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diff --git a/bench/statistics.py b/bench/statistics.py new file mode 100644 index 0000000..25a26d4 --- /dev/null +++ b/bench/statistics.py @@ -0,0 +1,595 @@ +## Module statistics.py +## +## Copyright (c) 2013 Steven D'Aprano <steve+python@pearwood.info>. +## +## 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 +## +## http://www.apache.org/licenses/LICENSE-2.0 +## +## 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. + + +""" +Basic statistics module. + +This module provides functions for calculating statistics of data, including +averages, variance, and standard deviation. + +Calculating averages +-------------------- + +================== ============================================= +Function Description +================== ============================================= +mean Arithmetic mean (average) of data. +median Median (middle value) of data. +median_low Low median of data. +median_high High median of data. +median_grouped Median, or 50th percentile, of grouped data. +mode Mode (most common value) of data. +================== ============================================= + +Calculate the arithmetic mean ("the average") of data: + +>>> mean([-1.0, 2.5, 3.25, 5.75]) +2.625 + + +Calculate the standard median of discrete data: + +>>> median([2, 3, 4, 5]) +3.5 + + +Calculate the median, or 50th percentile, of data grouped into class intervals +centred on the data values provided. E.g. if your data points are rounded to +the nearest whole number: + +>>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS +2.8333333333... + +This should be interpreted in this way: you have two data points in the class +interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in +the class interval 3.5-4.5. The median of these data points is 2.8333... + + +Calculating variability or spread +--------------------------------- + +================== ============================================= +Function Description +================== ============================================= +pvariance Population variance of data. +variance Sample variance of data. +pstdev Population standard deviation of data. +stdev Sample standard deviation of data. +================== ============================================= + +Calculate the standard deviation of sample data: + +>>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS +4.38961843444... + +If you have previously calculated the mean, you can pass it as the optional +second argument to the four "spread" functions to avoid recalculating it: + +>>> data = [1, 2, 2, 4, 4, 4, 5, 6] +>>> mu = mean(data) +>>> pvariance(data, mu) +2.5 + + +Exceptions +---------- + +A single exception is defined: StatisticsError is a subclass of ValueError. + +""" + +__all__ = [ 'StatisticsError', + 'pstdev', 'pvariance', 'stdev', 'variance', + 'median', 'median_low', 'median_high', 'median_grouped', + 'mean', 'mode', + ] + + +import collections +import math + +from fractions import Fraction +from decimal import Decimal + + +# === Exceptions === + +class StatisticsError(ValueError): + pass + + +# === Private utilities === + +def _sum(data, start=0): + """_sum(data [, start]) -> value + + Return a high-precision sum of the given numeric data. If optional + argument ``start`` is given, it is added to the total. If ``data`` is + empty, ``start`` (defaulting to 0) is returned. + + + Examples + -------- + + >>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75) + 11.0 + + Some sources of round-off error will be avoided: + + >>> _sum([1e50, 1, -1e50] * 1000) # Built-in sum returns zero. + 1000.0 + + Fractions and Decimals are also supported: + + >>> from fractions import Fraction as F + >>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)]) + Fraction(63, 20) + + >>> from decimal import Decimal as D + >>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")] + >>> _sum(data) + Decimal('0.6963') + + Mixed types are currently treated as an error, except that int is + allowed. + """ + # We fail as soon as we reach a value that is not an int or the type of + # the first value which is not an int. E.g. _sum([int, int, float, int]) + # is okay, but sum([int, int, float, Fraction]) is not. + allowed_types = set([int, type(start)]) + n, d = _exact_ratio(start) + partials = {d: n} # map {denominator: sum of numerators} + # Micro-optimizations. + exact_ratio = _exact_ratio + partials_get = partials.get + # Add numerators for each denominator. + for x in data: + _check_type(type(x), allowed_types) + n, d = exact_ratio(x) + partials[d] = partials_get(d, 0) + n + # Find the expected result type. If allowed_types has only one item, it + # will be int; if it has two, use the one which isn't int. + assert len(allowed_types) in (1, 2) + if len(allowed_types) == 1: + assert allowed_types.pop() is int + T = int + else: + T = (allowed_types - set([int])).pop() + if None in partials: + assert issubclass(T, (float, Decimal)) + assert not math.isfinite(partials[None]) + return T(partials[None]) + total = Fraction() + for d, n in sorted(partials.items()): + total += Fraction(n, d) + if issubclass(T, int): + assert total.denominator == 1 + return T(total.numerator) + if issubclass(T, Decimal): + return T(total.numerator)/total.denominator + return T(total) + + +def _check_type(T, allowed): + if T not in allowed: + if len(allowed) == 1: + allowed.add(T) + else: + types = ', '.join([t.__name__ for t in allowed] + [T.__name__]) + raise TypeError("unsupported mixed types: %s" % types) + + +def _exact_ratio(x): + """Convert Real number x exactly to (numerator, denominator) pair. + + >>> _exact_ratio(0.25) + (1, 4) + + x is expected to be an int, Fraction, Decimal or float. + """ + try: + try: + # int, Fraction + return (x.numerator, x.denominator) + except AttributeError: + # float + try: + return x.as_integer_ratio() + except AttributeError: + # Decimal + try: + return _decimal_to_ratio(x) + except AttributeError: + msg = "can't convert type '{}' to numerator/denominator" + raise TypeError(msg.format(type(x).__name__)) from None + except (OverflowError, ValueError): + # INF or NAN + if __debug__: + # Decimal signalling NANs cannot be converted to float :-( + if isinstance(x, Decimal): + assert not x.is_finite() + else: + assert not math.isfinite(x) + return (x, None) + + +# FIXME This is faster than Fraction.from_decimal, but still too slow. +def _decimal_to_ratio(d): + """Convert Decimal d to exact integer ratio (numerator, denominator). + + >>> from decimal import Decimal + >>> _decimal_to_ratio(Decimal("2.6")) + (26, 10) + + """ + sign, digits, exp = d.as_tuple() + if exp in ('F', 'n', 'N'): # INF, NAN, sNAN + assert not d.is_finite() + raise ValueError + num = 0 + for digit in digits: + num = num*10 + digit + if exp < 0: + den = 10**-exp + else: + num *= 10**exp + den = 1 + if sign: + num = -num + return (num, den) + + +def _counts(data): + # Generate a table of sorted (value, frequency) pairs. + table = collections.Counter(iter(data)).most_common() + if not table: + return table + # Extract the values with the highest frequency. + maxfreq = table[0][1] + for i in range(1, len(table)): + if table[i][1] != maxfreq: + table = table[:i] + break + return table + + +# === Measures of central tendency (averages) === + +def mean(data): + """Return the sample arithmetic mean of data. + + >>> mean([1, 2, 3, 4, 4]) + 2.8 + + >>> from fractions import Fraction as F + >>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)]) + Fraction(13, 21) + + >>> from decimal import Decimal as D + >>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")]) + Decimal('0.5625') + + If ``data`` is empty, StatisticsError will be raised. + """ + if iter(data) is data: + data = list(data) + n = len(data) + if n < 1: + raise StatisticsError('mean requires at least one data point') + return _sum(data)/n + + +# FIXME: investigate ways to calculate medians without sorting? Quickselect? +def median(data): + """Return the median (middle value) of numeric data. + + When the number of data points is odd, return the middle data point. + When the number of data points is even, the median is interpolated by + taking the average of the two middle values: + + >>> median([1, 3, 5]) + 3 + >>> median([1, 3, 5, 7]) + 4.0 + + """ + data = sorted(data) + n = len(data) + if n == 0: + raise StatisticsError("no median for empty data") + if n%2 == 1: + return data[n//2] + else: + i = n//2 + return (data[i - 1] + data[i])/2 + + +def median_low(data): + """Return the low median of numeric data. + + When the number of data points is odd, the middle value is returned. + When it is even, the smaller of the two middle values is returned. + + >>> median_low([1, 3, 5]) + 3 + >>> median_low([1, 3, 5, 7]) + 3 + + """ + data = sorted(data) + n = len(data) + if n == 0: + raise StatisticsError("no median for empty data") + if n%2 == 1: + return data[n//2] + else: + return data[n//2 - 1] + + +def median_high(data): + """Return the high median of data. + + When the number of data points is odd, the middle value is returned. + When it is even, the larger of the two middle values is returned. + + >>> median_high([1, 3, 5]) + 3 + >>> median_high([1, 3, 5, 7]) + 5 + + """ + data = sorted(data) + n = len(data) + if n == 0: + raise StatisticsError("no median for empty data") + return data[n//2] + + +def median_grouped(data, interval=1): + """"Return the 50th percentile (median) of grouped continuous data. + + >>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5]) + 3.7 + >>> median_grouped([52, 52, 53, 54]) + 52.5 + + This calculates the median as the 50th percentile, and should be + used when your data is continuous and grouped. In the above example, + the values 1, 2, 3, etc. actually represent the midpoint of classes + 0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in + class 3.5-4.5, and interpolation is used to estimate it. + + Optional argument ``interval`` represents the class interval, and + defaults to 1. Changing the class interval naturally will change the + interpolated 50th percentile value: + + >>> median_grouped([1, 3, 3, 5, 7], interval=1) + 3.25 + >>> median_grouped([1, 3, 3, 5, 7], interval=2) + 3.5 + + This function does not check whether the data points are at least + ``interval`` apart. + """ + data = sorted(data) + n = len(data) + if n == 0: + raise StatisticsError("no median for empty data") + elif n == 1: + return data[0] + # Find the value at the midpoint. Remember this corresponds to the + # centre of the class interval. + x = data[n//2] + for obj in (x, interval): + if isinstance(obj, (str, bytes)): + raise TypeError('expected number but got %r' % obj) + try: + L = x - interval/2 # The lower limit of the median interval. + except TypeError: + # Mixed type. For now we just coerce to float. + L = float(x) - float(interval)/2 + cf = data.index(x) # Number of values below the median interval. + # FIXME The following line could be more efficient for big lists. + f = data.count(x) # Number of data points in the median interval. + return L + interval*(n/2 - cf)/f + + +def mode(data): + """Return the most common data point from discrete or nominal data. + + ``mode`` assumes discrete data, and returns a single value. This is the + standard treatment of the mode as commonly taught in schools: + + >>> mode([1, 1, 2, 3, 3, 3, 3, 4]) + 3 + + This also works with nominal (non-numeric) data: + + >>> mode(["red", "blue", "blue", "red", "green", "red", "red"]) + 'red' + + If there is not exactly one most common value, ``mode`` will raise + StatisticsError. + """ + # Generate a table of sorted (value, frequency) pairs. + table = _counts(data) + if len(table) == 1: + return table[0][0] + elif table: + raise StatisticsError( + 'no unique mode; found %d equally common values' % len(table) + ) + else: + raise StatisticsError('no mode for empty data') + + +# === Measures of spread === + +# See http://mathworld.wolfram.com/Variance.html +# http://mathworld.wolfram.com/SampleVariance.html +# http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance +# +# Under no circumstances use the so-called "computational formula for +# variance", as that is only suitable for hand calculations with a small +# amount of low-precision data. It has terrible numeric properties. +# +# See a comparison of three computational methods here: +# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/ + +def _ss(data, c=None): + """Return sum of square deviations of sequence data. + + If ``c`` is None, the mean is calculated in one pass, and the deviations + from the mean are calculated in a second pass. Otherwise, deviations are + calculated from ``c`` as given. Use the second case with care, as it can + lead to garbage results. + """ + if c is None: + c = mean(data) + ss = _sum((x-c)**2 for x in data) + # The following sum should mathematically equal zero, but due to rounding + # error may not. + ss -= _sum((x-c) for x in data)**2/len(data) + assert not ss < 0, 'negative sum of square deviations: %f' % ss + return ss + + +def variance(data, xbar=None): + """Return the sample variance of data. + + data should be an iterable of Real-valued numbers, with at least two + values. The optional argument xbar, if given, should be the mean of + the data. If it is missing or None, the mean is automatically calculated. + + Use this function when your data is a sample from a population. To + calculate the variance from the entire population, see ``pvariance``. + + Examples: + + >>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5] + >>> variance(data) + 1.3720238095238095 + + If you have already calculated the mean of your data, you can pass it as + the optional second argument ``xbar`` to avoid recalculating it: + + >>> m = mean(data) + >>> variance(data, m) + 1.3720238095238095 + + This function does not check that ``xbar`` is actually the mean of + ``data``. Giving arbitrary values for ``xbar`` may lead to invalid or + impossible results. + + Decimals and Fractions are supported: + + >>> from decimal import Decimal as D + >>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) + Decimal('31.01875') + + >>> from fractions import Fraction as F + >>> variance([F(1, 6), F(1, 2), F(5, 3)]) + Fraction(67, 108) + + """ + if iter(data) is data: + data = list(data) + n = len(data) + if n < 2: + raise StatisticsError('variance requires at least two data points') + ss = _ss(data, xbar) + return ss/(n-1) + + +def pvariance(data, mu=None): + """Return the population variance of ``data``. + + data should be an iterable of Real-valued numbers, with at least one + value. The optional argument mu, if given, should be the mean of + the data. If it is missing or None, the mean is automatically calculated. + + Use this function to calculate the variance from the entire population. + To estimate the variance from a sample, the ``variance`` function is + usually a better choice. + + Examples: + + >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25] + >>> pvariance(data) + 1.25 + + If you have already calculated the mean of the data, you can pass it as + the optional second argument to avoid recalculating it: + + >>> mu = mean(data) + >>> pvariance(data, mu) + 1.25 + + This function does not check that ``mu`` is actually the mean of ``data``. + Giving arbitrary values for ``mu`` may lead to invalid or impossible + results. + + Decimals and Fractions are supported: + + >>> from decimal import Decimal as D + >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) + Decimal('24.815') + + >>> from fractions import Fraction as F + >>> pvariance([F(1, 4), F(5, 4), F(1, 2)]) + Fraction(13, 72) + + """ + if iter(data) is data: + data = list(data) + n = len(data) + if n < 1: + raise StatisticsError('pvariance requires at least one data point') + ss = _ss(data, mu) + return ss/n + + +def stdev(data, xbar=None): + """Return the square root of the sample variance. + + See ``variance`` for arguments and other details. + + >>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) + 1.0810874155219827 + + """ + var = variance(data, xbar) + try: + return var.sqrt() + except AttributeError: + return math.sqrt(var) + + +def pstdev(data, mu=None): + """Return the square root of the population variance. + + See ``pvariance`` for arguments and other details. + + >>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75]) + 0.986893273527251 + + """ + var = pvariance(data, mu) + try: + return var.sqrt() + except AttributeError: + return math.sqrt(var) |