"""Random variable generators.
generate random permutation
distributions on the real line:
------------------------------
distributions on the circle (angles 0 to 2pi)
---------------------------------------------
General notes on the underlying Mersenne Twister core generator:
* The period is 2**19937-1.
* It is one of the most extensively tested generators in existence.
* Without a direct way to compute N steps forward, the semantics of
jumpahead(n) are weakened to simply jump to another distant state and rely
on the large period to avoid overlapping sequences.
* The random() method is implemented in C, executes in a single Python step,
and is, therefore, threadsafe.
from __future__ import division
from warnings import warn as _warn
from types import MethodType as _MethodType, BuiltinMethodType as _BuiltinMethodType
from math import log as _log, exp as _exp, pi as _pi, e as _e, ceil as _ceil
from math import sqrt as _sqrt, acos as _acos, cos as _cos, sin as _sin
from os import urandom as _urandom
from binascii import hexlify as _hexlify
import hashlib as _hashlib
__all__ = ["Random","seed","random","uniform","randint","choice","sample",
"randrange","shuffle","normalvariate","lognormvariate",
"expovariate","vonmisesvariate","gammavariate","triangular",
"gauss","betavariate","paretovariate","weibullvariate",
"getstate","setstate","jumpahead", "WichmannHill", "getrandbits",
NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0)
SG_MAGICCONST = 1.0 + _log(4.5)
BPF = 53 # Number of bits in a float
# Translated by Guido van Rossum from C source provided by
# Adrian Baddeley. Adapted by Raymond Hettinger for use with
# the Mersenne Twister and os.urandom() core generators.
class Random(_random.Random):
"""Random number generator base class used by bound module functions.
Used to instantiate instances of Random to get generators that don't
share state. Especially useful for multi-threaded programs, creating
a different instance of Random for each thread, and using the jumpahead()
method to ensure that the generated sequences seen by each thread don't
Class Random can also be subclassed if you want to use a different basic
generator of your own devising: in that case, override the following
methods: random(), seed(), getstate(), setstate() and jumpahead().
Optionally, implement a getrandbits() method so that randrange() can cover
arbitrarily large ranges.
VERSION = 3 # used by getstate/setstate
def __init__(self, x=None):
"""Initialize an instance.
Optional argument x controls seeding, as for Random.seed().
"""Initialize internal state from hashable object.
None or no argument seeds from current time or from an operating
system specific randomness source if available.
If a is not None or an int or long, hash(a) is used instead.
# Seed with enough bytes to span the 19937 bit
# state space for the Mersenne Twister
a = long(_hexlify(_urandom(2500)), 16)
except NotImplementedError:
a = long(time.time() * 256) # use fractional seconds
super(Random, self).seed(a)
"""Return internal state; can be passed to setstate() later."""
return self.VERSION, super(Random, self).getstate(), self.gauss_next
def setstate(self, state):
"""Restore internal state from object returned by getstate()."""
version, internalstate, self.gauss_next = state
super(Random, self).setstate(internalstate)
version, internalstate, self.gauss_next = state
# In version 2, the state was saved as signed ints, which causes
# inconsistencies between 32/64-bit systems. The state is
# really unsigned 32-bit ints, so we convert negative ints from
# version 2 to positive longs for version 3.
internalstate = tuple( long(x) % (2**32) for x in internalstate )
super(Random, self).setstate(internalstate)
raise ValueError("state with version %s passed to "
"Random.setstate() of version %s" %
"""Change the internal state to one that is likely far away
from the current state. This method will not be in Py3.x,
so it is better to simply reseed.
# The super.jumpahead() method uses shuffling to change state,
# so it needs a large and "interesting" n to work with. Here,
# we use hashing to create a large n for the shuffle.
s = repr(n) + repr(self.getstate())
n = int(_hashlib.new('sha512', s).hexdigest(), 16)
super(Random, self).jumpahead(n)
## ---- Methods below this point do not need to be overridden when
## ---- subclassing for the purpose of using a different core generator.
## -------------------- pickle support -------------------
def __getstate__(self): # for pickle
def __setstate__(self, state): # for pickle
return self.__class__, (), self.getstate()
## -------------------- integer methods -------------------
def randrange(self, start, stop=None, step=1, _int=int, _maxwidth=1L<<BPF):
"""Choose a random item from range(start, stop[, step]).
This fixes the problem with randint() which includes the
endpoint; in Python this is usually not what you want.
# This code is a bit messy to make it fast for the
# common case while still doing adequate error checking.
raise ValueError, "non-integer arg 1 for randrange()"
return self._randbelow(istart)
return _int(self.random() * istart)
raise ValueError, "empty range for randrange()"
# stop argument supplied.
raise ValueError, "non-integer stop for randrange()"
if step == 1 and width > 0:
# int(istart + self.random()*width)
# instead would be incorrect. For example, consider istart
# = -2 and istop = 0. Then the guts would be in
# -2.0 to 0.0 exclusive on both ends (ignoring that random()
# might return 0.0), and because int() truncates toward 0, the
# final result would be -1 or 0 (instead of -2 or -1).
# istart + int(self.random()*width)
# would also be incorrect, for a subtler reason: the RHS
# can return a long, and then randrange() would also return
# a long, but we're supposed to return an int (for backward
return _int(istart + self._randbelow(width))
return _int(istart + _int(self.random()*width))
raise ValueError, "empty range for randrange() (%d,%d, %d)" % (istart, istop, width)
# Non-unit step argument supplied.
raise ValueError, "non-integer step for randrange()"
n = (width + istep - 1) // istep
n = (width + istep + 1) // istep
raise ValueError, "zero step for randrange()"
raise ValueError, "empty range for randrange()"
return istart + istep*self._randbelow(n)
return istart + istep*_int(self.random() * n)
"""Return random integer in range [a, b], including both end points.
return self.randrange(a, b+1)
def _randbelow(self, n, _log=_log, _int=int, _maxwidth=1L<<BPF,
_Method=_MethodType, _BuiltinMethod=_BuiltinMethodType):
"""Return a random int in the range [0,n)
Handles the case where n has more bits than returned
by a single call to the underlying generator.
getrandbits = self.getrandbits
# Only call self.getrandbits if the original random() builtin method
# has not been overridden or if a new getrandbits() was supplied.
# This assures that the two methods correspond.
if type(self.random) is _BuiltinMethod or type(getrandbits) is _Method:
k = _int(1.00001 + _log(n-1, 2.0)) # 2**k > n-1 > 2**(k-2)
_warn("Underlying random() generator does not supply \n"
"enough bits to choose from a population range this large")
return _int(self.random() * n)
## -------------------- sequence methods -------------------
"""Choose a random element from a non-empty sequence."""
return seq[int(self.random() * len(seq))] # raises IndexError if seq is empty
def shuffle(self, x, random=None):
"""x, random=random.random -> shuffle list x in place; return None.
Optional arg random is a 0-argument function returning a random
float in [0.0, 1.0); by default, the standard random.random.
for i in reversed(xrange(1, len(x))):
# pick an element in x[:i+1] with which to exchange x[i]
j = _int(random() * (i+1))
def sample(self, population, k):
"""Chooses k unique random elements from a population sequence.
Returns a new list containing elements from the population while
leaving the original population unchanged. The resulting list is
in selection order so that all sub-slices will also be valid random
samples. This allows raffle winners (the sample) to be partitioned
into grand prize and second place winners (the subslices).
Members of the population need not be hashable or unique. If the
population contains repeats, then each occurrence is a possible
To choose a sample in a range of integers, use xrange as an argument.
This is especially fast and space efficient for sampling from a
large population: sample(xrange(10000000), 60)
# Sampling without replacement entails tracking either potential
# selections (the pool) in a list or previous selections in a set.
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small set and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# set and it doesn't suffer from frequent reselections.
raise ValueError("sample larger than population")
setsize = 21 # size of a small set minus size of an empty list
setsize += 4 ** _ceil(_log(k * 3, 4)) # table size for big sets
if n <= setsize or hasattr(population, "keys"):
# An n-length list is smaller than a k-length set, or this is a
# mapping type so the other algorithm wouldn't work.
for i in xrange(k): # invariant: non-selected at [0,n-i)
j = _int(random() * (n-i))
pool[j] = pool[n-i-1] # move non-selected item into vacancy
selected_add = selected.add
result[i] = population[j]
except (TypeError, KeyError): # handle (at least) sets
if isinstance(population, list):
return self.sample(tuple(population), k)
## -------------------- real-valued distributions -------------------
## -------------------- uniform distribution -------------------
"Get a random number in the range [a, b) or [a, b] depending on rounding."
return a + (b-a) * self.random()
## -------------------- triangular --------------------
def triangular(self, low=0.0, high=1.0, mode=None):
"""Triangular distribution.
Continuous distribution bounded by given lower and upper limits,
and having a given mode value in-between.
http://en.wikipedia.org/wiki/Triangular_distribution
c = 0.5 if mode is None else (mode - low) / (high - low)
except ZeroDivisionError:
return low + (high - low) * (u * c) ** 0.5
## -------------------- normal distribution --------------------
def normalvariate(self, mu, sigma):
mu is the mean, and sigma is the standard deviation.
# mu = mean, sigma = standard deviation
# Uses Kinderman and Monahan method. Reference: Kinderman,
# A.J. and Monahan, J.F., "Computer generation of random
# variables using the ratio of uniform deviates", ACM Trans
# Math Software, 3, (1977), pp257-260.
z = NV_MAGICCONST*(u1-0.5)/u2
## -------------------- lognormal distribution --------------------
def lognormvariate(self, mu, sigma):
"""Log normal distribution.
If you take the natural logarithm of this distribution, you'll get a
normal distribution with mean mu and standard deviation sigma.
mu can have any value, and sigma must be greater than zero.
return _exp(self.normalvariate(mu, sigma))
## -------------------- exponential distribution --------------------
def expovariate(self, lambd):
"""Exponential distribution.
lambd is 1.0 divided by the desired mean. It should be
nonzero. (The parameter would be called "lambda", but that is
a reserved word in Python.) Returned values range from 0 to
positive infinity if lambd is positive, and from negative
infinity to 0 if lambd is negative.
# lambd: rate lambd = 1/mean
# ('lambda' is a Python reserved word)
# we use 1-random() instead of random() to preclude the
# possibility of taking the log of zero.
return -_log(1.0 - self.random())/lambd
## -------------------- von Mises distribution --------------------
def vonmisesvariate(self, mu, kappa):
"""Circular data distribution.
mu is the mean angle, expressed in radians between 0 and 2*pi, and
kappa is the concentration parameter, which must be greater than or
equal to zero. If kappa is equal to zero, this distribution reduces
to a uniform random angle over the range 0 to 2*pi.
# mu: mean angle (in radians between 0 and 2*pi)
# kappa: concentration parameter kappa (>= 0)
# if kappa = 0 generate uniform random angle
# Based upon an algorithm published in: Fisher, N.I.,
# "Statistical Analysis of Circular Data", Cambridge
# University Press, 1993.
# Thanks to Magnus Kessler for a correction to the
# implementation of step 4.
r = s + _sqrt(1.0 + s * s)
if u2 < 1.0 - d * d or u2 <= (1.0 - d) * _exp(d):
f = (q + z) / (1.0 + q * z)
theta = (mu + _acos(f)) % TWOPI
theta = (mu - _acos(f)) % TWOPI
## -------------------- gamma distribution --------------------
def gammavariate(self, alpha, beta):
"""Gamma distribution. Not the gamma function!
Conditions on the parameters are alpha > 0 and beta > 0.
The probability distribution function is:
x ** (alpha - 1) * math.exp(-x / beta)
pdf(x) = --------------------------------------
math.gamma(alpha) * beta ** alpha
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