def log_z(self, beta=1., ensembles=None):
"""
Use trapezoidal rule to evaluate the partition function.
"""
from numpy import array, multiply, reshape
is_float = False
if type(beta) == float:
beta = reshape(array(beta), (-1,))
is_float = True
x = self._ex[0, 1:] - self._ex[0, :-1]
y = self._ex[0]
for i in range(1, self._ex.shape[0]):
x = multiply.outer(x, self._ex[i, 1:] - self._ex[i, :-1])
y = multiply.outer(y, self._ex[i])
y = -multiply.outer(beta, y) + self._log_g
y = reshape(array([y.T[1:], y.T[:-1]]), (2, -1))
y = log_sum_exp(y, 0) - log(2)
y = reshape(y, (-1, len(beta))).T + log(x)
log_z = log_sum_exp(y.T, 0)
if is_float:
return float(log_z)
else:
return log_z
def log_z(self, beta=1., ensembles=None):
"""
Use trapezoidal rule to evaluate the partition function.
"""
from numpy import array, multiply, reshape
is_float = False
if type(beta) == float:
beta = reshape(array(beta), (-1, ))
is_float = True
x = self._ex[0, 1:] - self._ex[0, :-1]
y = self._ex[0]
for i in range(1, self._ex.shape[0]):
x = multiply.outer(x, self._ex[i, 1:] - self._ex[i, :-1])
y = multiply.outer(y, self._ex[i])
y = -multiply.outer(beta, y) + self._log_g
y = reshape(array([y.T[1:], y.T[:-1]]), (2, -1))
y = log_sum_exp(y, 0) - log(2)
y = reshape(y, (-1, len(beta))).T + log(x)
log_z = log_sum_exp(y.T, 0)
if is_float:
return float(log_z)
else:
return log_z
def estimate_Z(self, beta=1.):
prob = - 0.5 * beta * self.delta / self.sigma**2 \
- self.D * beta * self.M * np.log(self.sigma)
if not self.sequential_prior:
prob += log(self.w)
prob = np.exp((prob.T - log_sum_exp(prob.T, 0)).T)
for n in range(self.N):
self.Z[n,:] = np.random.multinomial(1, prob[n])
else:
a = log(self.w)
b = log((1-self.w)/(self.K-1))
for n in range(self.N):
p = prob[n]
if n > 1:
p += self.Z[n-1] * a + (1-self.Z[n-1]) * b
p = np.exp(p - log_sum_exp(p))
self.Z[n,:] = np.random.multinomial(1, p)
def log_z(self, n=500, integration='simpson'):
"""
Calculate the log partion function.
"""
from numpy import pi, linspace, max
from csb.numeric import log, exp
if integration == 'simpson':
from csb.numeric import simpson_2d
x = linspace(0., 2 * pi, 2 * n + 1)
dx = x[1] - x[0]
f = -self.beta * self.energy(x)
f_max = max(f)
f -= f_max
I = simpson_2d(exp(f))
return log(I) + f_max + 2 * log(dx)
elif integration == 'trapezoidal':
from csb.numeric import trapezoidal_2d
x = linspace(0., 2 * pi, n)
dx = x[1] - x[0]
f = -self.beta * self.energy(x)
f_max = max(f)
f -= f_max
I = trapezoidal_2d(exp(f))
return log(I) + f_max + 2 * log(dx)
else:
raise NotImplementedError(
'Choose from trapezoidal and simpson-rule Integration')
def log_Z(self, beta=1.):
from csb.numeric import log
from scipy.special import gammainc, gammaln
return log(0.5 * self.d) + log(gammainc(0.5 * self.d, 0.5 * self.k)) + \
gammaln(0.5 * self.d) + (0.5 * self.d) * (log(2) - log(self.k))
def log_Z(self, beta=1.):
from csb.numeric import log
from scipy.special import gammainc, gammaln
return log(0.5 * self.d) + log(gammainc(0.5 * self.d, 0.5 * self.k)) + \
gammaln(0.5 * self.d) + (0.5 * self.d) * (log(2) - log(self.k))
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
y = log(data).mean() - log((data ** -1).mean())
alpha = inv_digamma_minus_log(numpy.clip(y,
- 1e308,
- 1e-300))
alpha = abs(alpha)
beta = numpy.clip(alpha /
(data ** (-1)).mean(),
1e-100, 1e100)
pdf.alpha, pdf.beta = alpha, beta
return pdf
def log_z(self, n=500, integration='simpson'):
"""
Calculate the log partion function.
"""
from numpy import pi, linspace, max
from csb.numeric import log, exp
if integration == 'simpson':
from csb.numeric import simpson_2d
x = linspace(0., 2 * pi, 2 * n + 1)
dx = x[1] - x[0]
f = -self.beta * self.energy(x)
f_max = max(f)
f -= f_max
I = simpson_2d(exp(f))
return log(I) + f_max + 2 * log(dx)
elif integration == 'trapezoidal':
from csb.numeric import trapezoidal_2d
x = linspace(0., 2 * pi, n)
dx = x[1] - x[0]
f = -self.beta * self.energy(x)
f_max = max(f)
f -= f_max
I = trapezoidal_2d(exp(f))
return log(I) + f_max + 2 * log(dx)
else:
raise NotImplementedError(
'Choose from trapezoidal and simpson-rule Integration')
def log_prob(self, x):
a = self['a']
b = self['b']
p = self['p']
lz = 0.5 * p * (log(a) - log(b)) - log(2 * scipy.special.kv(p, sqrt(a * b)))
return lz + (p - 1) * log(x) - 0.5 * (a * x + b / x)
def log_prob(self, x):
from numpy.linalg import det
mu = self.mu
S = self.sigma
D = len(mu)
q = self.__q(x)
return -0.5 * (D * log(2 * pi) + log(abs(det(S)))) - 0.5 * q ** 2
def testLog(self):
from csb.numeric import log, LOG_MAX, LOG_MIN
from numpy import log as ref_log
x = np.linspace(LOG_MIN, LOG_MAX, 1000000)
self.assertTrue((log(x) == ref_log(x)).all())
self.assertEqual(log(10 * LOG_MAX), log(LOG_MAX))
self.assertEqual(log(0.1 * LOG_MIN), log(LOG_MIN))
def log_prob(self, x):
from numpy.linalg import det
mu = self.mu
S = self.sigma
D = len(mu)
q = self.__q(x)
return -0.5 * (D * log(2 * pi) + log(abs(det(S)))) - 0.5 * q**2
def log_likelihood_reduced(self):
"""
Log-likelihood of the marginalized model (no auxiliary indicator variables)
@rtype: float
"""
from csb.numeric import log, log_sum_exp
s_sq = (self.sigma**2).clip(1e-300, 1e300)
log_p = log(self.w) - 0.5 * \
(self.delta / s_sq + self.dimension * log(2 * numpy.pi * s_sq))
return log_sum_exp(log_p.T).sum()
def energy(self, raw_energies):
from numpy import isinf
if isinf(self.beta):
m = (raw_energies >= self.e_max).astype('f')
return - m * log(0.)
else:
x = 1 + exp(self.beta * (raw_energies - self.e_max))
return log(x)
def energy(self, raw_energies):
from numpy import isinf
if isinf(self.beta):
m = (raw_energies >= self.e_max).astype('f')
return -m * log(0.)
else:
x = 1 + exp(self.beta * (raw_energies - self.e_max))
return log(x)
def estimate_scales(self, beta=1.0):
"""
Update scales from current model and samples
@param beta: inverse temperature
@type beta: float
"""
from csb.numeric import log, log_sum_exp, exp
s_sq = (self.sigma ** 2).clip(1e-300, 1e300)
Z = (log(self.w) - 0.5 * (self.delta / s_sq + self.dimension * log(s_sq))) * beta
self.scales = exp(Z.T - log_sum_exp(Z.T))
def log_likelihood_reduced(self):
"""
Log-likelihood of the marginalized model (no auxiliary indicator variables)
@rtype: float
"""
from csb.numeric import log, log_sum_exp
s_sq = (self.sigma ** 2).clip(1e-300, 1e300)
log_p = log(self.w) - 0.5 * \
(self.delta / s_sq + self.dimension * log(2 * numpy.pi * s_sq))
return log_sum_exp(log_p.T).sum()
def log_prob(self, x):
a = self['a']
b = self['b']
p = self['p']
lz = 0.5 * p * (log(a) - log(b)) - log(
2 * scipy.special.kv(p, sqrt(a * b)))
return lz + (p - 1) * log(x) - 0.5 * (a * x + b / x)
def estimate_scales(self, beta=1.0):
"""
Update scales from current model and samples
@param beta: inverse temperature
@type beta: float
"""
from csb.numeric import log, log_sum_exp, exp
s_sq = (self.sigma**2).clip(1e-300, 1e300)
Z = (log(self.w) - 0.5 *
(self.delta / s_sq + self.dimension * log(s_sq))) * beta
self.scales = exp(Z.T - log_sum_exp(Z.T))
def log_prob(self, x):
mu = self.mu
scale = self.shape
x = numpy.array(x)
if numpy.min(x) <= 0:
raise ValueError('InverseGaussian is defined for x > 0')
y = -0.5 * scale * (x - mu) ** 2 / (mu ** 2 * x)
z = 0.5 * (log(scale) - log(2 * pi * x ** 3))
return z + y
def estimate(self, context, data):
mu = mean(data)
logmean = mean(log(data))
a = 0.5 / (log(mu) - logmean)
for dummy in range(self.n_iter):
a = inv_psi(logmean - log(mu) + log(a))
return Gamma(a, a / mu)
def log_prob(self, x):
mu = self.mu
scale = self.shape
x = numpy.array(x)
if numpy.min(x) <= 0:
raise ValueError('InverseGaussian is defined for x > 0')
y = -0.5 * scale * (x - mu)**2 / (mu**2 * x)
z = 0.5 * (log(scale) - log(2 * pi * x**3))
return z + y
def log_prob(self, x):
from csb.numeric import log_sum_exp
dim = self._d
s = self.scales
log_p = numpy.squeeze(-numpy.multiply.outer(x * x, 0.5 * s)) + \
numpy.squeeze(dim * 0.5 * (log(s) - log(2 * numpy.pi)))
if self._prior is not None:
log_p += numpy.squeeze(self._prior.log_prob(s))
return log_sum_exp(log_p.T, 0)
def estimate(self, context, data):
mu = mean(data)
logmean = mean(log(data))
a = 0.5 / (log(mu) - logmean)
for dummy in range(self.n_iter):
a = inv_psi(logmean - log(mu) + log(a))
return Gamma(a, a / mu)
def log_likelihood(self):
"""
Log-likelihood of the extended model (with indicators)
@rtype: float
"""
from csb.numeric import log
from numpy import pi, sum
n = self.scales.sum(1)
N = self.dimension
Z = self.scales.T
s_sq = (self.sigma**2).clip(1e-300, 1e300)
return sum(n * log(self.w)) - 0.5 * \
(sum(Z * self.delta / s_sq) + N * sum(n * log(2 * pi * s_sq)) + sum(log(s_sq)))
def log_likelihood(self):
"""
Log-likelihood of the extended model (with indicators)
@rtype: float
"""
from csb.numeric import log
from numpy import pi, sum
n = self.scales.sum(1)
N = self.dimension
Z = self.scales.T
s_sq = (self.sigma ** 2).clip(1e-300, 1e300)
return sum(n * log(self.w)) - 0.5 * \
(sum(Z * self.delta / s_sq) + N * sum(n * log(2 * pi * s_sq)) + sum(log(s_sq)))
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
## sufficient statistics
h = harmonic_mean(numpy.clip(data, 1e-308, 1e308))
g = geometric_mean(numpy.clip(data, 1e-308, 1e308))
n = len(data)
samples = []
a = numpy.mean(1 / data)
v = numpy.std(1 / data) ** 2
beta = a / v
alpha = beta * a
for i in range(self.n_samples):
## sample alpha with ARS
logp = ARSPosteriorAlpha(n * (log(beta) - log(g)) - context.hyper_alpha.beta,
context.hyper_alpha.alpha - 1., n)
ars = csb.statistics.ars.ARS(logp)
ars.initialize(logp.initial_values()[:2], z0=0.)
alpha = numpy.abs(ars.sample())
if alpha is None:
raise ValueError("Sampling failed")
## sample beta from Gamma distribution
beta = numpy.random.gamma(n * alpha + context.hyper_beta.alpha, \
1 / (n / h + context.hyper_beta.beta))
samples.append((alpha, beta))
pdf.alpha, pdf.beta = samples[-1]
return pdf
def BIC(self):
"""
Bayesian information criterion, calculated as
BIC = M * ln(sigma_e^2) + K * ln(M)
@rtype: float
"""
from numpy import log
n = self.M
k = self.K
error_variance = sum(self.sigma**2 * self.w)
return n * log(error_variance) + k * log(n)
def BIC(self):
"""
Bayesian information criterion, calculated as
BIC = M * ln(sigma_e^2) + K * ln(M)
@rtype: float
"""
from numpy import log
n = self.M
k = self.K
error_variance = sum(self.sigma ** 2 * self.w)
return n * log(error_variance) + k * log(n)
def estimate(self, n_iter=10000, tol=1e-10):
e_ij = numpy.array(
[ensemble.energy(self._e) for ensemble in self._ensembles]).T
f = self._f
log_n = log(self._n)
self._L = []
for _i in range(n_iter):
## update density of states
log_g = -log_sum_exp((-e_ij - f + log_n).T, 0)
log_g -= log_sum_exp(log_g)
## update free energies
f = log_sum_exp((-e_ij.T + log_g).T, 0)
self._L.append((self._n * f).sum() - log_g.sum())
self._f = f
self._log_g = log_g
if self._stop_criterium(tol):
break
return f, log_g
def testTruncatedGamma(self):
alpha = 2.
beta = 1.
x_min = 0.1
x_max = 5.
x = truncated_gamma(10000, alpha, beta, x_min, x_max)
self.assertTrue((x <= x_max).all())
self.assertTrue((x >= x_min).all())
hy, hx = density(x, 100)
hx = 0.5 * (hx[1:] + hx[:-1])
hy = hy.astype('d')
with warnings.catch_warnings(record=True) as warning:
warnings.simplefilter("always")
hy /= (hx[1] - hx[0]) * hy.sum()
self.assertLessEqual(len(warning), 1)
if len(warning) == 1:
warning = warning[0]
self.assertEqual(warning.category, RuntimeWarning)
self.assertTrue(str(warning.message).startswith('divide by zero encountered'))
x = numpy.linspace(x_min, x_max, 1000)
p = (alpha - 1) * log(x) - beta * x
p -= log_sum_exp(p)
p = exp(p) / (x[1] - x[0])
def estimate(self, n_iter=10000, tol=1e-10):
e_ij = numpy.array([ensemble.energy(self._e)
for ensemble in self._ensembles]).T
f = self._f
log_n = log(self._n)
self._L = []
for _i in range(n_iter):
## update density of states
log_g = -log_sum_exp((-e_ij - f + log_n).T, 0)
log_g -= log_sum_exp(log_g)
## update free energies
f = log_sum_exp((-e_ij.T + log_g).T, 0)
self._L.append((self._n * f).sum() - log_g.sum())
self._f = f
self._log_g = log_g
if self._stop_criterium(tol):
break
return f, log_g
def inv_digamma_minus_log(y, tol=1e-10, n_iter=100):
"""
Solve y = psi(alpha) - log(alpha) for alpha by fixed point
integration.
"""
if y >= -log(6.):
x = 1 / (2 * (1 - exp(y)))
else:
x = 1.e-10
for _i in range(n_iter):
z = approx_psi(x) - log(x) - y
if abs(z) < tol:
break
x -= x * z / (x * d_approx_psi(x) - 1)
x = abs(x)
return x
def testTruncatedGamma(self):
alpha = 2.
beta = 1.
x_min = 0.1
x_max = 5.
x = truncated_gamma(10000, alpha, beta, x_min, x_max)
self.assertTrue((x <= x_max).all())
self.assertTrue((x >= x_min).all())
hy, hx = density(x, 100)
hx = 0.5 * (hx[1:] + hx[:-1])
hy = hy.astype('d')
with warnings.catch_warnings(record=True) as warning:
warnings.simplefilter("always")
hy /= (hx[1] - hx[0]) * hy.sum()
self.assertLessEqual(len(warning), 1)
if len(warning) == 1:
warning = warning[0]
self.assertEqual(warning.category, RuntimeWarning)
self.assertTrue(
str(warning.message).startswith(
'divide by zero encountered'))
x = numpy.linspace(x_min, x_max, 1000)
p = (alpha - 1) * log(x) - beta * x
p -= log_sum_exp(p)
p = exp(p) / (x[1] - x[0])
def log_prob(self, x):
mu = self.mu
beta = self.beta
z = (x - mu) / beta
return log(1. / beta) - z - exp(-z)
def testLogTrapezoidal(self):
from csb.numeric import log_trapezoidal, log
x = np.linspace(-100., 100, 1000)
y = -0.5 * x * x - log(np.sqrt(2 * np.pi))
self.assertTrue(abs(log_trapezoidal(y, x)) <= 1e-8)
def log_prob(self, x):
mu = self.mu
sigma = self.sigma
return log(
1.0 / sqrt(2 * pi * sigma**2)) - (x - mu)**2 / (2 * sigma**2)
def log_prob(self, x):
mu = self.mu
beta = self.beta
z = (x - mu) / beta
return log(1. / beta) - z - exp(-z)
def log_prob(self, x):
mu = self.mu
alpha = self.alpha
beta = self.beta
return log(beta / (2.0 * alpha)) - gammaln(1. / beta) - power(fabs(x - mu) / alpha, beta)
def __call__(self, x):
from scipy.special import gammaln
return self.a * x - \
self.n * gammaln(numpy.clip(x, 1e-308, 1e308)) + \
self.b * log(x), \
self.a - self.n * psi(x) + self.b / x
def log_prob(self, x):
mu = self.mu
alpha = self.alpha
beta = self.beta
return log(beta / (2.0 * alpha)) - gammaln(1. / beta) - power(
fabs(x - mu) / alpha, beta)
def energy(self, raw_energies):
q = self.q
e_min = self.e_min
if (q < 1 + 1e-10):
return raw_energies * q
else:
return log(1 + (raw_energies - e_min) * (q - 1)) * q / (q - 1) + e_min
def estimate(self, context, data):
"""
Generate samples from the posterior of alpha and beta.
For beta the posterior is a gamma distribution and analytically
acessible.
The posterior of alpha can not be expressed analytically and is
aproximated using adaptive rejection sampling.
"""
pdf = GammaPrior()
## sufficient statistics
a = numpy.mean(data)
b = exp(numpy.mean(log(data)))
v = numpy.std(data) ** 2
n = len(data)
beta = a / v
alpha = beta * a
samples = []
for _i in range(self.n_samples):
## sample beta from Gamma distribution
beta = numpy.random.gamma(n * alpha + context._hyper_beta.alpha,
1 / (n * a + context._hyper_beta.beta))
## sample alpha with ARS
logp = ARSPosteriorAlpha(n * log(beta * b)\
- context.hyper_alpha.beta,
context.hyper_alpha.alpha - 1., n)
ars = csb.statistics.ars.ARS(logp)
ars.initialize(logp.initial_values()[:2], z0=0.)
alpha = ars.sample()
if alpha is None:
raise ValueError("ARS failed")
samples.append((alpha, beta))
pdf.alpha, pdf.beta = samples[-1]
return pdf
def log_likelihood(self):
"""
returns the log-likelihood
"""
N = self.Z.sum(0)
L = - 0.5 * np.sum(self.Z * self.delta / self.sigma**2) \
- 0.5 * self.D * self.M * np.sum(N * log(2 * np.pi * self.sigma**2))
return L
def log_prior(self):
N = self.Z.sum(0)
p = -0.5 * np.sum(self.t**2) / self.sigma_t**2
p+= -0.5 * np.sum(self.Y**2) / self.sigma_Y**2
p+= np.sum((self.alpha_precision-1) * log(1/self.sigma**2) - self.beta_precision / self.sigma**2)
if not self.sequential_prior:
p += np.sum((N + self.alpha_w - 1) * np.log(self.w))
else:
l = self.membership
Q = np.sum(l[1:] == l[:-1])
p += (Q + self.alpha_w - 1) * log(self.w) + \
(self.N - Q - self.alpha_w - 1) * log(1-self.w)
return p
def energy(self, raw_energies):
q = self.q
e_min = self.e_min
if (q < 1 + 1e-10):
return raw_energies * q
else:
return log(1 + (raw_energies - e_min) *
(q - 1)) * q / (q - 1) + e_min
def log_g(self, normalize=True):
e_ij = numpy.array([ensemble.energy(self._e)
for ensemble in self._ensembles]).T
log_g = -log_sum_exp((-e_ij - self._f + log(self._n)).T, 0)
if normalize:
log_g -= log_sum_exp(log_g)
return log_g
def log_g(self, normalize=True):
e_ij = numpy.array(
[ensemble.energy(self._e) for ensemble in self._ensembles]).T
log_g = -log_sum_exp((-e_ij - self._f + log(self._n)).T, 0)
if normalize:
log_g -= log_sum_exp(log_g)
return log_g
def entropy(self, n=500):
"""
Calculate the entropy of the model.
@param n: number of integration points for numerical integration
@type n: integer
"""
from csb.numeric import trapezoidal_2d
from numpy import pi, linspace, max
from csb.numeric import log, exp
x = linspace(0., 2 * pi, n)
dx = x[1] - x[0]
f = -self.beta * self.energy(x)
f_max = max(f)
log_z = log(trapezoidal_2d(exp(f - f_max))) + f_max + 2 * log(dx)
average_energy = trapezoidal_2d(f * exp(f - f_max))\
* exp(f_max + 2 * log(dx) - log_z)
return -average_energy + log_z
