def incompleteBetaFunction(x,a,b):
lbeta = math.lgamma(a + b) - math.lgamma(a) - math.lgamma(b) \
+ a * math.log(x) + b * math.log(1.0 - x)
if (x < (a + 1)/(a + b + 2)):
return math.exp(lbeta) * contFractionBeta(a,b,x)/a
else:
return 1 - math.exp(lbeta) * contFractionBeta(b,a,1.-x)/b
def LogCombinations(x,y):
u"""Calculates the logarithm of a binomial coefficient.
This avoids overflows. Implemented with gamma functions for efficiency"""
result=lgamma(x+1)
result-=lgamma(y+1)
result-=lgamma(x-y+1)
return result
def compute_likelihood(document, model, phi, var_gamma):
likelihood = 0
digsum = 0
var_gamma_sum = 0
dig = [0 for x in range(model.num_topics)]
for k in range(0, model.num_topics):
dig[k] = digamma(var_gamma[k])
var_gamma_sum = var_gamma[k] + var_gamma_sum
digsum = digamma(var_gamma_sum)
likelihood = math.lgamma(model.alpha * model.num_topics) \
- model.num_topics * math.lgamma(model.alpha) \
- (math.lgamma(var_gamma_sum))
for k in range(0, model.num_topics):
likelihood += ((model.alpha - 1) * (dig[k] - digsum)
+ math.lgamma(var_gamma[k]) - (var_gamma[k] - 1)
* (dig[k] - digsum))
for n in range(0, document.unique_word_count):
if phi[n][k] > 0:
likelihood += document.word_counts[n] * \
(phi[n][k] * ((dig[k] - digsum)
- math.log(phi[n][k])
+ model.log_prob_w[k][document.words[n]]))
return likelihood
def log_likelihood(self, full=False):
ll = (math.lgamma(self.K * self.alpha) - math.lgamma(self.K * self.alpha + self.N)
+ sum(math.lgamma(self.alpha + self.count[k]) for k in xrange(self.K))
- self.K * math.lgamma(self.alpha))
if full:
ll += self.prior.log_likelihood()
return ll
def log_likelihood(self, full=False):
ll = (math.lgamma(self.alpha) - math.lgamma(self.alpha + self.total_customers)
+ sum(math.lgamma(c) for tables in self.tables.itervalues() for c in tables)
+ self.ntables * math.log(self.alpha))
if full:
ll += self.base.log_likelihood(full=True) + self.prior.log_likelihood()
return ll
def calc_full(n, alphas):
""" Calculate the log likelihood under DirMult distribution with alphas=avec, given data counts of nvec."""
lg_sum_alphas = math.lgamma(alphas.sum())
sum_lg_alphas = np.sum(scipy.special.gammaln(alphas))
lg_sum_alphas_n = math.lgamma(alphas.sum() + n.sum())
sum_lg_alphas_n = np.sum(scipy.special.gammaln(n+alphas))
return lg_sum_alphas - sum_lg_alphas - lg_sum_alphas_n + sum_lg_alphas_n
def incomplete_gamma(x, s):
r"""
This function computes the incomplete lower gamma function
using the series expansion:
.. math::
\gamma(x, s) = x^s \Gamma(s)e^{-x}\sum^\infty_{k=0}
\frac{x^k}{\Gamma(s + k + 1)}
This series will converge strongly because the Gamma
function grows factorially.
Because the Gamma function does grow so quickly, we can
run into numerical stability issues. To solve this we carry
out as much math as possible in the log domain to reduce
numerical error. This function matches the results from
scipy to numerical precision.
"""
if x < 0:
return 1
if x > 1e3:
return math.gamma(s)
log_gamma_s = math.lgamma(s)
log_x = log(x)
value = 0
for k in range(100):
log_num = (k + s)*log_x + (-x) + log_gamma_s
log_denom = math.lgamma(k + s + 1)
value += math.exp(log_num - log_denom)
return value
def tdens(self, n, X):
C = (1.0 + (X * X) / (n * 1.0))
h = math.lgamma((n + 1.0) / 2.0) - math.lgamma(n / 2.0)
h = math.exp(h)
h = h / math.sqrt(math.pi) / math.sqrt(n)
Result = h * (C ** (-((n / 2.0) + (1.0 / 2.0))))
return Result
def log_likelihood(self, full=False):
ll = (math.lgamma(self.K * self.alpha) - math.lgamma(self.K * self.alpha + self.N)
+ sum(math.lgamma(self.alpha + self.count[k]) for k in self.count)
- len(self.count) * math.lgamma(self.alpha)) # zero counts
if full:
ll += self.prior.log_likelihood()
return ll
def UpdateKappa(self, it):
for ii in xrange(self.T-1):
for jj in xrange(self.p):
new_kappa = self.kappa[ii][jj]+(2*np.ceil(2*np.random.random())-3)*(np.random.geometric(1.0/(1+np.exp(self.log_kappa_q[ii][jj])))-1)
if new_kappa < 0:
accept = 0
else:
lam1 = self.lambda_[jj] + 1.0*self.kappa[ii][jj]
gam1 = self.lambda_[jj]/self.mu[jj] + self.delta[jj]
loglike = lam1*np.log(gam1) - math.lgamma(lam1)+(lam1-1)*np.log(self.psi[ii+1][jj])
pnmean = self.psi[ii][jj] * self.delta[jj]
loglike = loglike + 1.0*self.kappa[ii][jj]*np.log(pnmean) - math.lgamma(1.0*self.kappa[ii][jj]+1)
lam1 = self.lambda_[jj] + 1.0*new_kappa
gam1 = self.lambda_[jj]/self.mu[jj] + self.delta[jj]
new_loglike = lam1*np.log(gam1) - math.lgamma(lam1)+(lam1-1)*np.log(self.psi[ii+1][jj])
pnmean = self.psi[ii][jj]*self.delta[jj]
new_loglike = new_loglike + new_kappa*np.log(pnmean)-math.lgamma(1.0*new_kappa+1)
log_accept = new_loglike - loglike
accept =1
if np.isnan(log_accept) or np.isinf(log_accept):
accept =0
elif log_accept <0:
accept = np.exp(log_accept)
self.kappa_accept = self.kappa_accept + accept
self.kappa_count = self.kappa_count +1
if np.random.random() < accept:
self.kappa[ii][jj] = new_kappa
self.log_kappa_q[ii][jj] = self.log_kappa_q[ii][jj] + 1.0/it**0.55*(accept-0.3)
def UpdateKappaSigmaSq(self,it):
for ii in xrange(self.T-1):
new_kappa_sigma_sq = self.kappa_sigma_sq[ii]+(2*np.ceil(2*np.random.random())-3)*(np.random.geometric(1.0/(1+np.exp(self.log_kappa_sigma_sqq[ii])))-1)
if new_kappa_sigma_sq <0:
accept = 0
else:
lam1 = 1.0*self.lambda_sigma + self.kappa_sigma_sq[ii]
gam1 = 1.0*self.lambda_sigma/self.mu_sigma + 1.0*self.rho_sigma/(1-self.rho_sigma)*self.lambda_sigma/self.mu_sigma
loglike = lam1*np.log(gam1)-math.lgamma(lam1)+(lam1-1)*np.log(self.sigma_sq[ii+1])
pnmean = self.sigma_sq[ii]*self.rho_sigma/(1-self.rho_sigma)*self.lambda_sigma/self.mu_sigma
loglike = loglike + self.kappa_sigma_sq[ii]*np.log(pnmean)- math.lgamma(1.0*self.kappa_sigma_sq[ii]+1)
lam1 = 1.0*self.lambda_sigma + new_kappa_sigma_sq
gam1 = 1.0*self.lambda_sigma/self.mu_sigma + self.rho_sigma/(1-self.rho_sigma)*self.lambda_sigma/self.mu_sigma
new_loglike = lam1*np.log(gam1)-math.lgamma(lam1)+(lam1-1)*np.log(self.sigma_sq[ii+1])
pnmean = self.sigma_sq[ii]*self.rho_sigma/(1-self.rho_sigma)*self.lambda_sigma/self.mu_sigma
new_loglike = new_loglike + new_kappa_sigma_sq*np.log(pnmean)-math.lgamma(1.0*new_kappa_sigma_sq+1)
log_accept = new_loglike - loglike
accept =1
if np.isnan(log_accept) or np.isinf(log_accept):
accept = 0
elif log_accept <0:
accept = np.exp(log_accept)
self.kappa_lambda_sigma_accept = self.kappa_lambda_sigma_accept + accept
self.kappa_sigma_sq_count = self.kappa_sigma_sq_count +1
if np.random.random()<accept :
self.kappa_sigma_sq[ii] = new_kappa_sigma_sq
self.log_kappa_sigma_sqq[ii] = self.log_kappa_sigma_sqq[ii]+1.0/it**0.55*(accept-0.3)
def sample_document(self, m):
z = self.corpus[m]["state"] # Step1: カウントを減らす
if z > 0:
self.topic_document_freq[z] -= 1
self.topic_document_sum -= 1
for v in self.corpus[m]["bag_of_words"]:
self.topic_word_freq[z][v] -= 1
self.topic_word_sum[z] -= 1
n_d_v = defaultdict(float) # Step2: 事後分布の計算
n_d = 0.0
for v in self.corpus[m]["bag_of_words"]:
n_d_v[v] += 1.0
n_d += 1.0
p_z = defaultdict(lambda: 0.0)
for z in xrange(1, self.K + 1):
p_z[z] = math.log((self.topic_document_freq[z] + self.alpha) / (self.topic_document_sum + self.alpha*self.K))
p_z[z] += (math.lgamma(self.topic_word_sum[z] + self.beta*self.V) - math.lgamma(self.topic_word_sum[z] + n_d + self.beta*self.V))
for v in n_d_v.iterkeys():
p_z[z] += (math.lgamma(self.topic_word_freq[z][v] + n_d_v[v] + self.beta) - math.lgamma(self.topic_word_freq[z][v] + self.beta))
max_log = max(p_z.values()) # オーバーフロー対策
for z in p_z:
p_z[z] = math.exp(p_z[z] - max_log)
new_z = self.sample_one(p_z) # Step3: サンプル
self.corpus[m]["state"] = new_z # Step4: カウントを増やす
self.topic_document_freq[new_z] += 1
self.topic_document_sum += 1
for v in self.corpus[m]["bag_of_words"]:
self.topic_word_freq[new_z][v] += 1
self.topic_word_sum[new_z] += 1
def log_Beta(alphas):
"""
the beta function is a product-of-gammas over a gamma-of-sum
the gamma function generalizes factorials
tested against wolfram alpha
"""
#return product(map(gamma,alphas)) / gamma(sum(alphas))
return sum(lgamma(alpha) for alpha in alphas) - lgamma(sum(alphas))
def theta_likelihood(theta, S, J):
S += prior_s
J += prior_j
#If any of the values are 0 or negative return likelihood that will get rejected
if theta <= 0 or S <= 0 or J <= 0:
return 10000000
else:
return -(S * math.log(theta) + math.lgamma(theta) - math.lgamma(theta + J))
def multiTLogPDF(x,mu,Sigma,nu,p):
part1 = math.lgamma( 0.5 * (p + nu) )
part2 = - math.lgamma( 0.5 * nu ) - 0.5 * p * np.log( nu ) - 0.5 * p * np.log( np.pi )
part3 = - 0.5 * np.log( np.linalg.det(Sigma) )
part4 = - 0.5 * ( nu + p ) * np.log( 1.0 + nu**(-1) * np.dot( np.dot( (x - mu), np.linalg.inv(Sigma) ), (x - mu) ) )
return part1 + part2 + part3 + part4
def logchoose(ni, ki):
try:
lgn1 = lgamma(ni + 1)
lgk1 = lgamma(ki + 1)
lgnk1 = lgamma(ni - ki + 1)
except ValueError:
raise ValueError
return lgn1 - (lgnk1 + lgk1)
def Bernstein(n, k):
"""Bernstein polynomial.
"""
# binom
coeff = exp(lgamma(1+n)-lgamma(1+k)-lgamma(1+n-k))
return lambda x: coeff*x**k*(1-x)**(n-k)
def __compute_factor(self):
self._factor = lgamma (self.community.J + 1)
phi = table(self.community.abund)
phi += [0] * int (max (self.community.abund) - len (phi))
for spe in xrange (self.community.S):
self._factor -= log (max (1, self.community.abund[spe]))
for spe in xrange (int(max(self.community.abund))):
self._factor -= lgamma (phi[spe] + 1)
def _ewens_theta_likelihood (self, theta):
'''
returns the likelihood of theta for a given dataset
'''
if theta < 0:
return float ('-inf')
return self.community.S * log(theta) + lgamma(theta) - lgamma(theta + self.community.J)
def gammaln(x): if str(type(x))=="<type 'numpy.ndarray'>": result=n.zeros(x.shape) for index,value in n.ndenumerate(x): result[index]=lgamma(value) return result elif str(type(x))=="<type 'numpy.float64'>" or str(type(x))=="<type 'float'>": return lgamma(x)
def p(n,m,p):
"""Probability of m success out of n events, where an individual event succeeds with probability P... Useful for calculating <H^hat(j|w)> in semantic_information fun\
ction below"""
try:
return exp(lgamma(n+1) - lgamma(n-m+1) - lgamma(m+1) + m*log(p) + (n-m)*log(1.0-p))
except:
print "WARNING: domain range errer...returning 0"
return 0
def log_upsilon(n, counts):
k=0
result=0
for index, c in np.ndenumerate(counts):
result = result + math.lgamma(c+1)
k = k + 1
result = result + math.lgamma(k) - math.lgamma(k+n)
return(result)
def probD(d,l):
"""probability of document under label l, under marginalized theta"""
global gammat,Ccounts
sumGammaTheta = sum([x+gammat for x in Ccounts[l]])
NA = sumGammaTheta+sum(d.values())
res = math.lgamma(sumGammaTheta)-math.lgamma(NA)
for (wId,wCount) in d.iteritems():
res = res + (math.lgamma(wCount+gammat+Ccounts[l][wId])-math.lgamma(gammat+Ccounts[l][wId]))
return res
def psi(x): h=0.1e-5 if str(type(x))=="<type 'numpy.ndarray'>": result=n.zeros(x.shape) for index,value in n.ndenumerate(x): result[index]=(lgamma(value+h/2)-lgamma(value-h/2))/h return result else: return (lgamma(x+h/2)-lgamma(x-h/2))/h
def _log_likelihood(self, *tables):
tables = [t for ts in tables for t in ts]
ntables = len(tables)
ncustomers = sum(c for _, c in tables)
crp_ll = (math.lgamma(self.alpha) - math.lgamma(self.alpha + ncustomers)
+ sum(math.lgamma(c) for _, c in tables)
+ ntables * math.log(self.alpha))
base_ll = self.base.log_likelihood()
return crp_ll+base_ll, crp_ll, base_ll
def logMultinomial(hist): S = 0 for cat in categories: S += hist[cat] logB = 0 for cat in categories: logB += math.lgamma(hist[cat] + 1) logB -= math.lgamma(S + 1) return logB
def MSR(dataVec, priorVec):
prob = 0.0
prob = math.lgamma(priorVec.sum()) - math.lgamma(priorVec.sum() + dataVec.size)
x = dataVec.value_counts()
numLevels = x.size
for xLevel in xrange(0, numLevels):
prob = prob + math.lgamma(priorVec[xLevel] + x[x.index == (xLevel)]) - math.lgamma(priorVec[xLevel])
return prob
def incompleteBetaFunction(x,a,b):
try:
lbeta = math.lgamma(a + b) - math.lgamma(a) - math.lgamma(b) \
+ a * math.log(x) + b * math.log(1.0 - x)
except ValueError:
lbeta = float("nan")
if (x < (a + 1)/(a + b + 2)):
return math.exp(lbeta) * contFractionBeta(a,b,x)/a
else:
return 1 - math.exp(lbeta) * contFractionBeta(b,a,1.-x)/b
def logProb():
"""calculate log-probability of sampler state"""
global thetas,pi,gammaPi,Ccounts,sparsity,gammaTheta,documents,labels
#label probabilities, marginalized out
res = math.lgamma(sum(gammaPi)) - math.lgamma(sum([x+y for (x,y) in zip(labelCounts(-1),gammaPi)])) + sum([math.lgamma(labelCounts(-1)[i]+gammaPi[i]) - math.lgamma(gammaPi[i]) for i in range(K())])
for l in range(K()):
res = res + dirichletProb(thetas[l],gammaTheta) #word-distribution probabilities
for (doc,label) in zip(documents,labels):
res = res + probD(doc,label) #data-probabilities
return res
def _log_likelihood_heterozygous(cls, mean_depth, allele_depth1, allele_depth2, total_depth,
error_rate, allele_length1, allele_length2, non_zeros1, non_zeros2):
return sum([
-mean_depth * (1 + 0.5 * (allele_length1 + allele_length2 - non_zeros1 - non_zeros2)),
(allele_depth1 + allele_depth2) * math.log(0.5 * mean_depth),
-math.lgamma(allele_depth1 + 1),
-math.lgamma(allele_depth2 + 1),
(total_depth - allele_depth1 - allele_depth2) * math.log(error_rate),
(non_zeros1 + non_zeros2) * math.log(1 - poisson.pmf(0, 0.5 * mean_depth)),
])
def falin(x):
return -2 * math.sin(x) + (math.e**x) - math.lgamma(2) / 2**x
def fb(x):
return math.lgamma(x - 1) + math.cos(x - 1)
math.cos(math.radians(60)) math.tan(math.radians(60)) math.asin(math.radians(60)) math.acos(math.radians(60)) math.atan(math.radians(60)) math.atan2(math.radians(60)) math.hypot(math.radians(60)) ######################################## math.factorial(12) == math.gamma(13) True math.factorial(12) 479001600 math.gamma(13) 479001600.0 math.factorial(35) == math.gamma(36) False math.factorial(35) 10333147966386144929666651337523200000000 math.gamma(36) 1.0333147966386145e+40 ######################################## math.lgamma(45) == math.log(math.gamma(45)) True math.log(math.gamma(45)) 125.3172711493569 math.lgamma(45) 125.3172711493569 ########################################
def choose(n, k):
if k>n: out=float("-inf")
elif k==0: out=0
elif k==n: out=0
else: out=lgamma(n+1)-(lgamma(k+1)+lgamma(n-k+1))
return(out)
def lgamma(self):
self.result = False
self.current = math.lgamma(float(txtDisplay.get()))
self.display(self.current)