| """Random variable generators. |
| |
| integers |
| -------- |
| uniform within range |
| |
| sequences |
| --------- |
| pick random element |
| generate random permutation |
| |
| distributions on the real line: |
| ------------------------------ |
| uniform |
| 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. However, |
| you can instantiate a different instance of Random() in each thread to get |
| generators that don't share state, then use .setstate() and .jumpahead() to |
| move the generators to disjoint segments of the full period. For example, |
| |
| def create_generators(num, delta, firstseed=None): |
| ""\"Return list of num distinct generators. |
| Each generator has its own unique segment of delta elements from |
| Random.random()'s full period. |
| Seed the first generator with optional arg firstseed (default is |
| None, to seed from current time). |
| ""\" |
| |
| from random import Random |
| g = Random(firstseed) |
| result = [g] |
| for i in range(num - 1): |
| laststate = g.getstate() |
| g = Random() |
| g.setstate(laststate) |
| g.jumpahead(delta) |
| result.append(g) |
| return result |
| |
| gens = create_generators(10, 1000000) |
| |
| That creates 10 distinct generators, which can be passed out to 10 distinct |
| threads. The generators don't share state so can be called safely in |
| parallel. So long as no thread calls its g.random() more than a million |
| times (the second argument to create_generators), the sequences seen by |
| each thread will not overlap. |
| |
| The period of the underlying Wichmann-Hill generator is 6,953,607,871,644, |
| and that limits how far this technique can be pushed. |
| |
| Just for fun, note that since we know the period, .jumpahead() can also be |
| used to "move backward in time": |
| |
| >>> g = Random(42) # arbitrary |
| >>> g.random() |
| 0.25420336316883324 |
| >>> g.jumpahead(6953607871644L - 1) # move *back* one |
| >>> g.random() |
| 0.25420336316883324 |
| """ |
| # XXX The docstring sucks. |
| |
| 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 |
| |
| __all__ = ["Random","seed","random","uniform","randint","choice", |
| "randrange","shuffle","normalvariate","lognormvariate", |
| "cunifvariate","expovariate","vonmisesvariate","gammavariate", |
| "stdgamma","gauss","betavariate","paretovariate","weibullvariate", |
| "getstate","setstate","jumpahead","whseed"] |
| |
| 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)) |
| |
| NV_MAGICCONST = 4 * _exp(-0.5)/_sqrt(2.0) |
| _verify('NV_MAGICCONST', 1.71552776992141) |
| |
| TWOPI = 2.0*_pi |
| _verify('TWOPI', 6.28318530718) |
| |
| LOG4 = _log(4.0) |
| _verify('LOG4', 1.38629436111989) |
| |
| SG_MAGICCONST = 1.0 + _log(4.5) |
| _verify('SG_MAGICCONST', 2.50407739677627) |
| |
| del _verify |
| |
| # Translated by Guido van Rossum from C source provided by |
| # Adrian Baddeley. |
| |
| class Random: |
| |
| VERSION = 1 # used by getstate/setstate |
| |
| def __init__(self, x=None): |
| """Initialize an instance. |
| |
| Optional argument x controls seeding, as for Random.seed(). |
| """ |
| |
| self.seed(x) |
| self.gauss_next = None |
| |
| ## -------------------- core generator ------------------- |
| |
| # Specific to Wichmann-Hill generator. Subclasses wishing to use a |
| # different core generator should override the seed(), random(), |
| # getstate(), setstate() and jumpahead() methods. |
| |
| 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 type(a) not in (type(3), type(3L)): |
| 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 |
| |
| 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) == type(0): |
| 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) |
| |
| 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) |
| |
| ## ---- 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 |
| return self.getstate() |
| |
| def __setstate__(self, state): # for pickle |
| self.setstate(state) |
| |
| ## -------------------- 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' and 'default' arguments. |
| """ |
| |
| # This code is a bit messy to make it fast for the |
| # common case while still doing adequate error checking |
| istart = int(start) |
| if istart != start: |
| raise ValueError, "non-integer arg 1 for randrange()" |
| if stop is default: |
| if istart > 0: |
| return int(self.random() * istart) |
| raise ValueError, "empty range for randrange()" |
| istop = int(stop) |
| if istop != stop: |
| raise ValueError, "non-integer stop for randrange()" |
| if step == 1: |
| if istart < istop: |
| return istart + int(self.random() * |
| (istop - istart)) |
| raise ValueError, "empty range for randrange()" |
| istep = int(step) |
| if istep != step: |
| raise ValueError, "non-integer step for randrange()" |
| if istep > 0: |
| n = (istop - istart + istep - 1) / istep |
| elif istep < 0: |
| n = (istop - istart + istep + 1) / istep |
| else: |
| raise ValueError, "zero step for randrange()" |
| |
| if n <= 0: |
| raise ValueError, "empty range for randrange()" |
| return istart + istep*int(self.random() * n) |
| |
| def randint(self, a, b): |
| """Return random integer in range [a, b], including both end points. |
| |
| (Deprecated; use randrange(a, b+1).) |
| """ |
| |
| return self.randrange(a, b+1) |
| |
| ## -------------------- sequence methods ------------------- |
| |
| def choice(self, seq): |
| """Choose a random element from a non-empty sequence.""" |
| return seq[int(self.random() * len(seq))] |
| |
| 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. |
| """ |
| |
| if random is None: |
| random = self.random |
| 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] |
| |
| ## -------------------- real-valued distributions ------------------- |
| |
| ## -------------------- uniform distribution ------------------- |
| |
| def uniform(self, a, b): |
| """Get a random number in the range [a, b).""" |
| return a + (b-a) * self.random() |
| |
| ## -------------------- normal distribution -------------------- |
| |
| def normalvariate(self, 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. |
| |
| random = self.random |
| 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(self, mu, sigma): |
| return _exp(self.normalvariate(mu, sigma)) |
| |
| ## -------------------- circular uniform -------------------- |
| |
| def cunifvariate(self, mean, arc): |
| # mean: mean angle (in radians between 0 and pi) |
| # arc: range of distribution (in radians between 0 and pi) |
| |
| return (mean + arc * (self.random() - 0.5)) % _pi |
| |
| ## -------------------- exponential distribution -------------------- |
| |
| def expovariate(self, lambd): |
| # lambd: rate lambd = 1/mean |
| # ('lambda' is a Python reserved word) |
| |
| random = self.random |
| u = random() |
| while u <= 1e-7: |
| u = random() |
| return -_log(u)/lambd |
| |
| ## -------------------- von Mises distribution -------------------- |
| |
| def vonmisesvariate(self, 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. |
| |
| random = self.random |
| 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 -------------------- |
| |
| def gammavariate(self, alpha, beta): |
| # beta times standard gamma |
| ainv = _sqrt(2.0 * alpha - 1.0) |
| return beta * self.stdgamma(alpha, ainv, alpha - LOG4, alpha + ainv) |
| |
| def stdgamma(self, alpha, ainv, bbb, ccc): |
| # ainv = sqrt(2 * alpha - 1) |
| # bbb = alpha - log(4) |
| # ccc = alpha + ainv |
| |
| random = self.random |
| 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) -------------------- |
| |
| def gauss(self, 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.) |
| |
| random = self.random |
| z = self.gauss_next |
| self.gauss_next = None |
| if z is None: |
| x2pi = random() * TWOPI |
| g2rad = _sqrt(-2.0 * _log(1.0 - random())) |
| z = _cos(x2pi) * g2rad |
| self.gauss_next = _sin(x2pi) * g2rad |
| |
| return mu + z*sigma |
| |
| ## -------------------- beta -------------------- |
| ## See |
| ## 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) |
| ## return z/(y+z) |
| ## |
| ## was dead wrong, and how it probably got that way. |
| |
| def betavariate(self, alpha, beta): |
| # 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.) |
| if y == 0: |
| return 0.0 |
| else: |
| return y / (y + self.gammavariate(beta, 1.)) |
| |
| ## -------------------- Pareto -------------------- |
| |
| def paretovariate(self, alpha): |
| # Jain, pg. 495 |
| |
| u = self.random() |
| return 1.0 / pow(u, 1.0/alpha) |
| |
| ## -------------------- Weibull -------------------- |
| |
| def weibullvariate(self, alpha, beta): |
| # Jain, pg. 499; bug fix courtesy Bill Arms |
| |
| u = self.random() |
| return alpha * pow(-_log(u), 1.0/beta) |
| |
| ## -------------------- test program -------------------- |
| |
| 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) |
| |
| 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)') |
| |
| # Test jumpahead. |
| s = getstate() |
| jumpahead(N) |
| r1 = random() |
| # now do it the slow way |
| setstate(s) |
| for i in range(N): |
| random() |
| r2 = random() |
| if r1 != r2: |
| raise ValueError("jumpahead test failed " + `(N, r1, r2)`) |
| |
| # Create one instance, seeded from current time, and export its methods |
| # as module-level functions. The functions are not threadsafe, and state |
| # is shared 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 |
| shuffle = _inst.shuffle |
| normalvariate = _inst.normalvariate |
| lognormvariate = _inst.lognormvariate |
| cunifvariate = _inst.cunifvariate |
| expovariate = _inst.expovariate |
| vonmisesvariate = _inst.vonmisesvariate |
| gammavariate = _inst.gammavariate |
| stdgamma = _inst.stdgamma |
| gauss = _inst.gauss |
| betavariate = _inst.betavariate |
| paretovariate = _inst.paretovariate |
| weibullvariate = _inst.weibullvariate |
| getstate = _inst.getstate |
| setstate = _inst.setstate |
| jumpahead = _inst.jumpahead |
| whseed = _inst.whseed |
| |
| if __name__ == '__main__': |
| _test() |