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distributions.py
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distributions.py
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import theano.tensor as tt
import theano
theano.config.optimizer = 'None'
theano.config.exception_verbosity = 'high'
import numpy as np
import pymc3 as mc
from ops import CachedLogDet, OrdLogDet
from theano.tensor import slinalg
from scipy import stats
def get_dense_cov(mu, tau):
cov = tt.nlinalg.matrix_inverse(tau)
return cov
from pymc3.distributions.distribution import draw_values, generate_samples
class SAR_Error(mc.Continuous):
"""
Adapted from pg. 63 of Anselin 1988, the logp for a SAR_error Model is
-N/2(ln(pi * sigma^2)) + ln|A| \
- .5*sigma^{-2}*(y - X\beta)'A'A(y-X\beta)
where A is I - rho W
Arguments
----------
mu : mean of the distribution
scale : univariate variance parameter
rho : autoregressive parameter
W : spatial weighting matrix
"""
def __init__(self, mu, scale, rho, W, method='LU',
*args, **kwargs):
super(SAR_Error, self).__init__(*args, **kwargs)
self.mean = self.median = self.mode = self.mu = mu= tt.as_tensor_variable(mu)
self.scale = scale
self.W = W
self.Ws = W.sparse
self.spld = CachedLogDet(self.W)
self.rho = rho
self.A = np.eye(W.n) - rho * W.sparse.toarray()
self.AtA = tt.dot(tt.transpose(self.A), self.A)
self.tau = tt.mul(self.AtA, self.scale **-2)
cov = get_dense_cov(self.mu, self.tau)
self.cov = tt.as_tensor_variable(cov)
def random(self, point=None, size=None):
mu, cov = draw_values([self.mu, self.cov], point=point)
def _random(mean, cov, size=None):
return stats.multivariate_normal.rvs(mean, cov,
None if size==mean.shape else size)
samples = generate_samples(_random, mean=mu, cov=cov,
dist_shape=self.shape, broadcast_shape=mu.shape,
size=size)
return samples
def logp(self, value):
"""
Adapted from pg. 63 of Anselin 1988, the logp for a SAR_error Model is
-N/2(ln(pi * sigma^2)) + ln|A| \
- .5*sigma^{-2}*(y - X\beta)'A'A(y-X\beta)
where A is I - rho W
"""
delta = value - self.mu
ld = self.spld(self.rho)
out = -self.W.n / 2.0 * tt.log(np.pi * self.scale)
out += ld
kern = tt.dot(self.AtA, delta)
#kern = tt.dot(ts.dense_from_sparse(taus), delta) # shape issue in MulSD.grad
#kern = ts.dot(taus, delta) # AssertionError in _is_sparse_variable(gz) in MulSD.grad
kern = tt.mul(delta, kern).sum()
kern *= self.scale**-2
kern /= 2.0
return out - kern
class SAR_Lag(mc.Continuous):
def __init__(self, mu, scale, rho, W, method='LU',
*args, **kwargs):
"""
Adapted from pg. 63 of Anselin 1988, the log likelihood for a lag model is:
-N/2(ln(pi * scale^2)) + ln|A| \
- .5*scale^{-2}*(Ay - mu)'(Ay-mu)
where A is I - rho W
Arguments
----------
mu : mean of the distribution
scale : univariate variance parameter
rho : autoregressive parameter
W : spatial weighting matrix
"""
super(SAR_Lag, self).__init__(*args, **kwargs)
self.mean = self.median = self.mode = self.mu = mu= tt.as_tensor_variable(mu)
self.scale = scale
self.W = W
self.Ws = W.sparse
self.spld = _log_det_dispatch[method.lower()](self.W)
self.rho = rho
self.A = np.eye(W.n) - rho * W.sparse.toarray()
def random(self, point=None, size=None):
raise NotImplementedError
# This needs to give Ai(X\beta + \epsilon)
mu, cov = draw_values([self.mu, self.cov], point=point)
def _random(mean, cov, size=None):
return stats.multivariate_normal.rvs(mean, cov,
None if size==mean.shape else size)
samples = generate_samples(_random, mean=mu, cov=cov,
dist_shape=self.shape, broadcast_shape=mu.shape,
size=size)
return samples
def logp(self, value):
"""
Adapted from pg. 63 of Anselin 1988, the logp for a SAR-Lag variable is:
-N/2(ln(pi * sigma^2)) + ln|A|
- .5*sigma^{-2}*(Ay - X\beta)'(Ay-X\beta)
where A is I - rho W
"""
delta = tt.dot(self.A, value) - self.mu
ld = self.spld(self.rho)
out = -self.W.n / 2.0 * tt.log(np.pi * self.scale)
out += ld
kern = tt.mul(delta, delta).sum()
kern *= self.scale**-2
kern *= .5
return out - kern
class SAR_Combo(mc.Continuous):
def __init__(self, mu, scale, rho, lambda_, W, M=None, method='Ord',
*args, **kwargs):
super(SAR_Combo, self).__init__(*args, **kwargs)
"""
Adapted from pg. 63 of Anselin 1988, the log likelihood for a combo model is:
-N/2(ln(pi * scale^2)) + ln|A| + ln|B| \
- .5*scale^{-2}*(Ay - mu)'B'B(Ay-mu)
where B is I - lambda_ M and A is I - rho W
Arguments
----------
mu : mean of the distribution
scale : univariate variance parameter
rho : autoregressive parameter for the endogenous Lag component
lambda_ : autoregressive parameter for the spatial error component
W : spatial weighting matrix for the endogenous Lag component
M : spatial weighting matrix for the spatial error component, assumed to be W if not provided.
"""
if M is None:
M = W
self.mean = self.median = self.mode = self.mu = mu= tt.as_tensor_variable(mu)
self.scale = scale
self.W = W
self.Ws = W.sparse
self.M = M
self.Ms = M.sparse
self.spldW = _log_det_dispatch[method.lower()](self.W)
self.spldM = _log_det_dispatch[method.lower()](self.M)
self.rho = rho
self.lambda_ = lambda_
self.A = np.eye(W.n) - rho * W.sparse.toarray()
self.B = np.eye(W.n) - lambda_ * M.sparse.toarray()
self.BtB = tt.dot(tt.transpose(self.B), self.B)
def random(self, point=None, size=None):
raise NotImplementedError
# This needs to give Ai(X\beta + (Bi)\epsilon)
mu, cov = draw_values([self.mu, self.cov], point=point)
def _random(mean, cov, size=None):
return stats.multivariate_normal.rvs(mean, cov,
None if size==mean.shape else size)
samples = generate_samples(_random, mean=mu, cov=cov,
dist_shape=self.shape, broadcast_shape=mu.shape,
size=size)
return samples
def logp(self, value):
delta = tt.dot(self.A, value) - self.mu
ldA = self.spld_A(self.rho)
ldB = self.spld_B(self.lambda_)
out = -self.W.n / 2.0 * tt.log(np.pi * self.scale)
out += ldA
out += ldB
kern = tt.dot(self.BtB, delta)
kern = tt.mul(delta, kern).sum()
kern *= self.scale**-2
kern *= .5
return out - ker
class SMA(mc.Continuous):
def __init__(self, mu, scale, rho, W, method='LU',
*args, **kwargs):
"""
the logp for a spatial moving average error structure is:
-N/2 log(pi * sigma^2) - log(|A|)
- .5 * sigma^{-2} * (Y - X\beta)(AAt)^{-1}(Y - X\beta)
where A is I + rho W
Arguments
----------
mu : mean of the distribution
scale : univariate variance parameter
rho : autoregressive parameter
W : spatial weighting matrix
"""
super(SMA, self).__init__(*args, **kwargs)
self.mean = self.median = self.mode = self.mu = mu= tt.as_tensor_variable(mu)
self.scale = scale
self.W = W
self.Ws = W.sparse
self.spld = _log_det_dispatch[method.lower()](self.W)
self.rho = rho
self.A = np.eye(W.n) + rho * W.sparse.toarray()
self.AAt = tt.dot(tt.transpose(self.A), self.A)
self.cov = tt.mul(self.AAt, self.scale **2)
self.cov = tt.as_tensor_variable(cov)
def random(self, point=None, size=None):
mu, cov = draw_values([self.mu, self.cov], point=point)
def _random(mean, cov, size=None):
return stats.multivariate_normal.rvs(mean, cov,
None if size==mean.shape else size)
samples = generate_samples(_random, mean=mu, cov=cov,
dist_shape=self.shape, broadcast_shape=mu.shape,
size=size)
return samples
def logp(self, value):
"""
the sparse cached log determinant assumes I - rho W = A, and computes
the log determinant of A wrt rho with cached W.
To get this right with the SMA, we need to use -rho in the logdet.
"""
delta = value - self.mu
ld = self.spld(-self.rho)
out = -self.W.n / 2.0 * tt.log(np.pi * self.scale)
out -= ld
kern = slinalg.solve(self.AAt, delta)
kern = tt.mul(delta, kern)
kern = kern.sum()
kern *= self.scale**-2
kern /= 2.0
return out - kern
class CAR(mc.Continuous):
def __init__(self, mu, scale, rho, W, method='LU', *args, **kwargs):
"""
the logp for a conditional autoregressive error structure is:
-N/2 log(pi * sigma^2) - log(|A|)
- .5 * sigma^{-2} * (Y - X\beta)(A)(Y - X\beta)
where A is D - rho W, and D is the degree matrix
for the graph contained in W
Arguments
----------
mu : mean of the distribution
scale : univariate variance parameter
rho : autoregressive parameter
W : spatial weighting matrix
"""
raise NotImplementedError('This class isnt ready to go yet')
self.mean = self.median = self.mode = self.mu = mu = tt.as_tensor_variable(mu)
self.scale = scale
self.W.transform = 'b'
self.D = np.diag(W.sparse.toarray().sum(axis=1))
self.Tau = (D - rho * W.sparse.toarray()) * scale **-2
self.spld = CachedLogDet(W)
def random(self, point=None, size=None):
raise NotImplementedError
mu, cov = draw_values([self.mu, self.cov], point=point)
def _random(mean, cov, size=None):
return stats.multivariate_normal.rvs(mean, cov,
None if size==mean.shape else size)
samples = generate_samples(_random, mean=mu, cov=cov,
dist_shape=self.shape, broadcast_shape=mu.shape,
size=size)
return samples
def logp(self, value):
delta = value - self.mu
#CAR has no squared structure in the covariance, so it retains |Omega|^{-1/2}
ld = self.spld(self.rho) * .5
out = - self.W.n / 2.0 * tt.log(np.pi * self.scale)
out -= ld
kern = tt.dot(self.Tau, delta)
kern = tt.mul(delta, kern).sum()
kern *= self.scale **-2
kern /= 2.0
return out - kern
_log_det_dispatch = dict(ord=OrdLogDet,
lu=CachedLogDet
)