3081d59f92
The default seed is time.time(). Multiplied by 256 before truncating so that fractional seconds are used. This way, two successive calls to random.seed() are much more likely to produce different sequences.
760 lines
25 KiB
Python
760 lines
25 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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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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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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* Without a direct way to compute N steps forward, the
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semantics of jumpahead(n) are weakened to simply jump
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to another distant state and rely on the large period
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to avoid overlapping sequences.
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* The random() method is implemented in C, executes in
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a single Python step, and is, therefore, threadsafe.
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"""
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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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from math import floor as _floor
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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",
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"gauss","betavariate","paretovariate","weibullvariate",
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"getstate","setstate","jumpahead"]
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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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# 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 core generator.
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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. Especially useful for multi-threaded programs, creating
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a different instance of Random for each thread, and using the jumpahead()
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method to ensure that the generated sequences seen by each thread don't
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overlap.
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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(), setstate() and jumpahead().
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"""
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VERSION = 2 # 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):
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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 used instead.
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"""
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if a is None:
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import time
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a = long(time.time() * 256) # use fractional seconds
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super(Random, self).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(Random, self).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 == 2:
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version, internalstate, self.gauss_next = state
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super(Random, self).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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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, 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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# 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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if step == 1 and istart < istop:
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# Note that
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# int(istart + self.random()*(istop - istart))
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# instead would be incorrect. For example, consider istart
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# = -2 and istop = 0. Then the guts would be in
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# -2.0 to 0.0 exclusive on both ends (ignoring that random()
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# might return 0.0), and because int() truncates toward 0, the
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# final result would be -1 or 0 (instead of -2 or -1).
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# istart + int(self.random()*(istop - istart))
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# would also be incorrect, for a subtler reason: the RHS
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# can return a long, and then randrange() would also return
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# a long, but we're supposed to return an int (for backward
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# compatibility).
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return int(istart + int(self.random()*(istop - istart)))
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if step == 1:
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raise ValueError, "empty range for randrange()"
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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 = (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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"""
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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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def sample(self, population, k):
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"""Chooses k unique random elements from a population sequence.
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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 xrange 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(xrange(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
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# dictionary.
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# When the number of selections is small compared to the population,
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# then tracking selections is efficient, requiring only a small
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# dictionary and an occasional reselection. For a larger number of
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# selections, the pool tracking method is preferred since the list takes
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# less space than the dictionary and it doesn't suffer from frequent
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# reselections.
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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"
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random = self.random
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_int = int
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result = [None] * k
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if n < 6 * k: # if n len list takes less space than a k len dict
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pool = list(population)
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for i in xrange(k): # invariant: non-selected at [0,n-i)
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j = _int(random() * (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 = {}
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for i in xrange(k):
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j = _int(random() * n)
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while j in selected:
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j = _int(random() * n)
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result[i] = selected[j] = population[j]
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return result
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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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"""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 True:
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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. (The parameter would be
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called "lambda", but that is a reserved word in Python.) Returned
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values range from 0 to positive infinity.
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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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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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"""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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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 True:
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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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"""Gamma distribution. Not the gamma function!
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Conditions on the parameters are alpha > 0 and beta > 0.
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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)
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bbb = alpha - LOG4
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ccc = alpha + ainv
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while True:
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u1 = random()
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if not 1e-7 < u1 < .9999999:
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continue
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u2 = 1.0 - 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 * beta
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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) * beta
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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 True:
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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 * beta
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## -------------------- Gauss (faster alternative) --------------------
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def gauss(self, mu, sigma):
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"""Gaussian distribution.
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mu is the mean, and sigma is the standard deviation. This is
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slightly faster than the normalvariate() function.
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Not thread-safe without a lock around calls.
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"""
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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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"""Beta distribution.
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Conditions on the parameters are alpha > -1 and beta} > -1.
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Returned values range between 0 and 1.
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"""
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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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"""Pareto distribution. alpha is the shape parameter."""
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# Jain, pg. 495
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u = 1.0 - 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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"""Weibull distribution.
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alpha is the scale parameter and beta is the shape parameter.
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"""
|
|
# Jain, pg. 499; bug fix courtesy Bill Arms
|
|
|
|
u = 1.0 - self.random()
|
|
return alpha * pow(-_log(u), 1.0/beta)
|
|
|
|
## -------------------- Wichmann-Hill -------------------
|
|
|
|
class WichmannHill(Random):
|
|
|
|
VERSION = 1 # used by getstate/setstate
|
|
|
|
def seed(self, a=None):
|
|
"""Initialize internal state from hashable object.
|
|
|
|
None or no argument seeds from current time.
|
|
|
|
If a is not None or an int or long, hash(a) is used instead.
|
|
|
|
If a is an int or long, a is used directly. Distinct values between
|
|
0 and 27814431486575L inclusive are guaranteed to yield distinct
|
|
internal states (this guarantee is specific to the default
|
|
Wichmann-Hill generator).
|
|
"""
|
|
|
|
if a is None:
|
|
# Initialize from current time
|
|
import time
|
|
a = long(time.time() * 256)
|
|
|
|
if not isinstance(a, (int, long)):
|
|
a = hash(a)
|
|
|
|
a, x = divmod(a, 30268)
|
|
a, y = divmod(a, 30306)
|
|
a, z = divmod(a, 30322)
|
|
self._seed = int(x)+1, int(y)+1, int(z)+1
|
|
|
|
self.gauss_next = None
|
|
|
|
def random(self):
|
|
"""Get the next random number in the range [0.0, 1.0)."""
|
|
|
|
# Wichman-Hill random number generator.
|
|
#
|
|
# Wichmann, B. A. & Hill, I. D. (1982)
|
|
# Algorithm AS 183:
|
|
# An efficient and portable pseudo-random number generator
|
|
# Applied Statistics 31 (1982) 188-190
|
|
#
|
|
# see also:
|
|
# Correction to Algorithm AS 183
|
|
# Applied Statistics 33 (1984) 123
|
|
#
|
|
# McLeod, A. I. (1985)
|
|
# A remark on Algorithm AS 183
|
|
# Applied Statistics 34 (1985),198-200
|
|
|
|
# This part is thread-unsafe:
|
|
# BEGIN CRITICAL SECTION
|
|
x, y, z = self._seed
|
|
x = (171 * x) % 30269
|
|
y = (172 * y) % 30307
|
|
z = (170 * z) % 30323
|
|
self._seed = x, y, z
|
|
# END CRITICAL SECTION
|
|
|
|
# Note: on a platform using IEEE-754 double arithmetic, this can
|
|
# never return 0.0 (asserted by Tim; proof too long for a comment).
|
|
return (x/30269.0 + y/30307.0 + z/30323.0) % 1.0
|
|
|
|
def getstate(self):
|
|
"""Return internal state; can be passed to setstate() later."""
|
|
return self.VERSION, self._seed, self.gauss_next
|
|
|
|
def setstate(self, state):
|
|
"""Restore internal state from object returned by getstate()."""
|
|
version = state[0]
|
|
if version == 1:
|
|
version, self._seed, self.gauss_next = state
|
|
else:
|
|
raise ValueError("state with version %s passed to "
|
|
"Random.setstate() of version %s" %
|
|
(version, self.VERSION))
|
|
|
|
def jumpahead(self, n):
|
|
"""Act as if n calls to random() were made, but quickly.
|
|
|
|
n is an int, greater than or equal to 0.
|
|
|
|
Example use: If you have 2 threads and know that each will
|
|
consume no more than a million random numbers, create two Random
|
|
objects r1 and r2, then do
|
|
r2.setstate(r1.getstate())
|
|
r2.jumpahead(1000000)
|
|
Then r1 and r2 will use guaranteed-disjoint segments of the full
|
|
period.
|
|
"""
|
|
|
|
if not n >= 0:
|
|
raise ValueError("n must be >= 0")
|
|
x, y, z = self._seed
|
|
x = int(x * pow(171, n, 30269)) % 30269
|
|
y = int(y * pow(172, n, 30307)) % 30307
|
|
z = int(z * pow(170, n, 30323)) % 30323
|
|
self._seed = x, y, z
|
|
|
|
def __whseed(self, x=0, y=0, z=0):
|
|
"""Set the Wichmann-Hill seed from (x, y, z).
|
|
|
|
These must be integers in the range [0, 256).
|
|
"""
|
|
|
|
if not type(x) == type(y) == type(z) == int:
|
|
raise TypeError('seeds must be integers')
|
|
if not (0 <= x < 256 and 0 <= y < 256 and 0 <= z < 256):
|
|
raise ValueError('seeds must be in range(0, 256)')
|
|
if 0 == x == y == z:
|
|
# Initialize from current time
|
|
import time
|
|
t = long(time.time() * 256)
|
|
t = int((t&0xffffff) ^ (t>>24))
|
|
t, x = divmod(t, 256)
|
|
t, y = divmod(t, 256)
|
|
t, z = divmod(t, 256)
|
|
# Zero is a poor seed, so substitute 1
|
|
self._seed = (x or 1, y or 1, z or 1)
|
|
|
|
self.gauss_next = None
|
|
|
|
def whseed(self, a=None):
|
|
"""Seed from hashable object's hash code.
|
|
|
|
None or no argument seeds from current time. It is not guaranteed
|
|
that objects with distinct hash codes lead to distinct internal
|
|
states.
|
|
|
|
This is obsolete, provided for compatibility with the seed routine
|
|
used prior to Python 2.1. Use the .seed() method instead.
|
|
"""
|
|
|
|
if a is None:
|
|
self.__whseed()
|
|
return
|
|
a = hash(a)
|
|
a, x = divmod(a, 256)
|
|
a, y = divmod(a, 256)
|
|
a, z = divmod(a, 256)
|
|
x = (x + a) % 256 or 1
|
|
y = (y + a) % 256 or 1
|
|
z = (z + a) % 256 or 1
|
|
self.__whseed(x, y, z)
|
|
|
|
## -------------------- test program --------------------
|
|
|
|
def _test_generator(n, funccall):
|
|
import time
|
|
print n, 'times', funccall
|
|
code = compile(funccall, funccall, 'eval')
|
|
total = 0.0
|
|
sqsum = 0.0
|
|
smallest = 1e10
|
|
largest = -1e10
|
|
t0 = time.time()
|
|
for i in range(n):
|
|
x = eval(code)
|
|
total += x
|
|
sqsum = sqsum + x*x
|
|
smallest = min(x, smallest)
|
|
largest = max(x, largest)
|
|
t1 = time.time()
|
|
print round(t1-t0, 3), 'sec,',
|
|
avg = total/n
|
|
stddev = _sqrt(sqsum/n - avg*avg)
|
|
print 'avg %g, stddev %g, min %g, max %g' % \
|
|
(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)')
|
|
|
|
# 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
|
|
randint = _inst.randint
|
|
choice = _inst.choice
|
|
randrange = _inst.randrange
|
|
sample = _inst.sample
|
|
shuffle = _inst.shuffle
|
|
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
|
|
jumpahead = _inst.jumpahead
|
|
|
|
if __name__ == '__main__':
|
|
_test()
|