def _compute_betas(y, x):
"""
compute MLE coefficients using iwls routine
Methods: p189, Iteratively (Re)weighted Least Squares (IWLS),
Fotheringham, A. S., Brunsdon, C., & Charlton, M. (2002).
Geographically weighted regression: the analysis of spatially varying relationships.
"""
xT = x.T
xtx = spdot(xT, x)
xtx_inv = la.inv(xtx)
xtx_inv = sp.csr_matrix(xtx_inv)
xTy = spdot(xT, y, array_out=False)
betas = spdot(xtx_inv, xTy)
return betas
def logp_lambda_prec(state, val):
"""
Compute the log likelihood of the upper-level spatial correlation parameter using
sparse operations and the precision matrix, rather than the covariance matrix.
"""
st = state
if (val < st.Lambda_min) or (val > st.Lambda_max):
return np.array([-np.inf])
PsiLambdai = st.Psi_2i(val, st.M, sparse=True)
logdet = -splogdet(PsiLambdai) #negative because precision
kernel = spdot(spdot(st.Alphas.T, PsiLambdai), st.Alphas) / st.Tau2
return -.5 * logdet - .5 * kernel + st.Log_Lambda0(val)
def logp_lambda_prec(state, val):
"""
Compute the log likelihood of the upper-level spatial correlation parameter using
sparse operations and the precision matrix, rather than the covariance matrix.
"""
st = state
if (val < st.Lambda_min) or (val > st.Lambda_max):
return np.array([-np.inf])
PsiLambdai = st.Psi_2i(val, st.M, sparse=True)
logdet = -splogdet(PsiLambdai) #negative because precision
kernel = spdot(spdot(st.Alphas.T, PsiLambdai), st.Alphas) / st.Tau2
return -.5 * logdet - .5 * kernel + st.Log_Lambda0(val)
def logp_rho_prec(state, val):
"""
This computes the logp of the spatial parameter using the precision, rather than the covariance. This results in fewer matrix operations in the case of a SE formulation, but not in an SMA formulation.
"""
st = state
#must truncate in logp otherwise sampling gets unstable
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRhoi = st.Psi_1i(val, st.W, sparse=True)
logdet = splogdet(PsiRhoi)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(spdot(eta.T, PsiRhoi), eta) / st.Sigma2
return .5*logdet -.5 * kernel + st.Log_Rho0(val) #since precision, no negative on ld
def logp_lambda_prec(state, val):
"""
The logp for upper level spatial parameters in this case has the same form
as a multivariate normal distribution, sampled over the variance matrix,
rather than over Y.
"""
st = state
#must truncate
if (val < st.Lambda_min) or (val > st.Lambda_max):
return np.array([-np.inf])
PsiLambdai = st.Psi_2i(val, st.M)
logdet = splogdet(PsiLambdai)
kernel = spdot(spdot(st.Alphas.T, PsiLambdai), st.Alphas) / st.Tau2
return .5*logdet - .5*kernel + st.Log_Lambda0(val)
def logp_rho_prec(state, val):
"""
Compute the log likelihood of the lower-level spatial correlation parameter using
sparse operations and the precision matrix, rather than the covariance matrix.
"""
st = state
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRhoi = st.Psi_1i(val, st.W, sparse=True)
logdet = -splogdet(PsiRhoi)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(spdot(eta.T, PsiRhoi), eta) / st.Sigma2
return -.5 * logdet - .5 * kernel + st.Log_Rho0(val)
def logp_rho_prec(state, val):
"""
This computes the logp of the spatial parameter using the precision, rather than the covariance. This results in fewer matrix operations in the case of a SE formulation, but not in an SMA formulation.
"""
st = state
#must truncate in logp otherwise sampling gets unstable
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRhoi = st.Psi_1i(val, st.W, sparse=True)
logdet = splogdet(PsiRhoi)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(spdot(eta.T, PsiRhoi), eta) / st.Sigma2
return .5 * logdet - .5 * kernel + st.Log_Rho0(
val) #since precision, no negative on ld
def logp_lambda_prec(state, val):
"""
The logp for upper level spatial parameters in this case has the same form
as a multivariate normal distribution, sampled over the variance matrix,
rather than over Y.
"""
st = state
#must truncate
if (val < st.Lambda_min) or (val > st.Lambda_max):
return np.array([-np.inf])
PsiLambdai = st.Psi_2i(val, st.M)
logdet = splogdet(PsiLambdai)
kernel = spdot(spdot(st.Alphas.T, PsiLambdai), st.Alphas) / st.Tau2
return .5 * logdet - .5 * kernel + st.Log_Lambda0(val)
def logp_rho_prec(state, val):
"""
Compute the log likelihood of the lower-level spatial correlation parameter using
sparse operations and the precision matrix, rather than the covariance matrix.
"""
st = state
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRhoi = st.Psi_1i(val, st.W, sparse=True)
logdet = -splogdet(PsiRhoi)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(spdot(eta.T, PsiRhoi), eta) / st.Sigma2
return -.5 * logdet - .5 * kernel + st.Log_Rho0(val)
def logp_rho_cov(state, val):
"""
The logp for lower-level spatial parameters in this case has the same
form as a multivariate normal distribution, sampled over the variance matrix, rather than over y.
"""
st = state
#must truncate in logp otherwise sampling gets unstable
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRho = st.Psi_1(val, st.W)
logdet = splogdet(PsiRho)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(eta.T, spsolve(PsiRho, eta)) / st.Sigma2
return -.5 * logdet - .5 * kernel + st.Log_Rho0(val)
def logp_rho_cov(state, val):
"""
The logp for lower-level spatial parameters in this case has the same
form as a multivariate normal distribution, sampled over the variance matrix, rather than over y.
"""
st = state
#must truncate in logp otherwise sampling gets unstable
if (val < st.Rho_min) or (val > st.Rho_max):
return np.array([-np.inf])
PsiRho = st.Psi_1(val, st.W)
logdet = splogdet(PsiRho)
eta = st.Y - st.XBetas - st.DeltaAlphas
kernel = spdot(eta.T, spsolve(PsiRho, eta)) / st.Sigma2
return -.5*logdet -.5 * kernel + st.Log_Rho0(val)
def normalized_cov_params(self):
return la.inv(spdot(self.w.T, self.w))
def _iteration(self):
st = self.state
### Sample the Beta conditional posterior
### P(beta | . ) \propto L(Y|.) \dot P(\beta)
### is
### N(Sb, S) where
### S = (X' Sigma^{-1}_Y X + S_0^{-1})^{-1}
### b = X' Sigma^{-1}_Y (Y - Delta Alphas) + S^{-1}\mu_0
covm_update = st.X.T.dot(st.X) / st.Sigma2
covm_update += st.Betas_cov0i
covm_update = la.inv(covm_update)
resids = st.Y - st.Delta.dot(st.Alphas)
XtSresids = st.X.T.dot(resids) / st.Sigma2
mean_update = XtSresids + st.Betas_cov0i.dot(st.Betas_mean0)
mean_update = np.dot(covm_update, mean_update)
st.Betas = chol_mvn(mean_update, covm_update)
st.XBetas = np.dot(st.X, st.Betas)
### Sample the Random Effect conditional posterior
### P( Alpha | . ) \propto L(Y|.) \dot P(Alpha | \lambda, Tau2)
### \dot P(Tau2) \dot P(\lambda)
### is
### N(Sb, S)
### Where
### S = (Delta'Sigma_Y^{-1}Delta + Sigma_Alpha^{-1})^{-1}
### b = (Delta'Sigma_Y^{-1}(Y - X\beta) + 0)
covm_update = st.Delta.T.dot(st.Delta) / st.Sigma2
covm_update += st.PsiLambdai / st.Tau2
covm_update = np.asarray(covm_update)
covm_update = la.inv(covm_update)
resids = st.Y - st.XBetas
mean_update = st.Delta.T.dot(resids) / st.Sigma2
mean_update = np.dot(covm_update, mean_update)
mean_update = np.asarray(mean_update)
st.Alphas = chol_mvn(mean_update, covm_update)
st.DeltaAlphas = np.dot(st.Delta, st.Alphas)
### Sample the Random Effect aspatial variance parameter
### P(Tau2 | .) \propto L(Y|.) \dot P(\Alpha | \lambda, Tau2)
### \dot P(Tau2) \dot P(\lambda)
### is
### IG(J/2 + a0, u'(\Psi(\lambda))^{-1}u * .5 + b0)
bn = spdot(st.Alphas.T, spdot(st.PsiLambdai,
st.Alphas)) * .5 + st.Tau2_b0
st.Tau2 = stats.invgamma.rvs(st.Tau2_an, scale=bn)
### Sample the response aspatial variance parameter
### P(Sigma2 | . ) \propto L(Y | .) \dot P(Sigma2)
### is
### IG(N/2 + a0, eta'Psi(\rho)^{-1}eta * .5 + b0)
### Where eta is the linear predictor, Y - X\beta + \DeltaAlphas
eta = st.Y - st.XBetas - st.DeltaAlphas
bn = eta.T.dot(eta) * .5 + st.Sigma2_b0
st.Sigma2 = stats.invgamma.rvs(st.Sigma2_an, scale=bn)
### P(Psi(\rho) | . ) \propto L(Y | .) \dot P(\rho)
### is
### |Psi(rho)|^{-1/2} exp(1/2(eta'Psi(rho)^{-1}eta * Sigma2^{-1})) * 1/(emax-emin)
st.Lambda = self.configs.Lambda(st)
st.PsiLambdai = st.Psi_2i(st.Lambda, st.M)
def iwls(y,
x,
family,
offset,
y_fix,
ini_betas=None,
tol=1.0e-8,
max_iter=200,
wi=None):
"""
Iteratively re-weighted least squares estimation routine
Parameters
----------
y : array
n*1, dependent variable
x : array
n*k, designs matrix of k independent variables
family : family object
probability models: Gaussian, Poisson, or Binomial
offset : array
n*1, the offset variable for each observation.
y_fix : array
n*1, the fixed intercept value of y for each observation
ini_betas : array
1*k, starting values for the k betas within the iteratively
weighted least squares routine
tol : float
tolerance for estimation convergence
max_iter : integer maximum number of iterations if convergence not met
wi : array
n*1, weights to transform observations from location i in GWR
Returns
-------
betas : array
k*1, estimated coefficients
mu : array
n*1, predicted y values
wx : array
n*1, final weights used for iwls for GLM
n_iter : integer
number of iterations that when iwls algorithm terminates
w : array
n*1, final weights used for iwls for GWR
z : array
iwls throughput
v : array
iwls throughput
xtx_inv_xt : array
iwls throughout to compute GWR hat matrix
[X'X]^-1 X'
"""
n_iter = 0
diff = 1.0e6
if ini_betas is None:
betas = np.zeros((x.shape[1], 1), np.float)
else:
betas = ini_betas
if isinstance(family, Binomial):
y = family.link._clean(y)
if isinstance(family, Poisson):
y_off = y / offset
y_off = family.starting_mu(y_off)
v = family.predict(y_off)
mu = family.starting_mu(y)
else:
mu = family.starting_mu(y)
v = family.predict(mu)
while diff > tol and n_iter < max_iter:
n_iter += 1
w = family.weights(mu)
z = v + (family.link.deriv(mu) * (y - mu))
w = np.sqrt(w)
if not isinstance(x, np.ndarray):
w = sp.csr_matrix(w)
z = sp.csr_matrix(z)
wx = spmultiply(x, w, array_out=False)
wz = spmultiply(z, w, array_out=False)
if wi is None:
n_betas = _compute_betas(wz, wx)
else:
n_betas, xtx_inv_xt = _compute_betas_gwr(wz, wx, wi)
v = spdot(x, n_betas)
mu = family.fitted(v)
if isinstance(family, Poisson):
mu = mu * offset
diff = min(abs(n_betas - betas))
betas = n_betas
if wi is None:
return betas, mu, wx, n_iter
else:
return betas, mu, v, w, z, xtx_inv_xt, n_iter
def _iteration(self):
st = self.state
### Sample the Beta conditional posterior
### P(beta | . ) \propto L(Y|.) \dot P(\beta)
### is
### N(Sb, S) where
### S = (X' Sigma^{-1}_Y X + S_0^{-1})^{-1}
### b = X' Sigma^{-1}_Y (Y - Delta Alphas) + S^{-1}\mu_0
covm_update = st.X.T.dot(st.X) / st.Sigma2
covm_update += st.Betas_cov0i
covm_update = la.inv(covm_update)
resids = st.Y - st.Delta.dot(st.Alphas)
XtSresids = st.X.T.dot(resids) / st.Sigma2
mean_update = XtSresids + st.Betas_cov0i.dot(st.Betas_mean0)
mean_update = np.dot(covm_update, mean_update)
st.Betas = chol_mvn(mean_update, covm_update)
st.XBetas = np.dot(st.X, st.Betas)
### Sample the Random Effect conditional posterior
### P( Alpha | . ) \propto L(Y|.) \dot P(Alpha | \lambda, Tau2)
### \dot P(Tau2) \dot P(\lambda)
### is
### N(Sb, S)
### Where
### S = (Delta'Sigma_Y^{-1}Delta + Sigma_Alpha^{-1})^{-1}
### b = (Delta'Sigma_Y^{-1}(Y - X\beta) + 0)
covm_update = st.Delta.T.dot(st.Delta) / st.Sigma2
covm_update += st.PsiLambdai / st.Tau2
covm_update = np.asarray(covm_update)
covm_update = la.inv(covm_update)
resids = st.Y - st.XBetas
mean_update = st.Delta.T.dot(resids) / st.Sigma2
mean_update = np.dot(covm_update, mean_update)
mean_update = np.asarray(mean_update)
st.Alphas = chol_mvn(mean_update, covm_update)
st.DeltaAlphas = np.dot(st.Delta, st.Alphas)
### Sample the Random Effect aspatial variance parameter
### P(Tau2 | .) \propto L(Y|.) \dot P(\Alpha | \lambda, Tau2)
### \dot P(Tau2) \dot P(\lambda)
### is
### IG(J/2 + a0, u'(\Psi(\lambda))^{-1}u * .5 + b0)
bn = spdot(st.Alphas.T, spdot(st.PsiLambdai, st.Alphas)) * .5 + st.Tau2_b0
st.Tau2 = stats.invgamma.rvs(st.Tau2_an, scale=bn)
### Sample the response aspatial variance parameter
### P(Sigma2 | . ) \propto L(Y | .) \dot P(Sigma2)
### is
### IG(N/2 + a0, eta'Psi(\rho)^{-1}eta * .5 + b0)
### Where eta is the linear predictor, Y - X\beta + \DeltaAlphas
eta = st.Y - st.XBetas - st.DeltaAlphas
bn = eta.T.dot(eta) * .5 + st.Sigma2_b0
st.Sigma2 = stats.invgamma.rvs(st.Sigma2_an, scale=bn)
### P(Psi(\rho) | . ) \propto L(Y | .) \dot P(\rho)
### is
### |Psi(rho)|^{-1/2} exp(1/2(eta'Psi(rho)^{-1}eta * Sigma2^{-1})) * 1/(emax-emin)
st.Lambda = self.configs.Lambda(st)
st.PsiLambdai = st.Psi_2i(st.Lambda, st.M)
def normalized_cov_params(self):
return la.inv(spdot(self.w.T, self.w))