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# # Author: Travis Oliphant 2002-2011 with contributions from # SciPy Developers 2004-2011 #
rv_discrete, _lazywhere, _ncx2_pdf, _ncx2_cdf, get_distribution_names)
r"""A binomial discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `binom` is:
.. math::
f(k) = \binom{n}{k} p^k (1-p)^{n-k}
for ``k`` in ``{0, 1,..., n}``.
`binom` takes ``n`` and ``p`` as shape parameters.
%(after_notes)s
%(example)s
""" return self._random_state.binomial(n, p, self._size)
self.b = n return (n >= 0) & (p >= 0) & (p <= 1)
k = floor(x) combiln = (gamln(n+1) - (gamln(k+1) + gamln(n-k+1))) return combiln + special.xlogy(k, p) + special.xlog1py(n-k, -p)
# binom.pmf(k) = choose(n, k) * p**k * (1-p)**(n-k) return exp(self._logpmf(x, n, p))
k = floor(x) vals = special.bdtr(k, n, p) return vals
k = floor(x) return special.bdtrc(k, n, p)
vals = ceil(special.bdtrik(q, n, p)) vals1 = np.maximum(vals - 1, 0) temp = special.bdtr(vals1, n, p) return np.where(temp >= q, vals1, vals)
q = 1.0 - p mu = n * p var = n * p * q g1, g2 = None, None if 's' in moments: g1 = (q - p) / sqrt(var) if 'k' in moments: g2 = (1.0 - 6*p*q) / var return mu, var, g1, g2
k = np.r_[0:n + 1] vals = self._pmf(k, n, p) return np.sum(entr(vals), axis=0)
r"""A Bernoulli discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `bernoulli` is:
.. math::
f(k) = \begin{cases}1-p &\text{if } k = 0\\ p &\text{if } k = 1\end{cases}
for :math:`k` in :math:`\{0, 1\}`.
`bernoulli` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
""" return binom_gen._rvs(self, 1, p)
return (p >= 0) & (p <= 1)
return binom._logpmf(x, 1, p)
# bernoulli.pmf(k) = 1-p if k = 0 # = p if k = 1 return binom._pmf(x, 1, p)
return binom._cdf(x, 1, p)
return binom._sf(x, 1, p)
return binom._ppf(q, 1, p)
return binom._stats(1, p)
return entr(p) + entr(1-p)
r"""A negative binomial discrete random variable.
%(before_notes)s
Notes ----- Negative binomial distribution describes a sequence of i.i.d. Bernoulli trials, repeated until a predefined, non-random number of successes occurs.
The probability mass function of the number of failures for `nbinom` is:
.. math::
f(k) = \binom{k+n-1}{n-1} p^n (1-p)^k
for :math:`k \ge 0`.
`nbinom` takes :math:`n` and :math:`p` as shape parameters where n is the number of successes, whereas p is the probability of a single success.
%(after_notes)s
%(example)s
""" return self._random_state.negative_binomial(n, p, self._size)
return (n > 0) & (p >= 0) & (p <= 1)
# nbinom.pmf(k) = choose(k+n-1, n-1) * p**n * (1-p)**k return exp(self._logpmf(x, n, p))
coeff = gamln(n+x) - gamln(x+1) - gamln(n) return coeff + n*log(p) + special.xlog1py(x, -p)
k = floor(x) return special.betainc(n, k+1, p)
# skip because special.nbdtrc doesn't work for 0<n<1 k = floor(x) return special.nbdtrc(k, n, p)
vals = ceil(special.nbdtrik(q, n, p)) vals1 = (vals-1).clip(0.0, np.inf) temp = self._cdf(vals1, n, p) return np.where(temp >= q, vals1, vals)
Q = 1.0 / p P = Q - 1.0 mu = n*P var = n*P*Q g1 = (Q+P)/sqrt(n*P*Q) g2 = (1.0 + 6*P*Q) / (n*P*Q) return mu, var, g1, g2
r"""A geometric discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `geom` is:
.. math::
f(k) = (1-p)^{k-1} p
for :math:`k \ge 1`.
`geom` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
""" return self._random_state.geometric(p, size=self._size)
return (p <= 1) & (p >= 0)
# geom.pmf(k) = (1-p)**(k-1)*p return np.power(1-p, k-1) * p
return special.xlog1py(k - 1, -p) + log(p)
k = floor(x) return -expm1(log1p(-p)*k)
return np.exp(self._logsf(x, p))
k = floor(x) return k*log1p(-p)
vals = ceil(log(1.0-q)/log(1-p)) temp = self._cdf(vals-1, p) return np.where((temp >= q) & (vals > 0), vals-1, vals)
mu = 1.0/p qr = 1.0-p var = qr / p / p g1 = (2.0-p) / sqrt(qr) g2 = np.polyval([1, -6, 6], p)/(1.0-p) return mu, var, g1, g2
r"""A hypergeometric discrete random variable.
The hypergeometric distribution models drawing objects from a bin. `M` is the total number of objects, `n` is total number of Type I objects. The random variate represents the number of Type I objects in `N` drawn without replacement from the total population.
%(before_notes)s
Notes ----- The symbols used to denote the shape parameters (`M`, `n`, and `N`) are not universally accepted. See the Examples for a clarification of the definitions used here.
The probability mass function is defined as,
.. math:: p(k, M, n, N) = \frac{\binom{n}{k} \binom{M - n}{N - k}} {\binom{M}{N}}
for :math:`k \in [\max(0, N - M + n), \min(n, N)]`, where the binomial coefficients are defined as,
.. math:: \binom{n}{k} \equiv \frac{n!}{k! (n - k)!}.
%(after_notes)s
Examples -------- >>> from scipy.stats import hypergeom >>> import matplotlib.pyplot as plt
Suppose we have a collection of 20 animals, of which 7 are dogs. Then if we want to know the probability of finding a given number of dogs if we choose at random 12 of the 20 animals, we can initialize a frozen distribution and plot the probability mass function:
>>> [M, n, N] = [20, 7, 12] >>> rv = hypergeom(M, n, N) >>> x = np.arange(0, n+1) >>> pmf_dogs = rv.pmf(x)
>>> fig = plt.figure() >>> ax = fig.add_subplot(111) >>> ax.plot(x, pmf_dogs, 'bo') >>> ax.vlines(x, 0, pmf_dogs, lw=2) >>> ax.set_xlabel('# of dogs in our group of chosen animals') >>> ax.set_ylabel('hypergeom PMF') >>> plt.show()
Instead of using a frozen distribution we can also use `hypergeom` methods directly. To for example obtain the cumulative distribution function, use:
>>> prb = hypergeom.cdf(x, M, n, N)
And to generate random numbers:
>>> R = hypergeom.rvs(M, n, N, size=10)
""" return self._random_state.hypergeometric(n, M-n, N, size=self._size)
cond = (M > 0) & (n >= 0) & (N >= 0) cond &= (n <= M) & (N <= M) self.a = np.maximum(N-(M-n), 0) self.b = np.minimum(n, N) return cond
tot, good = M, n bad = tot - good return betaln(good+1, 1) + betaln(bad+1,1) + betaln(tot-N+1, N+1)\ - betaln(k+1, good-k+1) - betaln(N-k+1,bad-N+k+1)\ - betaln(tot+1, 1)
# same as the following but numerically more precise # return comb(good, k) * comb(bad, N-k) / comb(tot, N) return exp(self._logpmf(k, M, n, N))
# tot, good, sample_size = M, n, N # "wikipedia".replace('N', 'M').replace('n', 'N').replace('K', 'n') M, n, N = 1.*M, 1.*n, 1.*N m = M - n p = n/M mu = N*p
var = m*n*N*(M - N)*1.0/(M*M*(M-1)) g1 = (m - n)*(M-2*N) / (M-2.0) * sqrt((M-1.0) / (m*n*N*(M-N)))
g2 = M*(M+1) - 6.*N*(M-N) - 6.*n*m g2 *= (M-1)*M*M g2 += 6.*n*N*(M-N)*m*(5.*M-6) g2 /= n * N * (M-N) * m * (M-2.) * (M-3.) return mu, var, g1, g2
k = np.r_[N - (M - n):min(n, N) + 1] vals = self.pmf(k, M, n, N) return np.sum(entr(vals), axis=0)
"""More precise calculation, 1 - cdf doesn't cut it.""" # This for loop is needed because `k` can be an array. If that's the # case, the sf() method makes M, n and N arrays of the same shape. We # therefore unpack all inputs args, so we can do the manual # integration. res = [] for quant, tot, good, draw in zip(k, M, n, N): # Manual integration over probability mass function. More accurate # than integrate.quad. k2 = np.arange(quant + 1, draw + 1) res.append(np.sum(self._pmf(k2, tot, good, draw))) return np.asarray(res)
""" More precise calculation than log(sf) """ res = [] for quant, tot, good, draw in zip(k, M, n, N): # Integration over probability mass function using logsumexp k2 = np.arange(quant + 1, draw + 1) res.append(logsumexp(self._logpmf(k2, tot, good, draw))) return np.asarray(res)
# FIXME: Fails _cdfvec r"""A Logarithmic (Log-Series, Series) discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `logser` is:
.. math::
f(k) = - \frac{p^k}{k \log(1-p)}
for :math:`k \ge 1`.
`logser` takes :math:`p` as shape parameter.
%(after_notes)s
%(example)s
""" # looks wrong for p>0.5, too few k=1 # trying to use generic is worse, no k=1 at all return self._random_state.logseries(p, size=self._size)
return (p > 0) & (p < 1)
# logser.pmf(k) = - p**k / (k*log(1-p)) return -np.power(p, k) * 1.0 / k / special.log1p(-p)
r = special.log1p(-p) mu = p / (p - 1.0) / r mu2p = -p / r / (p - 1.0)**2 var = mu2p - mu*mu mu3p = -p / r * (1.0+p) / (1.0 - p)**3 mu3 = mu3p - 3*mu*mu2p + 2*mu**3 g1 = mu3 / np.power(var, 1.5)
mu4p = -p / r * ( 1.0 / (p-1)**2 - 6*p / (p - 1)**3 + 6*p*p / (p-1)**4) mu4 = mu4p - 4*mu3p*mu + 6*mu2p*mu*mu - 3*mu**4 g2 = mu4 / var**2 - 3.0 return mu, var, g1, g2
r"""A Poisson discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `poisson` is:
.. math::
f(k) = \exp(-\mu) \frac{mu^k}{k!}
for :math:`k \ge 0`.
`poisson` takes :math:`\mu` as shape parameter.
%(after_notes)s
%(example)s
"""
# Override rv_discrete._argcheck to allow mu=0. return mu >= 0
return self._random_state.poisson(mu, self._size)
Pk = special.xlogy(k, mu) - gamln(k + 1) - mu return Pk
# poisson.pmf(k) = exp(-mu) * mu**k / k! return exp(self._logpmf(k, mu))
k = floor(x) return special.pdtr(k, mu)
k = floor(x) return special.pdtrc(k, mu)
vals = ceil(special.pdtrik(q, mu)) vals1 = np.maximum(vals - 1, 0) temp = special.pdtr(vals1, mu) return np.where(temp >= q, vals1, vals)
var = mu tmp = np.asarray(mu) mu_nonzero = tmp > 0 g1 = _lazywhere(mu_nonzero, (tmp,), lambda x: sqrt(1.0/x), np.inf) g2 = _lazywhere(mu_nonzero, (tmp,), lambda x: 1.0/x, np.inf) return mu, var, g1, g2
r"""A Planck discrete exponential random variable.
%(before_notes)s
Notes ----- The probability mass function for `planck` is:
.. math::
f(k) = (1-\exp(-\lambda)) \exp(-\lambda k)
for :math:`k \lambda \ge 0`.
`planck` takes :math:`\lambda` as shape parameter.
%(after_notes)s
%(example)s
""" self.a = np.where(lambda_ > 0, 0, -np.inf) self.b = np.where(lambda_ > 0, np.inf, 0) return lambda_ != 0
# planck.pmf(k) = (1-exp(-lambda_))*exp(-lambda_*k) fact = (1-exp(-lambda_)) return fact*exp(-lambda_*k)
k = floor(x) return 1-exp(-lambda_*(k+1))
return np.exp(self._logsf(x, lambda_))
k = floor(x) return -lambda_*(k+1)
vals = ceil(-1.0/lambda_ * log1p(-q)-1) vals1 = (vals-1).clip(self.a, np.inf) temp = self._cdf(vals1, lambda_) return np.where(temp >= q, vals1, vals)
mu = 1/(exp(lambda_)-1) var = exp(-lambda_)/(expm1(-lambda_))**2 g1 = 2*cosh(lambda_/2.0) g2 = 4+2*cosh(lambda_) return mu, var, g1, g2
l = lambda_ C = (1-exp(-l)) return l*exp(-l)/C - log(C)
r"""A Boltzmann (Truncated Discrete Exponential) random variable.
%(before_notes)s
Notes ----- The probability mass function for `boltzmann` is:
.. math::
f(k) = (1-\exp(-\lambda) \exp(-\lambda k)/(1-\exp(-\lambda N))
for :math:`k = 0,..., N-1`.
`boltzmann` takes :math:`\lambda` and :math:`N` as shape parameters.
%(after_notes)s
%(example)s
""" # boltzmann.pmf(k) = # (1-exp(-lambda_)*exp(-lambda_*k)/(1-exp(-lambda_*N)) fact = (1-exp(-lambda_))/(1-exp(-lambda_*N)) return fact*exp(-lambda_*k)
k = floor(x) return (1-exp(-lambda_*(k+1)))/(1-exp(-lambda_*N))
qnew = q*(1-exp(-lambda_*N)) vals = ceil(-1.0/lambda_ * log(1-qnew)-1) vals1 = (vals-1).clip(0.0, np.inf) temp = self._cdf(vals1, lambda_, N) return np.where(temp >= q, vals1, vals)
z = exp(-lambda_) zN = exp(-lambda_*N) mu = z/(1.0-z)-N*zN/(1-zN) var = z/(1.0-z)**2 - N*N*zN/(1-zN)**2 trm = (1-zN)/(1-z) trm2 = (z*trm**2 - N*N*zN) g1 = z*(1+z)*trm**3 - N**3*zN*(1+zN) g1 = g1 / trm2**(1.5) g2 = z*(1+4*z+z*z)*trm**4 - N**4 * zN*(1+4*zN+zN*zN) g2 = g2 / trm2 / trm2 return mu, var, g1, g2
longname='A truncated discrete exponential ')
r"""A uniform discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `randint` is:
.. math::
f(k) = \frac{1}{high - low}
for ``k = low, ..., high - 1``.
`randint` takes ``low`` and ``high`` as shape parameters.
%(after_notes)s
%(example)s
""" self.a = low self.b = high - 1 return (high > low)
# randint.pmf(k) = 1./(high - low) p = np.ones_like(k) / (high - low) return np.where((k >= low) & (k < high), p, 0.)
k = floor(x) return (k - low + 1.) / (high - low)
vals = ceil(q * (high - low) + low) - 1 vals1 = (vals - 1).clip(low, high) temp = self._cdf(vals1, low, high) return np.where(temp >= q, vals1, vals)
m2, m1 = np.asarray(high), np.asarray(low) mu = (m2 + m1 - 1.0) / 2 d = m2 - m1 var = (d*d - 1) / 12.0 g1 = 0.0 g2 = -6.0/5.0 * (d*d + 1.0) / (d*d - 1.0) return mu, var, g1, g2
"""An array of *size* random integers >= ``low`` and < ``high``.""" if self._size is not None: # Numpy's RandomState.randint() doesn't broadcast its arguments. # Use `broadcast_to()` to extend the shapes of low and high # up to self._size. Then we can use the numpy.vectorize'd # randint without needing to pass it a `size` argument. low = broadcast_to(low, self._size) high = broadcast_to(high, self._size) randint = np.vectorize(self._random_state.randint, otypes=[np.int_]) return randint(low, high)
return log(high - low)
'(random integer)')
# FIXME: problems sampling. r"""A Zipf discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `zipf` is:
.. math::
f(k, a) = \frac{1}{\zeta(a) k^a}
for :math:`k \ge 1`.
`zipf` takes :math:`a` as shape parameter.
%(after_notes)s
%(example)s
""" return self._random_state.zipf(a, size=self._size)
return a > 1
# zipf.pmf(k, a) = 1/(zeta(a) * k**a) Pk = 1.0 / special.zeta(a, 1) / k**a return Pk
return _lazywhere( a > n + 1, (a, n), lambda a, n: special.zeta(a - n, 1) / special.zeta(a, 1), np.inf)
r"""A Laplacian discrete random variable.
%(before_notes)s
Notes ----- The probability mass function for `dlaplace` is:
.. math::
f(k) = \tanh(a/2) \exp(-a |k|)
for :math:`a > 0`.
`dlaplace` takes :math:`a` as shape parameter.
%(after_notes)s
%(example)s
""" # dlaplace.pmf(k) = tanh(a/2) * exp(-a*abs(k)) return tanh(a/2.0) * exp(-a * abs(k))
k = floor(x) f = lambda k, a: 1.0 - exp(-a * k) / (exp(a) + 1) f2 = lambda k, a: exp(a * (k+1)) / (exp(a) + 1) return _lazywhere(k >= 0, (k, a), f=f, f2=f2)
const = 1 + exp(a) vals = ceil(np.where(q < 1.0 / (1 + exp(-a)), log(q*const) / a - 1, -log((1-q) * const) / a)) vals1 = vals - 1 return np.where(self._cdf(vals1, a) >= q, vals1, vals)
ea = exp(a) mu2 = 2.*ea/(ea-1.)**2 mu4 = 2.*ea*(ea**2+10.*ea+1.) / (ea-1.)**4 return 0., mu2, 0., mu4/mu2**2 - 3.
return a / sinh(a) - log(tanh(a/2.0))
name='dlaplace', longname='A discrete Laplacian')
r"""A Skellam discrete random variable.
%(before_notes)s
Notes ----- Probability distribution of the difference of two correlated or uncorrelated Poisson random variables.
Let :math:`k_1` and :math:`k_2` be two Poisson-distributed r.v. with expected values lam1 and lam2. Then, :math:`k_1 - k_2` follows a Skellam distribution with parameters :math:`\mu_1 = \lambda_1 - \rho \sqrt{\lambda_1 \lambda_2}` and :math:`\mu_2 = \lambda_2 - \rho \sqrt{\lambda_1 \lambda_2}`, where :math:`\rho` is the correlation coefficient between :math:`k_1` and :math:`k_2`. If the two Poisson-distributed r.v. are independent then :math:`\rho = 0`.
Parameters :math:`\mu_1` and :math:`\mu_2` must be strictly positive.
For details see: http://en.wikipedia.org/wiki/Skellam_distribution
`skellam` takes :math:`\mu_1` and :math:`\mu_2` as shape parameters.
%(after_notes)s
%(example)s
""" n = self._size return (self._random_state.poisson(mu1, n) - self._random_state.poisson(mu2, n))
px = np.where(x < 0, _ncx2_pdf(2*mu2, 2*(1-x), 2*mu1)*2, _ncx2_pdf(2*mu1, 2*(1+x), 2*mu2)*2) # ncx2.pdf() returns nan's for extremely low probabilities return px
x = floor(x) px = np.where(x < 0, _ncx2_cdf(2*mu2, -2*x, 2*mu1), 1-_ncx2_cdf(2*mu1, 2*(x+1), 2*mu2)) return px
mean = mu1 - mu2 var = mu1 + mu2 g1 = mean / sqrt((var)**3) g2 = 1 / var return mean, var, g1, g2
# Collect names of classes and objects in this module.
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