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774 lines
27 KiB
Python
774 lines
27 KiB
Python
"""Random variable generators.
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integers
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--------
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uniform within range
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sequences
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---------
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pick random element
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pick random sample
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pick weighted random sample
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generate random permutation
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distributions on the real line:
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------------------------------
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uniform
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triangular
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normal (Gaussian)
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lognormal
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negative exponential
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gamma
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beta
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pareto
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Weibull
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distributions on the circle (angles 0 to 2pi)
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---------------------------------------------
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circular uniform
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von Mises
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General notes on the underlying Mersenne Twister core generator:
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* The period is 2**19937-1.
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* It is one of the most extensively tested generators in existence.
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* The random() method is implemented in C, executes in a single Python step,
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and is, therefore, threadsafe.
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"""
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from warnings import warn as _warn
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from types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodType
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from math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceil
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from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin
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from os import urandom as _urandom
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from _collections_abc import Set as _Set, Sequence as _Sequence
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from hashlib import sha512 as _sha512
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import itertools as _itertools
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import bisect as _bisect
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import os as _os
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__all__ = ["Random","seed","random","uniform","randint","choice","sample",
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"randrange","shuffle","normalvariate","lognormvariate",
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"expovariate","vonmisesvariate","gammavariate","triangular",
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"gauss","betavariate","paretovariate","weibullvariate",
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"getstate","setstate", "getrandbits", "choices",
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"SystemRandom"]
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NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)
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TWOPI = 2.0*_pi
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LOG4 = _log(4.0)
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SG_MAGICCONST = 1.0 + _log(4.5)
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BPF = 53 # Number of bits in a float
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RECIP_BPF = 2**-BPF
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# Translated by Guido van Rossum from C source provided by
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# Adrian Baddeley. Adapted by Raymond Hettinger for use with
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# the Mersenne Twister and os.urandom() core generators.
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import _random
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class Random(_random.Random):
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"""Random number generator base class used by bound module functions.
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Used to instantiate instances of Random to get generators that don't
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share state.
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Class Random can also be subclassed if you want to use a different basic
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generator of your own devising: in that case, override the following
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methods: random(), seed(), getstate(), and setstate().
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Optionally, implement a getrandbits() method so that randrange()
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can cover arbitrarily large ranges.
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"""
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VERSION = 3 # used by getstate/setstate
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def __init__(self, x=None):
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"""Initialize an instance.
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Optional argument x controls seeding, as for Random.seed().
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"""
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self.seed(x)
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self.gauss_next = None
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def seed(self, a=None, version=2):
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"""Initialize internal state from hashable object.
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None or no argument seeds from current time or from an operating
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system specific randomness source if available.
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If *a* is an int, all bits are used.
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For version 2 (the default), all of the bits are used if *a* is a str,
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bytes, or bytearray. For version 1 (provided for reproducing random
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sequences from older versions of Python), the algorithm for str and
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bytes generates a narrower range of seeds.
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"""
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if version == 1 and isinstance(a, (str, bytes)):
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a = a.decode('latin-1') if isinstance(a, bytes) else a
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x = ord(a[0]) << 7 if a else 0
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for c in map(ord, a):
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x = ((1000003 * x) ^ c) & 0xFFFFFFFFFFFFFFFF
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x ^= len(a)
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a = -2 if x == -1 else x
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if version == 2 and isinstance(a, (str, bytes, bytearray)):
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if isinstance(a, str):
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a = a.encode()
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a += _sha512(a).digest()
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a = int.from_bytes(a, 'big')
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super().seed(a)
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self.gauss_next = None
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def getstate(self):
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"""Return internal state; can be passed to setstate() later."""
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return self.VERSION, super().getstate(), self.gauss_next
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def setstate(self, state):
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"""Restore internal state from object returned by getstate()."""
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version = state[0]
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if version == 3:
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version, internalstate, self.gauss_next = state
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super().setstate(internalstate)
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elif version == 2:
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version, internalstate, self.gauss_next = state
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# In version 2, the state was saved as signed ints, which causes
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# inconsistencies between 32/64-bit systems. The state is
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# really unsigned 32-bit ints, so we convert negative ints from
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# version 2 to positive longs for version 3.
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try:
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internalstate = tuple(x % (2**32) for x in internalstate)
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except ValueError as e:
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raise TypeError from e
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super().setstate(internalstate)
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else:
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raise ValueError("state with version %s passed to "
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"Random.setstate() of version %s" %
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(version, self.VERSION))
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## ---- Methods below this point do not need to be overridden when
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## ---- subclassing for the purpose of using a different core generator.
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## -------------------- pickle support -------------------
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# Issue 17489: Since __reduce__ was defined to fix #759889 this is no
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# longer called; we leave it here because it has been here since random was
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# rewritten back in 2001 and why risk breaking something.
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def __getstate__(self): # for pickle
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return self.getstate()
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def __setstate__(self, state): # for pickle
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self.setstate(state)
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def __reduce__(self):
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return self.__class__, (), self.getstate()
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## -------------------- integer methods -------------------
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def randrange(self, start, stop=None, step=1, _int=int):
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"""Choose a random item from range(start, stop[, step]).
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This fixes the problem with randint() which includes the
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endpoint; in Python this is usually not what you want.
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"""
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# This code is a bit messy to make it fast for the
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# common case while still doing adequate error checking.
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istart = _int(start)
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if istart != start:
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raise ValueError("non-integer arg 1 for randrange()")
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if stop is None:
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if istart > 0:
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return self._randbelow(istart)
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raise ValueError("empty range for randrange()")
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# stop argument supplied.
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istop = _int(stop)
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if istop != stop:
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raise ValueError("non-integer stop for randrange()")
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width = istop - istart
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if step == 1 and width > 0:
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return istart + self._randbelow(width)
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if step == 1:
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raise ValueError("empty range for randrange() (%d,%d, %d)" % (istart, istop, width))
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# Non-unit step argument supplied.
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istep = _int(step)
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if istep != step:
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raise ValueError("non-integer step for randrange()")
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if istep > 0:
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n = (width + istep - 1) // istep
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elif istep < 0:
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n = (width + istep + 1) // istep
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else:
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raise ValueError("zero step for randrange()")
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if n <= 0:
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raise ValueError("empty range for randrange()")
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return istart + istep*self._randbelow(n)
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def randint(self, a, b):
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"""Return random integer in range [a, b], including both end points.
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"""
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return self.randrange(a, b+1)
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def _randbelow(self, n, int=int, maxsize=1<<BPF, type=type,
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Method=_MethodType, BuiltinMethod=_BuiltinMethodType):
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"Return a random int in the range [0,n). Raises ValueError if n==0."
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random = self.random
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getrandbits = self.getrandbits
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# Only call self.getrandbits if the original random() builtin method
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# has not been overridden or if a new getrandbits() was supplied.
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if type(random) is BuiltinMethod or type(getrandbits) is Method:
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k = n.bit_length() # don't use (n-1) here because n can be 1
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r = getrandbits(k) # 0 <= r < 2**k
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while r >= n:
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r = getrandbits(k)
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return r
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# There's an overridden random() method but no new getrandbits() method,
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# so we can only use random() from here.
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if n >= maxsize:
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_warn("Underlying random() generator does not supply \n"
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"enough bits to choose from a population range this large.\n"
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"To remove the range limitation, add a getrandbits() method.")
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return int(random() * n)
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rem = maxsize % n
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limit = (maxsize - rem) / maxsize # int(limit * maxsize) % n == 0
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r = random()
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while r >= limit:
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r = random()
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return int(r*maxsize) % n
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## -------------------- sequence methods -------------------
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def choice(self, seq):
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"""Choose a random element from a non-empty sequence."""
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try:
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i = self._randbelow(len(seq))
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except ValueError:
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raise IndexError('Cannot choose from an empty sequence') from None
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return seq[i]
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def shuffle(self, x, random=None):
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"""Shuffle list x in place, and return None.
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Optional argument random is a 0-argument function returning a
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random float in [0.0, 1.0); if it is the default None, the
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standard random.random will be used.
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"""
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if random is None:
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randbelow = self._randbelow
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for i in reversed(range(1, len(x))):
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# pick an element in x[:i+1] with which to exchange x[i]
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j = randbelow(i+1)
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x[i], x[j] = x[j], x[i]
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else:
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_int = int
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for i in reversed(range(1, len(x))):
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# pick an element in x[:i+1] with which to exchange x[i]
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j = _int(random() * (i+1))
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x[i], x[j] = x[j], x[i]
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def sample(self, population, k):
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"""Chooses k unique random elements from a population sequence or set.
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Returns a new list containing elements from the population while
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leaving the original population unchanged. The resulting list is
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in selection order so that all sub-slices will also be valid random
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samples. This allows raffle winners (the sample) to be partitioned
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into grand prize and second place winners (the subslices).
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Members of the population need not be hashable or unique. If the
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population contains repeats, then each occurrence is a possible
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selection in the sample.
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To choose a sample in a range of integers, use range as an argument.
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This is especially fast and space efficient for sampling from a
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large population: sample(range(10000000), 60)
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"""
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# Sampling without replacement entails tracking either potential
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# selections (the pool) in a list or previous selections in a set.
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# When the number of selections is small compared to the
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# population, then tracking selections is efficient, requiring
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# only a small set and an occasional reselection. For
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# a larger number of selections, the pool tracking method is
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# preferred since the list takes less space than the
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# set and it doesn't suffer from frequent reselections.
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if isinstance(population, _Set):
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population = tuple(population)
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if not isinstance(population, _Sequence):
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raise TypeError("Population must be a sequence or set. For dicts, use list(d).")
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randbelow = self._randbelow
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n = len(population)
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if not 0 <= k <= n:
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raise ValueError("Sample larger than population or is negative")
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result = [None] * k
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setsize = 21 # size of a small set minus size of an empty list
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if k > 5:
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setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
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if n <= setsize:
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# An n-length list is smaller than a k-length set
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pool = list(population)
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for i in range(k): # invariant: non-selected at [0,n-i)
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j = randbelow(n-i)
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result[i] = pool[j]
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pool[j] = pool[n-i-1] # move non-selected item into vacancy
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else:
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selected = set()
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selected_add = selected.add
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for i in range(k):
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j = randbelow(n)
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while j in selected:
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j = randbelow(n)
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selected_add(j)
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result[i] = population[j]
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return result
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def choices(self, population, weights=None, *, cum_weights=None, k=1):
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"""Return a k sized list of population elements chosen with replacement.
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If the relative weights or cumulative weights are not specified,
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the selections are made with equal probability.
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"""
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random = self.random
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if cum_weights is None:
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if weights is None:
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_int = int
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total = len(population)
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return [population[_int(random() * total)] for i in range(k)]
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cum_weights = list(_itertools.accumulate(weights))
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elif weights is not None:
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raise TypeError('Cannot specify both weights and cumulative weights')
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if len(cum_weights) != len(population):
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raise ValueError('The number of weights does not match the population')
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bisect = _bisect.bisect
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total = cum_weights[-1]
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return [population[bisect(cum_weights, random() * total)] for i in range(k)]
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## -------------------- real-valued distributions -------------------
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## -------------------- uniform distribution -------------------
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def uniform(self, a, b):
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"Get a random number in the range [a, b) or [a, b] depending on rounding."
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return a + (b-a) * self.random()
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## -------------------- triangular --------------------
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def triangular(self, low=0.0, high=1.0, mode=None):
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"""Triangular distribution.
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Continuous distribution bounded by given lower and upper limits,
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and having a given mode value in-between.
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http://en.wikipedia.org/wiki/Triangular_distribution
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"""
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u = self.random()
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try:
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c = 0.5 if mode is None else (mode - low) / (high - low)
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except ZeroDivisionError:
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return low
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if u > c:
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u = 1.0 - u
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c = 1.0 - c
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low, high = high, low
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return low + (high - low) * _sqrt(u * c)
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## -------------------- normal distribution --------------------
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def normalvariate(self, mu, sigma):
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"""Normal distribution.
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mu is the mean, and sigma is the standard deviation.
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"""
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# mu = mean, sigma = standard deviation
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# Uses Kinderman and Monahan method. Reference: Kinderman,
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# A.J. and Monahan, J.F., "Computer generation of random
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# variables using the ratio of uniform deviates", ACM Trans
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# Math Software, 3, (1977), pp257-260.
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random = self.random
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while 1:
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u1 = random()
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u2 = 1.0 - random()
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z = NV_MAGICCONST*(u1-0.5)/u2
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zz = z*z/4.0
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if zz <= -_log(u2):
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break
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return mu + z*sigma
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## -------------------- lognormal distribution --------------------
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def lognormvariate(self, mu, sigma):
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"""Log normal distribution.
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If you take the natural logarithm of this distribution, you'll get a
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normal distribution with mean mu and standard deviation sigma.
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mu can have any value, and sigma must be greater than zero.
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"""
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return _exp(self.normalvariate(mu, sigma))
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## -------------------- exponential distribution --------------------
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def expovariate(self, lambd):
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"""Exponential distribution.
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lambd is 1.0 divided by the desired mean. It should be
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nonzero. (The parameter would be called "lambda", but that is
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a reserved word in Python.) Returned values range from 0 to
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positive infinity if lambd is positive, and from negative
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infinity to 0 if lambd is negative.
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"""
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# lambd: rate lambd = 1/mean
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# ('lambda' is a Python reserved word)
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# we use 1-random() instead of random() to preclude the
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# possibility of taking the log of zero.
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return -_log(1.0 - self.random())/lambd
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## -------------------- von Mises distribution --------------------
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def vonmisesvariate(self, mu, kappa):
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"""Circular data distribution.
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mu is the mean angle, expressed in radians between 0 and 2*pi, and
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kappa is the concentration parameter, which must be greater than or
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equal to zero. If kappa is equal to zero, this distribution reduces
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to a uniform random angle over the range 0 to 2*pi.
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"""
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# mu: mean angle (in radians between 0 and 2*pi)
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# kappa: concentration parameter kappa (>= 0)
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# if kappa = 0 generate uniform random angle
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# Based upon an algorithm published in: Fisher, N.I.,
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# "Statistical Analysis of Circular Data", Cambridge
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# University Press, 1993.
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# Thanks to Magnus Kessler for a correction to the
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# implementation of step 4.
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random = self.random
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if kappa <= 1e-6:
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return TWOPI * random()
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s = 0.5 / kappa
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r = s + _sqrt(1.0 + s * s)
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while 1:
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u1 = random()
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z = _cos(_pi * u1)
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d = z / (r + z)
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u2 = random()
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if u2 < 1.0 - d * d or u2 <= (1.0 - d) * _exp(d):
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break
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q = 1.0 / r
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f = (q + z) / (1.0 + q * z)
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u3 = random()
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if u3 > 0.5:
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theta = (mu + _acos(f)) % TWOPI
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else:
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theta = (mu - _acos(f)) % TWOPI
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return theta
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## -------------------- gamma distribution --------------------
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def gammavariate(self, alpha, beta):
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"""Gamma distribution. Not the gamma function!
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Conditions on the parameters are alpha > 0 and beta > 0.
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The probability distribution function is:
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x ** (alpha - 1) * math.exp(-x / beta)
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pdf(x) = --------------------------------------
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math.gamma(alpha) * beta ** alpha
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"""
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# alpha > 0, beta > 0, mean is alpha*beta, variance is alpha*beta**2
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# Warning: a few older sources define the gamma distribution in terms
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# of alpha > -1.0
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if alpha <= 0.0 or beta <= 0.0:
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raise ValueError('gammavariate: alpha and beta must be > 0.0')
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random = self.random
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if alpha > 1.0:
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# Uses R.C.H. Cheng, "The generation of Gamma
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# variables with non-integral shape parameters",
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# Applied Statistics, (1977), 26, No. 1, p71-74
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ainv = _sqrt(2.0 * alpha - 1.0)
|
|
bbb = alpha - LOG4
|
|
ccc = alpha + ainv
|
|
|
|
while 1:
|
|
u1 = random()
|
|
if not 1e-7 < u1 < .9999999:
|
|
continue
|
|
u2 = 1.0 - random()
|
|
v = _log(u1/(1.0-u1))/ainv
|
|
x = alpha*_exp(v)
|
|
z = u1*u1*u2
|
|
r = bbb+ccc*v-x
|
|
if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= _log(z):
|
|
return x * beta
|
|
|
|
elif alpha == 1.0:
|
|
# expovariate(1/beta)
|
|
u = random()
|
|
while u <= 1e-7:
|
|
u = random()
|
|
return -_log(u) * beta
|
|
|
|
else: # alpha is between 0 and 1 (exclusive)
|
|
|
|
# Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
|
|
|
|
while 1:
|
|
u = random()
|
|
b = (_e + alpha)/_e
|
|
p = b*u
|
|
if p <= 1.0:
|
|
x = p ** (1.0/alpha)
|
|
else:
|
|
x = -_log((b-p)/alpha)
|
|
u1 = random()
|
|
if p > 1.0:
|
|
if u1 <= x ** (alpha - 1.0):
|
|
break
|
|
elif u1 <= _exp(-x):
|
|
break
|
|
return x * beta
|
|
|
|
## -------------------- Gauss (faster alternative) --------------------
|
|
|
|
def gauss(self, mu, sigma):
|
|
"""Gaussian distribution.
|
|
|
|
mu is the mean, and sigma is the standard deviation. This is
|
|
slightly faster than the normalvariate() function.
|
|
|
|
Not thread-safe without a lock around calls.
|
|
|
|
"""
|
|
|
|
# When x and y are two variables from [0, 1), uniformly
|
|
# distributed, then
|
|
#
|
|
# cos(2*pi*x)*sqrt(-2*log(1-y))
|
|
# sin(2*pi*x)*sqrt(-2*log(1-y))
|
|
#
|
|
# are two *independent* variables with normal distribution
|
|
# (mu = 0, sigma = 1).
|
|
# (Lambert Meertens)
|
|
# (corrected version; bug discovered by Mike Miller, fixed by LM)
|
|
|
|
# Multithreading note: When two threads call this function
|
|
# simultaneously, it is possible that they will receive the
|
|
# same return value. The window is very small though. To
|
|
# avoid this, you have to use a lock around all calls. (I
|
|
# didn't want to slow this down in the serial case by using a
|
|
# lock here.)
|
|
|
|
random = self.random
|
|
z = self.gauss_next
|
|
self.gauss_next = None
|
|
if z is None:
|
|
x2pi = random() * TWOPI
|
|
g2rad = _sqrt(-2.0 * _log(1.0 - random()))
|
|
z = _cos(x2pi) * g2rad
|
|
self.gauss_next = _sin(x2pi) * g2rad
|
|
|
|
return mu + z*sigma
|
|
|
|
## -------------------- beta --------------------
|
|
## See
|
|
## http://mail.python.org/pipermail/python-bugs-list/2001-January/003752.html
|
|
## for Ivan Frohne's insightful analysis of why the original implementation:
|
|
##
|
|
## def betavariate(self, alpha, beta):
|
|
## # Discrete Event Simulation in C, pp 87-88.
|
|
##
|
|
## y = self.expovariate(alpha)
|
|
## z = self.expovariate(1.0/beta)
|
|
## return z/(y+z)
|
|
##
|
|
## was dead wrong, and how it probably got that way.
|
|
|
|
def betavariate(self, alpha, beta):
|
|
"""Beta distribution.
|
|
|
|
Conditions on the parameters are alpha > 0 and beta > 0.
|
|
Returned values range between 0 and 1.
|
|
|
|
"""
|
|
|
|
# This version due to Janne Sinkkonen, and matches all the std
|
|
# texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
|
|
y = self.gammavariate(alpha, 1.0)
|
|
if y == 0:
|
|
return 0.0
|
|
else:
|
|
return y / (y + self.gammavariate(beta, 1.0))
|
|
|
|
## -------------------- Pareto --------------------
|
|
|
|
def paretovariate(self, alpha):
|
|
"""Pareto distribution. alpha is the shape parameter."""
|
|
# Jain, pg. 495
|
|
|
|
u = 1.0 - self.random()
|
|
return 1.0 / u ** (1.0/alpha)
|
|
|
|
## -------------------- Weibull --------------------
|
|
|
|
def weibullvariate(self, alpha, beta):
|
|
"""Weibull distribution.
|
|
|
|
alpha is the scale parameter and beta is the shape parameter.
|
|
|
|
"""
|
|
# Jain, pg. 499; bug fix courtesy Bill Arms
|
|
|
|
u = 1.0 - self.random()
|
|
return alpha * (-_log(u)) ** (1.0/beta)
|
|
|
|
## --------------- Operating System Random Source ------------------
|
|
|
|
class SystemRandom(Random):
|
|
"""Alternate random number generator using sources provided
|
|
by the operating system (such as /dev/urandom on Unix or
|
|
CryptGenRandom on Windows).
|
|
|
|
Not available on all systems (see os.urandom() for details).
|
|
"""
|
|
|
|
def random(self):
|
|
"""Get the next random number in the range [0.0, 1.0)."""
|
|
return (int.from_bytes(_urandom(7), 'big') >> 3) * RECIP_BPF
|
|
|
|
def getrandbits(self, k):
|
|
"""getrandbits(k) -> x. Generates an int with k random bits."""
|
|
if k <= 0:
|
|
raise ValueError('number of bits must be greater than zero')
|
|
if k != int(k):
|
|
raise TypeError('number of bits should be an integer')
|
|
numbytes = (k + 7) // 8 # bits / 8 and rounded up
|
|
x = int.from_bytes(_urandom(numbytes), 'big')
|
|
return x >> (numbytes * 8 - k) # trim excess bits
|
|
|
|
def seed(self, *args, **kwds):
|
|
"Stub method. Not used for a system random number generator."
|
|
return None
|
|
|
|
def _notimplemented(self, *args, **kwds):
|
|
"Method should not be called for a system random number generator."
|
|
raise NotImplementedError('System entropy source does not have state.')
|
|
getstate = setstate = _notimplemented
|
|
|
|
## -------------------- test program --------------------
|
|
|
|
def _test_generator(n, func, args):
|
|
import time
|
|
print(n, 'times', func.__name__)
|
|
total = 0.0
|
|
sqsum = 0.0
|
|
smallest = 1e10
|
|
largest = -1e10
|
|
t0 = time.time()
|
|
for i in range(n):
|
|
x = func(*args)
|
|
total += x
|
|
sqsum = sqsum + x*x
|
|
smallest = min(x, smallest)
|
|
largest = max(x, largest)
|
|
t1 = time.time()
|
|
print(round(t1-t0, 3), 'sec,', end=' ')
|
|
avg = total/n
|
|
stddev = _sqrt(sqsum/n - avg*avg)
|
|
print('avg %g, stddev %g, min %g, max %g\n' % \
|
|
(avg, stddev, smallest, largest))
|
|
|
|
|
|
def _test(N=2000):
|
|
_test_generator(N, random, ())
|
|
_test_generator(N, normalvariate, (0.0, 1.0))
|
|
_test_generator(N, lognormvariate, (0.0, 1.0))
|
|
_test_generator(N, vonmisesvariate, (0.0, 1.0))
|
|
_test_generator(N, gammavariate, (0.01, 1.0))
|
|
_test_generator(N, gammavariate, (0.1, 1.0))
|
|
_test_generator(N, gammavariate, (0.1, 2.0))
|
|
_test_generator(N, gammavariate, (0.5, 1.0))
|
|
_test_generator(N, gammavariate, (0.9, 1.0))
|
|
_test_generator(N, gammavariate, (1.0, 1.0))
|
|
_test_generator(N, gammavariate, (2.0, 1.0))
|
|
_test_generator(N, gammavariate, (20.0, 1.0))
|
|
_test_generator(N, gammavariate, (200.0, 1.0))
|
|
_test_generator(N, gauss, (0.0, 1.0))
|
|
_test_generator(N, betavariate, (3.0, 3.0))
|
|
_test_generator(N, triangular, (0.0, 1.0, 1.0/3.0))
|
|
|
|
# Create one instance, seeded from current time, and export its methods
|
|
# as module-level functions. The functions share state across all uses
|
|
#(both in the user's code and in the Python libraries), but that's fine
|
|
# for most programs and is easier for the casual user than making them
|
|
# instantiate their own Random() instance.
|
|
|
|
_inst = Random()
|
|
seed = _inst.seed
|
|
random = _inst.random
|
|
uniform = _inst.uniform
|
|
triangular = _inst.triangular
|
|
randint = _inst.randint
|
|
choice = _inst.choice
|
|
randrange = _inst.randrange
|
|
sample = _inst.sample
|
|
shuffle = _inst.shuffle
|
|
choices = _inst.choices
|
|
normalvariate = _inst.normalvariate
|
|
lognormvariate = _inst.lognormvariate
|
|
expovariate = _inst.expovariate
|
|
vonmisesvariate = _inst.vonmisesvariate
|
|
gammavariate = _inst.gammavariate
|
|
gauss = _inst.gauss
|
|
betavariate = _inst.betavariate
|
|
paretovariate = _inst.paretovariate
|
|
weibullvariate = _inst.weibullvariate
|
|
getstate = _inst.getstate
|
|
setstate = _inst.setstate
|
|
getrandbits = _inst.getrandbits
|
|
|
|
if hasattr(_os, "fork"):
|
|
_os.register_at_fork(after_in_child=_inst.seed)
|
|
|
|
|
|
if __name__ == '__main__':
|
|
_test()
|