"""Random variable generators.
distributions on the real line:
------------------------------
normal (Gaussian)
lognormal
negative exponential
gamma
beta
distributions on the circle (angles 0 to 2pi)
---------------------------------------------
circular uniform
von Mises
Translated from anonymously contributed C/C++ source.
Multi-threading note: the random number generator used here is not
thread-safe; it is possible that two calls return the same random
value. See whrandom.py for more info.
"""
import whrandom
from whrandom import random,uniform,randint,choice,randrange# For export!
frommathlog,exp,pi,e,sqrt,acos,cos,sin
# Interfaces to replace remaining needs for importing whrandom
# XXX TO DO: make the distribution functions below into methods.
def makeseed(a=None):
"""Turn a hashable value into three seed values for whrandom.seed().
None or no argument returns (0, 0, 0), to seed from current time.
"""
if a is None:
return (0, 0, 0)
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
return (x, y, z)
def seed(a=None):
"""Seed the default generator from any hashable value.
None or no argument returns (0, 0, 0) to seed from current time.
"""
x, y, z = makeseed(a)
whrandom.seed(x, y, z)
class generator(whrandom.whrandom):
"""Random generator class."""
def __init__(self, a=None):
"""Constructor. Seed from current time or hashable value."""
self.seed(a)
def seed(self, a=None):
"""Seed the generator from current time or hashable value."""
x, y, z = makeseed(a)
whrandom.whrandom.seed(self, x, y, z)
def new_generator(a=None):
"""Return a new random generator instance."""
return generator(a)
# Housekeeping function to verify that magic constants have been
# computed correctly
def verify(name, expected):
computed = eval(name)
if abs(computed - expected) > 1e-7:
raise ValueError, \
'computed value for %s deviates too much (computed %g, expected %g)' % \
(name, computed, expected)
# -------------------- normal distribution --------------------
NV_MAGICCONST = 4*exp(-0.5)/sqrt(2.0)
verify('NV_MAGICCONST', 1.71552776992141)
def normalvariate(mu, sigma):
# 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.
while 1:
u1 = random()
u2 = random()
z = NV_MAGICCONST*(u1-0.5)/u2
zz = z*z/4.0
if zz <= -log(u2):
break
return mu+z*sigma
# -------------------- lognormal distribution --------------------
def lognormvariate(mu, sigma):
return exp(normalvariate(mu, sigma))
# -------------------- circular uniform --------------------
def cunifvariate(mean, arc):
# mean: mean angle (in radians between 0 and pi)
# arc: range of distribution (in radians between 0 and pi)
return (mean + arc * (random() - 0.5)) % pi
# -------------------- exponential distribution --------------------
def expovariate(lambd):
# lambd: rate lambd = 1/mean
# ('lambda' is a Python reserved word)
u = random()
while u <= 1e-7:
u = random()
return -log(u)/lambd
# -------------------- von Mises distribution --------------------
TWOPI = 2.0*pi
verify('TWOPI', 6.28318530718)
def vonmisesvariate(mu, kappa):
# 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.
if kappa <= 1e-6:
return TWOPI * random()
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)
while 1:
u1 = random()
z = cos(pi * u1)
f = (1.0 + r * z)/(r + z)
c = kappa * (r - f)
u2 = random()
if not (u2 >= c * (2.0 - c) and u2 > c * exp(1.0 - c)):
break
u3 = random()
if u3 > 0.5:
theta = (mu % TWOPI) + acos(f)
else:
theta = (mu % TWOPI) - acos(f)
return theta
# -------------------- gamma distribution --------------------
LOG4 = log(4.0)
verify('LOG4', 1.38629436111989)
def gammavariate(alpha, beta):
# beta times standard gamma
ainv = sqrt(2.0 * alpha - 1.0)
return beta * stdgamma(alpha, ainv, alpha - LOG4, alpha + ainv)
SG_MAGICCONST = 1.0 + log(4.5)
verify('SG_MAGICCONST', 2.50407739677627)
def stdgamma(alpha, ainv, bbb, ccc):
# ainv = sqrt(2 * alpha - 1)
# bbb = alpha - log(4)
# ccc = alpha + ainv
if alpha <= 0.0:
raise ValueError, 'stdgamma: alpha must be > 0.0'
if alpha > 1.0:
# Uses R.C.H. Cheng, "The generation of Gamma
# variables with non-integral shape parameters",
# Applied Statistics, (1977), 26, No. 1, p71-74
while 1:
u1 = random()
u2 = random()
v = log(u1/(1.0-u1))/ainv
x = alpha*exp(v)
z = u1*u1*u2
r = bbb+ccc*v-x
if r + SG_MAGICCONST - 4.5*z >= 0.0 or r >= log(z):
return x
elif alpha == 1.0:
# expovariate(1)
u = random()
while u <= 1e-7:
u = random()
return -log(u)
else: # alpha is between 0 and 1 (exclusive)
# Uses ALGORITHM GS of Statistical Computing - Kennedy & Gentle
while 1:
u = random()
b = (e + alpha)/e
p = b*u
if p <= 1.0:
x = pow(p, 1.0/alpha)
else:
# p > 1
x = -log((b-p)/alpha)
u1 = random()
if not (((p <= 1.0) and (u1 > exp(-x))) or
((p > 1) and (u1 > pow(x, alpha - 1.0)))):
break
return x
# -------------------- Gauss (faster alternative) --------------------
gauss_next = None
def gauss(mu, sigma):
# When x and y are two variables from [0, 1), uniformly
# distributed, then
#
# cos(2*pi*x)*sqrt(-2*log(1-y))
# sin(2*pi*x)*sqrt(-2*log(1-y))
#
# are two *independent* variables with normal distribution
# (mu = 0, sigma = 1).
# (Lambert Meertens)
# (corrected version; bug discovered by Mike Miller, fixed by LM)
# Multithreading note: When two threads call this function
# simultaneously, it is possible that they will receive the
# same return value. The window is very small though. To
# avoid this, you have to use a lock around all calls. (I
# didn't want to slow this down in the serial case by using a
# lock here.)
global gauss_next
z = gauss_next
gauss_next = None
if z is None:
x2pi = random() * TWOPI
g2rad = sqrt(-2.0 * log(1.0 - random()))
z = cos(x2pi) * g2rad
gauss_next = sin(x2pi) * g2rad
return mu + z*sigma
# -------------------- beta --------------------
def betavariate(alpha, beta):
# Discrete Event Simulation in C, pp 87-88.
y = expovariate(alpha)
z = expovariate(1.0/beta)
return z/(y+z)
# -------------------- Pareto --------------------
def paretovariate(alpha):
# Jain, pg. 495
u = random()
return 1.0 / pow(u, 1.0/alpha)
# -------------------- Weibull --------------------
def weibullvariate(alpha, beta):
# Jain, pg. 499; bug fix courtesy Bill Arms
u = random()
return alpha * pow(-log(u), 1.0/beta)
# -------------------- shuffle --------------------
# Not quite a random distribution, but a standard algorithm.
# This implementation due to Tim Peters.
def shuffle(x, random=random, 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 xrange(len(x)-1, 0, -1):
# pick an element in x[:i+1] with which to exchange x[i]
j = int(random() * (i+1))
x[i], x[j] = x[j], x[i]
# -------------------- test program --------------------
def test(N = 200):
print 'TWOPI =', TWOPI
print 'LOG4 =', LOG4
print 'NV_MAGICCONST =', NV_MAGICCONST
print 'SG_MAGICCONST =', SG_MAGICCONST
test_generator(N, 'random()')
test_generator(N, 'normalvariate(0.0, 1.0)')
test_generator(N, 'lognormvariate(0.0, 1.0)')
test_generator(N, 'cunifvariate(0.0, 1.0)')
test_generator(N, 'expovariate(1.0)')
test_generator(N, 'vonmisesvariate(0.0, 1.0)')
test_generator(N, 'gammavariate(0.5, 1.0)')
test_generator(N, 'gammavariate(0.9, 1.0)')
test_generator(N, 'gammavariate(1.0, 1.0)')
test_generator(N, 'gammavariate(2.0, 1.0)')
test_generator(N, 'gammavariate(20.0, 1.0)')
test_generator(N, 'gammavariate(200.0, 1.0)')
test_generator(N, 'gauss(0.0, 1.0)')
test_generator(N, 'betavariate(3.0, 3.0)')
test_generator(N, 'paretovariate(1.0)')
test_generator(N, 'weibullvariate(1.0, 1.0)')
def test_generator(n, funccall):
import time
print n, 'times', funccall
code = compile(funccall, funccall, 'eval')
sum = 0.0
sqsum = 0.0
smallest = 1e10
largest = -1e10
t0 = time.time()
for i in range(n):
x = eval(code)
sum = sum + x
sqsum = sqsum + x*x
smallest = min(x, smallest)
largest = max(x, largest)
t1 = time.time()
print round(t1-t0, 3), 'sec,',
avg = sum/n
stddev = sqrt(sqsum/n - avg*avg)
print 'avg %g, stddev %g, min %g, max %g' % \
(avg, stddev, smallest, largest)
if __name__ == '__main__':
test()
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