From c2bd7ad32a8914db740ff813c7f484683fe0fd39 Mon Sep 17 00:00:00 2001 From: ltcptgeneral <35508619+ltcptgeneral@users.noreply.github.com> Date: Sat, 17 Nov 2018 18:11:33 -0600 Subject: [PATCH] commit-2018/11/17@18:11CST --- __pycache__/analysis.cpython-37.pyc | Bin 13137 -> 18017 bytes analysis.py | 282 ++++++++++-- analysis.pyc | Bin 14201 -> 0 bytes analysis_test.py | 16 +- statistics.py | 669 ---------------------------- 5 files changed, 254 insertions(+), 713 deletions(-) delete mode 100644 analysis.pyc delete mode 100644 statistics.py diff --git a/__pycache__/analysis.cpython-37.pyc b/__pycache__/analysis.cpython-37.pyc index 2b29feb58f850c130044459e44029c6936ed257a..f1dd7180548b5775ba03796fc362908bfa36efd6 100644 GIT binary patch literal 18017 zcmdU1e~etmU4K7!-|qeJ`FuHFE^hkb#LlhbT+WX3%W)kianfI@liF#VUXyJ0cHZr+ zy|;Te@2!2e-9;#uindCjNI)wg6}wcFB0wN0Du@zVi1<~cq9OtBFGML)glLfv{1K4g z^O<@3c5l6xSW#NhJ-?ZGGxO$s=R4ouGtZBVjpYse7N`F0WB+r`Fuu#1?r$80Q@Fy| 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'stdev_z_split', - 'histo_analysis', #histo_analysis_old is intentionally left out as it has been depreciated + 'histo_analysis', #histo_analysis_old is intentionally left out as it has been depreciated since v 1.0.1.005 'poly_regression', 'r_squared', 'rms', 'basic_analysis', + #all statistics functions left out due to integration in other functions ] #now back to your regularly scheduled programming: -import statistics +#import statistics <-- statistics.py functions have been integrated into analysis.py as of v 1.0.3.002 import math import csv import functools @@ -44,10 +45,8 @@ import torch import scipy import matplotlib from sklearn import * - import collections import numbers - from fractions import Fraction from decimal import Decimal from itertools import groupby @@ -56,7 +55,7 @@ from bisect import bisect_left, bisect_right def _init_device (setting, arg): #initiates computation device for ANNs if setting == "cuda": - temp = setting + ":" + arg + temp = setting + ":" + str(arg) the_device_woman = torch.device(temp if torch.cuda.is_available() else "cpu") return the_device_woman #name that reference elif setting == "cpu": @@ -311,13 +310,13 @@ def load_csv(filepath): file_array = list(csv.reader(csvfile)) return file_array -def basic_stats(data, mode, arg): # data=array, mode = ['1d':1d_basic_stats, 'column':c_basic_stats, 'row':r_basic_stats], arg for mode 1 or mode 2 for column or row - - if mode == 'debug': +def basic_stats(data, method, arg): # data=array, mode = ['1d':1d_basic_stats, 'column':c_basic_stats, 'row':r_basic_stats], arg for mode 1 or mode 2 for column or row + + if method == 'debug': out = "basic_stats requires 3 args: data, mode, arg; where data is data to be analyzed, mode is an int from 0 - 2 depending on type of analysis (by column or by row) and is only applicable to 2d arrays (for 1d arrays use mode 1), and arg is row/column number for mode 1 or mode 2; function returns: [mean, median, mode, stdev, variance]" return out - if mode == "1d" or mode == 0: + if method == "1d" or method == 0: data_t = [] @@ -325,29 +324,29 @@ def basic_stats(data, mode, arg): # data=array, mode = ['1d':1d_basic_stats, 'co data_t.append(float(data[i])) - mean = statistics.mean(data_t) - median = statistics.median(data_t) + _mean = mean(data_t) + _median = median(data_t) try: - mode = statistics.mode(data_t) + _mode = mode(data_t) except: - mode = None + _mode = None try: - stdev = statistics.stdev(data) + _stdev = stdev(data_t) except: - stdev = None + _stdev = None try: - variance = statistics.variance(data_t) + _variance = variance(data_t) except: - variance = None + _variance = None - out = [mean, median, mode, stdev, variance] + out = [_mean, _median, _mode, _stdev, _variance] return out - elif mode == "column" or mode == 1: + elif method == "column" or method == 1: c_data = [] c_data_sorted = [] @@ -358,52 +357,52 @@ def basic_stats(data, mode, arg): # data=array, mode = ['1d':1d_basic_stats, 'co except: pass - mean = statistics.mean(c_data) - median = statistics.median(c_data) + _mean = mean(c_data) + _median = median(c_data) try: - mode = statistics.mode(c_data) + _mode = mode(c_data) except: - mode = None + _mode = None try: - stdev = statistics.stdev(c_data) + _stdev = stdev(c_data) except: - stdev = None + _stdev = None try: - variance = statistics.variance(c_data) + _variance = variance(c_data) except: - variance = None + _variance = None - out = [mean, median, mode, stdev, variance] + out = [_mean, _median, _mode, _stdev, _variance] return out - elif mode == "row" or mode == 2: + elif method == "row" or method == 2: r_data = [] for i in range(len(data[arg])): r_data.append(float(data[arg][i])) - mean = statistics.mean(r_data) - median = statistics.median(r_data) + _mean = mean(r_data) + _median = median(r_data) try: - mode = statistics.mode(r_data) + _mode = mode(r_data) except: - mode = None + _mode = None try: - stdev = statistics.stdev(r_data) + _stdev = stdev(r_data) except: - stdev = None + _stdev = None try: - variance = statistics.variance(r_data) + _variance = variance(r_data) except: - variance = None + _variance = None - out = [mean, median, mode, stdev, variance] + out = [_mean, _median, _mode, _stdev, _variance] return out else: - return ["mode_error", "mode_error"] + return ["ERROR: method error"] def z_score(point, mean, stdev): #returns z score with inputs of point, mean and standard deviation of spread score = (point - mean)/stdev @@ -427,7 +426,7 @@ def stdev_z_split(mean, stdev, delta, low_bound, high_bound): #returns n-th perc return z_split -def histo_analysis_old(hist_data): #note: depreciated +def histo_analysis_old(hist_data): #note: depreciated since v 1.0.1.005 if hist_data == 'debug': return['lower estimate (5%)', 'lower middle estimate (25%)', 'middle estimate (50%)', 'higher middle estimate (75%)', 'high estimate (95%)', 'standard deviation', 'note: this has been depreciated'] @@ -495,13 +494,15 @@ def histo_analysis(hist_data, delta, low_bound, high_bound): def poly_regression(x, y, power): - if x == "null": + if x == "null": #if x is 'null', then x will be filled with integer points between 1 and the size of y x = [] for i in range(len(y)): - x.append(i) + print(i) + + x.append(i+1) reg_eq = scipy.polyfit(x, y, deg = power) @@ -581,3 +582,198 @@ def basic_analysis(filepath): #assumes that rows are the independent variable an column_b_stats.append(basic_stats(data, "column", i)) return[row_b_stats, column_b_stats, row_histo] + +#statistics def below------------------------------------------------------------------------------------------------------------------------------------------------------ + +class StatisticsError(ValueError): + pass + +def _sum(data, start=0): + 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: + + total = partials[None] + assert not _isfinite(total) + else: + + 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): + + assert T is not bool, "initial type T is bool" + + if T is S: return T + + if S is int or S is bool: return T + if T is int: return S + + if issubclass(S, T): return S + if issubclass(T, S): return T + + if issubclass(T, int): return S + if issubclass(S, int): return T + + if issubclass(T, Fraction) and issubclass(S, float): + return S + if issubclass(T, float) and issubclass(S, Fraction): + return T + + msg = "don't know how to coerce %s and %s" + raise TypeError(msg % (T.__name__, S.__name__)) + +def _exact_ratio(x): + + try: + + if type(x) is float or type(x) is Decimal: + return x.as_integer_ratio() + try: + + return (x.numerator, x.denominator) + except AttributeError: + try: + + return x.as_integer_ratio() + except AttributeError: + + pass + except (OverflowError, ValueError): + + 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): + + if type(value) is T: + + return value + if issubclass(T, int) and value.denominator != 1: + T = float + try: + + return T(value) + except TypeError: + if issubclass(T, Decimal): + return T(value.numerator)/T(value.denominator) + else: + raise + +def _counts(data): + + table = collections.Counter(iter(data)).most_common() + if not table: + return table + + maxfreq = table[0][1] + for i in range(1, len(table)): + if table[i][1] != maxfreq: + table = table[:i] + break + return table + + +def _find_lteq(a, x): + + i = bisect_left(a, x) + if i != len(a) and a[i] == x: + return i + raise ValueError + + +def _find_rteq(a, l, 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'): + + for x in values: + if x < 0: + raise StatisticsError(errmsg) + yield x + +def mean(data): + + 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 median(data): + + 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 mode(data): + + table = _counts(data) + if len(table) == 1: + return table[0][0] + elif table: + raise StatisticsError( + 'no unique mode; found %d equally common values' % len(table) + ) + else: + raise StatisticsError('no mode for empty data') + +def _ss(data, c=None): + + if c is None: + c = mean(data) + T, total, count = _sum((x-c)**2 for x in data) + + 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): + + 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 stdev(data, xbar=None): + + var = variance(data, xbar) + try: + return var.sqrt() + except AttributeError: + return math.sqrt(var) diff --git a/analysis.pyc b/analysis.pyc deleted file mode 100644 index 0c992d6962f45a623892d601c984f61ab49e9bdd..0000000000000000000000000000000000000000 GIT binary patch literal 0 HcmV?d00001 literal 14201 zcmd^GTWlOx89p<+-iu@B>NGKF$TaE2Ns~60R#lVI^b(pBQL7yX(z;ED@yvKV>3YVV znRV>SMo2AEA1WabNIW1ef+xfSycG$lNKoDoLgE1=9*|IZppQsA@KV0-KQp^yXPq>5 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E.g. if your data points are rounded to -the nearest whole number: - ->>> median_grouped([2, 2, 3, 3, 3, 4]) #doctest: +ELLIPSIS -2.8333333333... - -This should be interpreted in this way: you have two data points in the class -interval 1.5-2.5, three data points in the class interval 2.5-3.5, and one in -the class interval 3.5-4.5. The median of these data points is 2.8333... - - -Calculating variability or spread ---------------------------------- - -================== ============================================= -Function Description -================== ============================================= -pvariance Population variance of data. -variance Sample variance of data. -pstdev Population standard deviation of data. -stdev Sample standard deviation of data. -================== ============================================= - -Calculate the standard deviation of sample data: - ->>> stdev([2.5, 3.25, 5.5, 11.25, 11.75]) #doctest: +ELLIPSIS -4.38961843444... - -If you have previously calculated the mean, you can pass it as the optional -second argument to the four "spread" functions to avoid recalculating it: - ->>> data = [1, 2, 2, 4, 4, 4, 5, 6] ->>> mu = mean(data) ->>> pvariance(data, mu) -2.5 - - -Exceptions ----------- - -A single exception is defined: StatisticsError is a subclass of ValueError. - -""" - -__all__ = [ 'StatisticsError', - 'pstdev', 'pvariance', 'stdev', 'variance', - 'median', 'median_low', 'median_high', 'median_grouped', - 'mean', 'mode', 'harmonic_mean', - ] - -import collections -import math -import numbers - -from fractions import Fraction -from decimal import Decimal -from itertools import groupby -from bisect import bisect_left, bisect_right - - - -# === 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) - (, Fraction(11, 1), 5) - - Some sources of round-off error will be avoided: - - # Built-in sum returns zero. - >>> _sum([1e50, 1, -1e50] * 1000) - (, 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)]) - (, 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) - (, 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 _counts(data): - # Generate a table of sorted (value, frequency) pairs. - table = collections.Counter(iter(data)).most_common() - if not table: - return table - # Extract the values with the highest frequency. - maxfreq = table[0][1] - for i in range(1, len(table)): - if table[i][1] != maxfreq: - table = table[:i] - break - return table - - -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 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 is not exactly one most common value, ``mode`` will raise - StatisticsError. - """ - # Generate a table of sorted (value, frequency) pairs. - table = _counts(data) - if len(table) == 1: - return table[0][0] - elif table: - raise StatisticsError( - 'no unique mode; found %d equally common values' % len(table) - ) - else: - raise StatisticsError('no mode for empty data') - - -# === Measures of spread === - -# See http://mathworld.wolfram.com/Variance.html -# http://mathworld.wolfram.com/SampleVariance.html -# http://en.wikipedia.org/wiki/Algorithms_for_calculating_variance -# -# Under no circumstances use the so-called "computational formula for -# variance", as that is only suitable for hand calculations with a small -# amount of low-precision data. It has terrible numeric properties. -# -# See a comparison of three computational methods here: -# http://www.johndcook.com/blog/2008/09/26/comparing-three-methods-of-computing-standard-deviation/ - -def _ss(data, c=None): - """Return sum of square deviations of sequence data. - - If ``c`` is None, the mean is calculated in one pass, and the deviations - from the mean are calculated in a second pass. Otherwise, deviations are - calculated from ``c`` as given. Use the second case with care, as it can - lead to garbage results. - """ - if c is None: - c = mean(data) - 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 an iterable of Real-valued numbers, with at least one - value. The optional argument mu, if given, should be the mean of - the data. If it is missing or None, the mean is automatically calculated. - - Use this function to calculate the variance from the entire population. - To estimate the variance from a sample, the ``variance`` function is - usually a better choice. - - Examples: - - >>> data = [0.0, 0.25, 0.25, 1.25, 1.5, 1.75, 2.75, 3.25] - >>> pvariance(data) - 1.25 - - If you have already calculated the mean of the data, you can pass it as - the optional second argument to avoid recalculating it: - - >>> mu = mean(data) - >>> pvariance(data, mu) - 1.25 - - This function does not check that ``mu`` is actually the mean of ``data``. - Giving arbitrary values for ``mu`` may lead to invalid or impossible - results. - - Decimals and Fractions are supported: - - >>> from decimal import Decimal as D - >>> pvariance([D("27.5"), D("30.25"), D("30.25"), D("34.5"), D("41.75")]) - Decimal('24.815') - - >>> from fractions import Fraction as F - >>> pvariance([F(1, 4), F(5, 4), F(1, 2)]) - Fraction(13, 72) - - """ - if iter(data) is data: - data = list(data) - n = len(data) - if n < 1: - raise StatisticsError('pvariance requires at least one data point') - 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)