bpo-38490: statistics: Add covariance, Pearson's correlation, and simple linear regression (#16813)

Co-authored-by: Tymoteusz Wołodźko <twolodzko+gitkraken@gmail.com

Doc/library/statistics.rst

@@ -68,6 +68,17 @@ tends to deviate from the typical or average values.
 :func:`variance`         Sample variance of data.
 =======================  =============================================
 
+Statistics for relations between two inputs
+-------------------------------------------
+
+These functions calculate statistics regarding relations between two inputs.
+
+=========================  =====================================================
+:func:`covariance`         Sample covariance for two variables.
+:func:`correlation`        Pearson's correlation coefficient for two variables.
+:func:`linear_regression`  Intercept and slope for simple linear regression.
+=========================  =====================================================
+
 
 Function details
 ----------------
@@ -566,6 +577,98 @@ However, for reading convenience, most of the examples show sorted sequences.
 
    .. versionadded:: 3.8
 
+.. function:: covariance(x, y, /)
+
+   Return the sample covariance of two inputs *x* and *y*. Covariance
+   is a measure of the joint variability of two inputs.
+
+   Both inputs must be of the same length (no less than two), otherwise
+   :exc:`StatisticsError` is raised.
+
+   Examples:
+
+   .. doctest::
+
+      >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
+      >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
+      >>> covariance(x, y)
+      0.75
+      >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
+      >>> covariance(x, z)
+      -7.5
+      >>> covariance(z, x)
+      -7.5
+
+   .. versionadded:: 3.10
+
+.. function:: correlation(x, y, /)
+
+   Return the `Pearson's correlation coefficient
+   <https://en.wikipedia.org/wiki/Pearson_correlation_coefficient>`_
+   for two inputs. Pearson's correlation coefficient *r* takes values
+   between -1 and +1. It measures the strength and direction of the linear
+   relationship, where +1 means very strong, positive linear relationship,
+   -1 very strong, negative linear relationship, and 0 no linear relationship.
+
+   Both inputs must be of the same length (no less than two), and need
+   not to be constant, otherwise :exc:`StatisticsError` is raised.
+
+   Examples:
+
+   .. doctest::
+
+      >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
+      >>> y = [9, 8, 7, 6, 5, 4, 3, 2, 1]
+      >>> correlation(x, x)
+      1.0
+      >>> correlation(x, y)
+      -1.0
+
+   .. versionadded:: 3.10
+
+.. function:: linear_regression(regressor, dependent_variable)
+
+   Return the intercept and slope of `simple linear regression
+   <https://en.wikipedia.org/wiki/Simple_linear_regression>`_
+   parameters estimated using ordinary least squares. Simple linear
+   regression describes relationship between *regressor* and
+   *dependent variable* in terms of linear function:
+
+      *dependent_variable = intercept + slope \* regressor + noise*
+
+   where ``intercept`` and ``slope`` are the regression parameters that are
+   estimated, and noise term is an unobserved random variable, for the
+   variability of the data that was not explained by the linear regression
+   (it is equal to the difference between prediction and the actual values
+   of dependent variable).
+
+   Both inputs must be of the same length (no less than two), and regressor
+   needs not to be constant, otherwise :exc:`StatisticsError` is raised.
+
+   For example, if we took the data on the data on `release dates of the Monty
+   Python films <https://en.wikipedia.org/wiki/Monty_Python#Films>`_, and used
+   it to predict the cumulative number of Monty Python films produced, we could
+   predict what would be the number of films they could have made till year
+   2019, assuming that they kept the pace.
+
+   .. doctest::
+
+      >>> year = [1971, 1975, 1979, 1982, 1983]
+      >>> films_total = [1, 2, 3, 4, 5]
+      >>> intercept, slope = linear_regression(year, films_total)
+      >>> round(intercept + slope * 2019)
+      16
+
+   We could also use it to "predict" how many Monty Python films existed when
+   Brian Cohen was born.
+
+   .. doctest::
+
+      >>> round(intercept + slope * 1)
+      -610
+
+   .. versionadded:: 3.10
+
 
 Exceptions
 ----------

Doc/whatsnew/3.10.rst

@@ -1051,6 +1051,14 @@ The :mod:`shelve` module now uses :data:`pickle.DEFAULT_PROTOCOL` by default
 instead of :mod:`pickle` protocol ``3`` when creating shelves.
 (Contributed by Zackery Spytz in :issue:`34204`.)
 
+statistics
+----------
+
+Added :func:`~statistics.covariance`, Pearson's
+:func:`~statistics.correlation`, and simple
+:func:`~statistics.linear_regression` functions.
+(Contributed by Tymoteusz Wołodźko in :issue:`38490`.)
+
 site
 ----
 

Lib/statistics.py

@@ -73,6 +73,30 @@ second argument to the four "spread" functions to avoid recalculating it:
 2.5
 
 
+Statistics for relations between two inputs
+-------------------------------------------
+
+==================  ====================================================
+Function            Description
+==================  ====================================================
+covariance          Sample covariance for two variables.
+correlation         Pearson's correlation coefficient for two variables.
+linear_regression   Intercept and slope for simple linear regression.
+==================  ====================================================
+
+Calculate covariance, Pearson's correlation, and simple linear regression
+for two inputs:
+
+>>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
+>>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
+>>> covariance(x, y)
+0.75
+>>> correlation(x, y)  #doctest: +ELLIPSIS
+0.31622776601...
+>>> linear_regression(x, y)  #doctest:
+LinearRegression(intercept=1.5, slope=0.1)
+
+
 Exceptions
 ----------
 
@@ -98,6 +122,9 @@ __all__ = [
     'quantiles',
     'stdev',
     'variance',
+    'correlation',
+    'covariance',
+    'linear_regression',
 ]
 
 import math
@@ -110,7 +137,7 @@ from itertools import groupby, repeat
 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
+from collections import Counter, namedtuple
 
 # === Exceptions ===
 
@@ -826,6 +853,113 @@ def pstdev(data, mu=None):
         return math.sqrt(var)
 
 
+# === Statistics for relations between two inputs ===
+
+# See https://en.wikipedia.org/wiki/Covariance
+#     https://en.wikipedia.org/wiki/Pearson_correlation_coefficient
+#     https://en.wikipedia.org/wiki/Simple_linear_regression
+
+
+def covariance(x, y, /):
+    """Covariance
+
+    Return the sample covariance of two inputs *x* and *y*. Covariance
+    is a measure of the joint variability of two inputs.
+
+    >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
+    >>> y = [1, 2, 3, 1, 2, 3, 1, 2, 3]
+    >>> covariance(x, y)
+    0.75
+    >>> z = [9, 8, 7, 6, 5, 4, 3, 2, 1]
+    >>> covariance(x, z)
+    -7.5
+    >>> covariance(z, x)
+    -7.5
+
+    """
+    n = len(x)
+    if len(y) != n:
+        raise StatisticsError('covariance requires that both inputs have same number of data points')
+    if n < 2:
+        raise StatisticsError('covariance requires at least two data points')
+    xbar = mean(x)
+    ybar = mean(y)
+    total = fsum((xi - xbar) * (yi - ybar) for xi, yi in zip(x, y))
+    return total / (n - 1)
+
+
+def correlation(x, y, /):
+    """Pearson's correlation coefficient
+
+    Return the Pearson's correlation coefficient for two inputs. Pearson's
+    correlation coefficient *r* takes values between -1 and +1. It measures the
+    strength and direction of the linear relationship, where +1 means very
+    strong, positive linear relationship, -1 very strong, negative linear
+    relationship, and 0 no linear relationship.
+
+    >>> x = [1, 2, 3, 4, 5, 6, 7, 8, 9]
+    >>> y = [9, 8, 7, 6, 5, 4, 3, 2, 1]
+    >>> correlation(x, x)
+    1.0
+    >>> correlation(x, y)
+    -1.0
+
+    """
+    n = len(x)
+    if len(y) != n:
+        raise StatisticsError('correlation requires that both inputs have same number of data points')
+    if n < 2:
+        raise StatisticsError('correlation requires at least two data points')
+    cov = covariance(x, y)
+    stdx = stdev(x)
+    stdy = stdev(y)
+    try:
+        return cov / (stdx * stdy)
+    except ZeroDivisionError:
+        raise StatisticsError('at least one of the inputs is constant')
+
+
+LinearRegression = namedtuple('LinearRegression', ['intercept', 'slope'])
+
+
+def linear_regression(regressor, dependent_variable, /):
+    """Intercept and slope for simple linear regression
+
+    Return the intercept and slope of simple linear regression
+    parameters estimated using ordinary least squares. Simple linear
+    regression describes relationship between *regressor* and
+    *dependent variable* in terms of linear function::
+
+        dependent_variable = intercept + slope * regressor + noise
+
+    where ``intercept`` and ``slope`` are the regression parameters that are
+    estimated, and noise term is an unobserved random variable, for the
+    variability of the data that was not explained by the linear regression
+    (it is equal to the difference between prediction and the actual values
+    of dependent variable).
+
+    The parameters are returned as a named tuple.
+
+    >>> regressor = [1, 2, 3, 4, 5]
+    >>> noise = NormalDist().samples(5, seed=42)
+    >>> dependent_variable = [2 + 3 * regressor[i] + noise[i] for i in range(5)]
+    >>> linear_regression(regressor, dependent_variable)  #doctest: +ELLIPSIS
+    LinearRegression(intercept=1.75684970486..., slope=3.09078914170...)
+
+    """
+    n = len(regressor)
+    if len(dependent_variable) != n:
+        raise StatisticsError('linear regression requires that both inputs have same number of data points')
+    if n < 2:
+        raise StatisticsError('linear regression requires at least two data points')
+    try:
+        slope = covariance(regressor, dependent_variable) / variance(regressor)
+    except ZeroDivisionError:
+        raise StatisticsError('regressor is constant')
+    intercept = mean(dependent_variable) - slope * mean(regressor)
+    return LinearRegression(intercept=intercept, slope=slope)
+
+
 ## Normal Distribution #####################################################
 
 

Lib/test/test_statistics.py

@@ -2407,6 +2407,84 @@ class TestQuantiles(unittest.TestCase):
             quantiles([10, None, 30], n=4)      # data is non-numeric
 
 
+class TestBivariateStatistics(unittest.TestCase):
+
+    def test_unequal_size_error(self):
+        for x, y in [
+            ([1, 2, 3], [1, 2]),
+            ([1, 2], [1, 2, 3]),
+        ]:
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.covariance(x, y)
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.correlation(x, y)
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.linear_regression(x, y)
+
+    def test_small_sample_error(self):
+        for x, y in [
+            ([], []),
+            ([], [1, 2,]),
+            ([1, 2,], []),
+            ([1,], [1,]),
+            ([1,], [1, 2,]),
+            ([1, 2,], [1,]),
+        ]:
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.covariance(x, y)
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.correlation(x, y)
+            with self.assertRaises(statistics.StatisticsError):
+                statistics.linear_regression(x, y)
+
+
+class TestCorrelationAndCovariance(unittest.TestCase):
+
+    def test_results(self):
+        for x, y, result in [
+            ([1, 2, 3], [1, 2, 3], 1),
+            ([1, 2, 3], [-1, -2, -3], -1),
+            ([1, 2, 3], [3, 2, 1], -1),
+            ([1, 2, 3], [1, 2, 1], 0),
+            ([1, 2, 3], [1, 3, 2], 0.5),
+        ]:
+            self.assertAlmostEqual(statistics.correlation(x, y), result)
+            self.assertAlmostEqual(statistics.covariance(x, y), result)
+
+    def test_different_scales(self):
+        x = [1, 2, 3]
+        y = [10, 30, 20]
+        self.assertAlmostEqual(statistics.correlation(x, y), 0.5)
+        self.assertAlmostEqual(statistics.covariance(x, y), 5)
+
+        y = [.1, .2, .3]
+        self.assertAlmostEqual(statistics.correlation(x, y), 1)
+        self.assertAlmostEqual(statistics.covariance(x, y), 0.1)
+
+
+class TestLinearRegression(unittest.TestCase):
+
+    def test_constant_input_error(self):
+        x = [1, 1, 1,]
+        y = [1, 2, 3,]
+        with self.assertRaises(statistics.StatisticsError):
+            statistics.linear_regression(x, y)
+
+    def test_results(self):
+        for x, y, true_intercept, true_slope in [
+            ([1, 2, 3], [0, 0, 0], 0, 0),
+            ([1, 2, 3], [1, 2, 3], 0, 1),
+            ([1, 2, 3], [100, 100, 100], 100, 0),
+            ([1, 2, 3], [12, 14, 16], 10, 2),
+            ([1, 2, 3], [-1, -2, -3], 0, -1),
+            ([1, 2, 3], [21, 22, 23], 20, 1),
+            ([1, 2, 3], [5.1, 5.2, 5.3], 5, 0.1),
+        ]:
+            intercept, slope = statistics.linear_regression(x, y)
+            self.assertAlmostEqual(intercept, true_intercept)
+            self.assertAlmostEqual(slope, true_slope)
+
+
 class TestNormalDist:
 
     # General note on precision: The pdf(), cdf(), and overlap() methods

Misc/ACKS

@@ -1927,6 +1927,7 @@ David Wolever
 Klaus-Juergen Wolf
 Dan Wolfe
 Richard Wolff
+Tymoteusz Wołodźko
 Adam Woodbeck
 William Woodruff
 Steven Work

Misc/NEWS.d/next/Library/2019-10-16-08-08-14.bpo-38490.QbDXEF.rst

@@ -0,0 +1 @@
+Covariance, Pearson's correlation, and simple linear regression functionality was added to statistics module. Patch by Tymoteusz Wołodźko.