def ODEMstep(M,x0,u,y,p,Pw,Pz,alpha_x,alpha_th,embed=0,gembed=1,mfts=False):
th = M.th;
# Fetch state estimates based on current parameter estimates
xp,xu,yp,yu = FEFstep(M,x0,u,y,p,alpha_x,embed,mfts);
# parameter gradients
dydth = - M.H(xu,u[:,1],th) - dotn(M.C(xu,u[:,1],th),M.F(xu,u[:,1],th));
dxdth = - M.F(xu,u[:,1],th) - dotn(M.A(xu,u[:,1],th),M.F(xu,u[:,0],th));
# Update gradient
dth = dotn(tp(dxdth),Pw,xu - xp) + dotn(tp(dydth),Pz,y - yp);
#Mean-field terms
if mfts:
dth += dWxdth(M,xu,u[:,1]);
# Calculate cost
J = qcf(y,yp,Pz,xu,xp,Pw);
# Update parameters
th = M.th - alpha_th*dth;
return J,th,xp,xu,yp,yu;
def FEF(M_t,x0,u_t,y_t,p,alpha,embed=0,mfts=False):
# Embed
# 0 embedded predictions
# 1 embedded derivatives
# 2 embedded history
N = len(u_t[0,:]);
# Fetch functions
A = M_t.A;
C = M_t.C;
f = M_t.f;
h = M_t.h;
# Initialize storage
xu_t = zeros((M_t.nx,N)); xu_t[:,0] = x0;
xp_t = zeros((M_t.nx,N)); xp_t[:,0] = x0;
yu_t = zeros((M_t.nu,N)); yu_t[:,0] = h(x0, u_t[:,0], M_t.th);
yp_t = zeros((M_t.ny,N)); yp_t[:,0] = h(x0, u_t[:,0], M_t.th);
# upper-shift matrix
Du = shiftmatrix(p=p,embed=embed,nx = int(M_t.nx/p),dt=1e-2);
#Fetch precisions
Pw = M_t.Q;
Pz = M_t.R;
# Run filter
for k in range(1,N):
# Predictions
xp_t[:,k] = f(xu_t[:,k-1], u_t[:,k-1], M_t.th);
yp_t[:,k] = h(xp_t[:,k], u_t[:,k], M_t.th);
xu_t[:,k] = dotn(Du,xu_t[:,k-1]);
# Errors
ex = xu_t[:,k] - xp_t[:,k];
ey = y_t[:,k] - yp_t[:,k];
# Error gradients
dexdx = Du - A(xu_t[:,k-1], u_t[:,k-1], M_t.th);
deydx = - C(xu_t[:,k], u_t[:,k], M_t.th);
# Free-Energy gradient
dFEdx = dotn(tp(dexdx),Pw,ex) + dotn(tp(deydx),Pz,ey);
# Mean-field terms if specified
if mfts:
dFEdx += dWthdx(M_t,xu_t[:,k-1],u_t[:,k-1]);
# Upddates
xu_t[:,k] = xu_t[:,k] - alpha*dFEdx
yu_t[:,k] = h(xu_t[:,k], u_t[:,k], M_t.th);
return xu_t,xp_t,yu_t,yp_t;
def KF(M,x0,u,y,th=[],Q=[],R=[]):
#if M.typ != 'ltiss':
# raise Exception('KF code for non-linear state-space (EKF) not complete');
if isempty(th):
th = M.th;
if isempty(Q):
Q = M.Q;
if isempty(R):
R = M.R;
A = M.A(zeros(M.nx),zeros(M.nu),th);
C = M.C(zeros(M.nx),zeros(M.nu),th);
#K,P,S = dare(A,C,Q,R);
nx = M.nx;
ny = M.ny;
N = len(u[0,:]);
K = zeros((nx,ny));
P = zeros((nx,nx));
S = zeros((ny,ny));
xp = zeros((nx,N)); xp[:,0] = x0;
xu = zeros((nx,N)); xu[:,0] = x0;
yp = zeros((ny,N)); yp[:,0] = M.h(xp[:,0],u[:,0],th);
yu = zeros((ny,N)); yu[:,0] = M.h(xu[:,0],u[:,0],th);
for k in range(1,N):
A[:,:] = M.A(xu[:,k-1],u[:,k-1],th);
C[:,:] = M.C(xu[:,k-1],u[:,k-1],th);
# Predict
xp[:,k] = M.f(xu[:,k-1],u[:,k-1],th);
yp[:,k] = M.h(xp[:,k ],u[:,k ],th);
P[:,:] = dotn(A,P,tp(A)) + M.Q;
# Update
S[:,:] = dotn(C,P,tp(C)) + M.R;
K[:,:] = dotn(P,tp(C),inv(S));
xu[:,k] = xp[:,k] + dotn(K,y[:,k] - yp[:,k]);
yu[:,k] = M.h(xu[:,k ],u[:,k ],th);
P[:,:] = dotn(eye(M.nx)-dotn(K,C),P);
return K,P,S,xp,xu,yp,yu;
def dare(A,C,Q,R,tol=1e-6,maxit=1000,convthresh=200):
nx = len(A[:,0]);
ny = len(C[:,0]);
P = zeros((nx,nx));
S = zeros((ny,ny));
K = zeros((nx,ny));
e1 = 1000;
J = 1000;
e2 = 0;
i = 0;
j = 0;
while i < maxit and j < convthresh:
S = dotn(C,P,tp(C)) + R;
P = dotn(A,P,tp(A)) + Q;
K = dotn(P,tp(C),inv(S));
P = dotn(eye(2)-dotn(K,C),P);
# Stopping criterion: J > tol in last convthresh iterations
J = P - dotn(A,P,tp(A)) + Q;
e2 = e1;
e1 = norm(K,2) + norm(P,2) + norm(S,2);
J = abs(e2-e1);
if j == 0:
if J < tol:
j = 1;
else:
if J < tol:
j +=1;
else:
j = 0;
i +=1;
return K,P,S;
def KFstep(M,xu,u,y,P):
th = M.th;
A = M.A(xu,u[:,0],th);
C = M.C(xu,u[:,0],th);
# Predict
xp = M.f(xu,u[:,0],th);
yp = M.h(xp,u[:,1],th);
P = dotn(A,P,tp(A)) + M.Q;
# Update
S = dotn(C,P,tp(C)) + M.R;
K = dotn(P,tp(C),inv(S));
xu = xp + dotn(K,y - yp);
P = dotn(eye(M.nx)-dotn(K,C),P);
yu = M.h(xu,u[:,1],th);
return K,P,S,xp,xu,yp,yu;
def FEFstep(M_t,x0,u_t,y_t,p,alpha,embed=0,mfts=False):
# Embed
# 0 embedded predictions
# 1 embedded derivatives
# 2 embedded history
# Fetch matrices
A = M_t.A(x0, u_t[0], M_t.th);
C = M_t.C(x0, u_t[0], M_t.th);
# upper-shift matrix
if embed == 1 or embed == 0:
Du = kron(eye(p,p,1),eye(int(M_t.nx/p)));
if embed == 1:
Du = expm(Du*M_t.dt);
elif embed == 2:
Du = kron(eye(p,p,-1),eye(int(M_t.nx/p)));
#Fetch precisions
Pw = M_t.Q;
Pz = M_t.R;
# Predictions
xp_t = M_t.f(x0, u_t[:,0], M_t.th);
yp_t = M_t.h(xp_t, u_t[:,1], M_t.th);
xu_t = dotn(Du,x0);
# Free-Energy gradient
dFEdx = dotn(tp(Du - A),Pw,xu_t - xp_t) + dotn(tp(- C),Pz,y_t - yp_t);
# Mean-field terms if specified
if mfts:
dFEdx += dWthdx(M_t,x0,u_t[:,0]);
# Upddates
xu_t = xu_t - alpha*dFEdx
yu_t = M_t.h(xu_t, u_t[:,1], M_t.th);
return xu_t,xp_t,yu_t,yp_t;
def ADEMstep(M,x0,u,y,p,Pw,Pz,alpha_x,alpha_th,embed=0,gembed=1,mfts=False,x='def'):
N = len(u[0,:]);
th = M.th;
# Fetch state estimates based on current parameter estimates
if x=='def':
xp,xu,yp,yu = FEF(M,x0,u,y,p,alpha_x,embed);
else:
D = shiftmatrix(p=p,embed=embed,nx = int(M.nx/p),dt=M.dt);
xu = zeros((M.nx,N));
xp = zeros((M.nx,N));
yu = zeros((M.ny,N));
yp = zeros((M.ny,N));
for k in range(1,N):
xu[:,k] = dotn(D,x[:,k-1]);
xp[:,k] = M.f( x[:,k-1],u[:,k-1],th);
yu[:,k] = M.h( x[:,k ],u[:,k ],th);
yp[:,k] = M.h(xp[:,k ],u[:,k ],th);
# Initialize gradients
dth = zeros((M.nth,1));
dxdth = zeros((M.nx,M.nth));
dydth = zeros((M.ny,M.nth));
ex = zeros((M.nx,1));
ey = zeros((M.ny,1));
# Pre-calculate all matrices (trade storage for efficiency)
F = zeros((M.nx,M.nth,N));
H = zeros((M.ny,M.nth,N));
A = zeros((M.nx,M.nx,N));
B = zeros((M.nx,M.nu,N));
C = zeros((M.ny,M.nx,N));
D = zeros((M.ny,M.nu,N));
if x=='def':
for k in range(0,N):
F[:,:,k] = M.F(xu[:,k],u[:,k],th);
H[:,:,k] = M.H(xu[:,k],u[:,k],th);
A[:,:,k] = M.A(xu[:,k],u[:,k],th);
B[:,:,k] = M.B(xu[:,k],u[:,k],th);
C[:,:,k] = M.C(xu[:,k],u[:,k],th);
D[:,:,k] = M.D(xu[:,k],u[:,k],th);
else:
for k in range(0,N):
F[:,:,k] = M.F(x[:,k],u[:,k],th);
H[:,:,k] = M.H(x[:,k],u[:,k],th);
A[:,:,k] = M.A(x[:,k],u[:,k],th);
B[:,:,k] = M.B(x[:,k],u[:,k],th);
C[:,:,k] = M.C(x[:,k],u[:,k],th);
D[:,:,k] = M.D(x[:,k],u[:,k],th);
# Calculate FE-to-parameter gradient
for k in range(0,N):
# Errors
ex[:,0] = xu[:,k] - xp[:,k];
ey[:,0] = y[:,k] - yu[:,k];
# To do: Should at some point it turn out that gradient-emnedding is
# necessary, then:
# # State-to-parameter gradients
# dxdth[:,:] = 0;
# imax = min(k,gembed);
# for i in range(1,imax):
# dxdth[:] = F[:,:,k-imax+i] + dotn(A[:,:,k-imax+i],dxdth)
# dydth[:,:] = - H[:,:,k] - dotn(C[:,:,k],dxdth);
# dxdth[:,:] = - F[:,:,k] - dotn(A[:,:,k],dxdth);
# But until then:
dydth[:,:] = - H[:,:,k]; #- dotn(C[:,:,k],F[:,:,k-1]);
dxdth[:,:] = - F[:,:,k]; #- dotn(A[:,:,k],F[:,:,k-1]);
# update gradient
dth[:,:] += dotn(tp(dxdth),Pw,ex) + dotn(tp(dydth),Pz,ey);
# Calculate cost
J = qcf(y,yu,Pz,xu,xp,Pw);
# Update parameters
th = M.th - alpha_th*dth[:,0];
return J,th,xp,xu,yp,yu;
def AEMstep(M,x0,u,y,Q,R,alpha=1,gembed=1):
"""
# 1-step Algebraical-gradient Expectation-Maximization
INPUTS
M Data-structure of Model class
x0 Initial hidden state - 1-dimensional array_like (list works)
u Input sequence nu x N numpy_array
y Output sequence ny x N numpy_array
Q State-noise covariance - nx x nx numpy_array
R Output-noise covariance - ny x ny numpy_array
alpha Updating gain - scalar float, int - (opt. def=1)
gembed Gradient-embedding order - scalar int - (opt. def=1)
OUTPUTS
J Cost (log-likelihood) - scalar float
K Kalman gain - nx x ny numpy_array
P State-error covariance estimate - nx x nx numpy_array
S Output-error covariance estimate - nx x nx numpy_array
th Updated parameter estimate - 1-dimensional numpy_array
xp Predicted hidden state estimates - nx x N numpy_array
xu Updated hidden state estimates - nx x N numpy_array
xp Predicted output estimates - ny x N numpy_array
x
"""
N = len(u[0,:]);
xu = zeros((M.nx,N));
xp = zeros((M.nx,N));
yu = zeros((M.ny,N));
yp = zeros((M.ny,N));
#th = zeros((M.nth,N));
th = M.th;
# P = zeros((2,2));
#
# for k in range(1,N):
#
#
# Nk = min(k,gembed);
# J,K,P,S,th[:,k],xp[:,k],xu[:,k],yp[:,k],yu[:,k] = OEMstep(M,u[:,k-Nk:k+1],\
# xu[:,k-Nk:k+1],xp[:,k-Nk:k+1],y[:,k-Nk:k+1],\
# yp[:,k-Nk:k+1],th[:,k-Nk:k+1],iQ,iR,P,alpha)
#
# M.th = th[:,k];
#
# J = ll(y,yp,iR,xu,xp,iQ);
#
# return J,K,P,S,th[:,-1],xp,xu,yp,yu;
#
# Fetch state estimates based on current parameter estimates
K,P,S,xp,xu,yp,yu = KF(M,x0,u,y,th);
iQ = inv(P);
iR = inv(S);
# Initialize gradients
dth = zeros(M.nth);
dxdth = zeros((M.nx,M.nth));
dydth = zeros((M.ny,M.nth));
# Pre-calculate all matrices (trade storage for efficiency)
F = zeros((M.nx,M.nth,N));
H = zeros((M.ny,M.nth,N));
A = zeros((M.nx,M.nx,N));
B = zeros((M.nx,M.nu,N));
C = zeros((M.ny,M.nx,N));
D = zeros((M.ny,M.nu,N));
for k in range(0,N):
F[:,:,k] = M.F(xu[:,k],u[:,k],th);
H[:,:,k] = M.H(xu[:,k],u[:,k],th);
A[:,:,k] = M.A(xu[:,k],u[:,k],th);
B[:,:,k] = M.B(xu[:,k],u[:,k],th);
C[:,:,k] = M.C(xu[:,k],u[:,k],th);
D[:,:,k] = M.D(xu[:,k],u[:,k],th);
# Calculate FE-to-parameter gradient
for k in range(0,N-1):
# Errors
ex = xu[:,k+1] - xp[:,k+1];
ey = y[:,k ] - yu[:,k ];
# State-to-parameter gradients
dxdth[:,:] = 0;
kmin = max(0,k-gembed);
for i in range(kmin,k):
dxdth[:,:] = F[:,:,i] + dotn(A[:,:,i],dxdth);
dydth[:,:] = - H[:,:,k] - dotn(C[:,:,k],dxdth);
dxdth[:,:] = - F[:,:,k] - dotn(A[:,:,k],dxdth);
# update gradient
dth[:] += dotn(tp(dxdth),iQ,ex) + dotn(tp(dydth),iR,ey);
# Update parameters
th[:] = th - alpha*dth;
# Calculate cost
J = ll(y,yp,iR,xu,xp,iQ);
return J,K,P,S,th,xp,xu,yp,yu;
A[1,0] = -k/m;
A[1,1] = -d/m;
A = eye(2) + dt*A;
B = zeros((2,1));
B[1,0] = 1/m;
B = dt*B;
C = zeros((1,2));
C[0,0] = 1;
D = zeros((1,1));
# Get noise signals
zeta = zeros((3,2)); zeta[:,1] = std; # Sufficient statistics (mn, std)
Q = diag(std[0:2]); Q = dotn(tp(Q),Q); # Covariance on state noise
R = diag([std[2]]); R = dotn(tp(R),R); # Covariance on soutput noise
t,W = Noise(dt,T,1,1,1,zeta,[],[],seed).getNoise();
w[:,:] = W[0:1,:]; # State noise
z[:,:] = W[2,:]; # Output noise
# Construct input signal (step at 2 sec.)
u[0,int(2/dt+1):] = 10;
# Simulate system
y[:,0] = dotn(C,x[:,0]) + dotn(D,u[:,0]) + z[:,0];
for k in range(1,N):
x[:,k] = dotn(A,x[:,k-1]) + dotn(B,u[:,k-1]) + w[:,k-1];
y[:,k] = dotn(C,x[:,k]) + dotn(D,u[:,k]) + z[:,k];
# General figure settings