNDarray([1.2961, 0.3362, 1.0739, 0.9820], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.std(a, 1)
DNDarray([1.2961, 0.3362, 1.0739, 0.9820], dtype=ht.float32, device=cpu:0, split=None)

var(x: heat.core.dndarray.DNDarray, axis: int | Tuple[int] | List[int] = None, ddof: int = 0, **kwargs: object) → heat.core.dndarray.DNDarray

Calculates and returns the variance of a DNDarray. If an axis is given, the variance will be taken in that direction.

Parameters:

x (DNDarray) – Array for which the variance is calculated for. The datatype of x must be a float
axis (None or int or iterable) – Axis which the std is taken in. Default None calculates std of all data items.
ddof (int, optional) – Delta Degrees of Freedom: the denominator implicitely used in the calculation is N - ddof, where N represents the number of elements. If ddof=1, the Bessel correction will be applied. Setting ddof>1 raises a NotImplementedError.

Notes

Split semantics when axis is an integer:

if axis=x.split, then var(x).split=None
if axis>split, then var(x).split = x.split
if axis<split, then var(x).split=x.split - 1

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.

Examples

>>> a = ht.random.randn(1,3)
>>> a
DNDarray([[-2.3589, -0.2073,  0.8806]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a)
DNDarray(1.8119, dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a, ddof=1)
DNDarray(2.7179, dtype=ht.float32, device=cpu:0, split=None)
>>> a = ht.random.randn(4,4)
>>> a
DNDarray([[-0.8523, -1.4982, -0.5848, -0.2554],
          [ 0.8458, -0.3125, -0.2430,  1.9016],
          [-0.6778, -0.3584, -1.5112,  0.6545],
          [-0.9161,  0.0168,  0.0462,  0.5964]], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a, 1)
DNDarray([0.2777, 1.0957, 0.8015, 0.3936], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a, 0)
DNDarray([0.7001, 0.4376, 0.4576, 0.7890], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a, 0, ddof=1)
DNDarray([0.7001, 0.4376, 0.4576, 0.7890], dtype=ht.float32, device=cpu:0, split=None)
>>> ht.var(a, 0, ddof=0)
DNDarray([0.7001, 0.4376, 0.4576, 0.7890], dtype=ht.float32, device=cpu:0, split=None)

right	order of bins	returned index i satisfies
False	increasing	bins[i-1] <= x < bins[i]
True	increasing	bins[i-1] < x <= bins[i]
False	decreasing	bins[i-1] > x >= bins[i]
True	decreasing	bins[i-1] >= x > bins[i]

heat.statistics

Module Contents

`heat.statistics`