# Copyright (c) 2013, Mahmoud Hashemi
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"""``statsutils`` provides tools aimed primarily at descriptive
statistics for data analysis, such as :func:`mean` (average),
:func:`median`, :func:`variance`, and many others,
The :class:`Stats` type provides all the main functionality of the
``statsutils`` module. A :class:`Stats` object wraps a given dataset,
providing all statistical measures as property attributes. These
attributes cache their results, which allows efficient computation of
multiple measures, as many measures rely on other measures. For
example, relative standard deviation (:attr:`Stats.rel_std_dev`)
relies on both the mean and standard deviation. The Stats object
caches those results so no rework is done.
The :class:`Stats` type's attributes have module-level counterparts for
convenience when the computation reuse advantages do not apply.
>>> stats = Stats(range(42))
>>> stats.mean
20.5
>>> mean(range(42))
20.5
Statistics is a large field, and ``statsutils`` is focused on a few
basic techniques that are useful in software. The following is a brief
introduction to those techniques. For a more in-depth introduction,
`Statistics for Software
<https://www.paypal-engineering.com/2016/04/11/statistics-for-software/>`_,
an article I wrote on the topic. It introduces key terminology vital
to effective usage of statistics.
Statistical moments
-------------------
Python programmers are probably familiar with the concept of the
*mean* or *average*, which gives a rough quantitiative middle value by
which a sample can be can be generalized. However, the mean is just
the first of four `moment`_-based measures by which a sample or
distribution can be measured.
The four `Standardized moments`_ are:
1. `Mean`_ - :func:`mean` - theoretical middle value
2. `Variance`_ - :func:`variance` - width of value dispersion
3. `Skewness`_ - :func:`skewness` - symmetry of distribution
4. `Kurtosis`_ - :func:`kurtosis` - "peakiness" or "long-tailed"-ness
For more information check out `the Moment article on Wikipedia`_.
.. _moment: https://en.wikipedia.org/wiki/Moment_(mathematics)
.. _Standardized moments: https://en.wikipedia.org/wiki/Standardized_moment
.. _Mean: https://en.wikipedia.org/wiki/Mean
.. _Variance: https://en.wikipedia.org/wiki/Variance
.. _Skewness: https://en.wikipedia.org/wiki/Skewness
.. _Kurtosis: https://en.wikipedia.org/wiki/Kurtosis
.. _the Moment article on Wikipedia: https://en.wikipedia.org/wiki/Moment_(mathematics)
Keep in mind that while these moments can give a bit more insight into
the shape and distribution of data, they do not guarantee a complete
picture. Wildly different datasets can have the same values for all
four moments, so generalize wisely.
Robust statistics
-----------------
Moment-based statistics are notorious for being easily skewed by
outliers. The whole field of robust statistics aims to mitigate this
dilemma. ``statsutils`` also includes several robust statistical methods:
* `Median`_ - The middle value of a sorted dataset
* `Trimean`_ - Another robust measure of the data's central tendency
* `Median Absolute Deviation`_ (MAD) - A robust measure of
variability, a natural counterpart to :func:`variance`.
* `Trimming`_ - Reducing a dataset to only the middle majority of
data is a simple way of making other estimators more robust.
.. _Median: https://en.wikipedia.org/wiki/Median
.. _Trimean: https://en.wikipedia.org/wiki/Trimean
.. _Median Absolute Deviation: https://en.wikipedia.org/wiki/Median_absolute_deviation
.. _Trimming: https://en.wikipedia.org/wiki/Trimmed_estimator
Online and Offline Statistics
-----------------------------
Unrelated to computer networking, `online`_ statistics involve
calculating statistics in a `streaming`_ fashion, without all the data
being available. The :class:`Stats` type is meant for the more
traditional offline statistics when all the data is available. For
pure-Python online statistics accumulators, look at the `Lithoxyl`_
system instrumentation package.
.. _Online: https://en.wikipedia.org/wiki/Online_algorithm
.. _streaming: https://en.wikipedia.org/wiki/Streaming_algorithm
.. _Lithoxyl: https://github.com/mahmoud/lithoxyl
"""
import bisect
from math import floor, ceil
from collections import Counter
class _StatsProperty:
def __init__(self, name, func):
self.name = name
self.func = func
self.internal_name = '_' + name
doc = func.__doc__ or ''
pre_doctest_doc, _, _ = doc.partition('>>>')
self.__doc__ = pre_doctest_doc
def __get__(self, obj, objtype=None):
if obj is None:
return self
if not obj.data:
return obj.default
try:
return getattr(obj, self.internal_name)
except AttributeError:
setattr(obj, self.internal_name, self.func(obj))
return getattr(obj, self.internal_name)
[docs]
class Stats:
"""The ``Stats`` type is used to represent a group of unordered
statistical datapoints for calculations such as mean, median, and
variance.
Args:
data (list): List or other iterable containing numeric values.
default (float): A value to be returned when a given
statistical measure is not defined. 0.0 by default, but
``float('nan')`` is appropriate for stricter applications.
use_copy (bool): By default Stats objects copy the initial
data into a new list to avoid issues with
modifications. Pass ``False`` to disable this behavior.
is_sorted (bool): Presorted data can skip an extra sorting
step for a little speed boost. Defaults to False.
"""
def __init__(self, data, default=0.0, use_copy=True, is_sorted=False):
self._use_copy = use_copy
self._is_sorted = is_sorted
if use_copy:
self.data = list(data)
else:
self.data = data
self.default = default
cls = self.__class__
self._prop_attr_names = [a for a in dir(self)
if isinstance(getattr(cls, a, None),
_StatsProperty)]
self._pearson_precision = 0
def __len__(self):
return len(self.data)
def __iter__(self):
return iter(self.data)
def _get_sorted_data(self):
"""When using a copy of the data, it's better to have that copy be
sorted, but we do it lazily using this method, in case no
sorted measures are used. I.e., if median is never called,
sorting would be a waste.
When not using a copy, it's presumed that all optimizations
are on the user.
"""
if not self._use_copy:
return sorted(self.data)
elif not self._is_sorted:
self.data.sort()
return self.data
[docs]
def clear_cache(self):
"""``Stats`` objects automatically cache intermediary calculations
that can be reused. For instance, accessing the ``std_dev``
attribute after the ``variance`` attribute will be
significantly faster for medium-to-large datasets.
If you modify the object by adding additional data points,
call this function to have the cached statistics recomputed.
"""
for attr_name in self._prop_attr_names:
attr_name = getattr(self.__class__, attr_name).internal_name
if not hasattr(self, attr_name):
continue
delattr(self, attr_name)
return
def _calc_count(self):
"""The number of items in this Stats object. Returns the same as
:func:`len` on a Stats object, but provided for pandas terminology
parallelism.
>>> Stats(range(20)).count
20
"""
return len(self.data)
count = _StatsProperty('count', _calc_count)
def _calc_mean(self):
"""
The arithmetic mean, or "average". Sum of the values divided by
the number of values.
>>> mean(range(20))
9.5
>>> mean(list(range(19)) + [949]) # 949 is an arbitrary outlier
56.0
"""
return sum(self.data, 0.0) / len(self.data)
mean = _StatsProperty('mean', _calc_mean)
def _calc_max(self):
"""
The maximum value present in the data.
>>> Stats([2, 1, 3]).max
3
"""
if self._is_sorted:
return self.data[-1]
return max(self.data)
max = _StatsProperty('max', _calc_max)
def _calc_min(self):
"""
The minimum value present in the data.
>>> Stats([2, 1, 3]).min
1
"""
if self._is_sorted:
return self.data[0]
return min(self.data)
min = _StatsProperty('min', _calc_min)
def _calc_median(self):
"""
The median is either the middle value or the average of the two
middle values of a sample. Compared to the mean, it's generally
more resilient to the presence of outliers in the sample.
>>> median([2, 1, 3])
2
>>> median(range(97))
48
>>> median(list(range(96)) + [1066]) # 1066 is an arbitrary outlier
48
"""
return self._get_quantile(self._get_sorted_data(), 0.5)
median = _StatsProperty('median', _calc_median)
def _calc_iqr(self):
"""Inter-quartile range (IQR) is the difference between the 75th
percentile and 25th percentile. IQR is a robust measure of
dispersion, like standard deviation, but safer to compare
between datasets, as it is less influenced by outliers.
>>> iqr([1, 2, 3, 4, 5])
2
>>> iqr(range(1001))
500
"""
return self.get_quantile(0.75) - self.get_quantile(0.25)
iqr = _StatsProperty('iqr', _calc_iqr)
def _calc_trimean(self):
"""The trimean is a robust measure of central tendency, like the
median, that takes the weighted average of the median and the
upper and lower quartiles.
>>> trimean([2, 1, 3])
2.0
>>> trimean(range(97))
48.0
>>> trimean(list(range(96)) + [1066]) # 1066 is an arbitrary outlier
48.0
"""
sorted_data = self._get_sorted_data()
gq = lambda q: self._get_quantile(sorted_data, q)
return (gq(0.25) + (2 * gq(0.5)) + gq(0.75)) / 4.0
trimean = _StatsProperty('trimean', _calc_trimean)
def _calc_variance(self):
"""\
Variance is the average of the squares of the difference between
each value and the mean.
>>> variance(range(97))
784.0
"""
global mean # defined elsewhere in this file
return mean(self._get_pow_diffs(2))
variance = _StatsProperty('variance', _calc_variance)
def _calc_std_dev(self):
"""\
Standard deviation. Square root of the variance.
>>> std_dev(range(97))
28.0
"""
return self.variance ** 0.5
std_dev = _StatsProperty('std_dev', _calc_std_dev)
def _calc_median_abs_dev(self):
"""\
Median Absolute Deviation is a robust measure of statistical
dispersion: http://en.wikipedia.org/wiki/Median_absolute_deviation
>>> median_abs_dev(range(97))
24.0
"""
global median # defined elsewhere in this file
sorted_vals = sorted(self.data)
x = float(median(sorted_vals))
return median([abs(x - v) for v in sorted_vals])
median_abs_dev = _StatsProperty('median_abs_dev', _calc_median_abs_dev)
mad = median_abs_dev # convenience
def _calc_rel_std_dev(self):
"""\
Standard deviation divided by the absolute value of the average.
http://en.wikipedia.org/wiki/Relative_standard_deviation
>>> print('%1.3f' % rel_std_dev(range(97)))
0.583
"""
abs_mean = abs(self.mean)
if abs_mean:
return self.std_dev / abs_mean
else:
return self.default
rel_std_dev = _StatsProperty('rel_std_dev', _calc_rel_std_dev)
def _calc_skewness(self):
"""\
Indicates the asymmetry of a curve. Positive values mean the bulk
of the values are on the left side of the average and vice versa.
http://en.wikipedia.org/wiki/Skewness
See the module docstring for more about statistical moments.
>>> skewness(range(97)) # symmetrical around 48.0
0.0
>>> left_skewed = skewness(list(range(97)) + list(range(10)))
>>> right_skewed = skewness(list(range(97)) + list(range(87, 97)))
>>> round(left_skewed, 3), round(right_skewed, 3)
(0.114, -0.114)
"""
data, s_dev = self.data, self.std_dev
if len(data) > 1 and s_dev > 0:
return (sum(self._get_pow_diffs(3)) /
float((len(data) - 1) * (s_dev ** 3)))
else:
return self.default
skewness = _StatsProperty('skewness', _calc_skewness)
def _calc_kurtosis(self):
"""\
Indicates how much data is in the tails of the distribution. The
result is always positive, with the normal "bell-curve"
distribution having a kurtosis of 3.
http://en.wikipedia.org/wiki/Kurtosis
See the module docstring for more about statistical moments.
>>> kurtosis(range(9))
1.99125
With a kurtosis of 1.99125, [0, 1, 2, 3, 4, 5, 6, 7, 8] is more
centrally distributed than the normal curve.
"""
data, s_dev = self.data, self.std_dev
if len(data) > 1 and s_dev > 0:
return (sum(self._get_pow_diffs(4)) /
float((len(data) - 1) * (s_dev ** 4)))
else:
return 0.0
kurtosis = _StatsProperty('kurtosis', _calc_kurtosis)
def _calc_pearson_type(self):
precision = self._pearson_precision
skewness = self.skewness
kurtosis = self.kurtosis
beta1 = skewness ** 2.0
beta2 = kurtosis * 1.0
# TODO: range checks?
c0 = (4 * beta2) - (3 * beta1)
c1 = skewness * (beta2 + 3)
c2 = (2 * beta2) - (3 * beta1) - 6
if round(c1, precision) == 0:
if round(beta2, precision) == 3:
return 0 # Normal
else:
if beta2 < 3:
return 2 # Symmetric Beta
elif beta2 > 3:
return 7
elif round(c2, precision) == 0:
return 3 # Gamma
else:
k = c1 ** 2 / (4 * c0 * c2)
if k < 0:
return 1 # Beta
raise RuntimeError('missed a spot')
pearson_type = _StatsProperty('pearson_type', _calc_pearson_type)
@staticmethod
def _get_quantile(sorted_data, q):
data, n = sorted_data, len(sorted_data)
idx = q / 1.0 * (n - 1)
idx_f, idx_c = int(floor(idx)), int(ceil(idx))
if idx_f == idx_c:
return data[idx_f]
return (data[idx_f] * (idx_c - idx)) + (data[idx_c] * (idx - idx_f))
[docs]
def get_quantile(self, q):
"""Get a quantile from the dataset. Quantiles are floating point
values between ``0.0`` and ``1.0``, with ``0.0`` representing
the minimum value in the dataset and ``1.0`` representing the
maximum. ``0.5`` represents the median:
>>> Stats(range(100)).get_quantile(0.5)
49.5
"""
q = float(q)
if not 0.0 <= q <= 1.0:
raise ValueError('expected q between 0.0 and 1.0, not %r' % q)
elif not self.data:
return self.default
return self._get_quantile(self._get_sorted_data(), q)
[docs]
def get_zscore(self, value):
"""Get the z-score for *value* in the group. If the standard deviation
is 0, 0 inf or -inf will be returned to indicate whether the value is
equal to, greater than or below the group's mean.
"""
mean = self.mean
if self.std_dev == 0:
if value == mean:
return 0
if value > mean:
return float('inf')
if value < mean:
return float('-inf')
return (float(value) - mean) / self.std_dev
[docs]
def trim_relative(self, amount=0.15):
"""A utility function used to cut a proportion of values off each end
of a list of values. This has the effect of limiting the
effect of outliers.
Args:
amount (float): A value between 0.0 and 0.5 to trim off of
each side of the data.
.. note:
This operation modifies the data in-place. It does not
make or return a copy.
"""
trim = float(amount)
if not 0.0 <= trim < 0.5:
raise ValueError('expected amount between 0.0 and 0.5, not %r'
% trim)
size = len(self.data)
size_diff = int(size * trim)
if size_diff == 0.0:
return
self.data = self._get_sorted_data()[size_diff:-size_diff]
self.clear_cache()
def _get_pow_diffs(self, power):
"""
A utility function used for calculating statistical moments.
"""
m = self.mean
return [(v - m) ** power for v in self.data]
def _get_bin_bounds(self, count=None, with_max=False):
if not self.data:
return [0.0] # TODO: raise?
data = self.data
len_data, min_data, max_data = len(data), min(data), max(data)
if len_data < 4:
if not count:
count = len_data
dx = (max_data - min_data) / float(count)
bins = [min_data + (dx * i) for i in range(count)]
elif count is None:
# freedman algorithm for fixed-width bin selection
q25, q75 = self.get_quantile(0.25), self.get_quantile(0.75)
dx = 2 * (q75 - q25) / (len_data ** (1 / 3.0))
bin_count = max(1, int(ceil((max_data - min_data) / dx)))
bins = [min_data + (dx * i) for i in range(bin_count + 1)]
bins = [b for b in bins if b < max_data]
else:
dx = (max_data - min_data) / float(count)
bins = [min_data + (dx * i) for i in range(count)]
if with_max:
bins.append(float(max_data))
return bins
[docs]
def get_histogram_counts(self, bins=None, **kw):
"""Produces a list of ``(bin, count)`` pairs comprising a histogram of
the Stats object's data, using fixed-width bins. See
:meth:`Stats.format_histogram` for more details.
Args:
bins (int): maximum number of bins, or list of
floating-point bin boundaries. Defaults to the output of
Freedman's algorithm.
bin_digits (int): Number of digits used to round down the
bin boundaries. Defaults to 1.
The output of this method can be stored and/or modified, and
then passed to :func:`statsutils.format_histogram_counts` to
achieve the same text formatting as the
:meth:`~Stats.format_histogram` method. This can be useful for
snapshotting over time.
"""
bin_digits = int(kw.pop('bin_digits', 1))
if kw:
raise TypeError('unexpected keyword arguments: %r' % kw.keys())
if not bins:
bins = self._get_bin_bounds()
else:
try:
bin_count = int(bins)
except TypeError:
try:
bins = [float(x) for x in bins]
except Exception:
raise ValueError('bins expected integer bin count or list'
' of float bin boundaries, not %r' % bins)
if self.min < bins[0]:
bins = [self.min] + bins
else:
bins = self._get_bin_bounds(bin_count)
# floor and ceil really should have taken ndigits, like round()
round_factor = 10.0 ** bin_digits
bins = [floor(b * round_factor) / round_factor for b in bins]
bins = sorted(set(bins))
idxs = [bisect.bisect(bins, d) - 1 for d in self.data]
count_map = Counter(idxs)
bin_counts = [(b, count_map.get(i, 0)) for i, b in enumerate(bins)]
return bin_counts
[docs]
def describe(self, quantiles=None, format=None):
"""Provides standard summary statistics for the data in the Stats
object, in one of several convenient formats.
Args:
quantiles (list): A list of numeric values to use as
quantiles in the resulting summary. All values must be
0.0-1.0, with 0.5 representing the median. Defaults to
``[0.25, 0.5, 0.75]``, representing the standard
quartiles.
format (str): Controls the return type of the function,
with one of three valid values: ``"dict"`` gives back
a :class:`dict` with the appropriate keys and
values. ``"list"`` is a list of key-value pairs in an
order suitable to pass to an OrderedDict or HTML
table. ``"text"`` converts the values to text suitable
for printing, as seen below.
Here is the information returned by a default ``describe``, as
presented in the ``"text"`` format:
>>> stats = Stats(range(1, 8))
>>> print(stats.describe(format='text'))
count: 7
mean: 4.0
std_dev: 2.0
mad: 2.0
min: 1
0.25: 2.5
0.5: 4
0.75: 5.5
max: 7
For more advanced descriptive statistics, check out my blog
post on the topic `Statistics for Software
<https://www.paypal-engineering.com/2016/04/11/statistics-for-software/>`_.
"""
if format is None:
format = 'dict'
elif format not in ('dict', 'list', 'text'):
raise ValueError('invalid format for describe,'
' expected one of "dict"/"list"/"text", not %r'
% format)
quantiles = quantiles or [0.25, 0.5, 0.75]
q_items = []
for q in quantiles:
q_val = self.get_quantile(q)
q_items.append((str(q), q_val))
items = [('count', self.count),
('mean', self.mean),
('std_dev', self.std_dev),
('mad', self.mad),
('min', self.min)]
items.extend(q_items)
items.append(('max', self.max))
if format == 'dict':
ret = dict(items)
elif format == 'list':
ret = items
elif format == 'text':
ret = '\n'.join(['{}{}'.format((label + ':').ljust(10), val)
for label, val in items])
return ret
[docs]
def describe(data, quantiles=None, format=None):
"""A convenience function to get standard summary statistics useful
for describing most data. See :meth:`Stats.describe` for more
details.
>>> print(describe(range(7), format='text'))
count: 7
mean: 3.0
std_dev: 2.0
mad: 2.0
min: 0
0.25: 1.5
0.5: 3
0.75: 4.5
max: 6
See :meth:`Stats.format_histogram` for another very useful
summarization that uses textual visualization.
"""
return Stats(data).describe(quantiles=quantiles, format=format)
def _get_conv_func(attr_name):
def stats_helper(data, default=0.0):
return getattr(Stats(data, default=default, use_copy=False),
attr_name)
return stats_helper
for attr_name, attr in list(Stats.__dict__.items()):
if isinstance(attr, _StatsProperty):
if attr_name in ('max', 'min', 'count'): # don't shadow builtins
continue
if attr_name in ('mad',): # convenience aliases
continue
func = _get_conv_func(attr_name)
func.__doc__ = attr.func.__doc__
globals()[attr_name] = func
delattr(Stats, '_calc_' + attr_name)
# cleanup
del attr
del attr_name
del func