random.py

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        """
        # 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 = 1.0 - 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):
        """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))

## -------------------- circular uniform --------------------

    def cunifvariate(self, mean, arc):
        """Circular uniform distribution.

        mean is the mean angle, and arc is the range of the distribution,
        centered around the mean angle.  Both values must be expressed in
        radians.  Returned values range between mean - arc/2 and
        mean + arc/2 and are normalized to between 0 and pi.

        Deprecated in version 2.3.  Use:
            (mean + arc * (Random.random() - 0.5)) % Math.pi

        """
        # 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):
        """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)

        random = self.random
        u = random()
        while u <= 1e-7:
            u = random()
        return -_log(u)/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.

        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):
        """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
        # of alpha > -1.0
        if alpha <= 0.0 or beta <= 0.0:
            raise ValueError, 'gammavariate: alpha and beta must be > 0.0'

        random = self.random
        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

            ainv = _sqrt(2.0 * alpha - 1.0)
            bbb = alpha - LOG4
            ccc = alpha + ainv

            while 1:
                u1 = random()
                if not 1e-7 < u1 < .9999999:
                    continue
                u2 = 1.0 - 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 * beta

        elif alpha == 1.0:
            # expovariate(1)
            u = random()
            while u <= 1e-7:
                u = random()
            return -_log(u) * beta

        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 * beta


    def stdgamma(self, alpha, ainv, bbb, ccc):
        # This method was (and shall remain) undocumented.
        # This method is deprecated
        # for the following reasons:
        # 1. Returns same as .gammavariate(alpha, 1.0)
        # 2. Requires caller to provide 3 extra arguments
        #    that are functions of alpha anyway
        # 3. Can't be used for alpha < 0.5

        # ainv = sqrt(2 * alpha - 1)
        # bbb = alpha - log(4)
        # ccc = alpha + ainv
        import warnings
        warnings.warn("The stdgamma function is deprecated; "
                      "use gammavariate() instead",
                      DeprecationWarning)
        return self.gammavariate(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
        # 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):
        """Beta distribution.

        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.)
        if y == 0:
            return 0.0
        else:
            return y / (y + self.gammavariate(beta, 1.))

## -------------------- Pareto --------------------

    def paretovariate(self, alpha):
        """Pareto distribution.  alpha is the shape parameter."""
        # Jain, pg. 495

        u = 1.0 - self.random()
        return 1.0 / pow(u, 1.0/alpha)

## -------------------- Weibull --------------------

    def weibullvariate(self, alpha, beta):
        """Weibull distribution.

        alpha is the scale parameter and beta is the shape parameter.

        """
        # Jain, pg. 499; bug fix courtesy Bill Arms

        u = 1.0 - 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=20000):
    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(0.01, 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)')
    _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()