text: # -*- coding: UTF-8 -*-
  ##  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.
  
  
  u"""
  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.
  
  """
  
  from __future__ import division
  __all__ = [ u'StatisticsError',
              u'pstdev', u'pvariance', u'stdev', u'variance',
              u'median',  u'median_low', u'median_high', u'median_grouped',
              u'mean', u'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):
      u"""_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 = u', '.join([t.__name__ for t in allowed] + [T.__name__])
              raise TypeError(u"unsupported mixed types: %s" % types)
  
  
  def _exact_ratio(x):
      u"""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 = u"can't convert type '{}' to numerator/denominator"
                      raise TypeError(msg.format(type(x).__name__))
      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):
      u"""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 (u'F', u'n', u'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 xrange(1, len(table)):
          if table[i][1] != maxfreq:
              table = table[:i]
              break
      return table
  
  
  # === Measures of central tendency (averages) ===
  
  def mean(data):
      u"""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(u'mean requires at least one data point')
      return _sum(data)/n
  
  
  # FIXME: investigate ways to calculate medians without sorting? Quickselect?
  def median(data):
      u"""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(u"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):
      u"""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(u"no median for empty data")
      if n%2 == 1:
          return data[n//2]
      else:
          return data[n//2 - 1]
  
  
  def median_high(data):
      u"""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(u"no median for empty data")
      return data[n//2]
  
  
  def median_grouped(data, interval=1):
      u""""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(u"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, (unicode, str)):
              raise TypeError(u'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):
      u"""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(
                  u'no unique mode; found %d equally common values' % len(table)
                  )
      else:
          raise StatisticsError(u'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):
      u"""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, u'negative sum of square deviations: %f' % ss
      return ss
  
  
  def variance(data, xbar=None):
      u"""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(u'variance requires at least two data points')
      ss = _ss(data, xbar)
      return ss/(n-1)
  
  
  def pvariance(data, mu=None):
      u"""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(u'pvariance requires at least one data point')
      ss = _ss(data, mu)
      return ss/n
  
  
  def stdev(data, xbar=None):
      u"""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):
      u"""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)