def energy_mode(A, b, m, S, sDyn):
"""
ENERGY MODE
[Description]
Energy for the double-well SDE, and related quantities (including gradients).
[Input]
A : variational linear parameters (N x 1).
b : variational offset parameters (N x 1).
m : narginal means (N x 1).
S : marginal variances (N x 1).
sDyn : structure containing additional parameters.
[Output]
Esde : total energy of the sde.
Ef : average drift (N x 1).
Edf : average differentiated drift (N x 1).
dEsde_dm : gradient of Esde w.r.t. the means (N x 1).
dEsde_dS : gradient of Esde w.r.t. the covariance (N x 1).
dEsde_dth : gradient of Esde w.r.t. the parameter theta.
dEsde_dSig: gradient of Esde w.r.t. the parameter Sigma.
NOTE: The equation numbers correspond to the paper:
@CONFERENCE{Archambeau2007b,
author = {Cedric Archambeau, Manfred Opper, Yuan Shen, Dan Cornford and
J. Shawe-Taylor},title = {Variational Inference for Diffusion Processes},
booktitle = {Annual Conference on Neural Information Processing Systems},
year = {2007}
}
Copyright (c) Michail D. Vrettas, PhD - November 2015.
Last Updated: November 2015.
"""
# Find the time step.
dt = sDyn['dt']
# Extract the drift parameter.
theta = sDyn['theta']
# Inverse noise variance.
SigInv = 1.0 / sDyn['Sig']
# Observation times.
idx = sDyn['obsX']
# Higher order Gaussian Moments.
Ex2 = momGauss(m, S, 2)
# Precompute these quantities only once.
Q1 = (theta - A)**2
Q2 = A * b
# Energy from the sDyn: Eq(7)
varQ = Ex2 * Q1 + 2 * m * (theta - A) * b + b**2
Esde = 0.5 * SigInv * mytrapz(varQ, dt, idx)
# Average drift.
Ef = -theta * m
# Average gradient of drift.
Edf = -theta * np.ones(m.shape)
# Gradients of Esde w.r.t. 'm' and 'S'.
dEsde_dm = SigInv * (m * (theta - A)**2 + theta * b - Q2)
dEsde_dS = 0.5 * SigInv * Q1
# Gradients of Esde w.r.t. 'Theta'.
dEsde_dth = SigInv * mytrapz(Ex2 * (theta - A) + m * b, dt, idx)
# Gradients of Esde w.r.t. 'Sigma'.
dEsde_dSig = -SigInv * Esde
# --->
return Esde, Ef, Edf, dEsde_dm, dEsde_dS, dEsde_dth, dEsde_dSig
def energy_mode(A, b, m, S, sDyn):
"""
ENERGY MODE:
[Description]
Energy for the stocastic Lorenz 63 DE (3 dimensional) and related quantities
(including gradients).
[Input]
A : variational linear parameters (N x D x D).
b : variational offset parameters (N x D).
m : narginal means (N x D).
S : marginal variances (N x D x D).
sDyn : structure containing additional parameters.
[Output]
Esde : total energy of the sde.
Ef : average drift (N x D).
Edf : average differentiated drift (N x D).
dEsde_dm : gradient of Esde w.r.t. the means (N x D).
dEsde_dS : gradient of Esde w.r.t. the covariance (N x D x D).
dEsde_dth : gradient of Esde w.r.t. the parameter theta.
dEsde_dSig: gradient of Esde w.r.t. the parameter Sigma.
NOTE: The equation numbers correspond to the paper:
@CONFERENCE{Archambeau2007b,
author = {Cedric Archambeau and Manfred Opper and Yuan Shen
and Dan Cornford and J. Shawe-Taylor},
title = {Variational Inference for Diffusion Processes},
booktitle = {Annual Conference on Neural Information Processing Systems},
year = {2007}
}
Copyright (c) Michail D. Vrettas, PhD - November 2015.
Last Updated: November 2015.
"""
# {N}umber of discretised points
N = sDyn['N']
# Time discretiastion step.
dt = sDyn['dt']
# Inverse System Noise.
SigInv = np.linalg.inv(sDyn['Sig'])
# Observation times.
idx = sDyn['obsX']
# Diagonal elements of inverse Sigma.
diagSigI = np.diag(SigInv)
# Energy from the sde.
Esde = np.zeros((N, 1), dtype='float64')
# Average drift.
Ef = np.zeros((N, 3), dtype='float64')
# Average gradient of drift.
Edf = np.zeros((N, 3, 3), dtype='float64')
# Gradients of Esde w.r.t. 'm' and 'S'.
dEsde_dm = np.zeros((N, 3), dtype='float64')
dEsde_dS = np.zeros((N, 3, 3), dtype='float64')
# Gradients of Esde w.r.t. 'Theta'.
dEsde_dth = np.zeros((N, 3), dtype='float64')
# Gradients of Esde w.r.t. 'Sigma'.
dEsde_dSig = np.zeros((N, 3), dtype='float64')
# Drift parameters.
vS, vR, vB = sDyn['theta']
# Compute the quantities iteratively.
for t in range(N):
# Get the values at time 't'.
At = A[t, :, :]
bt = b[t, :]
St = S[t, :, :]
mt = m[t, :]
# Compute the energy and the related gradients.
Efg, Edm, EdS = Energy_dm_dS(At, bt, mt, St, diagSigI, sDyn)
# Energy Esde(t):
Esde[t] = 0.5 * diagSigI.dot(Efg)
# Gradient dEsde(t)/dm(t):
dEsde_dm[t, :] = Edm
# Gradient dEsde(t)/dS(t):
dEsde_dS[t, :, :] = EdS
# Average drift: <f(Xt)>
Ef[t,:] = np.array([(vS*(mt[1] - mt[0])),\
(vR*mt[0] - mt[1] - St[2,0] - mt[0]*mt[2]),\
(St[1,0] + mt[0]*mt[1] - vB*mt[2])])
# Average gradient of drift: <Df(Xt)>
Edf[t,:,:] = np.array([[-vS, vS, 0], [(vR - mt[2]), -1, -mt[0]],\
[mt[1], mt[0], -vB]])
# Gradients of Esde w.r.t. 'Theta'.
dEsde_dth[t, :] = Efg_drift_theta(At, bt, mt, St, sDyn)
# Gradients of Esde w.r.t. 'Sigma'.
dEsde_dSig[t, :] = Efg
# ...
# Compute energy using numerical integration.
Esde = mytrapz(Esde, dt, idx)
# Final adjustments for the (hyper)-parameters.
dEsde_dth = diagSigI * mytrapz(dEsde_dth, dt, idx)
# Final adjustments for the System noise.
dEsde_dSig = -0.5 * SigInv.dot(np.diag(mytrapz(dEsde_dSig, dt,
idx))).dot(SigInv)
# --->
return Esde, Ef, Edf, dEsde_dm, dEsde_dS, dEsde_dth, dEsde_dSig
def energy_mode(A, b, m, S, sDyn):
"""
ENERGY MODE
[Description]
Energy for the double-well SDE and related quantities (including gradients).
[Input]
A : variational linear parameters (N x 1).
b : variational offset parameters (N x 1).
m : narginal means (N x 1).
S : marginal variances (N x 1).
sDyn : structure containing additional parameters.
[Output]
Esde : total energy of the sde.
Ef : average drift (N x 1).
Edf : average differentiated drift (N x 1).
dEsde_dm : gradient of Esde w.r.t. the means (N x 1).
dEsde_dS : gradient of Esde w.r.t. the covariance (N x 1).
dEsde_dth : gradient of Esde w.r.t. the parameter theta.
dEsde_dSig: gradient of Esde w.r.t. the parameter Sigma.
NOTE: The equation numbers correspond to the paper:
@CONFERENCE{Archambeau2007b,
author = {Cedric Archambeau and Manfred Opper and Yuan Shen
and Dan Cornford and J. Shawe-Taylor},
title = {Variational Inference for Diffusion Processes},
booktitle = {Annual Conference on Neural Information Processing Systems},
year = {2007}
}
Copyright (c) Michail D. Vrettas, PhD - November 2015.
Last Updated: November 2015.
"""
# Find the time step.
dt = sDyn['dt']
# Extract the drift parameter.
theta = sDyn['theta']
# Observation times.
idx = sDyn['obsX']
# Constant value.
c = 4.0 * theta + A
# Auxiliary constant.
c2 = c**2
# Iverse of noise variance.
SigInv = 1.0 / sDyn['Sig']
# Higher order Gaussian Moments.
Ex2 = momGauss(m, S, 2)
Ex3 = momGauss(m, S, 3)
Ex4 = momGauss(m, S, 4)
Ex6 = momGauss(m, S, 6)
# Energy from the sDyn: Eq(7)
varQ = 16.0 * Ex6 - 8.0 * c * Ex4 + 8.0 * b * Ex3 + c2 * Ex2 - 2.0 * b * c * m + b**2
Esde = 0.5 * SigInv * mytrapz(varQ, dt, idx)
# Average drift {i.e. Eq(20) : f(t,x) = 4*x*(theta -x^2) }.
Ef = 4.0 * (theta * m - Ex3)
# Average gradient of drift {i.e df(t,x)_dx = 4*theta - 12*x^2}.
Edf = 4.0 * (theta - 3 * Ex2)
# Derivatives of higher order Gaussian moments w.r.t. 'm' and 'S'.
Dm2 = momGauss(m, S, 2, 'Dm')
DS2 = momGauss(m, S, 2, 'DS')
# ---
Dm3 = momGauss(m, S, 3, 'Dm')
DS3 = momGauss(m, S, 3, 'DS')
# ---
Dm4 = momGauss(m, S, 4, 'Dm')
DS4 = momGauss(m, S, 4, 'DS')
# ---
Dm6 = momGauss(m, S, 6, 'Dm')
DS6 = momGauss(m, S, 6, 'DS')
# Gradients of Esde w.r.t. 'm' and 'S'.
dEsde_dm = 0.5 * SigInv * (16.0 * Dm6 - 8.0 * c * Dm4 + 8.0 * b * Dm3 +
c2 * Dm2 - 2.0 * b * c)
dEsde_dS = 0.5 * SigInv * (16.0 * DS6 - 8.0 * c * DS4 + 8.0 * b * DS3 +
c2 * DS2)
# Gradients of Esde w.r.t. 'Theta'.
dEsde_dth = 4.0 * SigInv * mytrapz(c * Ex2 - 4.0 * Ex4 - b * m, dt, idx)
# Gradients of Esde w.r.t. 'Sigma'.
dEsde_dSig = -Esde * SigInv
# --->
return Esde, Ef, Edf, dEsde_dm, dEsde_dS, dEsde_dth, dEsde_dSig