659 lines
21 KiB
Python
659 lines
21 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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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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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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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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Translated from anonymously contributed C/C++ source.
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Multi-threading note: the random number generator used here is not thread-
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safe; it is possible that two calls return the same random value. However,
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you can instantiate a different instance of Random() in each thread to get
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generators that don't share state, then use .setstate() and .jumpahead() to
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move the generators to disjoint segments of the full period. For example,
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def create_generators(num, delta, firstseed=None):
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""\"Return list of num distinct generators.
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Each generator has its own unique segment of delta elements from
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Random.random()'s full period.
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Seed the first generator with optional arg firstseed (default is
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None, to seed from current time).
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""\"
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from random import Random
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g = Random(firstseed)
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result = [g]
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for i in range(num - 1):
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laststate = g.getstate()
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g = Random()
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g.setstate(laststate)
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g.jumpahead(delta)
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result.append(g)
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return result
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gens = create_generators(10, 1000000)
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That creates 10 distinct generators, which can be passed out to 10 distinct
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threads. The generators don't share state so can be called safely in
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parallel. So long as no thread calls its g.random() more than a million
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times (the second argument to create_generators), the sequences seen by
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each thread will not overlap.
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The period of the underlying Wichmann-Hill generator is 6,953,607,871,644,
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and that limits how far this technique can be pushed.
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Just for fun, note that since we know the period, .jumpahead() can also be
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used to "move backward in time":
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>>> g = Random(42) # arbitrary
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>>> g.random()
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0.25420336316883324
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>>> g.jumpahead(6953607871644L - 1) # move *back* one
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>>> g.random()
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0.25420336316883324
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"""
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# XXX The docstring sucks.
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from math import log as _log, exp as _exp, pi as _pi, e as _e
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from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin
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def _verify(name, expected):
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computed = eval(name)
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if abs(computed - expected) > 1e-7:
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raise ValueError(
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"computed value for %s deviates too much "
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"(computed %g, expected %g)" % (name, computed, expected))
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NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)
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_verify('NV_MAGICCONST', 1.71552776992141)
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TWOPI = 2.0*_pi
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_verify('TWOPI', 6.28318530718)
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LOG4 = _log(4.0)
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_verify('LOG4', 1.38629436111989)
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SG_MAGICCONST = 1.0 + _log(4.5)
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_verify('SG_MAGICCONST', 2.50407739677627)
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del _verify
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# Translated by Guido van Rossum from C source provided by
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# Adrian Baddeley.
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class Random:
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VERSION = 1 # 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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## -------------------- core generator -------------------
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# Specific to Wichmann-Hill generator. Subclasses wishing to use a
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# different core generator should override the seed(), random(),
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# getstate(), setstate() and jumpahead() methods.
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def seed(self, a=None):
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"""Initialize internal state from hashable object.
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None or no argument seeds from current time.
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If a is not None or an int or long, hash(a) is instead.
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If a is an int or long, a is used directly. Distinct values between
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0 and 27814431486575L inclusive are guaranteed to yield distinct
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internal states (this guarantee is specific to the default
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Wichmann-Hill generator).
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"""
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if a is None:
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# Initialize from current time
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import time
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a = long(time.time() * 256)
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if type(a) not in (type(3), type(3L)):
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a = hash(a)
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a, x = divmod(a, 30268)
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a, y = divmod(a, 30306)
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a, z = divmod(a, 30322)
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self._seed = int(x)+1, int(y)+1, int(z)+1
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def random(self):
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"""Get the next random number in the range [0.0, 1.0)."""
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# Wichman-Hill random number generator.
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#
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# Wichmann, B. A. & Hill, I. D. (1982)
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# Algorithm AS 183:
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# An efficient and portable pseudo-random number generator
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# Applied Statistics 31 (1982) 188-190
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#
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# see also:
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# Correction to Algorithm AS 183
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# Applied Statistics 33 (1984) 123
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#
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# McLeod, A. I. (1985)
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# A remark on Algorithm AS 183
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# Applied Statistics 34 (1985),198-200
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# This part is thread-unsafe:
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# BEGIN CRITICAL SECTION
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x, y, z = self._seed
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x = (171 * x) % 30269
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y = (172 * y) % 30307
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z = (170 * z) % 30323
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self._seed = x, y, z
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# END CRITICAL SECTION
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# Note: on a platform using IEEE-754 double arithmetic, this can
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# never return 0.0 (asserted by Tim; proof too long for a comment).
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return (x/30269.0 + y/30307.0 + z/30323.0) % 1.0
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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, self._seed, 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 == 1:
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version, self._seed, self.gauss_next = state
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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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def jumpahead(self, n):
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"""Act as if n calls to random() were made, but quickly.
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n is an int, greater than or equal to 0.
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Example use: If you have 2 threads and know that each will
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consume no more than a million random numbers, create two Random
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objects r1 and r2, then do
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r2.setstate(r1.getstate())
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r2.jumpahead(1000000)
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Then r1 and r2 will use guaranteed-disjoint segments of the full
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period.
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"""
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if not n >= 0:
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raise ValueError("n must be >= 0")
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x, y, z = self._seed
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x = int(x * pow(171, n, 30269)) % 30269
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y = int(y * pow(172, n, 30307)) % 30307
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z = int(z * pow(170, n, 30323)) % 30323
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self._seed = x, y, z
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def __whseed(self, x=0, y=0, z=0):
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"""Set the Wichmann-Hill seed from (x, y, z).
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These must be integers in the range [0, 256).
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"""
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if not type(x) == type(y) == type(z) == type(0):
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raise TypeError('seeds must be integers')
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if not (0 <= x < 256 and 0 <= y < 256 and 0 <= z < 256):
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raise ValueError('seeds must be in range(0, 256)')
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if 0 == x == y == z:
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# Initialize from current time
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import time
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t = long(time.time() * 256)
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t = int((t&0xffffff) ^ (t>>24))
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t, x = divmod(t, 256)
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t, y = divmod(t, 256)
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t, z = divmod(t, 256)
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# Zero is a poor seed, so substitute 1
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self._seed = (x or 1, y or 1, z or 1)
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def whseed(self, a=None):
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"""Seed from hashable object's hash code.
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None or no argument seeds from current time. It is not guaranteed
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that objects with distinct hash codes lead to distinct internal
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states.
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This is obsolete, provided for compatibility with the seed routine
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used prior to Python 2.1. Use the .seed() method instead.
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"""
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if a is None:
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self.__whseed()
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return
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a = hash(a)
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a, x = divmod(a, 256)
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a, y = divmod(a, 256)
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a, z = divmod(a, 256)
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x = (x + a) % 256 or 1
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y = (y + a) % 256 or 1
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z = (z + a) % 256 or 1
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self.__whseed(x, y, z)
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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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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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## -------------------- integer methods -------------------
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def randrange(self, start, stop=None, step=1, int=int, default=None):
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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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Do not supply the 'int' and 'default' arguments.
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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 default:
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if istart > 0:
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return int(self.random() * istart)
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raise ValueError, "empty range for randrange()"
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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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if step == 1:
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if istart < istop:
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return istart + int(self.random() *
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(istop - istart))
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raise ValueError, "empty range for randrange()"
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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 = (istop - istart + istep - 1) / istep
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elif istep < 0:
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n = (istop - istart + 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*int(self.random() * 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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(Deprecated; use randrange(a, b+1).)
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"""
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return self.randrange(a, b+1)
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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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return seq[int(self.random() * len(seq))]
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def shuffle(self, x, random=None, int=int):
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"""x, random=random.random -> shuffle list x in place; return None.
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Optional arg random is a 0-argument function returning a random
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float in [0.0, 1.0); by default, the standard random.random.
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Note that for even rather small len(x), the total number of
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permutations of x is larger than the period of most random number
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generators; this implies that "most" permutations of a long
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sequence can never be generated.
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"""
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if random is None:
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random = self.random
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for i in xrange(len(x)-1, 0, -1):
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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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## -------------------- 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)."""
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return a + (b-a) * self.random()
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## -------------------- normal distribution --------------------
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def normalvariate(self, mu, sigma):
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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 = 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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return _exp(self.normalvariate(mu, sigma))
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## -------------------- circular uniform --------------------
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def cunifvariate(self, mean, arc):
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# mean: mean angle (in radians between 0 and pi)
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# arc: range of distribution (in radians between 0 and pi)
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return (mean + arc * (self.random() - 0.5)) % _pi
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## -------------------- exponential distribution --------------------
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def expovariate(self, lambd):
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# lambd: rate lambd = 1/mean
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# ('lambda' is a Python reserved word)
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random = self.random
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u = random()
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while u <= 1e-7:
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u = random()
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return -_log(u)/lambd
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## -------------------- von Mises distribution --------------------
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def vonmisesvariate(self, mu, kappa):
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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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a = 1.0 + _sqrt(1.0 + 4.0 * kappa * kappa)
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b = (a - _sqrt(2.0 * a))/(2.0 * kappa)
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r = (1.0 + b * b)/(2.0 * b)
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while 1:
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u1 = random()
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z = _cos(_pi * u1)
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f = (1.0 + r * z)/(r + z)
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c = kappa * (r - f)
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u2 = random()
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if not (u2 >= c * (2.0 - c) and u2 > c * _exp(1.0 - c)):
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break
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u3 = random()
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if u3 > 0.5:
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theta = (mu % TWOPI) + _acos(f)
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else:
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theta = (mu % TWOPI) - _acos(f)
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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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# beta times standard gamma
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ainv = _sqrt(2.0 * alpha - 1.0)
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return beta * self.stdgamma(alpha, ainv, alpha - LOG4, alpha + ainv)
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def stdgamma(self, alpha, ainv, bbb, ccc):
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# ainv = sqrt(2 * alpha - 1)
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# bbb = alpha - log(4)
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# ccc = alpha + ainv
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random = self.random
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if alpha <= 0.0:
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raise ValueError, 'stdgamma: alpha must be > 0.0'
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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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while 1:
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u1 = random()
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u2 = random()
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v = _log(u1/(1.0-u1))/ainv
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x = alpha*_exp(v)
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z = u1*u1*u2
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r = bbb+ccc*v-x
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if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= _log(z):
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return x
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elif alpha == 1.0:
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# expovariate(1)
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u = random()
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while u <= 1e-7:
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u = random()
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return -_log(u)
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else: # alpha is between 0 and 1 (exclusive)
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# Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
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while 1:
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u = random()
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b = (_e + alpha)/_e
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p = b*u
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if p <= 1.0:
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x = pow(p, 1.0/alpha)
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else:
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# p > 1
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x = -_log((b-p)/alpha)
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u1 = random()
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if not (((p <= 1.0) and (u1 > _exp(-x))) or
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((p > 1) and (u1 > pow(x, alpha - 1.0)))):
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break
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return x
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## -------------------- Gauss (faster alternative) --------------------
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def gauss(self, mu, sigma):
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# When x and y are two variables from [0, 1), uniformly
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# distributed, then
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#
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# cos(2*pi*x)*sqrt(-2*log(1-y))
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# sin(2*pi*x)*sqrt(-2*log(1-y))
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#
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# are two *independent* variables with normal distribution
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# (mu = 0, sigma = 1).
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# (Lambert Meertens)
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# (corrected version; bug discovered by Mike Miller, fixed by LM)
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# Multithreading note: When two threads call this function
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# simultaneously, it is possible that they will receive the
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# same return value. The window is very small though. To
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# avoid this, you have to use a lock around all calls. (I
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# didn't want to slow this down in the serial case by using a
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# lock here.)
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random = self.random
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z = self.gauss_next
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self.gauss_next = None
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if z is None:
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x2pi = random() * TWOPI
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g2rad = _sqrt(-2.0 * _log(1.0 - random()))
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z = _cos(x2pi) * g2rad
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self.gauss_next = _sin(x2pi) * g2rad
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return mu + z*sigma
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## -------------------- beta --------------------
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## See
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## http://sourceforge.net/bugs/?func=detailbug&bug_id=130030&group_id=5470
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## for Ivan Frohne's insightful analysis of why the original implementation:
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##
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## def betavariate(self, alpha, beta):
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## # Discrete Event Simulation in C, pp 87-88.
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##
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## y = self.expovariate(alpha)
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## z = self.expovariate(1.0/beta)
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## return z/(y+z)
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##
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## was dead wrong, and how it probably got that way.
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def betavariate(self, alpha, beta):
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# This version due to Janne Sinkkonen, and matches all the std
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# texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
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y = self.gammavariate(alpha, 1.)
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if y == 0:
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return 0.0
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else:
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return y / (y + self.gammavariate(beta, 1.))
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## -------------------- Pareto --------------------
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def paretovariate(self, alpha):
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# Jain, pg. 495
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u = self.random()
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return 1.0 / pow(u, 1.0/alpha)
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## -------------------- Weibull --------------------
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def weibullvariate(self, alpha, beta):
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# Jain, pg. 499; bug fix courtesy Bill Arms
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u = self.random()
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return alpha * pow(-_log(u), 1.0/beta)
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## -------------------- test program --------------------
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def _test_generator(n, funccall):
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import time
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print n, 'times', funccall
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code = compile(funccall, funccall, 'eval')
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sum = 0.0
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sqsum = 0.0
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smallest = 1e10
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largest = -1e10
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t0 = time.time()
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for i in range(n):
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x = eval(code)
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sum = sum + x
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sqsum = sqsum + x*x
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smallest = min(x, smallest)
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largest = max(x, largest)
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t1 = time.time()
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print round(t1-t0, 3), 'sec,',
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avg = sum/n
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stddev = _sqrt(sqsum/n - avg*avg)
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print 'avg %g, stddev %g, min %g, max %g' % \
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(avg, stddev, smallest, largest)
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def _test(N=200):
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print 'TWOPI =', TWOPI
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print 'LOG4 =', LOG4
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print 'NV_MAGICCONST =', NV_MAGICCONST
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print 'SG_MAGICCONST =', SG_MAGICCONST
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_test_generator(N, 'random()')
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_test_generator(N, 'normalvariate(0.0, 1.0)')
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_test_generator(N, 'lognormvariate(0.0, 1.0)')
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_test_generator(N, 'cunifvariate(0.0, 1.0)')
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_test_generator(N, 'expovariate(1.0)')
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_test_generator(N, 'vonmisesvariate(0.0, 1.0)')
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_test_generator(N, 'gammavariate(0.5, 1.0)')
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_test_generator(N, 'gammavariate(0.9, 1.0)')
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_test_generator(N, 'gammavariate(1.0, 1.0)')
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_test_generator(N, 'gammavariate(2.0, 1.0)')
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_test_generator(N, 'gammavariate(20.0, 1.0)')
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_test_generator(N, 'gammavariate(200.0, 1.0)')
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_test_generator(N, 'gauss(0.0, 1.0)')
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_test_generator(N, 'betavariate(3.0, 3.0)')
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_test_generator(N, 'paretovariate(1.0)')
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_test_generator(N, 'weibullvariate(1.0, 1.0)')
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# Test jumpahead.
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s = getstate()
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jumpahead(N)
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r1 = random()
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# now do it the slow way
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setstate(s)
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for i in range(N):
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random()
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r2 = random()
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if r1 != r2:
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raise ValueError("jumpahead test failed " + `(N, r1, r2)`)
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# Create one instance, seeded from current time, and export its methods
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# as module-level functions. The functions are not threadsafe, and state
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# is shared across all uses (both in the user's code and in the Python
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# libraries), but that's fine for most programs and is easier for the
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# casual user than making them instantiate their own Random() instance.
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_inst = Random()
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seed = _inst.seed
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random = _inst.random
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uniform = _inst.uniform
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randint = _inst.randint
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choice = _inst.choice
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randrange = _inst.randrange
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shuffle = _inst.shuffle
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normalvariate = _inst.normalvariate
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lognormvariate = _inst.lognormvariate
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cunifvariate = _inst.cunifvariate
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expovariate = _inst.expovariate
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vonmisesvariate = _inst.vonmisesvariate
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gammavariate = _inst.gammavariate
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stdgamma = _inst.stdgamma
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gauss = _inst.gauss
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betavariate = _inst.betavariate
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paretovariate = _inst.paretovariate
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weibullvariate = _inst.weibullvariate
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getstate = _inst.getstate
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setstate = _inst.setstate
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jumpahead = _inst.jumpahead
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whseed = _inst.whseed
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if __name__ == '__main__':
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_test()
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