| """ |
| 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. |
| fmean Fast, floating point arithmetic mean. |
| geometric_mean Geometric mean of data. |
| harmonic_mean Harmonic mean 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. |
| multimode List of modes (most common values of data). |
| quantiles Divide data into intervals with equal probability. |
| ================== ================================================== |
| |
| 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__ = [ |
| 'NormalDist', |
| 'StatisticsError', |
| 'fmean', |
| 'geometric_mean', |
| 'harmonic_mean', |
| 'mean', |
| 'median', |
| 'median_grouped', |
| 'median_high', |
| 'median_low', |
| 'mode', |
| 'multimode', |
| 'pstdev', |
| 'pvariance', |
| 'quantiles', |
| 'stdev', |
| 'variance', |
| ] |
| |
| import math |
| import numbers |
| import random |
| |
| from fractions import Fraction |
| from decimal import Decimal |
| from itertools import groupby |
| from bisect import bisect_left, bisect_right |
| from math import hypot, sqrt, fabs, exp, erf, tau, log, fsum |
| from operator import itemgetter |
| from collections import Counter |
| |
| # === Exceptions === |
| |
| class StatisticsError(ValueError): |
| pass |
| |
| |
| # === Private utilities === |
| |
| def _sum(data, start=0): |
| """_sum(data [, start]) -> (type, sum, count) |
| |
| Return a high-precision sum of the given numeric data as a fraction, |
| together with the type to be converted to and the count of items. |
| |
| 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) |
| (<class 'float'>, Fraction(11, 1), 5) |
| |
| Some sources of round-off error will be avoided: |
| |
| # Built-in sum returns zero. |
| >>> _sum([1e50, 1, -1e50] * 1000) |
| (<class 'float'>, Fraction(1000, 1), 3000) |
| |
| 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)]) |
| (<class 'fractions.Fraction'>, Fraction(63, 20), 4) |
| |
| >>> from decimal import Decimal as D |
| >>> data = [D("0.1375"), D("0.2108"), D("0.3061"), D("0.0419")] |
| >>> _sum(data) |
| (<class 'decimal.Decimal'>, Fraction(6963, 10000), 4) |
| |
| Mixed types are currently treated as an error, except that int is |
| allowed. |
| """ |
| count = 0 |
| n, d = _exact_ratio(start) |
| partials = {d: n} |
| partials_get = partials.get |
| T = _coerce(int, type(start)) |
| for typ, values in groupby(data, type): |
| T = _coerce(T, typ) # or raise TypeError |
| for n,d in map(_exact_ratio, values): |
| count += 1 |
| partials[d] = partials_get(d, 0) + n |
| if None in partials: |
| # The sum will be a NAN or INF. We can ignore all the finite |
| # partials, and just look at this special one. |
| total = partials[None] |
| assert not _isfinite(total) |
| else: |
| # Sum all the partial sums using builtin sum. |
| # FIXME is this faster if we sum them in order of the denominator? |
| total = sum(Fraction(n, d) for d, n in sorted(partials.items())) |
| return (T, total, count) |
| |
| |
| def _isfinite(x): |
| try: |
| return x.is_finite() # Likely a Decimal. |
| except AttributeError: |
| return math.isfinite(x) # Coerces to float first. |
| |
| |
| def _coerce(T, S): |
| """Coerce types T and S to a common type, or raise TypeError. |
| |
| Coercion rules are currently an implementation detail. See the CoerceTest |
| test class in test_statistics for details. |
| """ |
| # See http://bugs.python.org/issue24068. |
| assert T is not bool, "initial type T is bool" |
| # If the types are the same, no need to coerce anything. Put this |
| # first, so that the usual case (no coercion needed) happens as soon |
| # as possible. |
| if T is S: return T |
| # Mixed int & other coerce to the other type. |
| if S is int or S is bool: return T |
| if T is int: return S |
| # If one is a (strict) subclass of the other, coerce to the subclass. |
| if issubclass(S, T): return S |
| if issubclass(T, S): return T |
| # Ints coerce to the other type. |
| if issubclass(T, int): return S |
| if issubclass(S, int): return T |
| # Mixed fraction & float coerces to float (or float subclass). |
| if issubclass(T, Fraction) and issubclass(S, float): |
| return S |
| if issubclass(T, float) and issubclass(S, Fraction): |
| return T |
| # Any other combination is disallowed. |
| msg = "don't know how to coerce %s and %s" |
| raise TypeError(msg % (T.__name__, S.__name__)) |
| |
| |
| def _exact_ratio(x): |
| """Return Real number x to exact (numerator, denominator) pair. |
| |
| >>> _exact_ratio(0.25) |
| (1, 4) |
| |
| x is expected to be an int, Fraction, Decimal or float. |
| """ |
| try: |
| # Optimise the common case of floats. We expect that the most often |
| # used numeric type will be builtin floats, so try to make this as |
| # fast as possible. |
| if type(x) is float or type(x) is Decimal: |
| return x.as_integer_ratio() |
| try: |
| # x may be an int, Fraction, or Integral ABC. |
| return (x.numerator, x.denominator) |
| except AttributeError: |
| try: |
| # x may be a float or Decimal subclass. |
| return x.as_integer_ratio() |
| except AttributeError: |
| # Just give up? |
| pass |
| except (OverflowError, ValueError): |
| # float NAN or INF. |
| assert not _isfinite(x) |
| return (x, None) |
| msg = "can't convert type '{}' to numerator/denominator" |
| raise TypeError(msg.format(type(x).__name__)) |
| |
| |
| def _convert(value, T): |
| """Convert value to given numeric type T.""" |
| if type(value) is T: |
| # This covers the cases where T is Fraction, or where value is |
| # a NAN or INF (Decimal or float). |
| return value |
| if issubclass(T, int) and value.denominator != 1: |
| T = float |
| try: |
| # FIXME: what do we do if this overflows? |
| return T(value) |
| except TypeError: |
| if issubclass(T, Decimal): |
| return T(value.numerator)/T(value.denominator) |
| else: |
| raise |
| |
| |
| def _find_lteq(a, x): |
| 'Locate the leftmost value exactly equal to x' |
| i = bisect_left(a, x) |
| if i != len(a) and a[i] == x: |
| return i |
| raise ValueError |
| |
| |
| def _find_rteq(a, l, x): |
| 'Locate the rightmost value exactly equal to x' |
| i = bisect_right(a, x, lo=l) |
| if i != (len(a)+1) and a[i-1] == x: |
| return i-1 |
| raise ValueError |
| |
| |
| def _fail_neg(values, errmsg='negative value'): |
| """Iterate over values, failing if any are less than zero.""" |
| for x in values: |
| if x < 0: |
| raise StatisticsError(errmsg) |
| yield x |
| |
| |
| # === 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') |
| T, total, count = _sum(data) |
| assert count == n |
| return _convert(total/n, T) |
| |
| |
| def fmean(data): |
| """Convert data to floats and compute the arithmetic mean. |
| |
| This runs faster than the mean() function and it always returns a float. |
| If the input dataset is empty, it raises a StatisticsError. |
| |
| >>> fmean([3.5, 4.0, 5.25]) |
| 4.25 |
| """ |
| try: |
| n = len(data) |
| except TypeError: |
| # Handle iterators that do not define __len__(). |
| n = 0 |
| def count(iterable): |
| nonlocal n |
| for n, x in enumerate(iterable, start=1): |
| yield x |
| total = fsum(count(data)) |
| else: |
| total = fsum(data) |
| try: |
| return total / n |
| except ZeroDivisionError: |
| raise StatisticsError('fmean requires at least one data point') from None |
| |
| |
| def geometric_mean(data): |
| """Convert data to floats and compute the geometric mean. |
| |
| Raises a StatisticsError if the input dataset is empty, |
| if it contains a zero, or if it contains a negative value. |
| |
| No special efforts are made to achieve exact results. |
| (However, this may change in the future.) |
| |
| >>> round(geometric_mean([54, 24, 36]), 9) |
| 36.0 |
| """ |
| try: |
| return exp(fmean(map(log, data))) |
| except ValueError: |
| raise StatisticsError('geometric mean requires a non-empty dataset ' |
| ' containing positive numbers') from None |
| |
| |
| def harmonic_mean(data): |
| """Return the harmonic mean of data. |
| |
| The harmonic mean, sometimes called the subcontrary mean, is the |
| reciprocal of the arithmetic mean of the reciprocals of the data, |
| and is often appropriate when averaging quantities which are rates |
| or ratios, for example speeds. Example: |
| |
| Suppose an investor purchases an equal value of shares in each of |
| three companies, with P/E (price/earning) ratios of 2.5, 3 and 10. |
| What is the average P/E ratio for the investor's portfolio? |
| |
| >>> harmonic_mean([2.5, 3, 10]) # For an equal investment portfolio. |
| 3.6 |
| |
| Using the arithmetic mean would give an average of about 5.167, which |
| is too high. |
| |
| If ``data`` is empty, or any element is less than zero, |
| ``harmonic_mean`` will raise ``StatisticsError``. |
| """ |
| # For a justification for using harmonic mean for P/E ratios, see |
| # http://fixthepitch.pellucid.com/comps-analysis-the-missing-harmony-of-summary-statistics/ |
| # http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2621087 |
| if iter(data) is data: |
| data = list(data) |
| errmsg = 'harmonic mean does not support negative values' |
| n = len(data) |
| if n < 1: |
| raise StatisticsError('harmonic_mean requires at least one data point') |
| elif n == 1: |
| x = data[0] |
| if isinstance(x, (numbers.Real, Decimal)): |
| if x < 0: |
| raise StatisticsError(errmsg) |
| return x |
| else: |
| raise TypeError('unsupported type') |
| try: |
| T, total, count = _sum(1/x for x in _fail_neg(data, errmsg)) |
| except ZeroDivisionError: |
| return 0 |
| assert count == n |
| return _convert(n/total, T) |
| |
| |
| # 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 |
| |
| # Uses bisection search to search for x in data with log(n) time complexity |
| # Find the position of leftmost occurrence of x in data |
| l1 = _find_lteq(data, x) |
| # Find the position of rightmost occurrence of x in data[l1...len(data)] |
| # Assuming always l1 <= l2 |
| l2 = _find_rteq(data, l1, x) |
| cf = l1 |
| f = l2 - l1 + 1 |
| 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 are multiple modes with same frequency, return the first one |
| encountered: |
| |
| >>> mode(['red', 'red', 'green', 'blue', 'blue']) |
| 'red' |
| |
| If *data* is empty, ``mode``, raises StatisticsError. |
| |
| """ |
| data = iter(data) |
| pairs = Counter(data).most_common(1) |
| try: |
| return pairs[0][0] |
| except IndexError: |
| raise StatisticsError('no mode for empty data') from None |
| |
| |
| def multimode(data): |
| """Return a list of the most frequently occurring values. |
| |
| Will return more than one result if there are multiple modes |
| or an empty list if *data* is empty. |
| |
| >>> multimode('aabbbbbbbbcc') |
| ['b'] |
| >>> multimode('aabbbbccddddeeffffgg') |
| ['b', 'd', 'f'] |
| >>> multimode('') |
| [] |
| """ |
| counts = Counter(iter(data)).most_common() |
| maxcount, mode_items = next(groupby(counts, key=itemgetter(1)), (0, [])) |
| return list(map(itemgetter(0), mode_items)) |
| |
| |
| # Notes on methods for computing quantiles |
| # ---------------------------------------- |
| # |
| # There is no one perfect way to compute quantiles. Here we offer |
| # two methods that serve common needs. Most other packages |
| # surveyed offered at least one or both of these two, making them |
| # "standard" in the sense of "widely-adopted and reproducible". |
| # They are also easy to explain, easy to compute manually, and have |
| # straight-forward interpretations that aren't surprising. |
| |
| # The default method is known as "R6", "PERCENTILE.EXC", or "expected |
| # value of rank order statistics". The alternative method is known as |
| # "R7", "PERCENTILE.INC", or "mode of rank order statistics". |
| |
| # For sample data where there is a positive probability for values |
| # beyond the range of the data, the R6 exclusive method is a |
| # reasonable choice. Consider a random sample of nine values from a |
| # population with a uniform distribution from 0.0 to 100.0. The |
| # distribution of the third ranked sample point is described by |
| # betavariate(alpha=3, beta=7) which has mode=0.250, median=0.286, and |
| # mean=0.300. Only the latter (which corresponds with R6) gives the |
| # desired cut point with 30% of the population falling below that |
| # value, making it comparable to a result from an inv_cdf() function. |
| # The R6 exclusive method is also idempotent. |
| |
| # For describing population data where the end points are known to |
| # be included in the data, the R7 inclusive method is a reasonable |
| # choice. Instead of the mean, it uses the mode of the beta |
| # distribution for the interior points. Per Hyndman & Fan, "One nice |
| # property is that the vertices of Q7(p) divide the range into n - 1 |
| # intervals, and exactly 100p% of the intervals lie to the left of |
| # Q7(p) and 100(1 - p)% of the intervals lie to the right of Q7(p)." |
| |
| # If needed, other methods could be added. However, for now, the |
| # position is that fewer options make for easier choices and that |
| # external packages can be used for anything more advanced. |
| |
| def quantiles(data, *, n=4, method='exclusive'): |
| """Divide *data* into *n* continuous intervals with equal probability. |
| |
| Returns a list of (n - 1) cut points separating the intervals. |
| |
| Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. |
| Set *n* to 100 for percentiles which gives the 99 cuts points that |
| separate *data* in to 100 equal sized groups. |
| |
| The *data* can be any iterable containing sample. |
| The cut points are linearly interpolated between data points. |
| |
| If *method* is set to *inclusive*, *data* is treated as population |
| data. The minimum value is treated as the 0th percentile and the |
| maximum value is treated as the 100th percentile. |
| """ |
| if n < 1: |
| raise StatisticsError('n must be at least 1') |
| data = sorted(data) |
| ld = len(data) |
| if ld < 2: |
| raise StatisticsError('must have at least two data points') |
| if method == 'inclusive': |
| m = ld - 1 |
| result = [] |
| for i in range(1, n): |
| j = i * m // n |
| delta = i*m - j*n |
| interpolated = (data[j] * (n - delta) + data[j+1] * delta) / n |
| result.append(interpolated) |
| return result |
| if method == 'exclusive': |
| m = ld + 1 |
| result = [] |
| for i in range(1, n): |
| j = i * m // n # rescale i to m/n |
| j = 1 if j < 1 else ld-1 if j > ld-1 else j # clamp to 1 .. ld-1 |
| delta = i*m - j*n # exact integer math |
| interpolated = (data[j-1] * (n - delta) + data[j] * delta) / n |
| result.append(interpolated) |
| return result |
| raise ValueError(f'Unknown method: {method!r}') |
| |
| |
| # === 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) |
| T, total, count = _sum((x-c)**2 for x in data) |
| # The following sum should mathematically equal zero, but due to rounding |
| # error may not. |
| U, total2, count2 = _sum((x-c) for x in data) |
| assert T == U and count == count2 |
| total -= total2**2/len(data) |
| assert not total < 0, 'negative sum of square deviations: %f' % total |
| return (T, total) |
| |
| |
| 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') |
| T, ss = _ss(data, xbar) |
| return _convert(ss/(n-1), T) |
| |
| |
| def pvariance(data, mu=None): |
| """Return the population variance of ``data``. |
| |
| data should be a sequence or 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 |
| |
| 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') |
| T, ss = _ss(data, mu) |
| return _convert(ss/n, T) |
| |
| |
| 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) |
| |
| |
| ## Normal Distribution ##################################################### |
| |
| |
| def _normal_dist_inv_cdf(p, mu, sigma): |
| # There is no closed-form solution to the inverse CDF for the normal |
| # distribution, so we use a rational approximation instead: |
| # Wichura, M.J. (1988). "Algorithm AS241: The Percentage Points of the |
| # Normal Distribution". Applied Statistics. Blackwell Publishing. 37 |
| # (3): 477–484. doi:10.2307/2347330. JSTOR 2347330. |
| q = p - 0.5 |
| if fabs(q) <= 0.425: |
| r = 0.180625 - q * q |
| # Hash sum: 55.88319_28806_14901_4439 |
| num = (((((((2.50908_09287_30122_6727e+3 * r + |
| 3.34305_75583_58812_8105e+4) * r + |
| 6.72657_70927_00870_0853e+4) * r + |
| 4.59219_53931_54987_1457e+4) * r + |
| 1.37316_93765_50946_1125e+4) * r + |
| 1.97159_09503_06551_4427e+3) * r + |
| 1.33141_66789_17843_7745e+2) * r + |
| 3.38713_28727_96366_6080e+0) * q |
| den = (((((((5.22649_52788_52854_5610e+3 * r + |
| 2.87290_85735_72194_2674e+4) * r + |
| 3.93078_95800_09271_0610e+4) * r + |
| 2.12137_94301_58659_5867e+4) * r + |
| 5.39419_60214_24751_1077e+3) * r + |
| 6.87187_00749_20579_0830e+2) * r + |
| 4.23133_30701_60091_1252e+1) * r + |
| 1.0) |
| x = num / den |
| return mu + (x * sigma) |
| r = p if q <= 0.0 else 1.0 - p |
| r = sqrt(-log(r)) |
| if r <= 5.0: |
| r = r - 1.6 |
| # Hash sum: 49.33206_50330_16102_89036 |
| num = (((((((7.74545_01427_83414_07640e-4 * r + |
| 2.27238_44989_26918_45833e-2) * r + |
| 2.41780_72517_74506_11770e-1) * r + |
| 1.27045_82524_52368_38258e+0) * r + |
| 3.64784_83247_63204_60504e+0) * r + |
| 5.76949_72214_60691_40550e+0) * r + |
| 4.63033_78461_56545_29590e+0) * r + |
| 1.42343_71107_49683_57734e+0) |
| den = (((((((1.05075_00716_44416_84324e-9 * r + |
| 5.47593_80849_95344_94600e-4) * r + |
| 1.51986_66563_61645_71966e-2) * r + |
| 1.48103_97642_74800_74590e-1) * r + |
| 6.89767_33498_51000_04550e-1) * r + |
| 1.67638_48301_83803_84940e+0) * r + |
| 2.05319_16266_37758_82187e+0) * r + |
| 1.0) |
| else: |
| r = r - 5.0 |
| # Hash sum: 47.52583_31754_92896_71629 |
| num = (((((((2.01033_43992_92288_13265e-7 * r + |
| 2.71155_55687_43487_57815e-5) * r + |
| 1.24266_09473_88078_43860e-3) * r + |
| 2.65321_89526_57612_30930e-2) * r + |
| 2.96560_57182_85048_91230e-1) * r + |
| 1.78482_65399_17291_33580e+0) * r + |
| 5.46378_49111_64114_36990e+0) * r + |
| 6.65790_46435_01103_77720e+0) |
| den = (((((((2.04426_31033_89939_78564e-15 * r + |
| 1.42151_17583_16445_88870e-7) * r + |
| 1.84631_83175_10054_68180e-5) * r + |
| 7.86869_13114_56132_59100e-4) * r + |
| 1.48753_61290_85061_48525e-2) * r + |
| 1.36929_88092_27358_05310e-1) * r + |
| 5.99832_20655_58879_37690e-1) * r + |
| 1.0) |
| x = num / den |
| if q < 0.0: |
| x = -x |
| return mu + (x * sigma) |
| |
| |
| class NormalDist: |
| "Normal distribution of a random variable" |
| # https://en.wikipedia.org/wiki/Normal_distribution |
| # https://en.wikipedia.org/wiki/Variance#Properties |
| |
| __slots__ = { |
| '_mu': 'Arithmetic mean of a normal distribution', |
| '_sigma': 'Standard deviation of a normal distribution', |
| } |
| |
| def __init__(self, mu=0.0, sigma=1.0): |
| "NormalDist where mu is the mean and sigma is the standard deviation." |
| if sigma < 0.0: |
| raise StatisticsError('sigma must be non-negative') |
| self._mu = float(mu) |
| self._sigma = float(sigma) |
| |
| @classmethod |
| def from_samples(cls, data): |
| "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)) |
| |
| 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)] |
| |
| 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) |
| |
| def cdf(self, x): |
| "Cumulative distribution 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)))) |
| |
| def inv_cdf(self, p): |
| """Inverse cumulative distribution function. x : P(X <= x) = p |
| |
| Finds the value of the random variable such that the probability of |
| the variable being less than or equal to that value equals the given |
| probability. |
| |
| This function is also called the percent point function or quantile |
| function. |
| """ |
| if p <= 0.0 or p >= 1.0: |
| raise StatisticsError('p must be in the range 0.0 < p < 1.0') |
| if self._sigma <= 0.0: |
| raise StatisticsError('cdf() not defined when sigma at or below zero') |
| return _normal_dist_inv_cdf(p, self._mu, self._sigma) |
| |
| def quantiles(self, n=4): |
| """Divide into *n* continuous intervals with equal probability. |
| |
| Returns a list of (n - 1) cut points separating the intervals. |
| |
| Set *n* to 4 for quartiles (the default). Set *n* to 10 for deciles. |
| Set *n* to 100 for percentiles which gives the 99 cuts points that |
| separate the normal distribution in to 100 equal sized groups. |
| """ |
| return [self.inv_cdf(i / n) for i in range(1, n)] |
| |
| def overlap(self, other): |
| """Compute the overlapping coefficient (OVL) between two normal distributions. |
| |
| Measures the agreement between two normal probability distributions. |
| Returns a value between 0.0 and 1.0 giving the overlapping area in |
| the two underlying probability density functions. |
| |
| >>> N1 = NormalDist(2.4, 1.6) |
| >>> N2 = NormalDist(3.2, 2.0) |
| >>> N1.overlap(N2) |
| 0.8035050657330205 |
| """ |
| # See: "The overlapping coefficient as a measure of agreement between |
| # probability distributions and point estimation of the overlap of two |
| # normal densities" -- Henry F. Inman and Edwin L. Bradley Jr |
| # http://dx.doi.org/10.1080/03610928908830127 |
| if not isinstance(other, NormalDist): |
| raise TypeError('Expected another NormalDist instance') |
| X, Y = self, other |
| if (Y._sigma, Y._mu) < (X._sigma, X._mu): # sort to assure commutativity |
| X, Y = Y, X |
| X_var, Y_var = X.variance, Y.variance |
| if not X_var or not Y_var: |
| raise StatisticsError('overlap() not defined when sigma is zero') |
| dv = Y_var - X_var |
| dm = fabs(Y._mu - X._mu) |
| if not dv: |
| return 1.0 - erf(dm / (2.0 * X._sigma * sqrt(2.0))) |
| a = X._mu * Y_var - Y._mu * X_var |
| b = X._sigma * Y._sigma * sqrt(dm**2.0 + dv * log(Y_var / X_var)) |
| x1 = (a + b) / dv |
| x2 = (a - b) / dv |
| return 1.0 - (fabs(Y.cdf(x1) - X.cdf(x1)) + fabs(Y.cdf(x2) - X.cdf(x2))) |
| |
| @property |
| def mean(self): |
| "Arithmetic mean of the normal distribution." |
| return self._mu |
| |
| @property |
| def median(self): |
| "Return the median of the normal distribution" |
| return self._mu |
| |
| @property |
| def mode(self): |
| """Return the mode of the normal distribution |
| |
| The mode is the value x where which the probability density |
| function (pdf) takes its maximum value. |
| """ |
| return self._mu |
| |
| @property |
| def stdev(self): |
| "Standard deviation of the normal distribution." |
| return self._sigma |
| |
| @property |
| def variance(self): |
| "Square of the standard deviation." |
| return self._sigma ** 2.0 |
| |
| def __add__(x1, x2): |
| """Add a constant or another NormalDist instance. |
| |
| If *other* is a constant, translate mu by the constant, |
| leaving sigma unchanged. |
| |
| If *other* is a NormalDist, add both the means and the variances. |
| Mathematically, this works only if the two distributions are |
| independent or if they are jointly normally distributed. |
| """ |
| if isinstance(x2, NormalDist): |
| return NormalDist(x1._mu + x2._mu, hypot(x1._sigma, x2._sigma)) |
| return NormalDist(x1._mu + x2, x1._sigma) |
| |
| def __sub__(x1, x2): |
| """Subtract a constant or another NormalDist instance. |
| |
| If *other* is a constant, translate by the constant mu, |
| leaving sigma unchanged. |
| |
| If *other* is a NormalDist, subtract the means and add the variances. |
| Mathematically, this works only if the two distributions are |
| independent or if they are jointly normally distributed. |
| """ |
| 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): |
| """Multiply both mu and sigma by a constant. |
| |
| Used for rescaling, perhaps to change measurement units. |
| Sigma is scaled with the absolute value of the constant. |
| """ |
| return NormalDist(x1._mu * x2, x1._sigma * fabs(x2)) |
| |
| def __truediv__(x1, x2): |
| """Divide both mu and sigma by a constant. |
| |
| Used for rescaling, perhaps to change measurement units. |
| Sigma is scaled with the absolute value of the constant. |
| """ |
| return NormalDist(x1._mu / x2, x1._sigma / fabs(x2)) |
| |
| def __pos__(x1): |
| "Return a copy of the instance." |
| return NormalDist(x1._mu, x1._sigma) |
| |
| def __neg__(x1): |
| "Negates mu while keeping sigma the same." |
| return NormalDist(-x1._mu, x1._sigma) |
| |
| __radd__ = __add__ |
| |
| def __rsub__(x1, x2): |
| "Subtract a NormalDist from a constant or another NormalDist." |
| return -(x1 - x2) |
| |
| __rmul__ = __mul__ |
| |
| def __eq__(x1, x2): |
| "Two NormalDist objects are equal if their mu and sigma are both equal." |
| if not isinstance(x2, NormalDist): |
| return NotImplemented |
| return x1._mu == x2._mu and x1._sigma == x2._sigma |
| |
| def __hash__(self): |
| "NormalDist objects hash equal if their mu and sigma are both equal." |
| return hash((self._mu, self._sigma)) |
| |
| def __repr__(self): |
| return f'{type(self).__name__}(mu={self._mu!r}, sigma={self._sigma!r})' |
| |
| # If available, use C implementation |
| try: |
| from _statistics import _normal_dist_inv_cdf |
| except ImportError: |
| pass |
| |
| |
| 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 |
| import doctest |
| |
| g1 = NormalDist(10, 20) |
| g2 = NormalDist(-5, 25) |
| |
| # Test scaling by a constant |
| assert (g1 * 5 / 5).mean == g1.mean |
| assert (g1 * 5 / 5).stdev == g1.stdev |
| |
| 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.mean, G1.mean, rel_tol=0.01), (G1, G2) |
| assert isclose(G1.stdev, G2.stdev, 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) |
| |
| print(doctest.testmod()) |
