| # Copyright 2015 The Chromium Authors. All rights reserved. |
| # Use of this source code is governed by a BSD-style license that can be |
| # found in the LICENSE file. |
| |
| """General statistical or mathematical functions.""" |
| |
| import math |
| |
| from recipe_engine import recipe_api |
| |
| |
| class MathUtilsApi(recipe_api.RecipeApi): |
| |
| @staticmethod |
| def mean(values): |
| """Calculates the arithmetic mean of a list of values.""" |
| if not values: |
| raise ValueError('Trying to take the mean of an empty list.') |
| return float(math.fsum(values)) / len(values) |
| |
| @staticmethod |
| def variance(values): |
| """Calculates the sample variance.""" |
| if len(values) == 1: |
| return 0.0 |
| mean = MathUtilsApi.mean(values) |
| differences_from_mean = [float(x) - mean for x in values] |
| squared_differences = [float(x * x) for x in differences_from_mean] |
| _variance = math.fsum(squared_differences) / (len(values) - 1) |
| return _variance |
| |
| @staticmethod |
| def standard_deviation(values): |
| """Calculates the sample standard deviation of the given list of values.""" |
| return math.sqrt(MathUtilsApi.variance(values)) |
| |
| @staticmethod |
| def relative_change(before, after): |
| """Returns the relative change of before and after, relative to before. |
| |
| There are several different ways to define relative difference between |
| two numbers; sometimes it is defined as relative to the smaller number, |
| or to the mean of the two numbers. This version returns the difference |
| relative to the first of the two numbers. |
| |
| Args: |
| before: A number representing an earlier value. |
| after: Another number, representing a later value. |
| |
| Returns: |
| A non-negative floating point number; 0.1 represents a 10% change. |
| """ |
| if before == after: |
| return 0.0 |
| if before == 0: |
| return float('nan') |
| difference = after - before |
| return math.fabs(difference / before) |
| |
| @staticmethod |
| def pooled_standard_error(work_sets): |
| """Calculates the pooled sample standard error for a set of samples. |
| |
| Args: |
| work_sets: A collection of collections of numbers. |
| |
| Returns: |
| Pooled sample standard error. |
| """ |
| numerator = 0.0 |
| denominator1 = 0.0 |
| denominator2 = 0.0 |
| |
| for current_set in work_sets: |
| std_dev = MathUtilsApi.standard_deviation(current_set) |
| numerator += (len(current_set) - 1) * std_dev ** 2 |
| denominator1 += len(current_set) - 1 |
| if len(current_set) > 0: |
| denominator2 += 1.0 / len(current_set) |
| |
| if denominator1 == 0: |
| return 0.0 |
| |
| return math.sqrt(numerator / denominator1) * math.sqrt(denominator2) |
| |
| @staticmethod |
| def standard_error(values): |
| """Calculates the standard error of a list of values.""" |
| if len(values) <= 1: |
| return 0.0 |
| std_dev = MathUtilsApi.standard_deviation(values) |
| return std_dev / math.sqrt(len(values)) |
| |
| #Copied this from BisectResults |
| @staticmethod |
| def confidence_score(sample1, sample2, |
| accept_single_bad_or_good=False): |
| """Calculates a confidence score. |
| |
| This score is a percentage which represents our degree of confidence in the |
| proposition that the good results and bad results are distinct groups, and |
| their differences aren't due to chance alone. |
| |
| |
| Args: |
| sample1: A flat list of "good" result numbers. |
| sample2: A flat list of "bad" result numbers. |
| accept_single_bad_or_good: If True, computes confidence even if there is |
| just one bad or good revision, otherwise single good or bad revision |
| always returns 0.0 confidence. This flag will probably get away when |
| we will implement expanding the bisect range by one more revision for |
| such case. |
| |
| Returns: |
| A number in the range [0, 100]. |
| """ |
| # If there's only one item in either list, this means only one revision was |
| # classified good or bad; this isn't good enough evidence to make a |
| # decision. If an empty list was passed, that also implies zero confidence. |
| if not accept_single_bad_or_good: |
| if len(sample1) <= 1 or len(sample2) <= 1: |
| return 0.0 |
| |
| # If there were only empty lists in either of the lists (this is unexpected |
| # and normally shouldn't happen), then we also want to return 0. |
| if not sample1 or not sample2: |
| return 0.0 |
| |
| # The p-value is approximately the probability of obtaining the given set |
| # of good and bad values just by chance. |
| _, _, p_value = MathUtilsApi.welchs_t_test(sample1, sample2) |
| return 100.0 * (1.0 - p_value) |
| |
| @staticmethod |
| def welchs_t_test(sample1, sample2): |
| """Performs Welch's t-test on the two samples. |
| |
| Welch's t-test is an adaptation of Student's t-test which is used when the |
| two samples may have unequal variances. It is also an independent two-sample |
| t-test. |
| |
| Args: |
| sample1: A collection of numbers. |
| sample2: Another collection of numbers. |
| |
| Returns: |
| A 3-tuple (t-statistic, degrees of freedom, p-value). |
| """ |
| mean1 = MathUtilsApi.mean(sample1) |
| mean2 = MathUtilsApi.mean(sample2) |
| v1 = MathUtilsApi.variance(sample1) |
| v2 = MathUtilsApi.variance(sample2) |
| n1 = len(sample1) |
| n2 = len(sample2) |
| t = MathUtilsApi._t_value(mean1, mean2, v1, v2, n1, n2) |
| df = MathUtilsApi._degrees_of_freedom(v1, v2, n1, n2) |
| p = MathUtilsApi._lookup_p_value(t, df) |
| return t, df, p |
| |
| @staticmethod |
| def _t_value(mean1, mean2, v1, v2, n1, n2): |
| """Calculates a t-statistic value using the formula for Welch's t-test. |
| |
| The t value can be thought of as a signal-to-noise ratio; a higher t-value |
| tells you that the groups are more different. |
| |
| Args: |
| mean1: Mean of sample 1. |
| mean2: Mean of sample 2. |
| v1: Variance of sample 1. |
| v2: Variance of sample 2. |
| n1: Sample size of sample 1. |
| n2: Sample size of sample 2. |
| |
| Returns: |
| A t value, which may be negative or positive. |
| """ |
| if v1 == 0 and v2 == 0: |
| # The variance of both segments is zero. |
| |
| # 64-bit floats have a machine epsilon of about 2.2e-16. |
| if abs(mean1 - mean2) < 1e-12: |
| # If they have the same mean, return 0. This implies that the p-value |
| # is 1 and we fail to reject the null hypothesis. |
| # (SciPy returns NaN in this scenario.) |
| return 0.0 |
| else: |
| # The distributions are maximally different; return a large t-value. |
| return 1000.0 |
| return (mean1 - mean2) / (math.sqrt(v1 / n1 + v2 / n2)) |
| |
| @staticmethod |
| def _degrees_of_freedom(v1, v2, n1, n2): |
| """Calculates degrees of freedom using the Welch-Satterthwaite formula. |
| |
| Degrees of freedom is a measure of sample size. For other types of tests, |
| degrees of freedom is sometimes N - 1, where N is the sample size. However, |
| |
| Args: |
| v1: Variance of sample 1. |
| v2: Variance of sample 2. |
| n1: Size of sample 1. |
| n2: Size of sample 2. |
| |
| Returns: |
| An estimate of degrees of freedom. Must be at least 1.0. |
| """ |
| # When there's no variance in either sample, return 1. |
| if v1 == 0 and v2 == 0: |
| return 1 |
| # If the sample size is too small, also return the minimum (1). |
| if n1 <= 1 or n2 <= 1: |
| return 1 # pragma: no cover |
| df = (((v1 / n1 + v2 / n2) ** 2) / |
| ((v1 ** 2) / ((n1 ** 2) * (n1 - 1)) + |
| (v2 ** 2) / ((n2 ** 2) * (n2 - 1)))) |
| return max(1, df) |
| |
| # Below is a hard-coded table for looking up p-values. |
| # |
| # Normally, p-values are calculated based on the t-distribution formula. |
| # Looking up pre-calculated values is a less accurate but less complicated |
| # alternative. |
| # |
| # Reference: http://www.sjsu.edu/faculty/gerstman/StatPrimer/t-table.pdf |
| |
| # A list of p-values for a two-tailed test. The entries correspond to to |
| # entries in the rows of the table below. |
| TWO_TAIL = [1, 0.20, 0.10, 0.05, 0.02, 0.01, 0.005, 0.002, 0.001] |
| |
| # A map of degrees of freedom to lists of t-values. The index of the t-value |
| # can be used to look up the corresponding p-value. |
| TABLE = { |
| 1: [0, 3.078, 6.314, 12.706, 31.820, 63.657, 127.321, 318.309, 636.619], |
| 2: [0, 1.886, 2.920, 4.303, 6.965, 9.925, 14.089, 22.327, 31.599], |
| 3: [0, 1.638, 2.353, 3.182, 4.541, 5.841, 7.453, 10.215, 12.924], |
| 4: [0, 1.533, 2.132, 2.776, 3.747, 4.604, 5.598, 7.173, 8.610], |
| 5: [0, 1.476, 2.015, 2.571, 3.365, 4.032, 4.773, 5.893, 6.869], |
| 6: [0, 1.440, 1.943, 2.447, 3.143, 3.707, 4.317, 5.208, 5.959], |
| 7: [0, 1.415, 1.895, 2.365, 2.998, 3.499, 4.029, 4.785, 5.408], |
| 8: [0, 1.397, 1.860, 2.306, 2.897, 3.355, 3.833, 4.501, 5.041], |
| 9: [0, 1.383, 1.833, 2.262, 2.821, 3.250, 3.690, 4.297, 4.781], |
| 10: [0, 1.372, 1.812, 2.228, 2.764, 3.169, 3.581, 4.144, 4.587], |
| 11: [0, 1.363, 1.796, 2.201, 2.718, 3.106, 3.497, 4.025, 4.437], |
| 12: [0, 1.356, 1.782, 2.179, 2.681, 3.055, 3.428, 3.930, 4.318], |
| 13: [0, 1.350, 1.771, 2.160, 2.650, 3.012, 3.372, 3.852, 4.221], |
| 14: [0, 1.345, 1.761, 2.145, 2.625, 2.977, 3.326, 3.787, 4.140], |
| 15: [0, 1.341, 1.753, 2.131, 2.602, 2.947, 3.286, 3.733, 4.073], |
| 16: [0, 1.337, 1.746, 2.120, 2.584, 2.921, 3.252, 3.686, 4.015], |
| 17: [0, 1.333, 1.740, 2.110, 2.567, 2.898, 3.222, 3.646, 3.965], |
| 18: [0, 1.330, 1.734, 2.101, 2.552, 2.878, 3.197, 3.610, 3.922], |
| 19: [0, 1.328, 1.729, 2.093, 2.539, 2.861, 3.174, 3.579, 3.883], |
| 20: [0, 1.325, 1.725, 2.086, 2.528, 2.845, 3.153, 3.552, 3.850], |
| 21: [0, 1.323, 1.721, 2.080, 2.518, 2.831, 3.135, 3.527, 3.819], |
| 22: [0, 1.321, 1.717, 2.074, 2.508, 2.819, 3.119, 3.505, 3.792], |
| 23: [0, 1.319, 1.714, 2.069, 2.500, 2.807, 3.104, 3.485, 3.768], |
| 24: [0, 1.318, 1.711, 2.064, 2.492, 2.797, 3.090, 3.467, 3.745], |
| 25: [0, 1.316, 1.708, 2.060, 2.485, 2.787, 3.078, 3.450, 3.725], |
| 26: [0, 1.315, 1.706, 2.056, 2.479, 2.779, 3.067, 3.435, 3.707], |
| 27: [0, 1.314, 1.703, 2.052, 2.473, 2.771, 3.057, 3.421, 3.690], |
| 28: [0, 1.313, 1.701, 2.048, 2.467, 2.763, 3.047, 3.408, 3.674], |
| 29: [0, 1.311, 1.699, 2.045, 2.462, 2.756, 3.038, 3.396, 3.659], |
| 30: [0, 1.310, 1.697, 2.042, 2.457, 2.750, 3.030, 3.385, 3.646], |
| 31: [0, 1.309, 1.695, 2.040, 2.453, 2.744, 3.022, 3.375, 3.633], |
| 32: [0, 1.309, 1.694, 2.037, 2.449, 2.738, 3.015, 3.365, 3.622], |
| 33: [0, 1.308, 1.692, 2.035, 2.445, 2.733, 3.008, 3.356, 3.611], |
| 34: [0, 1.307, 1.691, 2.032, 2.441, 2.728, 3.002, 3.348, 3.601], |
| 35: [0, 1.306, 1.690, 2.030, 2.438, 2.724, 2.996, 3.340, 3.591], |
| 36: [0, 1.306, 1.688, 2.028, 2.434, 2.719, 2.991, 3.333, 3.582], |
| 37: [0, 1.305, 1.687, 2.026, 2.431, 2.715, 2.985, 3.326, 3.574], |
| 38: [0, 1.304, 1.686, 2.024, 2.429, 2.712, 2.980, 3.319, 3.566], |
| 39: [0, 1.304, 1.685, 2.023, 2.426, 2.708, 2.976, 3.313, 3.558], |
| 40: [0, 1.303, 1.684, 2.021, 2.423, 2.704, 2.971, 3.307, 3.551], |
| 42: [0, 1.302, 1.682, 2.018, 2.418, 2.698, 2.963, 3.296, 3.538], |
| 44: [0, 1.301, 1.680, 2.015, 2.414, 2.692, 2.956, 3.286, 3.526], |
| 46: [0, 1.300, 1.679, 2.013, 2.410, 2.687, 2.949, 3.277, 3.515], |
| 48: [0, 1.299, 1.677, 2.011, 2.407, 2.682, 2.943, 3.269, 3.505], |
| 50: [0, 1.299, 1.676, 2.009, 2.403, 2.678, 2.937, 3.261, 3.496], |
| 60: [0, 1.296, 1.671, 2.000, 2.390, 2.660, 2.915, 3.232, 3.460], |
| 70: [0, 1.294, 1.667, 1.994, 2.381, 2.648, 2.899, 3.211, 3.435], |
| 80: [0, 1.292, 1.664, 1.990, 2.374, 2.639, 2.887, 3.195, 3.416], |
| 90: [0, 1.291, 1.662, 1.987, 2.369, 2.632, 2.878, 3.183, 3.402], |
| 100: [0, 1.290, 1.660, 1.984, 2.364, 2.626, 2.871, 3.174, 3.391], |
| 120: [0, 1.289, 1.658, 1.980, 2.358, 2.617, 2.860, 3.160, 3.373], |
| 150: [0, 1.287, 1.655, 1.976, 2.351, 2.609, 2.849, 3.145, 3.357], |
| 200: [0, 1.286, 1.652, 1.972, 2.345, 2.601, 2.839, 3.131, 3.340], |
| 300: [0, 1.284, 1.650, 1.968, 2.339, 2.592, 2.828, 3.118, 3.323], |
| 500: [0, 1.283, 1.648, 1.965, 2.334, 2.586, 2.820, 3.107, 3.310], |
| } |
| |
| @staticmethod |
| def _lookup_p_value(t, df): |
| """Looks up a p-value in a t-distribution table. |
| |
| Args: |
| t: A t statistic value; the result of a t-test. |
| df: Number of degrees of freedom. |
| |
| Returns: |
| A p-value, which represents the likelihood of obtaining a result at least |
| as extreme as the one observed just by chance (the null hypothesis). |
| """ |
| assert df >= 1, 'Degrees of freedom must be positive' |
| |
| # We ignore the negative sign on the t-value because our null hypothesis |
| # is that the two samples are the same; our alternative hypothesis is that |
| # the second sample is lesser OR greater than the first. |
| t = abs(t) |
| |
| def greatest_smaller(nums, target): |
| """Returns the largest number that is <= the target number.""" |
| lesser_equal = [n for n in nums if n <= target] |
| assert lesser_equal, 'No number in number list <= target.' |
| return max(lesser_equal) |
| |
| df_key = greatest_smaller(MathUtilsApi.TABLE.keys(), df) |
| t_table_row = MathUtilsApi.TABLE[df_key] |
| approximate_t_value = greatest_smaller(t_table_row, t) |
| t_value_index = t_table_row.index(approximate_t_value) |
| |
| return MathUtilsApi.TWO_TAIL[t_value_index] |
