def update_hyper(n, pi, alpha, sigma, tau, u, modelparam, mcmcparam):
abs_pi = pi.size
if modelparam['estimate_alpha']:
alpha_a = modelparam['alpha_a']
alpha_b = modelparam['alpha_b']
alpha = np.random.gamma(alpha_a + abs_pi,
1. / (alpha_b + ((u + tau) ** sigma
- tau ** sigma) / sigma))
if modelparam['estimate_sigma']:
# std = np.sqrt(1. / 4.)
std = 0.1
prop_sigma = 1 - np.random.lognormal(log(1 - sigma), std)
log_rate = log_density_sigma(alpha, prop_sigma, tau, u, abs_pi, n, pi) \
+ lognorm.logpdf(1 - sigma, std, scale=(1 - prop_sigma)) \
- log_density_sigma(alpha, sigma, tau, u, abs_pi, n, pi) \
- lognorm.logpdf(1 - prop_sigma, std, scale=(1 - sigma))
if np.isnan(log_rate):
log_rate = -np.Inf
if np.isinf(prop_sigma):
log_rate = -np.Inf
rate = min(1, np.exp(log_rate))
if np.random.random() < rate:
sigma = prop_sigma
if modelparam['estimate_tau']:
tau_a = modelparam['tau_a']
tau_b = modelparam['tau_b']
std = np.sqrt(1. / 4.)
logtau = log(tau)
prop_logtau = np.random.normal(logtau, std)
# compute acceptance probability
log_rate = log_density_tau(prop_logtau, alpha, sigma, u, n, abs_pi, tau_a,
tau_b) + norm.logpdf(logtau, prop_logtau, std) \
- log_density_tau(logtau, alpha, sigma, u, n, abs_pi, tau_a,
tau_b) - norm.logpdf(prop_logtau, logtau, std)
if np.isnan(log_rate):
log_rate = -np.Inf
if np.isinf(exp(prop_logtau)) or np.isnan(exp(prop_logtau)):
log_rate = -np.Inf
rate = min(1, np.exp(log_rate))
if np.random.random() < rate:
tau = exp(prop_logtau)
return alpha, sigma, tau
def generate(max_time, n_sequences, filename='stationary_renewal'):
times, nll = [], []
for _ in range(n_sequences):
s = np.sqrt(np.log(6*6+1))
mu = -s*s/2
tau = lognorm.rvs(s=s, scale=np.exp(mu), size=1000)
lpdf = lognorm.logpdf(tau, s=s, scale=np.exp(mu))
T = tau.cumsum()
T = T[T < max_time]
lpdf = lpdf[:len(T)]
score = -np.sum(lpdf)
times.append(T)
nll.append(score)
if filename is not None:
mean_number_items = sum(len(t) for t in times) / len(times)
nll = [n/mean_number_items for n in nll]
np.savez(f'{dataset_dir}/{filename}.npz', arrival_times=times, nll=nll, t_max=max_time, mean_number_items=mean_number_items)
else:
return times
def nll(arrival_times, std=6):
"""Negative log-likelihood of a renewal process.
Conditional density f*(t) is lognormal with given std.
"""
s = np.sqrt(np.log(std**2 + 1))
mu = -0.5 * s * s
inter_times = get_inter_times(arrival_times)
log_probs = lognorm.logpdf(inter_times, s=s, scale=np.exp(mu))
return -np.mean(log_probs)
def log_liklihood(rich, M, a,b,sigma):
p = 0
#p-= np.sum(((b*np.log(M)+np.log(a)-np.log(rich))**2)/(2*sigma**2)+np.log(sigma*rich))
#redefine A to be intercept at center rather than 0
#p-= np.sum(((b*(np.log(M)-13.5)+np.log(a)-np.log(rich))**2)/(2*sigma**2)+np.log(sigma*rich))
#p-= np.sum(((b*np.log(M)+a-np.log(rich))**2)/(2*sigma**2)+np.log(sigma*rich))#See Above
#p-= np.sum(((b*np.log(M-13.5)+a-np.log(rich))**2)/(2*sigma**2)+np.log(sigma*rich))#See Above
p+= np.sum(lognorm.logpdf(rich, sigma, loc = (b*(np.log(M)-offset)+np.log(a))))
return p
def lnprior_lognorm(p):
"""
Log-normal prior
"""
# note - for parameter definitions see
# http://nbviewer.ipython.org/url/xweb.geos.ed.ac.uk/~jsteven5/blog/lognormal_distributions.ipynb
mu = 7.3 # Mean of log(X)
sigma = 0.9 # Standard deviation of log(X)
shape = sigma # Scipy's shape parameter
scale = np.exp(mu) # Scipy's scale parameter
ret = lognorm.logpdf(p, shape, loc=0, scale=scale)
ret = ret.sum()
return ret
def mcmc(tree,max_steps,chain):
ne = 1
#generate list of times with k linages
time_linage = list()
internal_nodes_heights(tree.get_root(),time_linage)
time_linage.append(0)
time_linage.sort()
for i in range(1,len(time_linage)):
time_linage[i] = (time_linage[i] - time_linage[i-1])
del time_linage[0]
time_linage.reverse()
for i in range(max_steps):
u = np.random.uniform(-10,10) #random step
ne2 = ne + u
if(ne2 > 0): #handle by rejectioon
likelihood1 = 0
likelihood2 = 0
#generate log of the prior probability densities
prior1 = lognorm.logpdf(ne, s = 1.25,scale=np.exp(3))
prior2 = lognorm.logpdf(ne2, s = 1.25,scale=np.exp(3))
for k in range(2,len(tree.get_leaves())): #time interval with k linages
tk = time_linage[k-2]
b = (k*(k-1)*tk)/(2*ne)
c = (k*(k-1)*tk)/(2*ne2)
#compute the coalescent likelihood
likelihood1 += (-np.log(ne) - b)
likelihood2 += (-np.log(ne2) - c)
#obtain posterior distributions
post1 = likelihood1 + (prior1)
post2 = likelihood2 + (prior2)
post = post2 - post1
#compute acceptance probability
if post > -800 and post < 700: #handle overflow by reject large values
a = min(1,np.exp(post))
v = np.random.uniform(0,1)
if(a > v):
ne = ne2
chain.append(ne) #add to MCMC chain
def lnprior_Y0(Y0):
"""
Prior on detector state at t=0
"""
if Y0.min() <= 0.0:
return -np.inf
# note - for parameter definitions see
# http://nbviewer.ipython.org/url/xweb.geos.ed.ac.uk/~jsteven5/blog/lognormal_distributions.ipynb
sigma = np.log(2.0) # Standard deviation of log(X) - factor of two
shape = sigma # Scipy's shape parameter
scale = Y0_mu_prior # Scipy's scale parameter = np.exp( mean of log(X) )
ret = lognorm.logpdf(Y0, shape, loc=0, scale=scale)
ret = ret.sum()
return ret
def logprob(self, x):
"""
:param x: np.array
:return: float
"""
if np.any(x <= 0):
return -np.inf
x = np.sqrt(x)
dy_dx = 2.0 * x
return np.sum(lognorm.logpdf(x, s=self.scale,
scale=self.param)) - np.log(dy_dx)
def lnprior_lognorm_from_data(p, observed_data, n_window=10, sigma_factor=2, _cache=[]):
"""
Log-normal prior where the distribution parameters vary with time
"""
if len(_cache) == 0 or len(_cache[0]) != len(p):
mu, sigma = running_lognormal_parameters(p, observed_data,
n_window=n_window, sigma_factor=sigma_factor)
while len(_cache) > 0:
_cache.pop()
_cache.extend((mu,sigma))
else:
mu, sigma = _cache
shape = sigma # Scipy's shape parameter
scale = np.exp(mu) # Scipy's scale parameter
ret = lognorm.logpdf(p, shape, loc=0, scale=scale)
ret = ret.sum()
return ret
def logprob(self, x):
"""
:param x: np.array(nxm)
:return: float
"""
if len(x.shape) == 1:
x = x.reshape(len(x), 1)
if np.any(x < 0):
return -np.inf
llh = 0
for dim in xrange(self.dimension):
llh += np.sum(lognorm.logpdf(x[:, dim], s=self.scale[dim], scale=np.exp(self.mu[dim])))
return llh
def sampleproposal(B,D,K):
#sample pis from dirichlet(1,1...1,1)
pi_alpha = 0.1
pi_pr = np.random.dirichlet(pi_alpha* np.ones(K),size = B)
logpidensity =dirichlet.logpdf(np.transpose(pi_pr),pi_alpha*np.ones(K))
#sample mus from normal(0,25)
mu_std = 5
mu_pr = mu_std*np.random.randn(B,K,D)
logmudensity = np.sum(norm.logpdf(mu_pr, scale = mu_std),axis = (1,2))
#sample sigmas from lognormal(0,1)
sigma_pr = np.exp(np.random.randn(B,K,D))
logsigmadensity = np.sum(lognorm.logpdf(sigma_pr, s = 1),axis = (1,2))
logpropdensity = logpidensity + logmudensity + logsigmadensity
logpipriordensity = dirichlet.logpdf(np.transpose(pi_pr),np.ones(K))
logpriordensity = logpipriordensity + np.sum(norm.logpdf(mu_pr, scale = 1),axis = (1,2)) +logsigmadensity
return pi_pr,mu_pr,sigma_pr, logpriordensity, logpropdensity
def sigma_loglikelihood(sigma):
wp_loglikelihood = lognorm.logpdf(wp, s = sigma, scale = wp_pred_exp)
return -(np.sum(wp_loglikelihood))
