bpo-36324: Add inv_cdf() to statistics.NormalDist() (GH-12377)

Doc/library/statistics.rst

@@ -569,6 +569,18 @@ of applications in statistics.
        compute the probability that a random variable *X* will be less than or
        equal to *x*.  Mathematically, it is written ``P(X <= x)``.
 
+    .. method:: NormalDist.inv_cdf(p)
+
+       Compute the inverse cumulative distribution function, also known as the
+       `quantile function <https://en.wikipedia.org/wiki/Quantile_function>`_
+       or the `percent-point
+       <https://www.statisticshowto.datasciencecentral.com/inverse-distribution-function/>`_
+       function.  Mathematically, it is written ``x : P(X <= x) = p``.
+
+       Finds the value *x* of the random variable *X* such that the
+       probability of the variable being less than or equal to that value
+       equals the given probability *p*.
+
     .. method:: NormalDist.overlap(other)
 
        Compute the `overlapping coefficient (OVL)
@@ -628,6 +640,16 @@ rounding to the nearest whole number:
     >>> round(fraction * 100.0, 1)
     18.4
 
+Find the `quartiles <https://en.wikipedia.org/wiki/Quartile>`_ and `deciles
+<https://en.wikipedia.org/wiki/Decile>`_ for the SAT scores:
+
+.. doctest::
+
+    >>> [round(sat.inv_cdf(p)) for p in (0.25, 0.50, 0.75)]
+    [928, 1060, 1192]
+    >>> [round(sat.inv_cdf(p / 10)) for p in range(1, 10)]
+    [810, 896, 958, 1011, 1060, 1109, 1162, 1224, 1310]
+
 What percentage of men and women will have the same height in `two normally
 distributed populations with known means and standard deviations
 <http://www.usablestats.com/lessons/normal>`_?

Lib/statistics.py

@@ -745,6 +745,101 @@ class NormalDist:
             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')
+
+        # 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:
+            a0 = 3.38713_28727_96366_6080e+0
+            a1 = 1.33141_66789_17843_7745e+2
+            a2 = 1.97159_09503_06551_4427e+3
+            a3 = 1.37316_93765_50946_1125e+4
+            a4 = 4.59219_53931_54987_1457e+4
+            a5 = 6.72657_70927_00870_0853e+4
+            a6 = 3.34305_75583_58812_8105e+4
+            a7 = 2.50908_09287_30122_6727e+3
+            b1 = 4.23133_30701_60091_1252e+1
+            b2 = 6.87187_00749_20579_0830e+2
+            b3 = 5.39419_60214_24751_1077e+3
+            b4 = 2.12137_94301_58659_5867e+4
+            b5 = 3.93078_95800_09271_0610e+4
+            b6 = 2.87290_85735_72194_2674e+4
+            b7 = 5.22649_52788_52854_5610e+3
+            r = 0.180625 - q * q
+            num = (q * (((((((a7 * r + a6) * r + a5) * r + a4) * r + a3)
+                        * r + a2) * r + a1) * r + a0))
+            den = ((((((((b7 * r + b6) * r + b5) * r + b4) * r + b3)
+                        * r + b2) * r + b1) * r + 1.0))
+            x = num / den
+            return self.mu + (x * self.sigma)
+
+        r = p if q <= 0.0 else 1.0 - p
+        r = sqrt(-log(r))
+        if r <= 5.0:
+            c0 = 1.42343_71107_49683_57734e+0
+            c1 = 4.63033_78461_56545_29590e+0
+            c2 = 5.76949_72214_60691_40550e+0
+            c3 = 3.64784_83247_63204_60504e+0
+            c4 = 1.27045_82524_52368_38258e+0
+            c5 = 2.41780_72517_74506_11770e-1
+            c6 = 2.27238_44989_26918_45833e-2
+            c7 = 7.74545_01427_83414_07640e-4
+            d1 = 2.05319_16266_37758_82187e+0
+            d2 = 1.67638_48301_83803_84940e+0
+            d3 = 6.89767_33498_51000_04550e-1
+            d4 = 1.48103_97642_74800_74590e-1
+            d5 = 1.51986_66563_61645_71966e-2
+            d6 = 5.47593_80849_95344_94600e-4
+            d7 = 1.05075_00716_44416_84324e-9
+            r = r - 1.6
+            num = ((((((((c7 * r + c6) * r + c5) * r + c4) * r + c3)
+                      * r + c2) * r + c1) * r + c0))
+            den = ((((((((d7 * r + d6) * r + d5) * r + d4) * r + d3)
+                      * r + d2) * r + d1) * r + 1.0))
+        else:
+            e0 = 6.65790_46435_01103_77720e+0
+            e1 = 5.46378_49111_64114_36990e+0
+            e2 = 1.78482_65399_17291_33580e+0
+            e3 = 2.96560_57182_85048_91230e-1
+            e4 = 2.65321_89526_57612_30930e-2
+            e5 = 1.24266_09473_88078_43860e-3
+            e6 = 2.71155_55687_43487_57815e-5
+            e7 = 2.01033_43992_92288_13265e-7
+            f1 = 5.99832_20655_58879_37690e-1
+            f2 = 1.36929_88092_27358_05310e-1
+            f3 = 1.48753_61290_85061_48525e-2
+            f4 = 7.86869_13114_56132_59100e-4
+            f5 = 1.84631_83175_10054_68180e-5
+            f6 = 1.42151_17583_16445_88870e-7
+            f7 = 2.04426_31033_89939_78564e-15
+            r = r - 5.0
+            num = ((((((((e7 * r + e6) * r + e5) * r + e4) * r + e3)
+                      * r + e2) * r + e1) * r + e0))
+            den = ((((((((f7 * r + f6) * r + f5) * r + f4) * r + f3)
+                      * r + f2) * r + f1) * r + 1.0))
+
+        x = num / den
+        if q < 0.0:
+            x = -x
+        return self.mu + (x * self.sigma)
+
     def overlap(self, other):
         '''Compute the overlapping coefficient (OVL) between two normal distributions.
 

Lib/test/test_statistics.py

@@ -2174,6 +2174,69 @@ class TestNormalDist(unittest.TestCase):
         self.assertEqual(X.cdf(float('Inf')), 1.0)
         self.assertTrue(math.isnan(X.cdf(float('NaN'))))
 
+    def test_inv_cdf(self):
+        NormalDist = statistics.NormalDist
+
+        # Center case should be exact.
+        iq = NormalDist(100, 15)
+        self.assertEqual(iq.inv_cdf(0.50), iq.mean)
+
+        # Test versus a published table of known percentage points.
+        # See the second table at the bottom of the page here:
+        # http://people.bath.ac.uk/masss/tables/normaltable.pdf
+        Z = NormalDist()
+        pp = {5.0: (0.000, 1.645, 2.576, 3.291, 3.891,
+                    4.417, 4.892, 5.327, 5.731, 6.109),
+              2.5: (0.674, 1.960, 2.807, 3.481, 4.056,
+                    4.565, 5.026, 5.451, 5.847, 6.219),
+              1.0: (1.282, 2.326, 3.090, 3.719, 4.265,
+                    4.753, 5.199, 5.612, 5.998, 6.361)}
+        for base, row in pp.items():
+            for exp, x in enumerate(row, start=1):
+                p = base * 10.0 ** (-exp)
+                self.assertAlmostEqual(-Z.inv_cdf(p), x, places=3)
+                p = 1.0 - p
+                self.assertAlmostEqual(Z.inv_cdf(p), x, places=3)
+
+        # Match published example for MS Excel
+        # https://support.office.com/en-us/article/norm-inv-function-54b30935-fee7-493c-bedb-2278a9db7e13
+        self.assertAlmostEqual(NormalDist(40, 1.5).inv_cdf(0.908789), 42.000002)
+
+        # One million equally spaced probabilities
+        n = 2**20
+        for p in range(1, n):
+            p /= n
+            self.assertAlmostEqual(iq.cdf(iq.inv_cdf(p)), p)
+
+        # One hundred ever smaller probabilities to test tails out to
+        # extreme probabilities: 1 / 2**50 and (2**50-1) / 2 ** 50
+        for e in range(1, 51):
+            p = 2.0 ** (-e)
+            self.assertAlmostEqual(iq.cdf(iq.inv_cdf(p)), p)
+            p = 1.0 - p
+            self.assertAlmostEqual(iq.cdf(iq.inv_cdf(p)), p)
+
+        # Now apply cdf() first.  At six sigmas, the round-trip
+        # loses a lot of precision, so only check to 6 places.
+        for x in range(10, 190):
+            self.assertAlmostEqual(iq.inv_cdf(iq.cdf(x)), x, places=6)
+
+        # Error cases:
+        with self.assertRaises(statistics.StatisticsError):
+            iq.inv_cdf(0.0)                         # p is zero
+        with self.assertRaises(statistics.StatisticsError):
+            iq.inv_cdf(-0.1)                        # p under zero
+        with self.assertRaises(statistics.StatisticsError):
+            iq.inv_cdf(1.0)                         # p is one
+        with self.assertRaises(statistics.StatisticsError):
+            iq.inv_cdf(1.1)                         # p over one
+        with self.assertRaises(statistics.StatisticsError):
+            iq.sigma = 0.0                          # sigma is zero
+            iq.inv_cdf(0.5)
+        with self.assertRaises(statistics.StatisticsError):
+            iq.sigma = -0.1                         # sigma under zero
+            iq.inv_cdf(0.5)
+
     def test_overlap(self):
         NormalDist = statistics.NormalDist
 

Misc/NEWS.d/next/Library/2019-03-17-01-17-45.bpo-36324.dvNrRe.rst

+Add method to statistics.NormalDist for computing the inverse cumulative
+normal distribution.