File size: 65,095 Bytes
7885a28 |
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644 645 646 647 648 649 650 651 652 653 654 655 656 657 658 659 660 661 662 663 664 665 666 667 668 669 670 671 672 673 674 675 676 677 678 679 680 681 682 683 684 685 686 687 688 689 690 691 692 693 694 695 696 697 698 699 700 701 702 703 704 705 706 707 708 709 710 711 712 713 714 715 716 717 718 719 720 721 722 723 724 725 726 727 728 729 730 731 732 733 734 735 736 737 738 739 740 741 742 743 744 745 746 747 748 749 750 751 752 753 754 755 756 757 758 759 760 761 762 763 764 765 766 767 768 769 770 771 772 773 774 775 776 777 778 779 780 781 782 783 784 785 786 787 788 789 790 791 792 793 794 795 796 797 798 799 800 801 802 803 804 805 806 807 808 809 810 811 812 813 814 815 816 817 818 819 820 821 822 823 824 825 826 827 828 829 830 831 832 833 834 835 836 837 838 839 840 841 842 843 844 845 846 847 848 849 850 851 852 853 854 855 856 857 858 859 860 861 862 863 864 865 866 867 868 869 870 871 872 873 874 875 876 877 878 879 880 881 882 883 884 885 886 887 888 889 890 891 892 893 894 895 896 897 898 899 900 901 902 903 904 905 906 907 908 909 910 911 912 913 914 915 916 917 918 919 920 921 922 923 924 925 926 927 928 929 930 931 932 933 934 935 936 937 938 939 940 941 942 943 944 945 946 947 948 949 950 951 952 953 954 955 956 957 958 959 960 961 962 963 964 965 966 967 968 969 970 971 972 973 974 975 976 977 978 979 980 981 982 983 984 985 986 987 988 989 990 991 992 993 994 995 996 997 998 999 1000 1001 1002 1003 1004 1005 1006 1007 1008 1009 1010 1011 1012 1013 1014 1015 1016 1017 1018 1019 1020 1021 1022 1023 1024 1025 1026 1027 1028 1029 1030 1031 1032 1033 1034 1035 1036 1037 1038 1039 1040 1041 1042 1043 1044 1045 1046 1047 1048 1049 1050 1051 1052 1053 1054 1055 1056 1057 1058 1059 1060 1061 1062 1063 1064 1065 1066 1067 1068 1069 1070 1071 1072 1073 1074 1075 1076 1077 1078 1079 1080 1081 1082 1083 1084 1085 1086 1087 1088 1089 1090 1091 1092 1093 1094 1095 1096 1097 1098 1099 1100 1101 1102 1103 1104 1105 1106 1107 1108 1109 1110 1111 1112 1113 1114 1115 1116 1117 1118 1119 1120 1121 1122 1123 1124 1125 1126 1127 1128 1129 1130 1131 1132 1133 1134 1135 1136 1137 1138 1139 1140 1141 1142 1143 1144 1145 1146 1147 1148 1149 1150 1151 1152 1153 1154 1155 1156 1157 1158 1159 1160 1161 1162 1163 1164 1165 1166 1167 1168 1169 1170 1171 1172 1173 1174 1175 1176 1177 1178 1179 1180 1181 1182 1183 1184 1185 1186 1187 1188 1189 1190 1191 1192 1193 1194 1195 1196 1197 1198 1199 1200 1201 1202 1203 1204 1205 1206 1207 1208 1209 1210 1211 1212 1213 1214 1215 1216 1217 1218 1219 1220 1221 1222 1223 1224 1225 1226 1227 1228 1229 1230 1231 1232 1233 1234 1235 1236 1237 1238 1239 1240 1241 1242 1243 1244 1245 1246 1247 1248 1249 1250 1251 1252 1253 1254 1255 1256 1257 1258 1259 1260 1261 1262 1263 1264 1265 1266 1267 1268 1269 1270 1271 1272 1273 1274 1275 1276 1277 1278 1279 1280 1281 1282 1283 1284 1285 1286 1287 1288 1289 1290 1291 1292 1293 1294 1295 1296 1297 1298 1299 1300 1301 1302 1303 1304 1305 1306 1307 1308 1309 1310 1311 1312 1313 1314 1315 1316 1317 1318 1319 1320 1321 1322 1323 1324 1325 1326 1327 1328 1329 1330 1331 1332 1333 1334 1335 1336 1337 1338 1339 1340 1341 1342 1343 1344 1345 1346 1347 1348 1349 1350 1351 1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374 1375 1376 1377 1378 1379 1380 1381 1382 1383 1384 1385 1386 1387 1388 1389 1390 1391 1392 1393 1394 1395 1396 1397 1398 1399 1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417 1418 1419 1420 1421 1422 1423 1424 1425 1426 1427 1428 1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453 1454 1455 1456 1457 1458 1459 1460 1461 1462 1463 1464 1465 1466 1467 1468 1469 1470 1471 1472 1473 1474 1475 1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530 1531 1532 1533 1534 1535 1536 1537 1538 1539 1540 1541 1542 1543 1544 1545 1546 1547 1548 1549 1550 1551 1552 1553 1554 1555 1556 1557 1558 1559 1560 1561 1562 1563 1564 1565 1566 1567 1568 1569 1570 1571 1572 1573 1574 1575 1576 1577 1578 1579 1580 1581 1582 1583 1584 1585 1586 1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607 1608 1609 1610 1611 1612 1613 1614 1615 1616 1617 1618 1619 1620 1621 1622 1623 1624 1625 1626 1627 1628 1629 1630 1631 1632 1633 1634 1635 1636 1637 1638 1639 1640 1641 1642 1643 1644 1645 1646 1647 1648 1649 1650 1651 1652 1653 1654 1655 1656 1657 1658 1659 1660 1661 1662 1663 1664 1665 1666 1667 1668 1669 1670 1671 1672 1673 1674 1675 1676 1677 1678 1679 1680 1681 1682 1683 1684 1685 1686 1687 1688 1689 1690 1691 1692 1693 1694 1695 1696 1697 1698 1699 1700 1701 1702 1703 1704 1705 1706 1707 1708 1709 1710 1711 1712 1713 1714 1715 1716 1717 1718 1719 1720 1721 1722 1723 1724 1725 1726 1727 1728 1729 1730 1731 1732 1733 1734 1735 1736 1737 1738 1739 1740 1741 1742 1743 1744 1745 1746 1747 1748 1749 1750 1751 1752 1753 1754 1755 1756 1757 1758 1759 1760 1761 1762 1763 1764 1765 1766 1767 1768 1769 1770 1771 1772 1773 1774 1775 1776 1777 1778 1779 1780 1781 1782 1783 1784 1785 1786 1787 1788 1789 1790 1791 1792 1793 1794 1795 1796 1797 1798 1799 1800 1801 1802 1803 1804 1805 1806 1807 1808 1809 1810 1811 1812 1813 1814 1815 1816 1817 1818 1819 1820 1821 1822 1823 1824 1825 1826 1827 1828 1829 1830 1831 1832 1833 1834 1835 1836 1837 1838 1839 1840 1841 1842 1843 1844 1845 1846 1847 1848 1849 1850 1851 1852 1853 1854 1855 1856 1857 1858 1859 1860 1861 1862 1863 1864 1865 1866 1867 1868 1869 1870 1871 1872 1873 1874 1875 1876 1877 1878 1879 1880 1881 1882 1883 1884 1885 1886 1887 1888 1889 1890 1891 1892 1893 1894 1895 1896 1897 1898 1899 1900 1901 1902 1903 1904 1905 1906 1907 1908 1909 1910 1911 1912 1913 1914 1915 1916 1917 1918 1919 1920 1921 1922 1923 1924 1925 1926 1927 1928 1929 1930 1931 1932 1933 1934 1935 1936 1937 1938 1939 1940 1941 1942 1943 1944 1945 1946 1947 1948 1949 1950 1951 1952 1953 1954 1955 1956 1957 1958 1959 1960 1961 1962 1963 1964 1965 1966 1967 1968 1969 1970 1971 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017 2018 2019 2020 2021 2022 2023 2024 2025 2026 2027 2028 2029 2030 2031 2032 2033 2034 2035 2036 2037 2038 2039 2040 2041 2042 2043 2044 2045 2046 2047 2048 2049 2050 2051 2052 2053 2054 2055 2056 2057 2058 2059 2060 2061 2062 2063 2064 2065 2066 2067 2068 2069 2070 2071 2072 2073 2074 2075 2076 2077 2078 2079 2080 2081 2082 2083 2084 2085 2086 2087 2088 2089 2090 2091 2092 |
#
# Author: Travis Oliphant 2002-2011 with contributions from
# SciPy Developers 2004-2011
#
from functools import partial
from scipy import special
from scipy.special import entr, logsumexp, betaln, gammaln as gamln, zeta
from scipy._lib._util import _lazywhere, rng_integers
from scipy.interpolate import interp1d
from numpy import floor, ceil, log, exp, sqrt, log1p, expm1, tanh, cosh, sinh
import numpy as np
from ._distn_infrastructure import (rv_discrete, get_distribution_names,
_vectorize_rvs_over_shapes,
_ShapeInfo, _isintegral,
rv_discrete_frozen)
from ._biasedurn import (_PyFishersNCHypergeometric,
_PyWalleniusNCHypergeometric,
_PyStochasticLib3)
from ._stats_pythran import _poisson_binom
import scipy.special._ufuncs as scu
class binom_gen(rv_discrete):
r"""A binomial discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `binom` is:
.. math::
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
for :math:`k \in \{0, 1, \dots, n\}`, :math:`0 \leq p \leq 1`
`binom` takes :math:`n` and :math:`p` as shape parameters,
where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure.
This distribution uses routines from the Boost Math C++ library for
the computation of the ``pmf``, ``cdf``, ``sf``, ``ppf`` and ``isf``
methods. [1]_
%(after_notes)s
References
----------
.. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.
%(example)s
See Also
--------
hypergeom, nbinom, nhypergeom
"""
def _shape_info(self):
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("p", False, (0, 1), (True, True))]
def _rvs(self, n, p, size=None, random_state=None):
return random_state.binomial(n, p, size)
def _argcheck(self, n, p):
return (n >= 0) & _isintegral(n) & (p >= 0) & (p <= 1)
def _get_support(self, n, p):
return self.a, n
def _logpmf(self, x, n, p):
k = floor(x)
combiln = (gamln(n+1) - (gamln(k+1) + gamln(n-k+1)))
return combiln + special.xlogy(k, p) + special.xlog1py(n-k, -p)
def _pmf(self, x, n, p):
# binom.pmf(k) = choose(n, k) * p**k * (1-p)**(n-k)
return scu._binom_pmf(x, n, p)
def _cdf(self, x, n, p):
k = floor(x)
return scu._binom_cdf(k, n, p)
def _sf(self, x, n, p):
k = floor(x)
return scu._binom_sf(k, n, p)
def _isf(self, x, n, p):
return scu._binom_isf(x, n, p)
def _ppf(self, q, n, p):
return scu._binom_ppf(q, n, p)
def _stats(self, n, p, moments='mv'):
mu = n * p
var = mu - n * np.square(p)
g1, g2 = None, None
if 's' in moments:
pq = p - np.square(p)
npq_sqrt = np.sqrt(n * pq)
t1 = np.reciprocal(npq_sqrt)
t2 = (2.0 * p) / npq_sqrt
g1 = t1 - t2
if 'k' in moments:
pq = p - np.square(p)
npq = n * pq
t1 = np.reciprocal(npq)
t2 = 6.0/n
g2 = t1 - t2
return mu, var, g1, g2
def _entropy(self, n, p):
k = np.r_[0:n + 1]
vals = self._pmf(k, n, p)
return np.sum(entr(vals), axis=0)
binom = binom_gen(name='binom')
class bernoulli_gen(binom_gen):
r"""A Bernoulli discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `bernoulli` is:
.. math::
f(k) = \begin{cases}1-p &\text{if } k = 0\\
p &\text{if } k = 1\end{cases}
for :math:`k` in :math:`\{0, 1\}`, :math:`0 \leq p \leq 1`
`bernoulli` takes :math:`p` as shape parameter,
where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("p", False, (0, 1), (True, True))]
def _rvs(self, p, size=None, random_state=None):
return binom_gen._rvs(self, 1, p, size=size, random_state=random_state)
def _argcheck(self, p):
return (p >= 0) & (p <= 1)
def _get_support(self, p):
# Overrides binom_gen._get_support!x
return self.a, self.b
def _logpmf(self, x, p):
return binom._logpmf(x, 1, p)
def _pmf(self, x, p):
# bernoulli.pmf(k) = 1-p if k = 0
# = p if k = 1
return binom._pmf(x, 1, p)
def _cdf(self, x, p):
return binom._cdf(x, 1, p)
def _sf(self, x, p):
return binom._sf(x, 1, p)
def _isf(self, x, p):
return binom._isf(x, 1, p)
def _ppf(self, q, p):
return binom._ppf(q, 1, p)
def _stats(self, p):
return binom._stats(1, p)
def _entropy(self, p):
return entr(p) + entr(1-p)
bernoulli = bernoulli_gen(b=1, name='bernoulli')
class betabinom_gen(rv_discrete):
r"""A beta-binomial discrete random variable.
%(before_notes)s
Notes
-----
The beta-binomial distribution is a binomial distribution with a
probability of success `p` that follows a beta distribution.
The probability mass function for `betabinom` is:
.. math::
f(k) = \binom{n}{k} \frac{B(k + a, n - k + b)}{B(a, b)}
for :math:`k \in \{0, 1, \dots, n\}`, :math:`n \geq 0`, :math:`a > 0`,
:math:`b > 0`, where :math:`B(a, b)` is the beta function.
`betabinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.
References
----------
.. [1] https://en.wikipedia.org/wiki/Beta-binomial_distribution
%(after_notes)s
.. versionadded:: 1.4.0
See Also
--------
beta, binom
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("a", False, (0, np.inf), (False, False)),
_ShapeInfo("b", False, (0, np.inf), (False, False))]
def _rvs(self, n, a, b, size=None, random_state=None):
p = random_state.beta(a, b, size)
return random_state.binomial(n, p, size)
def _get_support(self, n, a, b):
return 0, n
def _argcheck(self, n, a, b):
return (n >= 0) & _isintegral(n) & (a > 0) & (b > 0)
def _logpmf(self, x, n, a, b):
k = floor(x)
combiln = -log(n + 1) - betaln(n - k + 1, k + 1)
return combiln + betaln(k + a, n - k + b) - betaln(a, b)
def _pmf(self, x, n, a, b):
return exp(self._logpmf(x, n, a, b))
def _stats(self, n, a, b, moments='mv'):
e_p = a / (a + b)
e_q = 1 - e_p
mu = n * e_p
var = n * (a + b + n) * e_p * e_q / (a + b + 1)
g1, g2 = None, None
if 's' in moments:
g1 = 1.0 / sqrt(var)
g1 *= (a + b + 2 * n) * (b - a)
g1 /= (a + b + 2) * (a + b)
if 'k' in moments:
g2 = (a + b).astype(e_p.dtype)
g2 *= (a + b - 1 + 6 * n)
g2 += 3 * a * b * (n - 2)
g2 += 6 * n ** 2
g2 -= 3 * e_p * b * n * (6 - n)
g2 -= 18 * e_p * e_q * n ** 2
g2 *= (a + b) ** 2 * (1 + a + b)
g2 /= (n * a * b * (a + b + 2) * (a + b + 3) * (a + b + n))
g2 -= 3
return mu, var, g1, g2
betabinom = betabinom_gen(name='betabinom')
class nbinom_gen(rv_discrete):
r"""A negative binomial discrete random variable.
%(before_notes)s
Notes
-----
Negative binomial distribution describes a sequence of i.i.d. Bernoulli
trials, repeated until a predefined, non-random number of successes occurs.
The probability mass function of the number of failures for `nbinom` is:
.. math::
f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k
for :math:`k \ge 0`, :math:`0 < p \leq 1`
`nbinom` takes :math:`n` and :math:`p` as shape parameters where :math:`n`
is the number of successes, :math:`p` is the probability of a single
success, and :math:`1-p` is the probability of a single failure.
Another common parameterization of the negative binomial distribution is
in terms of the mean number of failures :math:`\mu` to achieve :math:`n`
successes. The mean :math:`\mu` is related to the probability of success
as
.. math::
p = \frac{n}{n + \mu}
The number of successes :math:`n` may also be specified in terms of a
"dispersion", "heterogeneity", or "aggregation" parameter :math:`\alpha`,
which relates the mean :math:`\mu` to the variance :math:`\sigma^2`,
e.g. :math:`\sigma^2 = \mu + \alpha \mu^2`. Regardless of the convention
used for :math:`\alpha`,
.. math::
p &= \frac{\mu}{\sigma^2} \\
n &= \frac{\mu^2}{\sigma^2 - \mu}
This distribution uses routines from the Boost Math C++ library for
the computation of the ``pmf``, ``cdf``, ``sf``, ``ppf``, ``isf``
and ``stats`` methods. [1]_
%(after_notes)s
References
----------
.. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.
%(example)s
See Also
--------
hypergeom, binom, nhypergeom
"""
def _shape_info(self):
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("p", False, (0, 1), (True, True))]
def _rvs(self, n, p, size=None, random_state=None):
return random_state.negative_binomial(n, p, size)
def _argcheck(self, n, p):
return (n > 0) & (p > 0) & (p <= 1)
def _pmf(self, x, n, p):
# nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k
return scu._nbinom_pmf(x, n, p)
def _logpmf(self, x, n, p):
coeff = gamln(n+x) - gamln(x+1) - gamln(n)
return coeff + n*log(p) + special.xlog1py(x, -p)
def _cdf(self, x, n, p):
k = floor(x)
return scu._nbinom_cdf(k, n, p)
def _logcdf(self, x, n, p):
k = floor(x)
k, n, p = np.broadcast_arrays(k, n, p)
cdf = self._cdf(k, n, p)
cond = cdf > 0.5
def f1(k, n, p):
return np.log1p(-special.betainc(k + 1, n, 1 - p))
# do calc in place
logcdf = cdf
with np.errstate(divide='ignore'):
logcdf[cond] = f1(k[cond], n[cond], p[cond])
logcdf[~cond] = np.log(cdf[~cond])
return logcdf
def _sf(self, x, n, p):
k = floor(x)
return scu._nbinom_sf(k, n, p)
def _isf(self, x, n, p):
with np.errstate(over='ignore'): # see gh-17432
return scu._nbinom_isf(x, n, p)
def _ppf(self, q, n, p):
with np.errstate(over='ignore'): # see gh-17432
return scu._nbinom_ppf(q, n, p)
def _stats(self, n, p):
return (
scu._nbinom_mean(n, p),
scu._nbinom_variance(n, p),
scu._nbinom_skewness(n, p),
scu._nbinom_kurtosis_excess(n, p),
)
nbinom = nbinom_gen(name='nbinom')
class betanbinom_gen(rv_discrete):
r"""A beta-negative-binomial discrete random variable.
%(before_notes)s
Notes
-----
The beta-negative-binomial distribution is a negative binomial
distribution with a probability of success `p` that follows a
beta distribution.
The probability mass function for `betanbinom` is:
.. math::
f(k) = \binom{n + k - 1}{k} \frac{B(a + n, b + k)}{B(a, b)}
for :math:`k \ge 0`, :math:`n \geq 0`, :math:`a > 0`,
:math:`b > 0`, where :math:`B(a, b)` is the beta function.
`betanbinom` takes :math:`n`, :math:`a`, and :math:`b` as shape parameters.
References
----------
.. [1] https://en.wikipedia.org/wiki/Beta_negative_binomial_distribution
%(after_notes)s
.. versionadded:: 1.12.0
See Also
--------
betabinom : Beta binomial distribution
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("a", False, (0, np.inf), (False, False)),
_ShapeInfo("b", False, (0, np.inf), (False, False))]
def _rvs(self, n, a, b, size=None, random_state=None):
p = random_state.beta(a, b, size)
return random_state.negative_binomial(n, p, size)
def _argcheck(self, n, a, b):
return (n >= 0) & _isintegral(n) & (a > 0) & (b > 0)
def _logpmf(self, x, n, a, b):
k = floor(x)
combiln = -np.log(n + k) - betaln(n, k + 1)
return combiln + betaln(a + n, b + k) - betaln(a, b)
def _pmf(self, x, n, a, b):
return exp(self._logpmf(x, n, a, b))
def _stats(self, n, a, b, moments='mv'):
# reference: Wolfram Alpha input
# BetaNegativeBinomialDistribution[a, b, n]
def mean(n, a, b):
return n * b / (a - 1.)
mu = _lazywhere(a > 1, (n, a, b), f=mean, fillvalue=np.inf)
def var(n, a, b):
return (n * b * (n + a - 1.) * (a + b - 1.)
/ ((a - 2.) * (a - 1.)**2.))
var = _lazywhere(a > 2, (n, a, b), f=var, fillvalue=np.inf)
g1, g2 = None, None
def skew(n, a, b):
return ((2 * n + a - 1.) * (2 * b + a - 1.)
/ (a - 3.) / sqrt(n * b * (n + a - 1.) * (b + a - 1.)
/ (a - 2.)))
if 's' in moments:
g1 = _lazywhere(a > 3, (n, a, b), f=skew, fillvalue=np.inf)
def kurtosis(n, a, b):
term = (a - 2.)
term_2 = ((a - 1.)**2. * (a**2. + a * (6 * b - 1.)
+ 6. * (b - 1.) * b)
+ 3. * n**2. * ((a + 5.) * b**2. + (a + 5.)
* (a - 1.) * b + 2. * (a - 1.)**2)
+ 3 * (a - 1.) * n
* ((a + 5.) * b**2. + (a + 5.) * (a - 1.) * b
+ 2. * (a - 1.)**2.))
denominator = ((a - 4.) * (a - 3.) * b * n
* (a + b - 1.) * (a + n - 1.))
# Wolfram Alpha uses Pearson kurtosis, so we subtract 3 to get
# scipy's Fisher kurtosis
return term * term_2 / denominator - 3.
if 'k' in moments:
g2 = _lazywhere(a > 4, (n, a, b), f=kurtosis, fillvalue=np.inf)
return mu, var, g1, g2
betanbinom = betanbinom_gen(name='betanbinom')
class geom_gen(rv_discrete):
r"""A geometric discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `geom` is:
.. math::
f(k) = (1-p)^{k-1} p
for :math:`k \ge 1`, :math:`0 < p \leq 1`
`geom` takes :math:`p` as shape parameter,
where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure.
Note that when drawing random samples, the probability of observations that exceed
``np.iinfo(np.int64).max`` increases rapidly as $p$ decreases below $10^{-17}$. For
$p < 10^{-20}$, almost all observations would exceed the maximum ``int64``; however,
the output dtype is always ``int64``, so these values are clipped to the maximum.
%(after_notes)s
See Also
--------
planck
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("p", False, (0, 1), (True, True))]
def _rvs(self, p, size=None, random_state=None):
res = random_state.geometric(p, size=size)
# RandomState.geometric can wrap around to negative values; make behavior
# consistent with Generator.geometric by replacing with maximum integer.
max_int = np.iinfo(res.dtype).max
return np.where(res < 0, max_int, res)
def _argcheck(self, p):
return (p <= 1) & (p > 0)
def _pmf(self, k, p):
return np.power(1-p, k-1) * p
def _logpmf(self, k, p):
return special.xlog1py(k - 1, -p) + log(p)
def _cdf(self, x, p):
k = floor(x)
return -expm1(log1p(-p)*k)
def _sf(self, x, p):
return np.exp(self._logsf(x, p))
def _logsf(self, x, p):
k = floor(x)
return k*log1p(-p)
def _ppf(self, q, p):
vals = ceil(log1p(-q) / log1p(-p))
temp = self._cdf(vals-1, p)
return np.where((temp >= q) & (vals > 0), vals-1, vals)
def _stats(self, p):
mu = 1.0/p
qr = 1.0-p
var = qr / p / p
g1 = (2.0-p) / sqrt(qr)
g2 = np.polyval([1, -6, 6], p)/(1.0-p)
return mu, var, g1, g2
def _entropy(self, p):
return -np.log(p) - np.log1p(-p) * (1.0-p) / p
geom = geom_gen(a=1, name='geom', longname="A geometric")
class hypergeom_gen(rv_discrete):
r"""A hypergeometric discrete random variable.
The hypergeometric distribution models drawing objects from a bin.
`M` is the total number of objects, `n` is total number of Type I objects.
The random variate represents the number of Type I objects in `N` drawn
without replacement from the total population.
%(before_notes)s
Notes
-----
The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not
universally accepted. See the Examples for a clarification of the
definitions used here.
The probability mass function is defined as,
.. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}}
{\binom{M}{N}}
for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial
coefficients are defined as,
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
This distribution uses routines from the Boost Math C++ library for
the computation of the ``pmf``, ``cdf``, ``sf`` and ``stats`` methods. [1]_
%(after_notes)s
References
----------
.. [1] The Boost Developers. "Boost C++ Libraries". https://www.boost.org/.
Examples
--------
>>> import numpy as np
>>> from scipy.stats import hypergeom
>>> import matplotlib.pyplot as plt
Suppose we have a collection of 20 animals, of which 7 are dogs. Then if
we want to know the probability of finding a given number of dogs if we
choose at random 12 of the 20 animals, we can initialize a frozen
distribution and plot the probability mass function:
>>> [M, n, N] = [20, 7, 12]
>>> rv = hypergeom(M, n, N)
>>> x = np.arange(0, n+1)
>>> pmf_dogs = rv.pmf(x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, pmf_dogs, 'bo')
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
>>> ax.set_xlabel('# of dogs in our group of chosen animals')
>>> ax.set_ylabel('hypergeom PMF')
>>> plt.show()
Instead of using a frozen distribution we can also use `hypergeom`
methods directly. To for example obtain the cumulative distribution
function, use:
>>> prb = hypergeom.cdf(x, M, n, N)
And to generate random numbers:
>>> R = hypergeom.rvs(M, n, N, size=10)
See Also
--------
nhypergeom, binom, nbinom
"""
def _shape_info(self):
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("N", True, (0, np.inf), (True, False))]
def _rvs(self, M, n, N, size=None, random_state=None):
return random_state.hypergeometric(n, M-n, N, size=size)
def _get_support(self, M, n, N):
return np.maximum(N-(M-n), 0), np.minimum(n, N)
def _argcheck(self, M, n, N):
cond = (M > 0) & (n >= 0) & (N >= 0)
cond &= (n <= M) & (N <= M)
cond &= _isintegral(M) & _isintegral(n) & _isintegral(N)
return cond
def _logpmf(self, k, M, n, N):
tot, good = M, n
bad = tot - good
result = (betaln(good+1, 1) + betaln(bad+1, 1) + betaln(tot-N+1, N+1) -
betaln(k+1, good-k+1) - betaln(N-k+1, bad-N+k+1) -
betaln(tot+1, 1))
return result
def _pmf(self, k, M, n, N):
return scu._hypergeom_pmf(k, n, N, M)
def _cdf(self, k, M, n, N):
return scu._hypergeom_cdf(k, n, N, M)
def _stats(self, M, n, N):
M, n, N = 1. * M, 1. * n, 1. * N
m = M - n
# Boost kurtosis_excess doesn't return the same as the value
# computed here.
g2 = M * (M + 1) - 6. * N * (M - N) - 6. * n * m
g2 *= (M - 1) * M * M
g2 += 6. * n * N * (M - N) * m * (5. * M - 6)
g2 /= n * N * (M - N) * m * (M - 2.) * (M - 3.)
return (
scu._hypergeom_mean(n, N, M),
scu._hypergeom_variance(n, N, M),
scu._hypergeom_skewness(n, N, M),
g2,
)
def _entropy(self, M, n, N):
k = np.r_[N - (M - n):min(n, N) + 1]
vals = self.pmf(k, M, n, N)
return np.sum(entr(vals), axis=0)
def _sf(self, k, M, n, N):
return scu._hypergeom_sf(k, n, N, M)
def _logsf(self, k, M, n, N):
res = []
for quant, tot, good, draw in zip(*np.broadcast_arrays(k, M, n, N)):
if (quant + 0.5) * (tot + 0.5) < (good - 0.5) * (draw - 0.5):
# Less terms to sum if we calculate log(1-cdf)
res.append(log1p(-exp(self.logcdf(quant, tot, good, draw))))
else:
# Integration over probability mass function using logsumexp
k2 = np.arange(quant + 1, draw + 1)
res.append(logsumexp(self._logpmf(k2, tot, good, draw)))
return np.asarray(res)
def _logcdf(self, k, M, n, N):
res = []
for quant, tot, good, draw in zip(*np.broadcast_arrays(k, M, n, N)):
if (quant + 0.5) * (tot + 0.5) > (good - 0.5) * (draw - 0.5):
# Less terms to sum if we calculate log(1-sf)
res.append(log1p(-exp(self.logsf(quant, tot, good, draw))))
else:
# Integration over probability mass function using logsumexp
k2 = np.arange(0, quant + 1)
res.append(logsumexp(self._logpmf(k2, tot, good, draw)))
return np.asarray(res)
hypergeom = hypergeom_gen(name='hypergeom')
class nhypergeom_gen(rv_discrete):
r"""A negative hypergeometric discrete random variable.
Consider a box containing :math:`M` balls:, :math:`n` red and
:math:`M-n` blue. We randomly sample balls from the box, one
at a time and *without* replacement, until we have picked :math:`r`
blue balls. `nhypergeom` is the distribution of the number of
red balls :math:`k` we have picked.
%(before_notes)s
Notes
-----
The symbols used to denote the shape parameters (`M`, `n`, and `r`) are not
universally accepted. See the Examples for a clarification of the
definitions used here.
The probability mass function is defined as,
.. math:: f(k; M, n, r) = \frac{{{k+r-1}\choose{k}}{{M-r-k}\choose{n-k}}}
{{M \choose n}}
for :math:`k \in [0, n]`, :math:`n \in [0, M]`, :math:`r \in [0, M-n]`,
and the binomial coefficient is:
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
It is equivalent to observing :math:`k` successes in :math:`k+r-1`
samples with :math:`k+r`'th sample being a failure. The former
can be modelled as a hypergeometric distribution. The probability
of the latter is simply the number of failures remaining
:math:`M-n-(r-1)` divided by the size of the remaining population
:math:`M-(k+r-1)`. This relationship can be shown as:
.. math:: NHG(k;M,n,r) = HG(k;M,n,k+r-1)\frac{(M-n-(r-1))}{(M-(k+r-1))}
where :math:`NHG` is probability mass function (PMF) of the
negative hypergeometric distribution and :math:`HG` is the
PMF of the hypergeometric distribution.
%(after_notes)s
Examples
--------
>>> import numpy as np
>>> from scipy.stats import nhypergeom
>>> import matplotlib.pyplot as plt
Suppose we have a collection of 20 animals, of which 7 are dogs.
Then if we want to know the probability of finding a given number
of dogs (successes) in a sample with exactly 12 animals that
aren't dogs (failures), we can initialize a frozen distribution
and plot the probability mass function:
>>> M, n, r = [20, 7, 12]
>>> rv = nhypergeom(M, n, r)
>>> x = np.arange(0, n+2)
>>> pmf_dogs = rv.pmf(x)
>>> fig = plt.figure()
>>> ax = fig.add_subplot(111)
>>> ax.plot(x, pmf_dogs, 'bo')
>>> ax.vlines(x, 0, pmf_dogs, lw=2)
>>> ax.set_xlabel('# of dogs in our group with given 12 failures')
>>> ax.set_ylabel('nhypergeom PMF')
>>> plt.show()
Instead of using a frozen distribution we can also use `nhypergeom`
methods directly. To for example obtain the probability mass
function, use:
>>> prb = nhypergeom.pmf(x, M, n, r)
And to generate random numbers:
>>> R = nhypergeom.rvs(M, n, r, size=10)
To verify the relationship between `hypergeom` and `nhypergeom`, use:
>>> from scipy.stats import hypergeom, nhypergeom
>>> M, n, r = 45, 13, 8
>>> k = 6
>>> nhypergeom.pmf(k, M, n, r)
0.06180776620271643
>>> hypergeom.pmf(k, M, n, k+r-1) * (M - n - (r-1)) / (M - (k+r-1))
0.06180776620271644
See Also
--------
hypergeom, binom, nbinom
References
----------
.. [1] Negative Hypergeometric Distribution on Wikipedia
https://en.wikipedia.org/wiki/Negative_hypergeometric_distribution
.. [2] Negative Hypergeometric Distribution from
http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Negativehypergeometric.pdf
"""
def _shape_info(self):
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("r", True, (0, np.inf), (True, False))]
def _get_support(self, M, n, r):
return 0, n
def _argcheck(self, M, n, r):
cond = (n >= 0) & (n <= M) & (r >= 0) & (r <= M-n)
cond &= _isintegral(M) & _isintegral(n) & _isintegral(r)
return cond
def _rvs(self, M, n, r, size=None, random_state=None):
@_vectorize_rvs_over_shapes
def _rvs1(M, n, r, size, random_state):
# invert cdf by calculating all values in support, scalar M, n, r
a, b = self.support(M, n, r)
ks = np.arange(a, b+1)
cdf = self.cdf(ks, M, n, r)
ppf = interp1d(cdf, ks, kind='next', fill_value='extrapolate')
rvs = ppf(random_state.uniform(size=size)).astype(int)
if size is None:
return rvs.item()
return rvs
return _rvs1(M, n, r, size=size, random_state=random_state)
def _logpmf(self, k, M, n, r):
cond = ((r == 0) & (k == 0))
result = _lazywhere(~cond, (k, M, n, r),
lambda k, M, n, r:
(-betaln(k+1, r) + betaln(k+r, 1) -
betaln(n-k+1, M-r-n+1) + betaln(M-r-k+1, 1) +
betaln(n+1, M-n+1) - betaln(M+1, 1)),
fillvalue=0.0)
return result
def _pmf(self, k, M, n, r):
# same as the following but numerically more precise
# return comb(k+r-1, k) * comb(M-r-k, n-k) / comb(M, n)
return exp(self._logpmf(k, M, n, r))
def _stats(self, M, n, r):
# Promote the datatype to at least float
# mu = rn / (M-n+1)
M, n, r = 1.*M, 1.*n, 1.*r
mu = r*n / (M-n+1)
var = r*(M+1)*n / ((M-n+1)*(M-n+2)) * (1 - r / (M-n+1))
# The skew and kurtosis are mathematically
# intractable so return `None`. See [2]_.
g1, g2 = None, None
return mu, var, g1, g2
nhypergeom = nhypergeom_gen(name='nhypergeom')
# FIXME: Fails _cdfvec
class logser_gen(rv_discrete):
r"""A Logarithmic (Log-Series, Series) discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `logser` is:
.. math::
f(k) = - \frac{p^k}{k \log(1-p)}
for :math:`k \ge 1`, :math:`0 < p < 1`
`logser` takes :math:`p` as shape parameter,
where :math:`p` is the probability of a single success
and :math:`1-p` is the probability of a single failure.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("p", False, (0, 1), (True, True))]
def _rvs(self, p, size=None, random_state=None):
# looks wrong for p>0.5, too few k=1
# trying to use generic is worse, no k=1 at all
return random_state.logseries(p, size=size)
def _argcheck(self, p):
return (p > 0) & (p < 1)
def _pmf(self, k, p):
# logser.pmf(k) = - p**k / (k*log(1-p))
return -np.power(p, k) * 1.0 / k / special.log1p(-p)
def _stats(self, p):
r = special.log1p(-p)
mu = p / (p - 1.0) / r
mu2p = -p / r / (p - 1.0)**2
var = mu2p - mu*mu
mu3p = -p / r * (1.0+p) / (1.0 - p)**3
mu3 = mu3p - 3*mu*mu2p + 2*mu**3
g1 = mu3 / np.power(var, 1.5)
mu4p = -p / r * (
1.0 / (p-1)**2 - 6*p / (p - 1)**3 + 6*p*p / (p-1)**4)
mu4 = mu4p - 4*mu3p*mu + 6*mu2p*mu*mu - 3*mu**4
g2 = mu4 / var**2 - 3.0
return mu, var, g1, g2
logser = logser_gen(a=1, name='logser', longname='A logarithmic')
class poisson_gen(rv_discrete):
r"""A Poisson discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `poisson` is:
.. math::
f(k) = \exp(-\mu) \frac{\mu^k}{k!}
for :math:`k \ge 0`.
`poisson` takes :math:`\mu \geq 0` as shape parameter.
When :math:`\mu = 0`, the ``pmf`` method
returns ``1.0`` at quantile :math:`k = 0`.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("mu", False, (0, np.inf), (True, False))]
# Override rv_discrete._argcheck to allow mu=0.
def _argcheck(self, mu):
return mu >= 0
def _rvs(self, mu, size=None, random_state=None):
return random_state.poisson(mu, size)
def _logpmf(self, k, mu):
Pk = special.xlogy(k, mu) - gamln(k + 1) - mu
return Pk
def _pmf(self, k, mu):
# poisson.pmf(k) = exp(-mu) * mu**k / k!
return exp(self._logpmf(k, mu))
def _cdf(self, x, mu):
k = floor(x)
return special.pdtr(k, mu)
def _sf(self, x, mu):
k = floor(x)
return special.pdtrc(k, mu)
def _ppf(self, q, mu):
vals = ceil(special.pdtrik(q, mu))
vals1 = np.maximum(vals - 1, 0)
temp = special.pdtr(vals1, mu)
return np.where(temp >= q, vals1, vals)
def _stats(self, mu):
var = mu
tmp = np.asarray(mu)
mu_nonzero = tmp > 0
g1 = _lazywhere(mu_nonzero, (tmp,), lambda x: sqrt(1.0/x), np.inf)
g2 = _lazywhere(mu_nonzero, (tmp,), lambda x: 1.0/x, np.inf)
return mu, var, g1, g2
poisson = poisson_gen(name="poisson", longname='A Poisson')
class planck_gen(rv_discrete):
r"""A Planck discrete exponential random variable.
%(before_notes)s
Notes
-----
The probability mass function for `planck` is:
.. math::
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)
for :math:`k \ge 0` and :math:`\lambda > 0`.
`planck` takes :math:`\lambda` as shape parameter. The Planck distribution
can be written as a geometric distribution (`geom`) with
:math:`p = 1 - \exp(-\lambda)` shifted by ``loc = -1``.
%(after_notes)s
See Also
--------
geom
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("lambda", False, (0, np.inf), (False, False))]
def _argcheck(self, lambda_):
return lambda_ > 0
def _pmf(self, k, lambda_):
return -expm1(-lambda_)*exp(-lambda_*k)
def _cdf(self, x, lambda_):
k = floor(x)
return -expm1(-lambda_*(k+1))
def _sf(self, x, lambda_):
return exp(self._logsf(x, lambda_))
def _logsf(self, x, lambda_):
k = floor(x)
return -lambda_*(k+1)
def _ppf(self, q, lambda_):
vals = ceil(-1.0/lambda_ * log1p(-q)-1)
vals1 = (vals-1).clip(*(self._get_support(lambda_)))
temp = self._cdf(vals1, lambda_)
return np.where(temp >= q, vals1, vals)
def _rvs(self, lambda_, size=None, random_state=None):
# use relation to geometric distribution for sampling
p = -expm1(-lambda_)
return random_state.geometric(p, size=size) - 1.0
def _stats(self, lambda_):
mu = 1/expm1(lambda_)
var = exp(-lambda_)/(expm1(-lambda_))**2
g1 = 2*cosh(lambda_/2.0)
g2 = 4+2*cosh(lambda_)
return mu, var, g1, g2
def _entropy(self, lambda_):
C = -expm1(-lambda_)
return lambda_*exp(-lambda_)/C - log(C)
planck = planck_gen(a=0, name='planck', longname='A discrete exponential ')
class boltzmann_gen(rv_discrete):
r"""A Boltzmann (Truncated Discrete Exponential) random variable.
%(before_notes)s
Notes
-----
The probability mass function for `boltzmann` is:
.. math::
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k) / (1-\exp(-\lambda N))
for :math:`k = 0,..., N-1`.
`boltzmann` takes :math:`\lambda > 0` and :math:`N > 0` as shape parameters.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("lambda_", False, (0, np.inf), (False, False)),
_ShapeInfo("N", True, (0, np.inf), (False, False))]
def _argcheck(self, lambda_, N):
return (lambda_ > 0) & (N > 0) & _isintegral(N)
def _get_support(self, lambda_, N):
return self.a, N - 1
def _pmf(self, k, lambda_, N):
# boltzmann.pmf(k) =
# (1-exp(-lambda_)*exp(-lambda_*k)/(1-exp(-lambda_*N))
fact = (1-exp(-lambda_))/(1-exp(-lambda_*N))
return fact*exp(-lambda_*k)
def _cdf(self, x, lambda_, N):
k = floor(x)
return (1-exp(-lambda_*(k+1)))/(1-exp(-lambda_*N))
def _ppf(self, q, lambda_, N):
qnew = q*(1-exp(-lambda_*N))
vals = ceil(-1.0/lambda_ * log(1-qnew)-1)
vals1 = (vals-1).clip(0.0, np.inf)
temp = self._cdf(vals1, lambda_, N)
return np.where(temp >= q, vals1, vals)
def _stats(self, lambda_, N):
z = exp(-lambda_)
zN = exp(-lambda_*N)
mu = z/(1.0-z)-N*zN/(1-zN)
var = z/(1.0-z)**2 - N*N*zN/(1-zN)**2
trm = (1-zN)/(1-z)
trm2 = (z*trm**2 - N*N*zN)
g1 = z*(1+z)*trm**3 - N**3*zN*(1+zN)
g1 = g1 / trm2**(1.5)
g2 = z*(1+4*z+z*z)*trm**4 - N**4 * zN*(1+4*zN+zN*zN)
g2 = g2 / trm2 / trm2
return mu, var, g1, g2
boltzmann = boltzmann_gen(name='boltzmann', a=0,
longname='A truncated discrete exponential ')
class randint_gen(rv_discrete):
r"""A uniform discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `randint` is:
.. math::
f(k) = \frac{1}{\texttt{high} - \texttt{low}}
for :math:`k \in \{\texttt{low}, \dots, \texttt{high} - 1\}`.
`randint` takes :math:`\texttt{low}` and :math:`\texttt{high}` as shape
parameters.
%(after_notes)s
Examples
--------
>>> import numpy as np
>>> from scipy.stats import randint
>>> import matplotlib.pyplot as plt
>>> fig, ax = plt.subplots(1, 1)
Calculate the first four moments:
>>> low, high = 7, 31
>>> mean, var, skew, kurt = randint.stats(low, high, moments='mvsk')
Display the probability mass function (``pmf``):
>>> x = np.arange(low - 5, high + 5)
>>> ax.plot(x, randint.pmf(x, low, high), 'bo', ms=8, label='randint pmf')
>>> ax.vlines(x, 0, randint.pmf(x, low, high), colors='b', lw=5, alpha=0.5)
Alternatively, the distribution object can be called (as a function) to
fix the shape and location. This returns a "frozen" RV object holding the
given parameters fixed.
Freeze the distribution and display the frozen ``pmf``:
>>> rv = randint(low, high)
>>> ax.vlines(x, 0, rv.pmf(x), colors='k', linestyles='-',
... lw=1, label='frozen pmf')
>>> ax.legend(loc='lower center')
>>> plt.show()
Check the relationship between the cumulative distribution function
(``cdf``) and its inverse, the percent point function (``ppf``):
>>> q = np.arange(low, high)
>>> p = randint.cdf(q, low, high)
>>> np.allclose(q, randint.ppf(p, low, high))
True
Generate random numbers:
>>> r = randint.rvs(low, high, size=1000)
"""
def _shape_info(self):
return [_ShapeInfo("low", True, (-np.inf, np.inf), (False, False)),
_ShapeInfo("high", True, (-np.inf, np.inf), (False, False))]
def _argcheck(self, low, high):
return (high > low) & _isintegral(low) & _isintegral(high)
def _get_support(self, low, high):
return low, high-1
def _pmf(self, k, low, high):
# randint.pmf(k) = 1./(high - low)
p = np.ones_like(k) / (np.asarray(high, dtype=np.int64) - low)
return np.where((k >= low) & (k < high), p, 0.)
def _cdf(self, x, low, high):
k = floor(x)
return (k - low + 1.) / (high - low)
def _ppf(self, q, low, high):
vals = ceil(q * (high - low) + low) - 1
vals1 = (vals - 1).clip(low, high)
temp = self._cdf(vals1, low, high)
return np.where(temp >= q, vals1, vals)
def _stats(self, low, high):
m2, m1 = np.asarray(high), np.asarray(low)
mu = (m2 + m1 - 1.0) / 2
d = m2 - m1
var = (d*d - 1) / 12.0
g1 = 0.0
g2 = -6.0/5.0 * (d*d + 1.0) / (d*d - 1.0)
return mu, var, g1, g2
def _rvs(self, low, high, size=None, random_state=None):
"""An array of *size* random integers >= ``low`` and < ``high``."""
if np.asarray(low).size == 1 and np.asarray(high).size == 1:
# no need to vectorize in that case
return rng_integers(random_state, low, high, size=size)
if size is not None:
# NumPy's RandomState.randint() doesn't broadcast its arguments.
# Use `broadcast_to()` to extend the shapes of low and high
# up to size. Then we can use the numpy.vectorize'd
# randint without needing to pass it a `size` argument.
low = np.broadcast_to(low, size)
high = np.broadcast_to(high, size)
randint = np.vectorize(partial(rng_integers, random_state),
otypes=[np.dtype(int)])
return randint(low, high)
def _entropy(self, low, high):
return log(high - low)
randint = randint_gen(name='randint', longname='A discrete uniform '
'(random integer)')
# FIXME: problems sampling.
class zipf_gen(rv_discrete):
r"""A Zipf (Zeta) discrete random variable.
%(before_notes)s
See Also
--------
zipfian
Notes
-----
The probability mass function for `zipf` is:
.. math::
f(k, a) = \frac{1}{\zeta(a) k^a}
for :math:`k \ge 1`, :math:`a > 1`.
`zipf` takes :math:`a > 1` as shape parameter. :math:`\zeta` is the
Riemann zeta function (`scipy.special.zeta`)
The Zipf distribution is also known as the zeta distribution, which is
a special case of the Zipfian distribution (`zipfian`).
%(after_notes)s
References
----------
.. [1] "Zeta Distribution", Wikipedia,
https://en.wikipedia.org/wiki/Zeta_distribution
%(example)s
Confirm that `zipf` is the large `n` limit of `zipfian`.
>>> import numpy as np
>>> from scipy.stats import zipf, zipfian
>>> k = np.arange(11)
>>> np.allclose(zipf.pmf(k, a), zipfian.pmf(k, a, n=10000000))
True
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (1, np.inf), (False, False))]
def _rvs(self, a, size=None, random_state=None):
return random_state.zipf(a, size=size)
def _argcheck(self, a):
return a > 1
def _pmf(self, k, a):
k = k.astype(np.float64)
# zipf.pmf(k, a) = 1/(zeta(a) * k**a)
Pk = 1.0 / special.zeta(a, 1) * k**-a
return Pk
def _munp(self, n, a):
return _lazywhere(
a > n + 1, (a, n),
lambda a, n: special.zeta(a - n, 1) / special.zeta(a, 1),
np.inf)
zipf = zipf_gen(a=1, name='zipf', longname='A Zipf')
def _gen_harmonic_gt1(n, a):
"""Generalized harmonic number, a > 1"""
# See https://en.wikipedia.org/wiki/Harmonic_number; search for "hurwitz"
return zeta(a, 1) - zeta(a, n+1)
def _gen_harmonic_leq1(n, a):
"""Generalized harmonic number, a <= 1"""
if not np.size(n):
return n
n_max = np.max(n) # loop starts at maximum of all n
out = np.zeros_like(a, dtype=float)
# add terms of harmonic series; starting from smallest to avoid roundoff
for i in np.arange(n_max, 0, -1, dtype=float):
mask = i <= n # don't add terms after nth
out[mask] += 1/i**a[mask]
return out
def _gen_harmonic(n, a):
"""Generalized harmonic number"""
n, a = np.broadcast_arrays(n, a)
return _lazywhere(a > 1, (n, a),
f=_gen_harmonic_gt1, f2=_gen_harmonic_leq1)
class zipfian_gen(rv_discrete):
r"""A Zipfian discrete random variable.
%(before_notes)s
See Also
--------
zipf
Notes
-----
The probability mass function for `zipfian` is:
.. math::
f(k, a, n) = \frac{1}{H_{n,a} k^a}
for :math:`k \in \{1, 2, \dots, n-1, n\}`, :math:`a \ge 0`,
:math:`n \in \{1, 2, 3, \dots\}`.
`zipfian` takes :math:`a` and :math:`n` as shape parameters.
:math:`H_{n,a}` is the :math:`n`:sup:`th` generalized harmonic
number of order :math:`a`.
The Zipfian distribution reduces to the Zipf (zeta) distribution as
:math:`n \rightarrow \infty`.
%(after_notes)s
References
----------
.. [1] "Zipf's Law", Wikipedia, https://en.wikipedia.org/wiki/Zipf's_law
.. [2] Larry Leemis, "Zipf Distribution", Univariate Distribution
Relationships. http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
%(example)s
Confirm that `zipfian` reduces to `zipf` for large `n`, ``a > 1``.
>>> import numpy as np
>>> from scipy.stats import zipf, zipfian
>>> k = np.arange(11)
>>> np.allclose(zipfian.pmf(k, a=3.5, n=10000000), zipf.pmf(k, a=3.5))
True
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (True, False)),
_ShapeInfo("n", True, (0, np.inf), (False, False))]
def _argcheck(self, a, n):
# we need np.asarray here because moment (maybe others) don't convert
return (a >= 0) & (n > 0) & (n == np.asarray(n, dtype=int))
def _get_support(self, a, n):
return 1, n
def _pmf(self, k, a, n):
k = k.astype(np.float64)
return 1.0 / _gen_harmonic(n, a) * k**-a
def _cdf(self, k, a, n):
return _gen_harmonic(k, a) / _gen_harmonic(n, a)
def _sf(self, k, a, n):
k = k + 1 # # to match SciPy convention
# see http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
return ((k**a*(_gen_harmonic(n, a) - _gen_harmonic(k, a)) + 1)
/ (k**a*_gen_harmonic(n, a)))
def _stats(self, a, n):
# see # see http://www.math.wm.edu/~leemis/chart/UDR/PDFs/Zipf.pdf
Hna = _gen_harmonic(n, a)
Hna1 = _gen_harmonic(n, a-1)
Hna2 = _gen_harmonic(n, a-2)
Hna3 = _gen_harmonic(n, a-3)
Hna4 = _gen_harmonic(n, a-4)
mu1 = Hna1/Hna
mu2n = (Hna2*Hna - Hna1**2)
mu2d = Hna**2
mu2 = mu2n / mu2d
g1 = (Hna3/Hna - 3*Hna1*Hna2/Hna**2 + 2*Hna1**3/Hna**3)/mu2**(3/2)
g2 = (Hna**3*Hna4 - 4*Hna**2*Hna1*Hna3 + 6*Hna*Hna1**2*Hna2
- 3*Hna1**4) / mu2n**2
g2 -= 3
return mu1, mu2, g1, g2
zipfian = zipfian_gen(a=1, name='zipfian', longname='A Zipfian')
class dlaplace_gen(rv_discrete):
r"""A Laplacian discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for `dlaplace` is:
.. math::
f(k) = \tanh(a/2) \exp(-a |k|)
for integers :math:`k` and :math:`a > 0`.
`dlaplace` takes :math:`a` as shape parameter.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("a", False, (0, np.inf), (False, False))]
def _pmf(self, k, a):
# dlaplace.pmf(k) = tanh(a/2) * exp(-a*abs(k))
return tanh(a/2.0) * exp(-a * abs(k))
def _cdf(self, x, a):
k = floor(x)
def f(k, a):
return 1.0 - exp(-a * k) / (exp(a) + 1)
def f2(k, a):
return exp(a * (k + 1)) / (exp(a) + 1)
return _lazywhere(k >= 0, (k, a), f=f, f2=f2)
def _ppf(self, q, a):
const = 1 + exp(a)
vals = ceil(np.where(q < 1.0 / (1 + exp(-a)),
log(q*const) / a - 1,
-log((1-q) * const) / a))
vals1 = vals - 1
return np.where(self._cdf(vals1, a) >= q, vals1, vals)
def _stats(self, a):
ea = exp(a)
mu2 = 2.*ea/(ea-1.)**2
mu4 = 2.*ea*(ea**2+10.*ea+1.) / (ea-1.)**4
return 0., mu2, 0., mu4/mu2**2 - 3.
def _entropy(self, a):
return a / sinh(a) - log(tanh(a/2.0))
def _rvs(self, a, size=None, random_state=None):
# The discrete Laplace is equivalent to the two-sided geometric
# distribution with PMF:
# f(k) = (1 - alpha)/(1 + alpha) * alpha^abs(k)
# Reference:
# https://www.sciencedirect.com/science/
# article/abs/pii/S0378375804003519
# Furthermore, the two-sided geometric distribution is
# equivalent to the difference between two iid geometric
# distributions.
# Reference (page 179):
# https://pdfs.semanticscholar.org/61b3/
# b99f466815808fd0d03f5d2791eea8b541a1.pdf
# Thus, we can leverage the following:
# 1) alpha = e^-a
# 2) probability_of_success = 1 - alpha (Bernoulli trial)
probOfSuccess = -np.expm1(-np.asarray(a))
x = random_state.geometric(probOfSuccess, size=size)
y = random_state.geometric(probOfSuccess, size=size)
return x - y
dlaplace = dlaplace_gen(a=-np.inf,
name='dlaplace', longname='A discrete Laplacian')
class poisson_binom_gen(rv_discrete):
r"""A Poisson Binomial discrete random variable.
%(before_notes)s
See Also
--------
binom
Notes
-----
The probability mass function for `poisson_binom` is:
.. math::
f(k; p_1, p_2, ..., p_n) = \sum_{A \in F_k} \prod_{i \in A} p_i \prod_{j \in A^C} 1 - p_j
where :math:`k \in \{0, 1, \dots, n-1, n\}`, :math:`F_k` is the set of all
subsets of :math:`k` integers that can be selected :math:`\{0, 1, \dots, n-1, n\}`,
and :math:`A^C` is the complement of a set :math:`A`.
`poisson_binom` accepts a single array argument ``p`` for shape parameters
:math:`0 ≤ p_i ≤ 1`, where the last axis corresponds with the index :math:`i` and
any others are for batch dimensions. Broadcasting behaves according to the usual
rules except that the last axis of ``p`` is ignored. Instances of this class do
not support serialization/unserialization.
%(after_notes)s
References
----------
.. [1] "Poisson binomial distribution", Wikipedia,
https://en.wikipedia.org/wiki/Poisson_binomial_distribution
.. [2] Biscarri, William, Sihai Dave Zhao, and Robert J. Brunner. "A simple and
fast method for computing the Poisson binomial distribution function".
Computational Statistics & Data Analysis 122 (2018) 92-100.
:doi:`10.1016/j.csda.2018.01.007`
%(example)s
""" # noqa: E501
def _shape_info(self):
# message = 'Fitting is not implemented for this distribution."
# raise NotImplementedError(message)
return []
def _argcheck(self, *args):
p = np.stack(args, axis=0)
conds = (0 <= p) & (p <= 1)
return np.all(conds, axis=0)
def _rvs(self, *args, size=None, random_state=None):
# convenient to work along the last axis here to avoid interference with `size`
p = np.stack(args, axis=-1)
# Size passed by the user is the *shape of the returned array*, so it won't
# contain the length of the last axis of p.
size = (p.shape if size is None else
(size, 1) if np.isscalar(size) else tuple(size) + (1,))
size = np.broadcast_shapes(p.shape, size)
return bernoulli._rvs(p, size=size, random_state=random_state).sum(axis=-1)
def _get_support(self, *args):
return 0, len(args)
def _pmf(self, k, *args):
k = np.atleast_1d(k).astype(np.int64)
k, *args = np.broadcast_arrays(k, *args)
args = np.asarray(args, dtype=np.float64)
return _poisson_binom(k, args, 'pmf')
def _cdf(self, k, *args):
k = np.atleast_1d(k).astype(np.int64)
k, *args = np.broadcast_arrays(k, *args)
args = np.asarray(args, dtype=np.float64)
return _poisson_binom(k, args, 'cdf')
def _stats(self, *args, **kwds):
p = np.stack(args, axis=0)
mean = np.sum(p, axis=0)
var = np.sum(p * (1-p), axis=0)
return (mean, var, None, None)
def __call__(self, *args, **kwds):
return poisson_binomial_frozen(self, *args, **kwds)
poisson_binom = poisson_binom_gen(name='poisson_binom', longname='A Poisson binomial',
shapes='p')
# The _parse_args methods don't work with vector-valued shape parameters, so we rewrite
# them. Note that `p` is accepted as an array with the index `i` of `p_i` corresponding
# with the last axis; we return it as a tuple (p_1, p_2, ..., p_n) so that it looks
# like `n` scalar (or arrays of scalar-valued) shape parameters to the infrastructure.
def _parse_args_rvs(self, p, loc=0, size=None):
return tuple(np.moveaxis(p, -1, 0)), loc, 1.0, size
def _parse_args_stats(self, p, loc=0, moments='mv'):
return tuple(np.moveaxis(p, -1, 0)), loc, 1.0, moments
def _parse_args(self, p, loc=0):
return tuple(np.moveaxis(p, -1, 0)), loc, 1.0
# The infrastructure manually binds these methods to the instance, so
# we can only override them by manually binding them, too.
_pb_obj, _pb_cls = poisson_binom, poisson_binom_gen # shorter names (for PEP8)
poisson_binom._parse_args_rvs = _parse_args_rvs.__get__(_pb_obj, _pb_cls)
poisson_binom._parse_args_stats = _parse_args_stats.__get__(_pb_obj, _pb_cls)
poisson_binom._parse_args = _parse_args.__get__(_pb_obj, _pb_cls)
class poisson_binomial_frozen(rv_discrete_frozen):
# copied from rv_frozen; we just need to bind the `_parse_args` methods
def __init__(self, dist, *args, **kwds): # verbatim
self.args = args # verbatim
self.kwds = kwds # verbatim
# create a new instance # verbatim
self.dist = dist.__class__(**dist._updated_ctor_param()) # verbatim
# Here is the only modification
self.dist._parse_args_rvs = _parse_args_rvs.__get__(_pb_obj, _pb_cls)
self.dist._parse_args_stats = _parse_args_stats.__get__(_pb_obj, _pb_cls)
self.dist._parse_args = _parse_args.__get__(_pb_obj, _pb_cls)
shapes, _, _ = self.dist._parse_args(*args, **kwds) # verbatim
self.a, self.b = self.dist._get_support(*shapes) # verbatim
def expect(self, func=None, lb=None, ub=None, conditional=False, **kwds):
a, loc, scale = self.dist._parse_args(*self.args, **self.kwds)
# Here's the modification: we pass all args (including `loc`) into the `args`
# parameter of `expect` so the shape only goes through `_parse_args` once.
return self.dist.expect(func, self.args, loc, lb, ub, conditional, **kwds)
class skellam_gen(rv_discrete):
r"""A Skellam discrete random variable.
%(before_notes)s
Notes
-----
Probability distribution of the difference of two correlated or
uncorrelated Poisson random variables.
Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with
expected values :math:`\lambda_1` and :math:`\lambda_2`. Then,
:math:`k_1 - k_2` follows a Skellam distribution with parameters
:math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and
:math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where
:math:`\rho` is the correlation coefficient between :math:`k_1` and
:math:`k_2`. If the two Poisson-distributed r.v. are independent then
:math:`\rho = 0`.
Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.
For details see: https://en.wikipedia.org/wiki/Skellam_distribution
`skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("mu1", False, (0, np.inf), (False, False)),
_ShapeInfo("mu2", False, (0, np.inf), (False, False))]
def _rvs(self, mu1, mu2, size=None, random_state=None):
n = size
return (random_state.poisson(mu1, n) -
random_state.poisson(mu2, n))
def _pmf(self, x, mu1, mu2):
with np.errstate(over='ignore'): # see gh-17432
px = np.where(x < 0,
scu._ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2,
scu._ncx2_pdf(2*mu1, 2*(1+x), 2*mu2)*2)
# ncx2.pdf() returns nan's for extremely low probabilities
return px
def _cdf(self, x, mu1, mu2):
x = floor(x)
with np.errstate(over='ignore'): # see gh-17432
px = np.where(x < 0,
scu._ncx2_cdf(2*mu2, -2*x, 2*mu1),
1 - scu._ncx2_cdf(2*mu1, 2*(x+1), 2*mu2))
return px
def _stats(self, mu1, mu2):
mean = mu1 - mu2
var = mu1 + mu2
g1 = mean / sqrt((var)**3)
g2 = 1 / var
return mean, var, g1, g2
skellam = skellam_gen(a=-np.inf, name="skellam", longname='A Skellam')
class yulesimon_gen(rv_discrete):
r"""A Yule-Simon discrete random variable.
%(before_notes)s
Notes
-----
The probability mass function for the `yulesimon` is:
.. math::
f(k) = \alpha B(k, \alpha+1)
for :math:`k=1,2,3,...`, where :math:`\alpha>0`.
Here :math:`B` refers to the `scipy.special.beta` function.
The sampling of random variates is based on pg 553, Section 6.3 of [1]_.
Our notation maps to the referenced logic via :math:`\alpha=a-1`.
For details see the wikipedia entry [2]_.
References
----------
.. [1] Devroye, Luc. "Non-uniform Random Variate Generation",
(1986) Springer, New York.
.. [2] https://en.wikipedia.org/wiki/Yule-Simon_distribution
%(after_notes)s
%(example)s
"""
def _shape_info(self):
return [_ShapeInfo("alpha", False, (0, np.inf), (False, False))]
def _rvs(self, alpha, size=None, random_state=None):
E1 = random_state.standard_exponential(size)
E2 = random_state.standard_exponential(size)
ans = ceil(-E1 / log1p(-exp(-E2 / alpha)))
return ans
def _pmf(self, x, alpha):
return alpha * special.beta(x, alpha + 1)
def _argcheck(self, alpha):
return (alpha > 0)
def _logpmf(self, x, alpha):
return log(alpha) + special.betaln(x, alpha + 1)
def _cdf(self, x, alpha):
return 1 - x * special.beta(x, alpha + 1)
def _sf(self, x, alpha):
return x * special.beta(x, alpha + 1)
def _logsf(self, x, alpha):
return log(x) + special.betaln(x, alpha + 1)
def _stats(self, alpha):
mu = np.where(alpha <= 1, np.inf, alpha / (alpha - 1))
mu2 = np.where(alpha > 2,
alpha**2 / ((alpha - 2.0) * (alpha - 1)**2),
np.inf)
mu2 = np.where(alpha <= 1, np.nan, mu2)
g1 = np.where(alpha > 3,
sqrt(alpha - 2) * (alpha + 1)**2 / (alpha * (alpha - 3)),
np.inf)
g1 = np.where(alpha <= 2, np.nan, g1)
g2 = np.where(alpha > 4,
alpha + 3 + ((11 * alpha**3 - 49 * alpha - 22) /
(alpha * (alpha - 4) * (alpha - 3))),
np.inf)
g2 = np.where(alpha <= 2, np.nan, g2)
return mu, mu2, g1, g2
yulesimon = yulesimon_gen(name='yulesimon', a=1)
class _nchypergeom_gen(rv_discrete):
r"""A noncentral hypergeometric discrete random variable.
For subclassing by nchypergeom_fisher_gen and nchypergeom_wallenius_gen.
"""
rvs_name = None
dist = None
def _shape_info(self):
return [_ShapeInfo("M", True, (0, np.inf), (True, False)),
_ShapeInfo("n", True, (0, np.inf), (True, False)),
_ShapeInfo("N", True, (0, np.inf), (True, False)),
_ShapeInfo("odds", False, (0, np.inf), (False, False))]
def _get_support(self, M, n, N, odds):
N, m1, n = M, n, N # follow Wikipedia notation
m2 = N - m1
x_min = np.maximum(0, n - m2)
x_max = np.minimum(n, m1)
return x_min, x_max
def _argcheck(self, M, n, N, odds):
M, n = np.asarray(M), np.asarray(n),
N, odds = np.asarray(N), np.asarray(odds)
cond1 = (M.astype(int) == M) & (M >= 0)
cond2 = (n.astype(int) == n) & (n >= 0)
cond3 = (N.astype(int) == N) & (N >= 0)
cond4 = odds > 0
cond5 = N <= M
cond6 = n <= M
return cond1 & cond2 & cond3 & cond4 & cond5 & cond6
def _rvs(self, M, n, N, odds, size=None, random_state=None):
@_vectorize_rvs_over_shapes
def _rvs1(M, n, N, odds, size, random_state):
length = np.prod(size)
urn = _PyStochasticLib3()
rv_gen = getattr(urn, self.rvs_name)
rvs = rv_gen(N, n, M, odds, length, random_state)
rvs = rvs.reshape(size)
return rvs
return _rvs1(M, n, N, odds, size=size, random_state=random_state)
def _pmf(self, x, M, n, N, odds):
x, M, n, N, odds = np.broadcast_arrays(x, M, n, N, odds)
if x.size == 0: # np.vectorize doesn't work with zero size input
return np.empty_like(x)
@np.vectorize
def _pmf1(x, M, n, N, odds):
urn = self.dist(N, n, M, odds, 1e-12)
return urn.probability(x)
return _pmf1(x, M, n, N, odds)
def _stats(self, M, n, N, odds, moments):
@np.vectorize
def _moments1(M, n, N, odds):
urn = self.dist(N, n, M, odds, 1e-12)
return urn.moments()
m, v = (_moments1(M, n, N, odds) if ("m" in moments or "v" in moments)
else (None, None))
s, k = None, None
return m, v, s, k
class nchypergeom_fisher_gen(_nchypergeom_gen):
r"""A Fisher's noncentral hypergeometric discrete random variable.
Fisher's noncentral hypergeometric distribution models drawing objects of
two types from a bin. `M` is the total number of objects, `n` is the
number of Type I objects, and `odds` is the odds ratio: the odds of
selecting a Type I object rather than a Type II object when there is only
one object of each type.
The random variate represents the number of Type I objects drawn if we
take a handful of objects from the bin at once and find out afterwards
that we took `N` objects.
%(before_notes)s
See Also
--------
nchypergeom_wallenius, hypergeom, nhypergeom
Notes
-----
Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
with parameters `N`, `n`, and `M` (respectively) as defined above.
The probability mass function is defined as
.. math::
p(x; M, n, N, \omega) =
\frac{\binom{n}{x}\binom{M - n}{N-x}\omega^x}{P_0},
for
:math:`x \in [x_l, x_u]`,
:math:`M \in {\mathbb N}`,
:math:`n \in [0, M]`,
:math:`N \in [0, M]`,
:math:`\omega > 0`,
where
:math:`x_l = \max(0, N - (M - n))`,
:math:`x_u = \min(N, n)`,
.. math::
P_0 = \sum_{y=x_l}^{x_u} \binom{n}{y}\binom{M - n}{N-y}\omega^y,
and the binomial coefficients are defined as
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
`nchypergeom_fisher` uses the BiasedUrn package by Agner Fog with
permission for it to be distributed under SciPy's license.
The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
universally accepted; they are chosen for consistency with `hypergeom`.
Note that Fisher's noncentral hypergeometric distribution is distinct
from Wallenius' noncentral hypergeometric distribution, which models
drawing a pre-determined `N` objects from a bin one by one.
When the odds ratio is unity, however, both distributions reduce to the
ordinary hypergeometric distribution.
%(after_notes)s
References
----------
.. [1] Agner Fog, "Biased Urn Theory".
https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf
.. [2] "Fisher's noncentral hypergeometric distribution", Wikipedia,
https://en.wikipedia.org/wiki/Fisher's_noncentral_hypergeometric_distribution
%(example)s
"""
rvs_name = "rvs_fisher"
dist = _PyFishersNCHypergeometric
nchypergeom_fisher = nchypergeom_fisher_gen(
name='nchypergeom_fisher',
longname="A Fisher's noncentral hypergeometric")
class nchypergeom_wallenius_gen(_nchypergeom_gen):
r"""A Wallenius' noncentral hypergeometric discrete random variable.
Wallenius' noncentral hypergeometric distribution models drawing objects of
two types from a bin. `M` is the total number of objects, `n` is the
number of Type I objects, and `odds` is the odds ratio: the odds of
selecting a Type I object rather than a Type II object when there is only
one object of each type.
The random variate represents the number of Type I objects drawn if we
draw a pre-determined `N` objects from a bin one by one.
%(before_notes)s
See Also
--------
nchypergeom_fisher, hypergeom, nhypergeom
Notes
-----
Let mathematical symbols :math:`N`, :math:`n`, and :math:`M` correspond
with parameters `N`, `n`, and `M` (respectively) as defined above.
The probability mass function is defined as
.. math::
p(x; N, n, M) = \binom{n}{x} \binom{M - n}{N-x}
\int_0^1 \left(1-t^{\omega/D}\right)^x\left(1-t^{1/D}\right)^{N-x} dt
for
:math:`x \in [x_l, x_u]`,
:math:`M \in {\mathbb N}`,
:math:`n \in [0, M]`,
:math:`N \in [0, M]`,
:math:`\omega > 0`,
where
:math:`x_l = \max(0, N - (M - n))`,
:math:`x_u = \min(N, n)`,
.. math::
D = \omega(n - x) + ((M - n)-(N-x)),
and the binomial coefficients are defined as
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
`nchypergeom_wallenius` uses the BiasedUrn package by Agner Fog with
permission for it to be distributed under SciPy's license.
The symbols used to denote the shape parameters (`N`, `n`, and `M`) are not
universally accepted; they are chosen for consistency with `hypergeom`.
Note that Wallenius' noncentral hypergeometric distribution is distinct
from Fisher's noncentral hypergeometric distribution, which models
take a handful of objects from the bin at once, finding out afterwards
that `N` objects were taken.
When the odds ratio is unity, however, both distributions reduce to the
ordinary hypergeometric distribution.
%(after_notes)s
References
----------
.. [1] Agner Fog, "Biased Urn Theory".
https://cran.r-project.org/web/packages/BiasedUrn/vignettes/UrnTheory.pdf
.. [2] "Wallenius' noncentral hypergeometric distribution", Wikipedia,
https://en.wikipedia.org/wiki/Wallenius'_noncentral_hypergeometric_distribution
%(example)s
"""
rvs_name = "rvs_wallenius"
dist = _PyWalleniusNCHypergeometric
nchypergeom_wallenius = nchypergeom_wallenius_gen(
name='nchypergeom_wallenius',
longname="A Wallenius' noncentral hypergeometric")
# Collect names of classes and objects in this module.
pairs = list(globals().copy().items())
_distn_names, _distn_gen_names = get_distribution_names(pairs, rv_discrete)
__all__ = _distn_names + _distn_gen_names
|