```#!/usr/bin/env python
"""Unit tests for special functions used in statistics.
"""
from cogent.util.unit_test import TestCase, main
from cogent.maths.stats.special import permutations, permutations_exact, \
ln_permutations, combinations, combinations_exact, \
ln_combinations, ln_binomial, log_one_minus, one_minus_exp, igami,\
ndtri, incbi, log1p

import math

__author__ = "Rob Knight"
__copyright__ = "Copyright 2007-2012, The Cogent Project"
__credits__ = ["Gavin Huttley", "Rob Knight", "Sandra Smit"]
__version__ = "1.5.3"
__maintainer__ = "Rob Knight"
__status__ = "Production"

class SpecialTests(TestCase):
"""Tests miscellaneous functions."""

def test_permutations(self):
"""permutations should return expected results"""
self.assertEqual(permutations(1,1), 1)
self.assertEqual(permutations(2,1), 2)
self.assertEqual(permutations(3,1), 3)
self.assertEqual(permutations(4,1), 4)
self.assertEqual(permutations(4,2), 12)
self.assertEqual(permutations(4,3), 24)
self.assertEqual(permutations(4,4), 24)
self.assertFloatEqual(permutations(300, 100), 3.8807387193009318e+239)

def test_permutations_errors(self):
"""permutations should raise errors on invalid input"""
self.assertRaises(IndexError,permutations,10,50)
self.assertRaises(IndexError,permutations,-1,50)
self.assertRaises(IndexError,permutations,10,-5)

def test_permutations_float(self):
"""permutations should use gamma function when floats as input"""
self.assertFloatEqual(permutations(1.0,1), 1)
self.assertFloatEqual(permutations(2,1.0), 2)
self.assertFloatEqual(permutations(3.0,1.0), 3)
self.assertFloatEqual(permutations(4.0,1), 4)
self.assertFloatEqual(permutations(4.0,2.0), 12)
self.assertFloatEqual(permutations(4.0,3.0), 24)
self.assertFloatEqual(permutations(4,4.0), 24)
self.assertFloatEqual(permutations(300, 100), 3.8807387193009318e+239)

def test_permutations_range(self):
"""permutations should increase gradually with increasing k"""
start = 5 #permuations(10,5) = 30240
end = 6 #permutations(10,6) = 151200
step = 0.1
lower_lim = 30240
upper_lim = 151200
previous_value = 30239.9999
while start <= end:
obs = permutations(10,start)
assert lower_lim <= obs <= upper_lim
assert obs > previous_value
previous_value = obs
start += step

def test_permutations_exact(self):
"""permutations_exact should return expected results"""
self.assertFloatEqual(permutations_exact(1,1), 1)
self.assertFloatEqual(permutations_exact(2,1), 2)
self.assertFloatEqual(permutations_exact(3,1), 3)
self.assertFloatEqual(permutations_exact(4,1), 4)
self.assertFloatEqual(permutations_exact(4,2), 12)
self.assertFloatEqual(permutations_exact(4,3), 24)
self.assertFloatEqual(permutations_exact(4,4), 24)
self.assertFloatEqual(permutations_exact(300,100),\
3.8807387193009318e239)

def test_ln_permutations(self):
"""ln_permutations should return expected results"""
self.assertFloatEqual(ln_permutations(1,1), math.log(1))
self.assertFloatEqual(ln_permutations(2,1), math.log(2))
self.assertFloatEqual(ln_permutations(3,1.0), math.log(3))
self.assertFloatEqual(ln_permutations(4,1), math.log(4))
self.assertFloatEqual(ln_permutations(4.0,2), math.log(12))
self.assertFloatEqual(ln_permutations(4,3.0), math.log(24))
self.assertFloatEqual(ln_permutations(4,4), math.log(24))
self.assertFloatEqual(ln_permutations(300.0,100),\
math.log(3.8807387193009318e239))

def test_combinations(self):
"""combinations should return expected results when int as input"""
self.assertEqual(combinations(1,1), 1)
self.assertEqual(combinations(2,1), 2)
self.assertEqual(combinations(3,1), 3)
self.assertEqual(combinations(4,1), 4)
self.assertEqual(combinations(4,2), 6)
self.assertEqual(combinations(4,3), 4)
self.assertEqual(combinations(4,4), 1)
self.assertEqual(combinations(20,4), 19*17*15)
self.assertFloatEqual(combinations(300, 100), 4.1582514632578812e+81)

def test_combinations_errors(self):
"""combinations should raise errors on invalid input"""
self.assertRaises(IndexError,combinations,10,50)
self.assertRaises(IndexError,combinations,-1,50)
self.assertRaises(IndexError,combinations,10,-5)

def test_combinations_float(self):
"""combinations should use gamma function when floats as input"""
self.assertFloatEqual(combinations(1.0,1.0), 1)
self.assertFloatEqual(combinations(2.0,1.0), 2)
self.assertFloatEqual(combinations(3.0,1.0), 3)
self.assertFloatEqual(combinations(4.0,1.0), 4)
self.assertFloatEqual(combinations(4.0,2), 6)
self.assertFloatEqual(combinations(4,3.0), 4)
self.assertFloatEqual(combinations(4.0,4.0), 1)
self.assertFloatEqual(combinations(20.0,4.0), 19*17*15)
self.assertFloatEqual(combinations(300,100.0),4.1582514632578812e81)

def test_combinations_range(self):
"""combinations should decrease gradually with increasing k"""
start = 5 #combinations(10,5) = 252
end = 6 #combinations(10,6) = 210
step = 0.1
lower_lim = 210
upper_lim = 252
previous_value = 252.00001
while start <= end:
obs = combinations(10,start)
assert lower_lim <= obs <= upper_lim
assert obs < previous_value
previous_value = obs
start += step

def test_combinations_exact(self):
"""combinations_exact should return expected results"""
self.assertEqual(combinations_exact(1,1), 1)
self.assertEqual(combinations_exact(2,1), 2)
self.assertEqual(combinations_exact(3,1), 3)
self.assertEqual(combinations_exact(4,1), 4)
self.assertEqual(combinations_exact(4,2), 6)
self.assertEqual(combinations_exact(4,3), 4)
self.assertEqual(combinations_exact(4,4), 1)
self.assertEqual(combinations_exact(20,4), 19*17*15)
self.assertFloatEqual(combinations_exact(300,100),4.1582514632578812e81)

def test_ln_combinations(self):
"""ln_combinations should return expected results"""
self.assertFloatEqual(ln_combinations(1,1), math.log(1))
self.assertFloatEqual(ln_combinations(2,1), math.log(2))
self.assertFloatEqual(ln_combinations(3,1), math.log(3))
self.assertFloatEqual(ln_combinations(4.0,1), math.log(4))
self.assertFloatEqual(ln_combinations(4,2.0), math.log(6))
self.assertFloatEqual(ln_combinations(4,3), math.log(4))
self.assertFloatEqual(ln_combinations(4,4.0), math.log(1))
self.assertFloatEqual(ln_combinations(20,4), math.log(19*17*15))
self.assertFloatEqual(ln_combinations(300,100),\
math.log(4.1582514632578812e+81))

def test_ln_binomial_integer(self):
"""ln_binomial should match R results for integer values"""
expected = {
(10,60,0.1): -3.247883,
(1, 1, 0.5): math.log(0.5),
(1, 1, 0.0000001): math.log(1e-07),
(1, 1, 0.9999999): math.log(0.9999999),
(3, 5, 0.75): math.log(0.2636719),
(0, 60, 0.5): math.log(8.673617e-19),
(129, 130, 0.5): math.log(9.550892e-38),
(299, 300, 0.099): math.log(1.338965e-298),
(9, 27, 0.0003): math.log(9.175389e-26),
(1032, 2050, 0.5): math.log(0.01679804),
}
for (key, value) in expected.items():
self.assertFloatEqualRel(ln_binomial(*key), value, 1e-4)

def test_ln_binomial_floats(self):
"""Binomial exact should match values from R for integer successes"""
expected = {
(18.3, 100, 0.2): (math.log(0.09089812), math.log(0.09807429)),
(2.7,1050,0.006): (math.log(0.03615498), math.log(0.07623827)),
(2.7,1050,0.06): (math.log(1.365299e-25), math.log(3.044327e-24)),
(2,100.5,0.6): (math.log(7.303533e-37), math.log(1.789727e-36)),
(0.2, 60, 0.5): (math.log(8.673617e-19), math.log(5.20417e-17)),
(.5,5,.3):(math.log(0.16807),math.log(0.36015)),
(10,100.5,.5):(math.log(7.578011e-18),math.log(1.365543e-17)),
}

for (key, value) in expected.items():
min_val, max_val = value
assert min_val < ln_binomial(*key) < max_val
#self.assertFloatEqualRel(binomial_exact(*key), value, 1e-4)

def test_ln_binomial_range(self):
"""ln_binomial should increase in a monotonically increasing region.
"""
start=0
end=1
step = 0.1
lower_lim = -1.783375-1e-4
upper_lim = -1.021235+1e-4
previous_value = -1.784
while start <= end:
obs = ln_binomial(start,5,.3)
assert lower_lim <= obs <= upper_lim
assert obs > previous_value
previous_value = obs
start += step

def test_log_one_minus_large(self):
"""log_one_minus_x should return math.log(1-x) if x is large"""
self.assertFloatEqual(log_one_minus(0.2), math.log(1-0.2))

def test_log_one_minus_small(self):
"""log_one_minus_x should return -x if x is small"""
self.assertFloatEqualRel(log_one_minus(1e-30), 1e-30)

def test_one_minus_exp_large(self):
"""one_minus_exp_x should return 1 - math.exp(x) if x is large"""
self.assertFloatEqual(one_minus_exp(0.2), 1-(math.exp(0.2)))

def test_one_minus_exp_small(self):
"""one_minus_exp_x should return -x if x is small"""
self.assertFloatEqual(one_minus_exp(1e-30), -1e-30)

def test_log1p(self):
"""log1p should give same results as cephes"""
p_s = [1e-10, 1e-5, 0.1, 0.8, 0.9, 0.95, 0.999, 0.9999999, 1, \
1.000000001, 1.01, 2]
exp = [
9.9999999995e-11,
9.99995000033e-06,
0.0953101798043,
0.587786664902,
0.641853886172,
0.667829372576,
0.692647055518,
0.69314713056,
0.69314718056,
0.69314718106,
0.698134722071,
1.09861228867,]
for p, e in zip(p_s, exp):
self.assertFloatEqual(log1p(p), e)

def test_igami(self):
"""igami should give same result as cephes implementation"""
a_vals = [1e-10, 1e-5, 0.5, 1, 10, 200]
y_vals = range(0,10,2)
obs = [igami(a, y/10.0) for a in a_vals for y in y_vals]
exp=[1.79769313486e+308,
0.0,
0.0,
0.0,
0.0,
1.79769313486e+308,
0.0,
0.0,
0.0,
0.0,
1.79769313486e+308,
0.821187207575,
0.3541631504,
0.137497948864,
0.0320923773337,
1.79769313486e+308,
1.60943791243,
0.916290731874,
0.510825623766,
0.223143551314,
1.79769313486e+308,
12.5187528198,
10.4756841889,
8.9044147366,
7.28921960854,
1.79769313486e+308,
211.794753362,
203.267574402,
196.108740945,
188.010915412,
]
for o, e in zip(obs, exp):
self.assertFloatEqual(o,e)

def test_ndtri(self):
"""ndtri should give same result as implementation in cephes"""
exp=[-1.79769313486e+308,
-2.32634787404,
-2.05374891063,
-1.88079360815,
-1.75068607125,
-1.64485362695,
-1.5547735946,
-1.47579102818,
-1.40507156031,
-1.34075503369,
-1.28155156554,
-1.22652812004,
-1.17498679207,
-1.12639112904,
-1.08031934081,
-1.03643338949,
-0.99445788321,
-0.954165253146,
-0.915365087843,
-0.877896295051,
-0.841621233573,
-0.806421247018,
-0.772193214189,
-0.738846849185,
-0.70630256284,
-0.674489750196,
-0.643345405393,
-0.612812991017,
-0.582841507271,
-0.553384719556,
-0.524400512708,
-0.495850347347,
-0.467698799115,
-0.439913165673,
-0.412463129441,
-0.385320466408,
-0.358458793251,
-0.331853346437,
-0.305480788099,
-0.279319034447,
-0.253347103136,
-0.227544976641,
-0.201893479142,
-0.176374164781,
-0.150969215497,
-0.125661346855,
-0.100433720511,
-0.0752698620998,
-0.0501535834647,
-0.0250689082587,
0.0,
0.0250689082587,
0.0501535834647,
0.0752698620998,
0.100433720511,
0.125661346855,
0.150969215497,
0.176374164781,
0.201893479142,
0.227544976641,
0.253347103136,
0.279319034447,
0.305480788099,
0.331853346437,
0.358458793251,
0.385320466408,
0.412463129441,
0.439913165673,
0.467698799115,
0.495850347347,
0.524400512708,
0.553384719556,
0.582841507271,
0.612812991017,
0.643345405393,
0.674489750196,
0.70630256284,
0.738846849185,
0.772193214189,
0.806421247018,
0.841621233573,
0.877896295051,
0.915365087843,
0.954165253146,
0.99445788321,
1.03643338949,
1.08031934081,
1.12639112904,
1.17498679207,
1.22652812004,
1.28155156554,
1.34075503369,
1.40507156031,
1.47579102818,
1.5547735946,
1.64485362695,
1.75068607125,
1.88079360815,
2.05374891063,
2.32634787404,
]
obs = [ndtri(i/100.0) for i in range(100)]
self.assertFloatEqual(obs, exp)

def test_incbi(self):
"""incbi results should match cephes libraries"""
aa_range = [0.1, 0.2, 0.5, 1, 2, 5]
bb_range = aa_range
yy_range = [0.1, 0.2, 0.5, 0.9]
exp = [
8.86928001193e-08,
9.08146855855e-05,
0.5,
0.999999911307,
4.39887474012e-09,
4.50443299194e-06,
0.0416524955556,
0.997881005025,
3.46456275553e-10,
3.54771169012e-07,
0.00337816430373,
0.732777808689,
1e-10,
1.024e-07,
0.0009765625,
0.3486784401,
3.85543289443e-11,
3.94796342545e-08,
0.000376636057552,
0.154915841005,
1.33210087225e-11,
1.36407136078e-08,
0.000130149552409,
0.056682323296,
0.00211899497509,
0.0646097657259,
0.958347504444,
0.999999995601,
0.000247764691908,
0.00788804962659,
0.5,
0.999752235308,
3.09753032747e-05,
0.000990813218262,
0.092990311753,
0.906714634947,
1e-05,
0.00032,
0.03125,
0.59049,
4.01878917904e-06,
0.000128614607219,
0.0126923538971,
0.309157452156,
1.41593162013e-06,
4.5316442592e-05,
0.00449136140034,
0.122896698096,
0.267222191311,
0.684264602461,
0.996621835696,
0.999999999654,
0.0932853650529,
0.321847764104,
0.907009688247,
0.999969024697,
0.0244717418524,
0.0954915028125,
0.5,
0.975528258148,
0.01,
0.04,
0.25,
0.81,
0.00445768188762,
0.0179929616503,
0.120614758428,
0.531877433474,
0.00165851285512,
0.00672409501831,
0.046687245337,
0.247272226803,
0.6513215599,
0.8926258176,
0.9990234375,
0.9999999999,
0.40951,
0.67232,
0.96875,
0.99999,
0.19,
0.36,
0.75,
0.99,
0.1,
0.2,
0.5,
0.9,
0.0513167019495,
0.105572809,
0.292893218813,
0.683772233983,
0.020851637639,
0.04364750021,
0.129449436704,
0.36904265552,
0.845084158995,
0.956946913164,
0.999623363942,
0.999999999961,
0.690842547844,
0.850620771098,
0.987307646103,
0.999995981211,
0.468122566526,
0.629849697132,
0.879385241572,
0.995542318112,
0.316227766017,
0.4472135955,
0.707106781187,
0.948683298051,
0.195800105659,
0.287140725417,
0.5,
0.804199894341,
0.0925952589131,
0.13988068827,
0.264449983296,
0.510316306551,
0.943317676704,
0.984896695084,
0.999869850448,
0.999999999987,
0.877103301904,
0.944441767096,
0.9955086386,
0.999998584068,
0.752727773197,
0.841546267738,
0.953312754663,
0.998341487145,
0.63095734448,
0.724779663678,
0.870550563296,
0.979148362361,
0.489683693449,
0.577552475154,
0.735550016704,
0.907404741087,
0.300968763593,
0.366086516536,
0.5,
0.699031236407,
]
i = 0
for a in aa_range:
for b in bb_range:
for y in yy_range:
result = incbi(a,b,y)
e = exp[i]
self.assertFloatEqual(e, result)
i += 1
#specific cases that failed elsewhere
self.assertFloatEqual(incbi(999,2,1e-10), 0.97399698104554944)

#execute tests if called from command line
if __name__ == '__main__':
main()

```