"""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 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
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
__all__
= ["Random","seed","random","uniform","randint","choice","sample",
"randrange","shuffle","normalvariate","lognormvariate",
"expovariate","vonmisesvariate","gammavariate",
"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 getrandombits() method so that randrange()
can cover arbitrarily large ranges.
VERSION
= 2 # 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.
a
= long(_hexlify(_urandom(16)), 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
)
raise ValueError("state with version %s passed to "
"Random.setstate() of version %s" %
## ---- 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, default
=None,
"""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.
Do not supply the 'int', 'default', and 'maxwidth' arguments.
# 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
+ 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, int=int):
"""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.
Note that for even rather small len(x), the total number of
permutations of x is larger than the period of most random number
generators; this implies that "most" permutations of a long
sequence can never be generated.
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
# When the number of selections is small compared to the
# population, then tracking selections is efficient, requiring
# only a small dictionary and an occasional reselection. For
# a larger number of selections, the pool tracking method is
# preferred since the list takes less space than the
# dictionary and it doesn't suffer from frequent reselections.
raise ValueError, "sample larger than population"
if n
< 6 * k
: # if n len list takes less space than a k len dict
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
n
> 0 and (population
[0], population
[n
//2], population
[n
-1])
except (TypeError, KeyError): # handle sets and dictionaries
population
= tuple(population
)
result
[i
] = selected
[j
] = population
[j
]
## -------------------- real-valued distributions -------------------
## -------------------- uniform distribution -------------------
"""Get a random number in the range [a, b)."""
return a
+ (b
-a
) * self
.random()
## -------------------- 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. (The parameter would be
called "lambda", but that is a reserved word in Python.) Returned
values range from 0 to positive infinity.
# lambd: rate lambd = 1/mean
# ('lambda' is a Python reserved word)
## -------------------- 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.
a
= 1.0 + _sqrt(1.0 + 4.0 * kappa
* kappa
)
b
= (a
- _sqrt(2.0 * a
))/(2.0 * kappa
)
r
= (1.0 + b
* b
)/(2.0 * b
)
f
= (1.0 + r
* z
)/(r
+ z
)
if u2
< c
* (2.0 - c
) or u2
<= c
* _exp(1.0 - c
):
theta
= (mu
% TWOPI
) + _acos(f
)
theta
= (mu
% TWOPI
) - _acos(f
)
## -------------------- gamma distribution --------------------
def gammavariate(self
, alpha
, beta
):
"""Gamma distribution. Not the gamma function!
Conditions on the parameters are alpha > 0 and beta > 0.
# alpha > 0, beta > 0, mean is alpha*beta, variance is alpha*beta**2
# Warning: a few older sources define the gamma distribution in terms
if alpha
<= 0.0 or beta
<= 0.0:
raise ValueError, 'gammavariate: alpha and beta must be > 0.0'
# Uses R.C.H. Cheng, "The generation of Gamma
# variables with non-integral shape parameters",
# Applied Statistics, (1977), 26, No. 1, p71-74
ainv
= _sqrt(2.0 * alpha
- 1.0)
if not 1e-7 < u1
< .9999999:
v
= _log(u1
/(1.0-u1
))/ainv
if r
+ SG_MAGICCONST
- 4.5*z
>= 0.0 or r
>= _log(z
):
else: # alpha is between 0 and 1 (exclusive)
# Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
if u1
<= x
** (alpha
- 1.0):
## -------------------- Gauss (faster alternative) --------------------
def gauss(self
, mu
, sigma
):
"""Gaussian distribution.
mu is the mean, and sigma is the standard deviation. This is
slightly faster than the normalvariate() function.
Not thread-safe without a lock around calls.
# When x and y are two variables from [0, 1), uniformly
# cos(2*pi*x)*sqrt(-2*log(1-y))
# sin(2*pi*x)*sqrt(-2*log(1-y))
# are two *independent* variables with normal distribution
# (corrected version; bug discovered by Mike Miller, fixed by LM)
# Multithreading note: When two threads call this function
# simultaneously, it is possible that they will receive the
# same return value. The window is very small though. To
# avoid this, you have to use a lock around all calls. (I
# didn't want to slow this down in the serial case by using a
g2rad
= _sqrt(-2.0 * _log(1.0 - random()))
self
.gauss_next
= _sin(x2pi
) * g2rad
## -------------------- beta --------------------
## http://sourceforge.net/bugs/?func=detailbug&bug_id=130030&group_id=5470
## for Ivan Frohne's insightful analysis of why the original implementation:
## def betavariate(self, alpha, beta):
## # Discrete Event Simulation in C, pp 87-88.
## y = self.expovariate(alpha)
## z = self.expovariate(1.0/beta)
## was dead wrong, and how it probably got that way.
def betavariate(self
, alpha
, beta
):
Conditions on the parameters are alpha > -1 and beta} > -1.
Returned values range between 0 and 1.
# This version due to Janne Sinkkonen, and matches all the std
# texts (e.g., Knuth Vol 2 Ed 3 pg 134 "the beta distribution").
y
= self
.gammavariate(alpha
, 1.)
return y
/ (y
+ self
.gammavariate(beta
, 1.))
## -------------------- Pareto --------------------
def paretovariate(self
, alpha
):
"""Pareto distribution. alpha is the shape parameter."""
return 1.0 / pow(u
, 1.0/alpha
)
## -------------------- Weibull --------------------
def weibullvariate(self
, alpha
, beta
):
alpha is the scale parameter and beta is the shape parameter.
# Jain, pg. 499; bug fix courtesy Bill Arms
return alpha
* pow(-_log(u
), 1.0/beta
)
## -------------------- Wichmann-Hill -------------------
class WichmannHill(Random
):
VERSION
= 1 # used by getstate/setstate
"""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.
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).
a
= long(_hexlify(_urandom(16)), 16)
except NotImplementedError:
a
= long(time
.time() * 256) # use fractional seconds
if not isinstance(a
, (int, long)):
self
._seed
= int(x
)+1, int(y
)+1, int(z
)+1
"""Get the next random number in the range [0.0, 1.0)."""
# Wichman-Hill random number generator.
# Wichmann, B. A. & Hill, I. D. (1982)
# An efficient and portable pseudo-random number generator
# Applied Statistics 31 (1982) 188-190
# Correction to Algorithm AS 183
# Applied Statistics 33 (1984) 123
# A remark on Algorithm AS 183
# Applied Statistics 34 (1985),198-200
# This part is thread-unsafe:
# 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
"""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
, self
._seed
, self
.gauss_next
= state
raise ValueError("state with version %s passed to "
"Random.setstate() of version %s" %
"""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())
Then r1 and r2 will use guaranteed-disjoint segments of the full
raise ValueError("n must be >= 0")
x
= int(x
* pow(171, n
, 30269)) % 30269
y
= int(y
* pow(172, n
, 30307)) % 30307
z
= int(z
* pow(170, n
, 30323)) % 30323
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)')
# Initialize from current time
t
= long(time
.time() * 256)
t
= int((t
&0xffffff) ^
(t
>>24))
# Zero is a poor seed, so substitute 1
self
._seed
= (x
or 1, y
or 1, z
or 1)
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
This is obsolete, provided for compatibility with the seed routine
used prior to Python 2.1. Use the .seed() method instead.
## --------------- Operating System Random Source ------------------
class SystemRandom(Random
):
"""Alternate random number generator using sources provided
by the operating system (such as /dev/urandom on Unix or
CryptGenRandom on Windows).
Not available on all systems (see os.urandom() for details).
"""Get the next random number in the range [0.0, 1.0)."""
return (long(_hexlify(_urandom(7)), 16) >> 3) * RECIP_BPF
def getrandbits(self
, k
):
"""getrandbits(k) -> x. Generates a long int with k random bits."""
raise ValueError('number of bits must be greater than zero')
raise TypeError('number of bits should be an integer')
bytes
= (k
+ 7) // 8 # bits / 8 and rounded up
x
= long(_hexlify(_urandom(bytes
)), 16)
return x
>> (bytes
* 8 - k
) # trim excess bits
def _stub(self
, *args
, **kwds
):
"Stub method. Not used for a system random number generator."
def _notimplemented(self
, *args
, **kwds
):
"Method should not be called for a system random number generator."
raise NotImplementedError('System entropy source does not have state.')
getstate
= setstate
= _notimplemented
## -------------------- test program --------------------
def _test_generator(n
, func
, args
):
print n
, 'times', func
.__name
__
smallest
= min(x
, smallest
)
largest
= max(x
, largest
)
print round(t1
-t0
, 3), 'sec,',
stddev
= _sqrt(sqsum
/n
- avg
*avg
)
print 'avg %g, stddev %g, min %g, max %g' % \
(avg
, stddev
, smallest
, largest
)
_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.
randrange
= _inst
.randrange
normalvariate
= _inst
.normalvariate
lognormvariate
= _inst
.lognormvariate
expovariate
= _inst
.expovariate
vonmisesvariate
= _inst
.vonmisesvariate
gammavariate
= _inst
.gammavariate
betavariate
= _inst
.betavariate
paretovariate
= _inst
.paretovariate
weibullvariate
= _inst
.weibullvariate
getstate
= _inst
.getstate
setstate
= _inst
.setstate
jumpahead
= _inst
.jumpahead
getrandbits
= _inst
.getrandbits
if __name__
== '__main__':