851 lines
26 KiB
Python
851 lines
26 KiB
Python
"""
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Basic statistics module.
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This module provides functions for calculating statistics of data, including
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averages, variance, and standard deviation.
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Calculating averages
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--------------------
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================== =============================================
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Function Description
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================== =============================================
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mean Arithmetic mean (average) of data.
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harmonic_mean Harmonic mean of data.
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median Median (middle value) of data.
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median_low Low median of data.
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median_high High median of data.
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median_grouped Median, or 50th percentile, of grouped data.
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mode Mode (most common value) of data.
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================== =============================================
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Calculate the arithmetic mean ("the average") of data:
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>>> mean([-1.0, 2.5, 3.25, 5.75])
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2.625
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Calculate the standard median of discrete data:
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>>> median([2, 3, 4, 5])
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3.5
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Calculate the median, or 50th percentile, of data grouped into class intervals
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centred on the data values provided. E.g. if your data points are rounded to
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the nearest whole number:
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>>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS
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2.8333333333...
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This should be interpreted in this way: you have two data points in the class
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interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in
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the class interval 3.5-4.5. The median of these data points is 2.8333...
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Calculating variability or spread
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---------------------------------
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================== =============================================
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Function Description
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================== =============================================
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pvariance Population variance of data.
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variance Sample variance of data.
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pstdev Population standard deviation of data.
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stdev Sample standard deviation of data.
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================== =============================================
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Calculate the standard deviation of sample data:
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>>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS
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4.38961843444...
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If you have previously calculated the mean, you can pass it as the optional
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second argument to the four "spread" functions to avoid recalculating it:
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>>> data = [1, 2, 2, 4, 4, 4, 5, 6]
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>>> mu = mean(data)
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>>> pvariance(data, mu)
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2.5
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Exceptions
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----------
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A single exception is defined: StatisticsError is a subclass of ValueError.
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"""
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__all__ = [ 'StatisticsError', 'NormalDist',
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'pstdev', 'pvariance', 'stdev', 'variance',
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'median', 'median_low', 'median_high', 'median_grouped',
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'mean', 'mode', 'harmonic_mean', 'fmean',
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]
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import collections
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import math
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import numbers
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import random
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from fractions import Fraction
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from decimal import Decimal
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from itertools import groupby
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from bisect import bisect_left, bisect_right
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from math import hypot, sqrt, fabs, exp, erf, tau
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# === Exceptions ===
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class StatisticsError(ValueError):
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pass
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# === Private utilities ===
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def _sum(data, start=0):
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"""_sum(data [, start]) -> (type, sum, count)
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Return a high-precision sum of the given numeric data as a fraction,
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together with the type to be converted to and the count of items.
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If optional argument ``start`` is given, it is added to the total.
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If ``data`` is empty, ``start`` (defaulting to 0) is returned.
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Examples
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--------
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>>> _sum([3, 2.25, 4.5, -0.5, 1.0], 0.75)
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(<class 'float'>, Fraction(11, 1), 5)
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Some sources of round-off error will be avoided:
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# Built-in sum returns zero.
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>>> _sum([1e50, 1, -1e50] * 1000)
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(<class 'float'>, Fraction(1000, 1), 3000)
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Fractions and Decimals are also supported:
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>>> from fractions import Fraction as F
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>>> _sum([F(2, 3), F(7, 5), F(1, 4), F(5, 6)])
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(<class 'fractions.Fraction'>, Fraction(63, 20), 4)
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>>> from decimal import Decimal as D
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>>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")]
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>>> _sum(data)
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(<class 'decimal.Decimal'>, Fraction(6963, 10000), 4)
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Mixed types are currently treated as an error, except that int is
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allowed.
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"""
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count = 0
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n, d = _exact_ratio(start)
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partials = {d: n}
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partials_get = partials.get
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T = _coerce(int, type(start))
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for typ, values in groupby(data, type):
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T = _coerce(T, typ) # or raise TypeError
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for n,d in map(_exact_ratio, values):
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count += 1
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partials[d] = partials_get(d, 0) + n
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if None in partials:
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# The sum will be a NAN or INF. We can ignore all the finite
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# partials, and just look at this special one.
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total = partials[None]
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assert not _isfinite(total)
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else:
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# Sum all the partial sums using builtin sum.
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# FIXME is this faster if we sum them in order of the denominator?
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total = sum(Fraction(n, d) for d, n in sorted(partials.items()))
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return (T, total, count)
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def _isfinite(x):
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try:
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return x.is_finite() # Likely a Decimal.
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except AttributeError:
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return math.isfinite(x) # Coerces to float first.
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def _coerce(T, S):
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"""Coerce types T and S to a common type, or raise TypeError.
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Coercion rules are currently an implementation detail. See the CoerceTest
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test class in test_statistics for details.
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"""
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# See http://bugs.python.org/issue24068.
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assert T is not bool, "initial type T is bool"
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# If the types are the same, no need to coerce anything. Put this
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# first, so that the usual case (no coercion needed) happens as soon
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# as possible.
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if T is S: return T
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# Mixed int & other coerce to the other type.
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if S is int or S is bool: return T
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if T is int: return S
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# If one is a (strict) subclass of the other, coerce to the subclass.
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if issubclass(S, T): return S
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if issubclass(T, S): return T
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# Ints coerce to the other type.
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if issubclass(T, int): return S
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if issubclass(S, int): return T
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# Mixed fraction & float coerces to float (or float subclass).
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if issubclass(T, Fraction) and issubclass(S, float):
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return S
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if issubclass(T, float) and issubclass(S, Fraction):
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return T
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# Any other combination is disallowed.
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msg = "don't know how to coerce %s and %s"
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raise TypeError(msg % (T.__name__, S.__name__))
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def _exact_ratio(x):
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"""Return Real number x to exact (numerator, denominator) pair.
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>>> _exact_ratio(0.25)
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(1, 4)
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x is expected to be an int, Fraction, Decimal or float.
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"""
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try:
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# Optimise the common case of floats. We expect that the most often
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# used numeric type will be builtin floats, so try to make this as
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# fast as possible.
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if type(x) is float or type(x) is Decimal:
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return x.as_integer_ratio()
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try:
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# x may be an int, Fraction, or Integral ABC.
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return (x.numerator, x.denominator)
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except AttributeError:
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try:
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# x may be a float or Decimal subclass.
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return x.as_integer_ratio()
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except AttributeError:
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# Just give up?
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pass
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except (OverflowError, ValueError):
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# float NAN or INF.
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assert not _isfinite(x)
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return (x, None)
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msg = "can't convert type '{}' to numerator/denominator"
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raise TypeError(msg.format(type(x).__name__))
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def _convert(value, T):
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"""Convert value to given numeric type T."""
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if type(value) is T:
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# This covers the cases where T is Fraction, or where value is
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# a NAN or INF (Decimal or float).
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return value
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if issubclass(T, int) and value.denominator != 1:
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T = float
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try:
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# FIXME: what do we do if this overflows?
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return T(value)
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except TypeError:
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if issubclass(T, Decimal):
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return T(value.numerator)/T(value.denominator)
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else:
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raise
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def _counts(data):
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# Generate a table of sorted (value, frequency) pairs.
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table = collections.Counter(iter(data)).most_common()
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if not table:
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return table
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# Extract the values with the highest frequency.
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maxfreq = table[0][1]
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for i in range(1, len(table)):
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if table[i][1] != maxfreq:
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table = table[:i]
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break
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return table
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def _find_lteq(a, x):
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'Locate the leftmost value exactly equal to x'
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i = bisect_left(a, x)
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if i != len(a) and a[i] == x:
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return i
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raise ValueError
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def _find_rteq(a, l, x):
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'Locate the rightmost value exactly equal to x'
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i = bisect_right(a, x, lo=l)
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if i != (len(a)+1) and a[i-1] == x:
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return i-1
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raise ValueError
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def _fail_neg(values, errmsg='negative value'):
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"""Iterate over values, failing if any are less than zero."""
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for x in values:
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if x < 0:
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raise StatisticsError(errmsg)
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yield x
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# === Measures of central tendency (averages) ===
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def mean(data):
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"""Return the sample arithmetic mean of data.
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>>> mean([1, 2, 3, 4, 4])
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2.8
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>>> from fractions import Fraction as F
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>>> mean([F(3, 7), F(1, 21), F(5, 3), F(1, 3)])
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Fraction(13, 21)
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>>> from decimal import Decimal as D
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>>> mean([D("0.5"), D("0.75"), D("0.625"), D("0.375")])
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Decimal('0.5625')
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If ``data`` is empty, StatisticsError will be raised.
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"""
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if iter(data) is data:
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data = list(data)
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n = len(data)
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if n < 1:
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raise StatisticsError('mean requires at least one data point')
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T, total, count = _sum(data)
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assert count == n
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return _convert(total/n, T)
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def fmean(data):
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""" Convert data to floats and compute the arithmetic mean.
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This runs faster than the mean() function and it always returns a float.
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The result is highly accurate but not as perfect as mean().
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If the input dataset is empty, it raises a StatisticsError.
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>>> fmean([3.5, 4.0, 5.25])
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4.25
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"""
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try:
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n = len(data)
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except TypeError:
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# Handle iterators that do not define __len__().
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n = 0
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def count(x):
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nonlocal n
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n += 1
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return x
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total = math.fsum(map(count, data))
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else:
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total = math.fsum(data)
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try:
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return total / n
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except ZeroDivisionError:
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raise StatisticsError('fmean requires at least one data point') from None
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def harmonic_mean(data):
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"""Return the harmonic mean of data.
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The harmonic mean, sometimes called the subcontrary mean, is the
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reciprocal of the arithmetic mean of the reciprocals of the data,
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and is often appropriate when averaging quantities which are rates
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or ratios, for example speeds. Example:
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Suppose an investor purchases an equal value of shares in each of
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three companies, with P/E (price/earning) ratios of 2.5, 3 and 10.
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What is the average P/E ratio for the investor's portfolio?
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>>> harmonic_mean([2.5, 3, 10]) # For an equal investment portfolio.
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3.6
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Using the arithmetic mean would give an average of about 5.167, which
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is too high.
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If ``data`` is empty, or any element is less than zero,
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``harmonic_mean`` will raise ``StatisticsError``.
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"""
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# For a justification for using harmonic mean for P/E ratios, see
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# http://fixthepitch.pellucid.com/comps-analysis-the-missing-harmony-of-summary-statistics/
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# http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2621087
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if iter(data) is data:
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data = list(data)
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errmsg = 'harmonic mean does not support negative values'
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n = len(data)
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if n < 1:
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raise StatisticsError('harmonic_mean requires at least one data point')
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elif n == 1:
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x = data[0]
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if isinstance(x, (numbers.Real, Decimal)):
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if x < 0:
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raise StatisticsError(errmsg)
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return x
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else:
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raise TypeError('unsupported type')
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try:
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T, total, count = _sum(1/x for x in _fail_neg(data, errmsg))
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except ZeroDivisionError:
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return 0
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assert count == n
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return _convert(n/total, T)
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# FIXME: investigate ways to calculate medians without sorting? Quickselect?
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def median(data):
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"""Return the median (middle value) of numeric data.
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When the number of data points is odd, return the middle data point.
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When the number of data points is even, the median is interpolated by
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taking the average of the two middle values:
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>>> median([1, 3, 5])
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3
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>>> median([1, 3, 5, 7])
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4.0
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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if n%2 == 1:
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return data[n//2]
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else:
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i = n//2
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return (data[i - 1] + data[i])/2
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def median_low(data):
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"""Return the low median of numeric data.
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When the number of data points is odd, the middle value is returned.
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When it is even, the smaller of the two middle values is returned.
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>>> median_low([1, 3, 5])
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3
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>>> median_low([1, 3, 5, 7])
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3
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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if n%2 == 1:
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return data[n//2]
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else:
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return data[n//2 - 1]
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def median_high(data):
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"""Return the high median of data.
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When the number of data points is odd, the middle value is returned.
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When it is even, the larger of the two middle values is returned.
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>>> median_high([1, 3, 5])
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3
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>>> median_high([1, 3, 5, 7])
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5
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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return data[n//2]
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def median_grouped(data, interval=1):
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"""Return the 50th percentile (median) of grouped continuous data.
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>>> median_grouped([1, 2, 2, 3, 4, 4, 4, 4, 4, 5])
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3.7
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>>> median_grouped([52, 52, 53, 54])
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52.5
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This calculates the median as the 50th percentile, and should be
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used when your data is continuous and grouped. In the above example,
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the values 1, 2, 3, etc. actually represent the midpoint of classes
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0.5-1.5, 1.5-2.5, 2.5-3.5, etc. The middle value falls somewhere in
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class 3.5-4.5, and interpolation is used to estimate it.
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Optional argument ``interval`` represents the class interval, and
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defaults to 1. Changing the class interval naturally will change the
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interpolated 50th percentile value:
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>>> median_grouped([1, 3, 3, 5, 7], interval=1)
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3.25
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>>> median_grouped([1, 3, 3, 5, 7], interval=2)
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3.5
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This function does not check whether the data points are at least
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``interval`` apart.
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"""
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data = sorted(data)
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n = len(data)
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if n == 0:
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raise StatisticsError("no median for empty data")
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elif n == 1:
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return data[0]
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# Find the value at the midpoint. Remember this corresponds to the
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# centre of the class interval.
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x = data[n//2]
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for obj in (x, interval):
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if isinstance(obj, (str, bytes)):
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raise TypeError('expected number but got %r' % obj)
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try:
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L = x - interval/2 # The lower limit of the median interval.
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except TypeError:
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# Mixed type. For now we just coerce to float.
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L = float(x) - float(interval)/2
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# Uses bisection search to search for x in data with log(n) time complexity
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# Find the position of leftmost occurrence of x in data
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l1 = _find_lteq(data, x)
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# Find the position of rightmost occurrence of x in data[l1...len(data)]
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# Assuming always l1 <= l2
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l2 = _find_rteq(data, l1, x)
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cf = l1
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f = l2 - l1 + 1
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return L + interval*(n/2 - cf)/f
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def mode(data):
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"""Return the most common data point from discrete or nominal data.
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``mode`` assumes discrete data, and returns a single value. This is the
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standard treatment of the mode as commonly taught in schools:
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>>> mode([1, 1, 2, 3, 3, 3, 3, 4])
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3
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This also works with nominal (non-numeric) data:
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>>> mode(["red", "blue", "blue", "red", "green", "red", "red"])
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'red'
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If there is not exactly one most common value, ``mode`` will raise
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StatisticsError.
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"""
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# Generate a table of sorted (value, frequency) pairs.
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table = _counts(data)
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if len(table) == 1:
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return table[0][0]
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elif table:
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raise StatisticsError(
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'no unique mode; found %d equally common values' % len(table)
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)
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else:
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raise StatisticsError('no mode for empty data')
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# === Measures of spread ===
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# See http://mathworld.wolfram.com/Variance.html
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# http://mathworld.wolfram.com/SampleVariance.html
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# http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance
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#
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# Under no circumstances use the so-called "computational formula for
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# variance", as that is only suitable for hand calculations with a small
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# amount of low-precision data. It has terrible numeric properties.
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#
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# See a comparison of three computational methods here:
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# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/
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def _ss(data, c=None):
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"""Return sum of square deviations of sequence data.
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If ``c`` is None, the mean is calculated in one pass, and the deviations
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from the mean are calculated in a second pass. Otherwise, deviations are
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calculated from ``c`` as given. Use the second case with care, as it can
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lead to garbage results.
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"""
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if c is None:
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c = mean(data)
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T, total, count = _sum((x-c)**2 for x in data)
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# The following sum should mathematically equal zero, but due to rounding
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# error may not.
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U, total2, count2 = _sum((x-c) for x in data)
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assert T == U and count == count2
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total -= total2**2/len(data)
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assert not total < 0, 'negative sum of square deviations: %f' % total
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return (T, total)
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def variance(data, xbar=None):
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"""Return the sample variance of data.
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data should be an iterable of Real-valued numbers, with at least two
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values. The optional argument xbar, if given, should be the mean of
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the data. If it is missing or None, the mean is automatically calculated.
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Use this function when your data is a sample from a population. To
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calculate the variance from the entire population, see ``pvariance``.
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Examples:
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>>> data = [2.75, 1.75, 1.25, 0.25, 0.5, 1.25, 3.5]
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>>> variance(data)
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1.3720238095238095
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If you have already calculated the mean of your data, you can pass it as
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the optional second argument ``xbar`` to avoid recalculating it:
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>>> m = mean(data)
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>>> variance(data, m)
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1.3720238095238095
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This function does not check that ``xbar`` is actually the mean of
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``data``. Giving arbitrary values for ``xbar`` may lead to invalid or
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impossible results.
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Decimals and Fractions are supported:
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>>> from decimal import Decimal as D
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>>> variance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
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Decimal('31.01875')
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>>> from fractions import Fraction as F
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>>> variance([F(1, 6), F(1, 2), F(5, 3)])
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Fraction(67, 108)
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"""
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if iter(data) is data:
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data = list(data)
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n = len(data)
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if n < 2:
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raise StatisticsError('variance requires at least two data points')
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T, ss = _ss(data, xbar)
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return _convert(ss/(n-1), T)
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def pvariance(data, mu=None):
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"""Return the population variance of ``data``.
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data should be an iterable of Real-valued numbers, with at least one
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value. The optional argument mu, if given, should be the mean of
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the data. If it is missing or None, the mean is automatically calculated.
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Use this function to calculate the variance from the entire population.
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To estimate the variance from a sample, the ``variance`` function is
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usually a better choice.
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Examples:
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>>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25]
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>>> pvariance(data)
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1.25
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If you have already calculated the mean of the data, you can pass it as
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the optional second argument to avoid recalculating it:
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>>> mu = mean(data)
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>>> pvariance(data, mu)
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1.25
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This function does not check that ``mu`` is actually the mean of ``data``.
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Giving arbitrary values for ``mu`` may lead to invalid or impossible
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results.
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Decimals and Fractions are supported:
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|
>>> from decimal import Decimal as D
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>>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")])
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Decimal('24.815')
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>>> from fractions import Fraction as F
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>>> pvariance([F(1, 4), F(5, 4), F(1, 2)])
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Fraction(13, 72)
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"""
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if iter(data) is data:
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data = list(data)
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n = len(data)
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if n < 1:
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raise StatisticsError('pvariance requires at least one data point')
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T, ss = _ss(data, mu)
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return _convert(ss/n, T)
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def stdev(data, xbar=None):
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"""Return the square root of the sample variance.
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See ``variance`` for arguments and other details.
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>>> stdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
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1.0810874155219827
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"""
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var = variance(data, xbar)
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try:
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return var.sqrt()
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except AttributeError:
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|
return math.sqrt(var)
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|
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def pstdev(data, mu=None):
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"""Return the square root of the population variance.
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See ``pvariance`` for arguments and other details.
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>>> pstdev([1.5, 2.5, 2.5, 2.75, 3.25, 4.75])
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0.986893273527251
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"""
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var = pvariance(data, mu)
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try:
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return var.sqrt()
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|
except AttributeError:
|
|
return math.sqrt(var)
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## Normal Distribution #####################################################
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class NormalDist:
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'Normal distribution of a random variable'
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# https://en.wikipedia.org/wiki/Normal_distribution
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# https://en.wikipedia.org/wiki/Variance#Properties
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__slots__ = ('mu', 'sigma')
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def __init__(self, mu=0.0, sigma=1.0):
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'NormalDist where mu is the mean and sigma is the standard deviation'
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if sigma < 0.0:
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raise StatisticsError('sigma must be non-negative')
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self.mu = mu
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self.sigma = sigma
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|
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@classmethod
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|
def from_samples(cls, data):
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|
'Make a normal distribution instance from sample data'
|
|
if not isinstance(data, (list, tuple)):
|
|
data = list(data)
|
|
xbar = fmean(data)
|
|
return cls(xbar, stdev(data, xbar))
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|
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def samples(self, n, seed=None):
|
|
'Generate *n* samples for a given mean and standard deviation'
|
|
gauss = random.gauss if seed is None else random.Random(seed).gauss
|
|
mu, sigma = self.mu, self.sigma
|
|
return [gauss(mu, sigma) for i in range(n)]
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|
|
def pdf(self, x):
|
|
'Probability density function: P(x <= X < x+dx) / dx'
|
|
variance = self.sigma ** 2.0
|
|
if not variance:
|
|
raise StatisticsError('pdf() not defined when sigma is zero')
|
|
return exp((x - self.mu)**2.0 / (-2.0*variance)) / sqrt(tau * variance)
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|
|
def cdf(self, x):
|
|
'Cumulative density function: P(X <= x)'
|
|
if not self.sigma:
|
|
raise StatisticsError('cdf() not defined when sigma is zero')
|
|
return 0.5 * (1.0 + erf((x - self.mu) / (self.sigma * sqrt(2.0))))
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|
|
@property
|
|
def variance(self):
|
|
'Square of the standard deviation'
|
|
return self.sigma ** 2.0
|
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|
|
def __add__(x1, x2):
|
|
if isinstance(x2, NormalDist):
|
|
return NormalDist(x1.mu + x2.mu, hypot(x1.sigma, x2.sigma))
|
|
return NormalDist(x1.mu + x2, x1.sigma)
|
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|
|
def __sub__(x1, x2):
|
|
if isinstance(x2, NormalDist):
|
|
return NormalDist(x1.mu - x2.mu, hypot(x1.sigma, x2.sigma))
|
|
return NormalDist(x1.mu - x2, x1.sigma)
|
|
|
|
def __mul__(x1, x2):
|
|
return NormalDist(x1.mu * x2, x1.sigma * fabs(x2))
|
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|
|
def __truediv__(x1, x2):
|
|
return NormalDist(x1.mu / x2, x1.sigma / fabs(x2))
|
|
|
|
def __pos__(x1):
|
|
return NormalDist(x1.mu, x1.sigma)
|
|
|
|
def __neg__(x1):
|
|
return NormalDist(-x1.mu, x1.sigma)
|
|
|
|
__radd__ = __add__
|
|
|
|
def __rsub__(x1, x2):
|
|
return -(x1 - x2)
|
|
|
|
__rmul__ = __mul__
|
|
|
|
def __eq__(x1, x2):
|
|
if not isinstance(x2, NormalDist):
|
|
return NotImplemented
|
|
return (x1.mu, x2.sigma) == (x2.mu, x2.sigma)
|
|
|
|
def __repr__(self):
|
|
return f'{type(self).__name__}(mu={self.mu!r}, sigma={self.sigma!r})'
|
|
|
|
|
|
if __name__ == '__main__':
|
|
|
|
# Show math operations computed analytically in comparsion
|
|
# to a monte carlo simulation of the same operations
|
|
|
|
from math import isclose
|
|
from operator import add, sub, mul, truediv
|
|
from itertools import repeat
|
|
|
|
g1 = NormalDist(10, 20)
|
|
g2 = NormalDist(-5, 25)
|
|
|
|
# Test scaling by a constant
|
|
assert (g1 * 5 / 5).mu == g1.mu
|
|
assert (g1 * 5 / 5).sigma == g1.sigma
|
|
|
|
n = 100_000
|
|
G1 = g1.samples(n)
|
|
G2 = g2.samples(n)
|
|
|
|
for func in (add, sub):
|
|
print(f'\nTest {func.__name__} with another NormalDist:')
|
|
print(func(g1, g2))
|
|
print(NormalDist.from_samples(map(func, G1, G2)))
|
|
|
|
const = 11
|
|
for func in (add, sub, mul, truediv):
|
|
print(f'\nTest {func.__name__} with a constant:')
|
|
print(func(g1, const))
|
|
print(NormalDist.from_samples(map(func, G1, repeat(const))))
|
|
|
|
const = 19
|
|
for func in (add, sub, mul):
|
|
print(f'\nTest constant with {func.__name__}:')
|
|
print(func(const, g1))
|
|
print(NormalDist.from_samples(map(func, repeat(const), G1)))
|
|
|
|
def assert_close(G1, G2):
|
|
assert isclose(G1.mu, G1.mu, rel_tol=0.01), (G1, G2)
|
|
assert isclose(G1.sigma, G2.sigma, rel_tol=0.01), (G1, G2)
|
|
|
|
X = NormalDist(-105, 73)
|
|
Y = NormalDist(31, 47)
|
|
s = 32.75
|
|
n = 100_000
|
|
|
|
S = NormalDist.from_samples([x + s for x in X.samples(n)])
|
|
assert_close(X + s, S)
|
|
|
|
S = NormalDist.from_samples([x - s for x in X.samples(n)])
|
|
assert_close(X - s, S)
|
|
|
|
S = NormalDist.from_samples([x * s for x in X.samples(n)])
|
|
assert_close(X * s, S)
|
|
|
|
S = NormalDist.from_samples([x / s for x in X.samples(n)])
|
|
assert_close(X / s, S)
|
|
|
|
S = NormalDist.from_samples([x + y for x, y in zip(X.samples(n),
|
|
Y.samples(n))])
|
|
assert_close(X + Y, S)
|
|
|
|
S = NormalDist.from_samples([x - y for x, y in zip(X.samples(n),
|
|
Y.samples(n))])
|
|
assert_close(X - Y, S)
|