def gen_probs(n, a, b):
r"""
Generate the vector of probabilities for the Beta-binomial
(n, a, b) distribution.
The Beta-binomial distribution takes the form
.. math::
p(k \,|\, n, a, b) =
{n \choose k} \frac{B(k + a, n - k + b)}{B(a, b)},
\qquad k = 0, \ldots, n
Parameters
----------
n : scalar(int)
First parameter to the Beta-binomial distribution
a : scalar(float)
Second parameter to the Beta-binomial distribution
b : scalar(float)
Third parameter to the Beta-binomial distribution
Returns
-------
probs: array_like(float)
Vector of probabilities over k
"""
probs = np.zeros(n+1)
for k in range(n+1):
probs[k] = binom(n, k) * beta(k + a, n - k + b) / beta(a, b)
return probs
def beta_binomial(k,n,a,b,multi_precission=False): """ Beta binomial function, returning the probability of k successes in n trials (given a p distribution beta of barameters a and b), and supporting multiprecission output. Parameters ---------- k : int, ndarray Successes. n : int, ndarray Trials. a, b : int,ndarray Parameters of the beta distribution. multi_precission : bool, optional Whether or not to use multiprecision floating-point output (default: False). Returns ------- p : int,ndarray Probability of k successes in n trials. Examples -------- >>> n = 80000 >>> k = 40000 >>> mp_comb(n, k) 7.0802212521852e+24079 """ if multi_precission: import mpmath as mp p = mp_comb(n,k) * mp.beta(k+a, n-k+b) / mp.beta(a,b) else: from scipy.special import beta from scipy.misc import comb p = comb(n,k) * beta(k+a, n-k+b) / beta(a,b) return p
def pdf(self):
r"""
Generate the vector of probabilities for the Beta-binomial
(n, a, b) distribution.
The Beta-binomial distribution takes the form
.. math::
p(k \,|\, n, a, b) =
{n \choose k} \frac{B(k + a, n - k + b)}{B(a, b)},
\qquad k = 0, \ldots, n,
where :math:`B` is the beta function.
Parameters
----------
n : scalar(int)
First parameter to the Beta-binomial distribution
a : scalar(float)
Second parameter to the Beta-binomial distribution
b : scalar(float)
Third parameter to the Beta-binomial distribution
Returns
-------
probs: array_like(float)
Vector of probabilities over k
"""
n, a, b = self.n, self.a, self.b
k = np.arange(n + 1)
probs = binom(n, k) * beta(k + a, n - k + b) / beta(a, b)
return probs
def probability_of_n_purchases_up_to_time(self, t, n):
"""
Compute the probability of
P( N(t) = n | model )
where N(t) is the number of repeat purchases a customer makes in t units of time.
"""
r, alpha, a, b = self._unload_params("r", "alpha", "a", "b")
first_term = (
special.beta(a, b + n)
/ special.beta(a, b)
* special.gamma(r + n)
/ special.gamma(r)
/ special.gamma(n + 1)
* (alpha / (alpha + t)) ** r
* (t / (alpha + t)) ** n
)
if n > 0:
finite_sum = np.sum(
[
special.gamma(r + j) / special.gamma(r) / special.gamma(j + 1) * (t / (alpha + t)) ** j
for j in range(0, n)
]
)
second_term = (
special.beta(a + 1, b + n - 1) / special.beta(a, b) * (1 - (alpha / (alpha + t)) ** r * finite_sum)
)
else:
second_term = 0
return first_term + second_term
def BetaBinomial_pdf(n, a, b):
''' BetaBinomial.
'''
k = np.arange(n + 1)
probs = binom(n, k) * beta(k + a, n - k + b) / beta(a, b)
return probs
def __init__(self, a=1.5, b=1.03, p=.2, q=.2, T=1):
self.p, self.q = p, q
# Scaling
self.a = 100 * a / T**.5
# Martingale restriction
self.b = np.ones_like(b) * ss.beta(p, q) \
/ ss.beta(p+1/self.a, q-1/self.a)
def deflect(self):
xi =np.sqrt((self.image_x-self.centroid[0])**2+(self.image_y-self.centroid[1])**2)/self.a
alpha = 2*self.kappa_0*self.a/xi * (sp.beta((self.n_outer-3)/2.0,(3-self.gamma)/2.0) \
- sp.beta((self.n_outer-3)/2.0, 3.0/2.0) * (1+xi**2)**((3-self.n_outer)/2.0)\
* sp.hyp2f1((self.n_outer-3)/2.0,self.gamma/2.0,self.n_outer/2.0,1/(1+xi**2)))
self.alpha_x = alpha*(self.image_x-self.centroid[0])/np.sqrt((self.image_x-self.centroid[0])**2+(self.image_y-self.centroid[1])**2)
self.alpha_y = alpha*(self.image_y-self.centroid[1])/np.sqrt((self.image_x-self.centroid[0])**2+(self.image_y-self.centroid[1])**2)
def beta_kl_divergence(a, b, aa, bb, samples=1000):
'''
Calculate the Kullback-Leibler divergence between to beta distributions,
defined by a and b and aa and bb respectively.
$KL(P || Q) = \int_{-infty}^{\infty} P \ln \frac{P}{Q}$
'''
return math.log(beta(aa, bb)/beta(a, b)) + (aa - a) * psi(a) + (bb - b) * psi(b) + (aa - a + bb - b) * psi(a+b)
def test_reduce(self):
phi1 = self.phi1.copy()
phi1.reduce([('x', 1)])
reduced_pdf1 = lambda y: (np.power(1, 1) * np.power(y, 2))/beta(1, y)
self.assertEqual(phi1.scope(), ['y'])
for inp in np.random.rand(4):
self.assertEqual(phi1.pdf(inp), reduced_pdf1(inp))
self.assertEqual(phi1.pdf(y=inp), reduced_pdf1(inp))
phi1 = self.phi1.reduce([('x', 1)], inplace=False)
self.assertEqual(phi1.scope(), ['y'])
for inp in np.random.rand(4):
self.assertEqual(phi1.pdf(inp), reduced_pdf1(inp))
self.assertEqual(phi1.pdf(y=inp), reduced_pdf1(inp))
phi2 = self.phi2.copy()
phi2.reduce([('x2', 7.213)])
reduced_pdf2 = lambda x1: multivariate_normal.pdf([x1, 7.213], [0, 0], [[1, 0], [0, 1]])
self.assertEqual(phi2.scope(), ['x1'])
for inp in np.random.rand(4):
self.assertEqual(phi2.pdf(inp), reduced_pdf2(inp))
self.assertEqual(phi2.pdf(x1=inp), reduced_pdf2(inp))
phi2 = self.phi2.reduce([('x2', 7.213)], inplace=False)
self.assertEqual(phi2.scope(), ['x1'])
for inp in np.random.rand(4):
self.assertEqual(phi2.pdf(inp), reduced_pdf2(inp))
self.assertEqual(phi2.pdf(x1=inp), reduced_pdf2(inp))
phi3 = self.phi3.copy()
phi3.reduce([('y', 0.112), ('z', 23)])
reduced_pdf4 = lambda x: 23*(np.power(x, 1)*np.power(0.112, 2))/beta(x, 0.112)
self.assertEqual(phi3.scope(), ['x'])
for inp in np.random.rand(4):
self.assertEqual(phi3.pdf(inp), reduced_pdf4(inp))
self.assertEqual(phi3.pdf(x=inp), reduced_pdf4(inp))
phi3 = self.phi3.copy()
phi3.reduce([('y', 0.112)])
reduced_pdf3 = lambda x, z: z*(np.power(x, 1)*np.power(0.112, 2))/beta(x, 0.112)
self.assertEqual(phi3.scope(), ['x', 'z'])
for inp in np.random.rand(4, 2):
self.assertEqual(phi3.pdf(inp[0], inp[1]), reduced_pdf3(inp[0], inp[1]))
self.assertEqual(phi3.pdf(x=inp[0], z=inp[1]), reduced_pdf3(inp[0], inp[1]))
phi3 = self.phi3.reduce([('y', 0.112)], inplace=False)
self.assertEqual(phi3.scope(), ['x', 'z'])
for inp in np.random.rand(4, 2):
self.assertEqual(phi3.pdf(inp[0], inp[1]), reduced_pdf3(inp[0], inp[1]))
self.assertEqual(phi3.pdf(x=inp[0], z=inp[1]), reduced_pdf3(inp[0], inp[1]))
self.assertEqual(phi3.pdf(inp[0], z=inp[1]), reduced_pdf3(inp[0], inp[1]))
phi3 = self.phi3.reduce([('y', 0.112), ('z', 23)], inplace=False)
self.assertEqual(phi3.scope(), ['x'])
for inp in np.random.rand(4):
self.assertEqual(phi3.pdf(inp), reduced_pdf4(inp))
self.assertEqual(phi3.pdf(x=inp), reduced_pdf4(inp))
def gen_probs(n, a, b):
"""
Generate and return the vector of probabilities for the Beta-binomial
(n, a, b) distribution.
"""
probs = np.zeros(n + 1)
for k in range(n + 1):
probs[k] = binom(n, k) * beta(k + a, n - k + b) / beta(a, b)
return probs
def _get_bayesian_prob(self):
alpha_a = 1 + self.success_count["A"]
beta_a = 1 + self.failure_count["A"]
alpha_b = 1 + self.success_count["B"]
beta_b = 1 + self.failure_count["B"]
return sum(
beta(alpha_a + i, beta_b + beta_a) * 1.0 /
((beta_b + i) * beta(1 + i, beta_b) * beta(alpha_a, beta_a))
for i in range(alpha_b)
)
def tukeylambda_kurtosis(lam):
"""Kurtosis of the Tukey Lambda distribution.
Parameters
----------
lam : array_like
The lambda values at which to compute the variance.
Returns
-------
v : ndarray
The variance. For lam < -0.25, the variance is not defined, so
np.nan is returned. For lam = 0.25, np.inf is returned.
"""
lam = np.asarray(lam)
shp = lam.shape
lam = np.atleast_1d(lam).astype(np.float64)
# For absolute values of lam less than threshold, use the Pade
# approximation.
threshold = 0.055
# Use masks to implement the conditional evaluation of the kurtosis.
# lambda < -0.25: kurtosis = nan
low_mask = lam < -0.25
# lambda == -0.25: kurtosis = inf
negqrtr_mask = lam == -0.25
# lambda near 0: use Pade approximation
small_mask = np.abs(lam) < threshold
# else the "regular" case: use the explicit formula.
reg_mask = ~(low_mask | negqrtr_mask | small_mask)
# Get the 'lam' values for the cases where they are needed.
small = lam[small_mask]
reg = lam[reg_mask]
# Compute the function for each case.
k = np.empty_like(lam)
k[low_mask] = np.nan
k[negqrtr_mask] = np.inf
if small.size > 0:
k[small_mask] = _tukeylambda_kurt_p(small) / _tukeylambda_kurt_q(small)
if reg.size > 0:
numer = (1.0 / (4 * reg + 1) - 4 * beta(3 * reg + 1, reg + 1) +
3 * beta(2 * reg + 1, 2 * reg + 1))
denom = 2 * (1.0/(2 * reg + 1) - beta(reg + 1, reg + 1))**2
k[reg_mask] = numer / denom - 3
# The return value will be a numpy array; resetting the shape ensures that
# if `lam` was a scalar, the return value is a 0-d array.
k.shape = shp
return k
def __init__(self, param, data):
"""Initialize the model.
"""
super().__init__(param, data)
[param_a, param_p, param_c] = self.param
self.param_p = param_p / 100
self.param_a = param_a * 100 / self.data['maturity']**(.5 - param_c)
self.param_q = .03
self.param_b = (scp.beta(self.param_p, self.param_q) /
scp.beta(self.param_p+1/self.param_a,
self.param_q-1/self.param_a))
def __init__(self, a=1.5, b=1.03, p=.2, q=.2, c=.0, T=1):
#super(ClassName, self).__init__()
self.N = np.max([np.array(a).size, np.array(b).size,
np.array(p).size, np.array(q).size,
np.array(T).size, np.array(c).size])
self.p = p / 100
self.q = q / 100
# Scaling
self.a = 100 * a / T**(.5 - c)
# Martingale restriction
self.b = np.ones_like(b) * ss.beta(p, q) \
/ ss.beta(p+1/self.a, q-1/self.a)
def log_likelihood(params): a = params[0] b = params[1] likelihood = 0. for i in range(len(x)): choose = float(factorial(n))/(factorial(x[i])*factorial(n-x[i])) print ">>>>>>>>>>choose is :",choose pi = (choose*special.beta(a+x[i], n+b-x[i])/special.beta(a,b))*y[i]/np.sum(y) print ">>>>>>>>>>>>each pi is:", pi likelihood += pi # print likelihood # return -np.sum(np.log(likelihood)) return -1.*likelihood
def beta_binomial(k, n, a, b):
"""The pmf/pdf of the Beta-binomial distribution.
Computation based on beta function.
See: http://en.wikipedia.org/wiki/Beta-binomial_distribution
and http://mathworld.wolfram.com/BetaBinomialDistribution.html
k = a vector of non-negative integers <= n
n = an integer
a = an array of non-negative real numbers
b = an array of non-negative real numbers
"""
return (comb(n, k) * beta(k+a, n-k+b) / beta(a,b)).prod(0)
def binomial(n, k):
""" Binomial coefficient
n!
c = ---------
(n-k)! k!
Parameters
----------
n : float
n of (n, k)
k : float
k of (n, k)
Returns
-------
c : float
Examples
--------
First 3 values of 4 th row of Pascal triangle
>>> [binomial(4, k) for k in range(3)]
[1.0, 4.0, 6.0]
"""
if n <= k or n == 0:
return 0.
elif k == 0:
return 1.
return 1./(beta(n-k+1, k+1)*(n+1))
def betaPrior(x, alpha, beta, left, right): Beta = sp.beta(alpha, beta) pf = np.zeros(len(x)) ylog = ( (1.0-alpha-beta) * np.log(right-left) - (sp.gammaln(alpha) + sp.gammaln(beta) - sp.gammaln(alpha+beta)) + (alpha-1.0) * np.log(x - left) + (beta-1.0) * np.log(right - x) ) pf = np.exp(ylog) return pf
def ibetam(a, b, x):
"""
Incomplete beta function defined as the Mathematica Beta[x, a, b]:
Beta[x, a, b] = Integral[t^(a - 1) * (1 - t)^(b - 1), {t, 0, x}]
This routine only works for (0 < x < 1) & (b > 0) as required by JAM.
"""
# V1.0: Michele Cappellari, Oxford, 01/APR/2008
# V2.0: Use Hypergeometric function for negative a or b.
# From equation (6.6.8) of Abramoviz & Stegun (1964)
# MC, Oxford, 04/APR/2008
# V3.0: Use recurrence relation of equation (26.5.16)
# from Abramoviz & Stegun (1964) for (a < 0) & (b > 0).
# See the online book here http://www.nr.com/aands/
# After suggestion by Gary Mamon. MC, Oxford, 16/APR/2009
a = a + 3e-7 # Perturb to avoid singularities in gamma and betainc
if np.all(a > 0):
ib = special.betainc(a, b, x)
else:
p = int(np.ceil(np.abs(np.min(a))))
tot = np.zeros((x.size, a.size))
for j in range(p): # Do NOT use gamma recurrence relation to avoid instabilities
tot += special.gamma(j + b + a)/special.gamma(j + 1 + a)*x**(j + a)
ib = tot*(1 - x)**b/special.gamma(b) + special.betainc(a + p, b, x)
return ib*special.beta(a, b)
def init_gam( y , x , c=c ):
"""Ininitial guess for gamma function. Uniform distribution"""
out = y**(c-1) * ( x - y )**(c-1) / ( x**(2*c-1) ) / beta( c , c )
out[y>x] = 0
return out
def ppi(b):
# parameters for geophys prior from Martins and Stedinger p. 740
if abs(b) < 0.5:
pp, qq = 6, 9
return ((0.5 + b) ** (pp - 1) * (0.5 - b) ** (qq - 1)) / beta(pp, qq)
else:
return 0
def probability_of_n_purchases_up_to_time(self, t, n):
"""
Compute the probability of
P( N(t) = n | model )
where N(t) is the number of repeat purchases a customer makes in t units of time.
"""
r, alpha, a, b = self._unload_params('r', 'alpha', 'a', 'b')
_j = np.arange(0, n)
first_term = beta(a, b + n + 1) / beta(a, b) * gamma(r + n) / gamma(r) / gamma(n + 1) * (alpha / (alpha + t)) ** r * (t / (alpha + t)) ** n
finite_sum = (gamma(r + _j) / gamma(r) / gamma(_j + 1) * (t / (alpha + t)) ** _j).sum()
second_term = beta(a + 1, b + n) / beta(a, b) * (1 - (alpha / (alpha + t)) ** r * finite_sum)
return first_term + second_term
def betaAvgPrior(x, alpha, beta, left, right): Beta = sp.beta(alpha, beta) pf = np.zeros(len(x)) for i in range(len(x)): ylog = ( (1.0-alpha-beta) * np.log(right-left) - (sp.gammaln(alpha) + sp.gammaln(beta) - sp.gammaln(alpha+beta)) + (alpha-1.0) * np.log(x[i] - left) + (beta-1.0) * np.log(right - x[i]) ) pf[i] = np.mean(np.exp(ylog)) return pf
def Dbeta(a=1.5,b=2.5): # the beta distribution
return Distr(
name='beta[a={0},b={1}]'.format(a,b),
dom=(0.,1.), domv=(1.e-10,1.-1.e-10),
mean=a/(a+b), std=sqrt(a*b/(a+b+1))/(a+b),
pdf=lambda x: x**(a-1.)*(1.-x)**(b-1.)/beta(a,b),
cdf=lambda x: betainc(a,b,x),
)
def tukeylambda_variance(lam):
"""Variance of the Tukey Lambda distribution.
Parameters
----------
lam : array_like
The lambda values at which to compute the variance.
Returns
-------
v : ndarray
The variance. For lam < -0.5, the variance is not defined, so
np.nan is returned. For lam = 0.5, np.inf is returned.
Notes
-----
In an interval around lambda=0, this function uses the [4,4] Pade
approximation to compute the variance. Otherwise it uses the standard
formula (http://en.wikipedia.org/wiki/Tukey_lambda_distribution). The
Pade approximation is used because the standard formula has a removable
discontinuity at lambda = 0, and does not produce accurate numerical
results near lambda = 0.
"""
lam = np.asarray(lam)
shp = lam.shape
lam = np.atleast_1d(lam).astype(np.float64)
# For absolute values of lam less than threshold, use the Pade
# approximation.
threshold = 0.075
# Play games with masks to implement the conditional evaluation of
# the distribution.
# lambda < -0.5: var = nan
low_mask = lam < -0.5
# lambda == -0.5: var = inf
neghalf_mask = lam == -0.5
# abs(lambda) < threshold: use Pade approximation
small_mask = np.abs(lam) < threshold
# else the "regular" case: use the explicit formula.
reg_mask = ~(low_mask | neghalf_mask | small_mask)
# Get the 'lam' values for the cases where they are needed.
small = lam[small_mask]
reg = lam[reg_mask]
# Compute the function for each case.
v = np.empty_like(lam)
v[low_mask] = np.nan
v[neghalf_mask] = np.inf
if small.size > 0:
# Use the Pade approximation near lambda = 0.
v[small_mask] = _tukeylambda_var_p(small) / _tukeylambda_var_q(small)
if reg.size > 0:
v[reg_mask] = (2.0 / reg**2) * (1.0 / (1.0 + 2 * reg) -
beta(reg + 1, reg + 1))
v.shape = shp
return v
def _sp_subvector_error_out_of_range(radius, dimensions, subdimensions):
dist = SubvectorLength(dimensions, subdimensions)
sq_r = radius * radius
normalization = 1.0 - dist.cdf(radius)
b = (dimensions - subdimensions) / 2.0
aligned_integral = beta(subdimensions / 2.0 + 1.0, b) * (1.0 - betainc(
subdimensions / 2.0 + 1.0, b, sq_r))
cross_integral = beta((subdimensions + 1) / 2.0, b) * (1.0 - betainc(
(subdimensions + 1) / 2.0, b, sq_r))
numerator = (sq_r * normalization + (
aligned_integral - 2.0 * radius * cross_integral) / beta(
subdimensions / 2.0, b))
with np.errstate(invalid='ignore'):
return np.where(
numerator > np.MachAr().eps,
numerator / normalization, np.zeros_like(normalization))
def gb2_density(self, arg):
"""Density of the return.
"""
return (self.param_a * arg**(self.param_a*self.param_p-1) /
(self.param_b**(self.param_a*self.param_p) *
scp.beta(self.param_p, self.param_q) *
(1 + (arg / self.param_b) ** self.param_a) **
(self.param_p+self.param_q)))
def BetaDistribution (xs, alpha=0, beta=1): ys = [] for x in xs: # 1/B(a,b) * x^a-1 * (1-x)^(b-1) y = (x ** (alpha-1)) * ((1-x) ** (beta-1)) #BetaInv = mt.factorial(alpha+beta-1) / ( mt.factorial(alpha-1)*mt.factorial(beta-1) ) BetaInv = 1.0 / sp.beta(alpha,beta) #print y, FuncBeta y *= BetaInv ys.append(y) return ys
def binomial_conf_interval(x, n, conf=0.95):
if n == 0:
left = random.random()*(1 - conf)
return left, left + conf
b = special.beta(x+1, n-x+1)
def f(left_a):
left, right = max(1e-8, special.betaincinv(x+1, n-x+1, left_a)), min(1-1e-8, special.betaincinv(x+1, n-x+1, left_a + conf))
top = right**(x+1) * (1-right)**(n-x+1) * left*(1-left) - left**(x+1) * (1-left)**(n-x+1) * right * (1-right)
bottom = (x - n*right)*left*(1-left) - (x - n*left)*right*(1-right)
return top/bottom/b
left_a = find_root(f, (1-conf)/2, bounds=(0, 1-conf))
return special.betaincinv(x+1, n-x+1, left_a), special.betaincinv(x+1, n-x+1, left_a + conf)
def gam( y , x , a=a, b=b):
"""True post-fragmentation density distribution
used for data generation."""
out = y**(a-1) * ( np.abs( x - y )**(b-1) ) / ( x**(a+b-1) ) / beta( a , b )
if type(x) == np.ndarray or type(y) == np.ndarray:
out[y>x] = 0
out[ np.isnan(out) ] = 0
out[ np.isinf(out) ] = 0
return out
a_prior, b_prior = 1, 1
Y = stats.bernoulli(0.7).rvs(20)
N1, N0 = Y.sum(), len(Y) - Y.sum()
a_post = a_prior + N1
b_post = b_prior + N0
prior_pred_dist, post_pred_dist = [], []
N = 20
for k in range(N + 1):
post_pred_dist.append(
comb(N, k) * beta(k + a_post, N - k + b_post) / beta(a_post, b_post))
prior_pred_dist.append(
comb(N, k) * beta(k + a_prior, N - k + b_prior) /
beta(a_prior, b_prior))
fig, ax = plt.subplots()
ax.bar(np.arange(N + 1), prior_pred_dist, align='center', color='grey')
ax.set_title(f"Prior predictive distribution", fontweight='bold')
ax.set_xlim(-1, 21)
ax.set_xticks(list(range(N + 1)))
ax.set_xticklabels(list(range(N + 1)))
ax.set_ylim(0, 0.15)
ax.set_xlabel("number of success")
fig, ax = plt.subplots()
ax.bar(np.arange(N + 1), post_pred_dist, align='center', color='grey')
def pdf(self, F, y, Y_metadata=None):
eF = safe_exp(F)
pdf = (y**(eF[:, 0] - 1)) * (
(1 - y)**(eF[:, 1] - 1)) / beta(eF[:, 0], eF[:, 1])
return pdf
def cox_snell(x):
return (special.betainc(2 / 3., 2 / 3., x) *
special.beta(2 / 3., 2 / 3.))
def pdf(a, b, c, aa, bb, x):
return c / sp.beta(
a / c, b) * aa / bb * math.pow(x / bb, a * aa - 1) * math.pow(
1 + math.pow(x / bb, aa), -a - 1) * math.pow(
1 - math.pow(1 - 1 / (1 + math.pow(x / bb, aa)), c), b - 1)
abs(x) np.absolute(x) theta=np.linspace(0, np.pi, 3) print(theta, np.sin(theta), np.cos(theta), np.tan(theta)) x=[-1,0,1] print(x, np.arcsin(x), np.arccos(x), np.arctan(x)) x=[0,0.001,0.01,0.1] np.expm1(x) np.log1p(x) from scipy import special x=[1,5,10] special.gamma(x) special.gammaln(x) special.beta(x,2) x=np.arange(5) y=np.empty(5) np.multiply(x,10,out=y) y y=np.zeros(10) np.power(2,x,out=y[::2]) y x=np.arange(1,6) np.add.reduce(x) np.multiply.reduce(x) np.add.accumulate(x) np.multiply.accumulate(x) np.multiply.outer(x,x)
def _pdf(self, x, c):
return np.power((1.0-x*x),c/2.0-1) / special.beta(0.5,c/2.0)
def _mom(self, k, a, b):
return special.beta(a + k, b) / special.beta(a, b)
def expect(d, N):
return N * beta(1 + 1. / d, N)
def _kumaraswamy_moment(a, b, n):
a = np.asarray(a)
b = np.asarray(b)
return b * sp_special.beta(1.0 + n / a, b)
def Sigma_Exact(x):
# Parameters of the Hypergeometric
a, b, c = 0.5, 2.0, 2.5
y = pow((np.sqrt(x + 1) - np.sqrt(x)), 4)
return sc.beta(2, 0.5) * sc.hyp2f1(a, b, c, y) / x
def Sigma_Large_pt(x):
# Asymptotic behavior of the FO cross section at low pt
return sc.beta(2, 0.5) / x
def _munp(self, n, c):
return (1-(n % 2))*special.beta((n+1.0)/2,c/2.0)
def _cdf_skip(self, x, c):
#error inspecial.hyp2f1 for some values see tickets 758, 759
return 0.5 + x/special.beta(0.5,c/2.0)* \
special.hyp2f1(0.5,1.0-c/2.0,1.5,x*x)
def _pdf(self, x, a, b):
return x**(a-1)*(1-x)**(b-1)/ \
special.beta(a, b)
def pdf(self, f, g, y, Y_metadata=None):
ef = np.exp(f)
eg = np.exp(g)
pdf = y**(ef-1) * (1-y)**(eg-1) / beta(ef, eg)
return pdf
#!/usr/bin/python3
## file: rejection_beta.py
import numpy as np
import numpy.random as nprd
import scipy.special as scisp
# 设定参数
alpha = 4
beta = 2
# beta分布密度函数
f=lambda x: 1/scisp.beta(alpha,beta)* \
x**(alpha-1) * (1-x)**(beta-1)
# 计算密度函数最大值
M = f((1 - alpha) / (2 - alpha - beta))
print(M)
# 随机抽两个均匀分布，一个为(0,1)，一个为(0,M)，抽500个
N = 500
x = nprd.random(N)
u = nprd.random(N) * M
# 挑出使得u<beta密度函数的x
accepted = [i for i in range(N) if u[i] <= f(x[i])]
rand_beta = x[accepted] #生成的Beta分布随机数
rand_U_selected = u[accepted]
# 画图
import matplotlib.pyplot as plt
# 设定图像大小和坐标范围
plt.rcParams['figure.figsize'] = (8.0, 5.0)
plt.xlim(0, 1)
plt.ylim(0, M + 0.1)
# 横线和竖线
x_grid = np.linspace(0, 1, 100) #(0,1)均匀的100个点
y_M = np.ones(100) * M
def _sf(self, x, alpha):
return x * special.beta(x, alpha + 1)
def beta_distribution(x, a, b):
return (1.0 / ss.beta(a,b)) * x**(a-1) * (1-x)**(b-1)
def _pmf(self, x, alpha):
return alpha * special.beta(x, alpha + 1)
def _cdf(self, x, alpha):
return 1 - x * special.beta(x, alpha + 1)