/
equations.py
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/
equations.py
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import autograd.scipy.special as special
import autograd.numpy as np
import wh, wp
import util
import gtilde
from scipy.integrate import quad
from scipy.stats import multivariate_normal
from autograd import primitive
import matplotlib.pyplot as plt
__nugget_scalar = 1e-7
__nugget = lambda n: __nugget_scalar * np.eye(n)
__verify = False
__euler_mascheroni = 0.57721566490153286060651209008240243104215933593992
from scipy.misc import logsumexp
def psi(x, gamma, alpha, trange):
tmin, tmax = trange
y = x
d = x.shape[0]
exp1 = ((np.pi * alpha / 4) ** (d/2)) * (gamma**2) * np.exp(-wh.sqdist(x,None) / (4*alpha))
xbar = 0.5 * (x.reshape(x.shape[0],x.shape[1],1)+y.reshape(y.shape[0],1,y.shape[1]))
d = special.erf((xbar-tmin) / np.sqrt(alpha)) - special.erf((xbar - tmax) / np.sqrt(alpha))
prodd = np.prod(d, axis=0)
rval = exp1 * prodd
rval = 0.5 * (rval + rval.T)
rval += 2 * gamma * __nugget_scalar
rval += np.eye(rval.shape[0]) * __nugget_scalar ** 2
return rval
def k(x, y, gamma, alpha):
"""kernel function"""
d2 = wh.sqdist(x,y)
rval = gamma * np.exp(-d2/ (2*alpha))
if y is None:
delta = np.eye(x.shape[1])
rval = 0.5 * (rval + rval.T)
else:
delta = (d2 <= 1e-7)
rval += (delta * __nugget_scalar)
return rval
def kdiag(x, gamma):
"""diagonal elements of the kernel matrix"""
return (gamma + __nugget_scalar) * np.ones(x.shape[1])
def kl_tril(L, m, Lzz,u):
"""KL divergence of q(u) and p(u)"""
M = L.shape[0]
traceterm = np.sum(np.linalg.solve(Lzz, L)**2)
mkmterm = np.sum(np.linalg.solve(Lzz,u-m)**2)
logdetk = 2 * np.sum(np.log(np.abs(np.diag(Lzz))))
logdets = 2 * np.sum(np.log(np.abs(np.diag(L))))
kl = 0.5 * (traceterm + logdetk - logdets - M + mkmterm)
if __verify:
S = L @ L.T
Kzz = Lzz @ Lzz.T
traceterm2 = np.trace(np.linalg.solve(Kzz, S))
mkmterm2 = np.dot((u-m).T,np.linalg.solve(Kzz, u-m))[0,0]
logdetk2 = np.log(np.linalg.det(Kzz))
logdets2 = np.log(np.linalg.det(S))
wh.assert_close(traceterm, traceterm2)
wh.assert_close(mkmterm, mkmterm2)
if not np.isinf(logdetk2) and not np.isnan(logdetk2):
wh.assert_close(logdetk, logdetk2, rtol=5e-2)
if not np.isinf(logdets2) and not np.isnan(logdets2):
wh.assert_close(logdets, logdets2, rtol=5e-2)
return kl
def expected_log_f2(mu, sigma):
"""expectation of log of f^2"""
assert mu.shape == sigma.shape, (mu.shape, sigma.shape)
return -gtilde.gtilde_ad( - (mu ** 2) / (2 * (sigma ** 2))), \
np.log((sigma ** 2) / 2), \
- __euler_mascheroni * np.ones(mu.shape, dtype=float)
def predictive(L, m, precomp, precomp_predictive):
"""mean and variance of q(f)"""
kzzinv = precomp.Lzzinv.T @ precomp.Lzzinv
kzzinv_m = kzzinv @ m
mu = (precomp_predictive.Kxz @ kzzinv_m).flatten()
sigmaa = precomp_predictive.Kxx_diag
Lzz_inv_Kzx = np.linalg.solve(precomp.Lzz, precomp_predictive.Kxz.T)
sigmab = np.sum(precomp_predictive.Kxz * np.linalg.solve(precomp.Lzz.T, Lzz_inv_Kzx).T, axis=1)
Kzz_inv_Kzx = np.linalg.solve(precomp.Lzz.T, Lzz_inv_Kzx)
sigmac = np.sum((L.T @ Kzz_inv_Kzx) ** 2, axis=0)
sigma2 = sigmaa - sigmab + sigmac
Sigmaa = precomp_predictive.Kxx
Sigmab = precomp_predictive.Kxz @ kzzinv @ precomp_predictive.Kxz.T
Sigmac = precomp_predictive.Kxz @ kzzinv @ L @ L.T @ kzzinv @ precomp_predictive.Kxz.T
Sigma = Sigmaa - Sigmab + Sigmac
wh.assert_close(np.diag(Sigma), sigma2, rtol=1e-3)
if __verify:
sigmab2 = np.diag(precomp_predictive.Kxz @ np.linalg.solve(precomp.Kzz, precomp_predictive.Kxz.T))
sigmac2 = np.diag(precomp_predictive.Kxz @ np.linalg.solve(precomp.Kzz, L @ L.T @ np.linalg.solve(precomp.Kzz,
precomp_predictive.Kxz.T)))
wh.assert_close(sigmab, sigmab2, rtol=1e-3)
wh.assert_close(sigmac, sigmac2, rtol=1e-3)
return mu, sigma2, Sigma
def precompute(z, gamma, alpha, run_time, x, taus):
""" calculating Kzz, Lzz and Psi"""
precomp = wh.RaisingDotDict()
taus = np.concatenate([tau.flatten() for tau in taus]).reshape((1, -1))
precomp.Kzz = k(z, None, gamma, alpha)
try:
precomp.Lzz = np.linalg.cholesky(precomp.Kzz)
precomp.Lzzinv = np.linalg.inv(precomp.Lzz)
precomp.Kzzinv = precomp.Lzzinv.T @ precomp.Lzzinv
except Exception as e:
print('gamma:', gamma, 'alpha:', alpha, 'Kzz:', precomp.Kzz)
raise
tmin = 0
exp = ((np.pi * alpha / 4) ** (1 / 2)) * (gamma ** 2) * np.exp(-wh.sqdist(z, None) / (4 * alpha))
zy = z
zbar = 0.5 * (z.reshape(z.shape[0], z.shape[1], 1) + zy.reshape(zy.shape[0], 1, zy.shape[1]))
dmin_array = special.erf((zbar - tmin) / np.sqrt(alpha))
dprod_sum = np.sum(
np.array([np.prod(dmin_array - special.erf((zbar - (run_time - x[i])) / np.sqrt(alpha)), axis=0)
for i in range(len(x))]), axis=0)
r = exp * dprod_sum
r = 0.5 * (r + r.T)
precomp.psi_sum = r + 2 * gamma * __nugget_scalar + np.eye(r.shape[0]) * __nugget_scalar ** 2
precomp.Kzzinv_psi_sum = precomp.Kzzinv@precomp.psi_sum
precomp.Kzzinv_psi_sum_Kzzinv = precomp.Kzzinv @ precomp.psi_sum @ precomp.Kzzinv
precomp.Kxz = k(taus, z, gamma, alpha)
precomp.Kzzinv_kzx = precomp.Kzzinv @ precomp.Kxz.T
precomp.sigmas = kdiag(taus, gamma)
return precomp
def precompute_predictive(x, z, gamma, alpha):
"""???"""
precomp = wh.RaisingDotDict()
precomp.Kxz = k(x, z, gamma, alpha)
precomp.Kxx = k(x, None, gamma, alpha)
precomp.Kxx_diag = kdiag(x, gamma)
return precomp
def update_pij(paramslin, taus, z, gamma, alpha, pij, kzzinv):
nx = len(taus)+1
params = util.unlinearise_params(paramslin, verbose=0)
kzzinv_m = kzzinv @ params.m
expEmu = params.scale*np.exp(special.digamma(params.shape))
for i in range(nx-1):
tau = taus[i]
Kxz = k(tau, z, gamma, alpha)
mutilde = (Kxz @ kzzinv_m).flatten()
sigmaa = kdiag(tau, gamma)
kzzinv_kzx = kzzinv @ Kxz.T
sigmab = np.sum(Kxz * kzzinv_kzx.T, axis=1)
sigmac = np.sum((params.L.T @ kzzinv_kzx) ** 2, axis=0)
sigmatilde = sigmaa - sigmab + sigmac
eqn19a, eqn19b, eqn19c = expected_log_f2(mutilde, np.sqrt(sigmatilde))
eqn19 = eqn19a + eqn19b + eqn19c
expeqn19 = np.exp(eqn19)
denom = expEmu + np.sum(expeqn19)
pij[i+1][0] = expEmu / denom
pij[i+1][1:tau.shape[1]+1] = expeqn19 / denom
return pij
def eqn15sum_numerical(paramslin, x,z,run_time, gamma, alpha):
params = util.unlinearise_params(paramslin, verbose=0,)
precomp = precompute(z, gamma, alpha, [0, run_time])
kzzinv = precomp.Lzzinv.T @ precomp.Lzzinv
kzzinv_m = kzzinv @ params.m
interp_x = np.linspace(0,run_time,4096).reshape((1,-1))
delta = interp_x[0,1]-interp_x[0,0]
kxz = k(interp_x,z, gamma, alpha)
mutilde2 = (kxz@kzzinv_m).flatten()**2
eqn15sum = 0
for i in range(x.shape[1]):
N = np.sum(interp_x[0] < (run_time - x[0, i]))
eqn15sum += np.sum([np.sum(mutilde2[:N-1]+mutilde2[1:N])])*delta/2
return eqn15sum
def eqn16sum_numerical(paramslin, x,z,run_time, gamma, alpha):
params = util.unlinearise_params(paramslin, verbose=0)
precomp = precompute(z, gamma, alpha, [0, run_time])
kzzinv = precomp.Lzzinv.T @ precomp.Lzzinv
interp_x = np.linspace(0, run_time, 5000).reshape((1, -1))
kxx = kdiag(interp_x, gamma)
delta = interp_x[0,1]-interp_x[0,0]
kxz = k(interp_x, z, gamma, alpha)
sigmatilde2 = kxx-np.sum(kxz*(kxz@kzzinv.T),axis=1) + np.sum((kxz@kzzinv@params.L)**2,axis=1)
eqn16sum = 0
for i in range(x.shape[1]):
N = np.sum(interp_x[0] < (run_time - x[0, i]))
eqn16sum += np.sum([np.sum(sigmatilde2[:N-1]+sigmatilde2[1:N])])*delta/2
return eqn16sum
def eqn19sum_numerical(paramslin, x,z, pij, run_time, gamma, alpha):
params = util.unlinearise_params(paramslin, verbose=0)
precomp = precompute(z, gamma, alpha, [0, run_time])
kzzinv = precomp.Lzzinv.T @ precomp.Lzzinv
kzzinv_m = kzzinv @ params.m
eqn19sum = 0
kzzinv_S_kzzinv = kzzinv @ params.L @ params.L.T @ kzzinv
mutilde_list = []
for i in range(1, x.shape[1]):
taus = np.array([x[0, i] - x[0, :i]])
kxx = kdiag(taus, gamma)
kxz = k(taus, z, gamma, alpha)
mutilde = (kxz @ kzzinv_m).flatten()
mutilde_list += mutilde.tolist()
sigmatilde = np.sqrt(np.diag(kxx - kxz @ kzzinv @ kxz.T + kxz @ kzzinv_S_kzzinv @ kxz.T))
assert mutilde.ndim == 1
assert sigmatilde.ndim == 1
for j in range(len(mutilde)):
mui = mutilde[j]
sigmai = sigmatilde[j]
interp_f = np.linspace(mui - 6*sigmai, mui + 6 * sigmai, 4096)
delta = interp_f[1] - interp_f[0]
e_log_f2 = multivariate_normal(mui, sigmai**2).pdf(interp_f) * np.log(interp_f ** 2) * delta
eqn19sum += pij[i, j + 1] * np.sum(e_log_f2)
return eqn19sum
def objective(paramslin, x, z, pij_flatten, pij0sum, run_time, taus, gamma, alpha, g0_params,precomp):
params = util.unlinearise_params(paramslin, verbose=0)
d, nz = z.shape
nx = x.shape[1]
kzzinv_m = precomp.Kzzinv @ params.m
s = params.L @ params.L.T+__nugget(params.L.shape[1])
eqn15sum = (params.m.T @ precomp.Kzzinv_psi_sum_Kzzinv @params.m)[0,0]
eqn16a = np.trace(precomp.Kzzinv_psi_sum)
eqn16b = np.trace(precomp.Kzzinv_psi_sum_Kzzinv @ s)
eqn16sum = gamma*np.sum((run_time-x[0])**d)-eqn16a + eqn16b
mutilde = (precomp.Kzzinv_kzx.T @ params.m).flatten()
sigmaa = precomp.sigmas
sigmab = np.sum(precomp.Kxz * precomp.Kzzinv_kzx.T, axis=1)
sigmac = np.sum((params.L.T @ precomp.Kzzinv_kzx) ** 2, axis=0)
sigmatilde = sigmaa - sigmab + sigmac
eqn19a, eqn19b, eqn19c = expected_log_f2(mutilde, np.sqrt(sigmatilde))
eqn19sum = -(eqn19c + eqn19a + eqn19b)@pij_flatten
kl_normal = kl_tril(params.L, params.m, precomp.Lzz, 0)
kl_g = kl_gamma(params.scale,params.shape, g0_params['scale'],g0_params['shape'])
total = kl_normal+kl_g+eqn15sum + eqn16sum + eqn19sum +run_time*params.shape*params.scale-\
pij0sum*(special.digamma(params.shape)+np.log(params.scale))
return total
def log_marginal_likelihood(paramslin, x, z, pij, pij_flatten, pij0sum, run_time,taus, gamma, alpha, precomp):
params = util.unlinearise_params(paramslin, verbose=0)
d, nz = z.shape
nx = x.shape[1]
s = params.L @ params.L.T+__nugget(params.L.shape[1])
eqn15sum = (params.m.T @ precomp.Kzzinv_psi_sum_Kzzinv @params.m)[0,0]
eqn16a = np.trace(precomp.Kzzinv_psi_sum)
eqn16b = np.trace(precomp.Kzzinv_psi_sum_Kzzinv @ s)
eqn16sum = gamma*np.sum((run_time-x[0])**d)-eqn16a + eqn16b
mutilde = (precomp.Kzzinv_kzx.T @ params.m).flatten()
sigmaa = precomp.sigmas
sigmab = np.sum(precomp.Kxz * precomp.Kzzinv_kzx.T, axis=1)
sigmac = np.sum((params.L.T @ precomp.Kzzinv_kzx) ** 2, axis=0)
sigmatilde = sigmaa - sigmab + sigmac
eqn19a, eqn19b, eqn19c = expected_log_f2(mutilde, np.sqrt(sigmatilde))
eqn19sum = -(eqn19c + eqn19a + eqn19b)@pij_flatten
ppij = pij[pij > 0]
total = eqn15sum + eqn16sum + eqn19sum + run_time * params.shape * params.scale - \
pij0sum*(special.digamma(params.shape) + np.log(params.scale)) + ppij @ np.log(ppij)
return -total
def kl_gamma(a,b,c,d):
# `b` and `d` are Gamma shape parameters and
# `a` and `c` are scale parameters.
# (All, therefore, must be positive.)
# copy from: https://stats.stackexchange.com/questions/11646/kullback-leibler-divergence-between-two-gamma-distributions
def I(a,b,c,d):
return -c*d/a -b*np.log(a) - special.gammaln(b) + (b-1)*(special.digamma(d) + np.log(c))
return I(c,d,c,d) - I(a,b,c,d)