def assimilate(self, HMM, xx, yy):
Dyn, Obs, chrono, X0, stats, N = \
HMM.Dyn, HMM.Obs, HMM.t, HMM.X0, self.stats, self.N
N1 = N - 1
step = 1 / N
cdf_grid = np.linspace(step / 2, 1 - step / 2, N)
R = Obs.noise
Rm12 = Obs.noise.C.sym_sqrt_inv
E = X0.sample(N)
stats.assess(0, E=E)
for k, kObs, t, dt in progbar(chrono.ticker):
E = Dyn(E, t - dt, dt)
E = add_noise(E, dt, Dyn.noise, self.fnoise_treatm)
if kObs is not None:
stats.assess(k, kObs, 'f', E=E)
y = yy[kObs]
inds = serial_inds(self.ordr, y, R, center(E)[0])
for _, j in enumerate(inds):
Eo = Obs(E, t)
xo = np.mean(Eo, 0)
Y = Eo - xo
mu = np.mean(E, 0)
A = E - mu
# Update j-th component of observed ensemble
dYf = Rm12[j, :] @ (y - Eo).T # NB: does Rm12 make sense?
Yj = Rm12[j, :] @ Y.T
Regr = A.T @ Yj / np.sum(Yj**2)
Sorted = np.argsort(dYf)
Revert = np.argsort(Sorted)
dYf = dYf[Sorted]
w = reweight(np.ones(N), innovs=dYf[:, None]) # Lklhd
w = w.clip(1e-10) # Avoid zeros in interp1
cw = w.cumsum()
cw /= cw[-1]
cw *= N1 / N
cdfs = np.minimum(np.maximum(cw[0], cdf_grid), cw[-1])
dhE = -dYf + np.interp(cdfs, cw, dYf)
dhE = dhE[Revert]
# Update state by regression
E += np.outer(-dhE, Regr)
E = post_process(E, self.infl, self.rot)
stats.assess(k, kObs, E=E)
def assimilate(self, HMM, xx, yy):
chrono, X0, stats = HMM.t, HMM.X0, self.stats
R, KObs = HMM.Obs.noise.C, HMM.t.KObs
Rm12 = R.sym_sqrt_inv
assert HMM.Dyn.noise.C == 0, (
"Q>0 not yet supported."
" See Sakov et al 2017: 'An iEnKF with mod. error'")
if self.bundle:
EPS = 1e-4 # Sakov/Boc use T=EPS*eye(N), with EPS=1e-4, but I ...
else:
EPS = 1.0 # ... prefer using T=EPS*T, yielding a conditional cloud shape
# Initial ensemble
E = X0.sample(self.N)
# Forward ensemble to kObs = 0 if Lag = 0
t = 0
k = 0
if self.Lag == 0:
for k, t, dt in chrono.cycle(kObs=0):
stats.assess(k - 1, None, 'u', E=E)
E = HMM.Dyn(E, t - dt, dt)
# Loop over DA windows (DAW).
for kObs in progbar(range(0, KObs + self.Lag + 1)):
kLag = kObs - self.Lag
DAW = range(max(0, kLag + 1), min(kObs, KObs) + 1)
# Assimilation (if ∃ "not-fully-assimlated" obs).
if kObs <= KObs:
E = iEnKS_update(self.upd_a, E, DAW, HMM, stats, EPS, yy[kObs],
(k, kObs, t), Rm12, self.xN, self.MDA,
(self.nIter, self.wtol))
E = post_process(E, self.infl, self.rot)
# Slide/shift DAW by propagating smoothed ('s') ensemble from [kLag].
if kLag >= 0:
stats.assess(chrono.kkObs[kLag], kLag, 's', E=E)
cycle_window = range(max(kLag + 1, 0),
min(max(kLag + 1 + 1, 0), KObs + 1))
for kCycle in cycle_window:
for k, t, dt in chrono.cycle(kCycle):
stats.assess(k - 1, None, 'u', E=E)
E = HMM.Dyn(E, t - dt, dt)
stats.assess(k, KObs, 'us', E=E)
def assimilate(self, HMM, xx, yy):
Dyn, Obs, chrono, X0, stats = HMM.Dyn, HMM.Obs, HMM.t, HMM.X0, self.stats
Rm12 = Obs.noise.C.sym_sqrt_inv
E = X0.sample(self.N)
stats.assess(0, E=E)
for k, kObs, t, dt in progbar(chrono.ticker):
E = Dyn(E, t - dt, dt)
E = add_noise(E, dt, Dyn.noise, self.fnoise_treatm)
if kObs is not None:
stats.assess(k, kObs, 'f', E=E)
mu = np.mean(E, 0)
A = E - mu
Eo = Obs(E, t)
xo = np.mean(Eo, 0)
YR = (Eo - xo) @ Rm12.T
yR = (yy[kObs] - xo) @ Rm12.T
state_batches, obs_taperer = Obs.localizer(
self.loc_rad, 'x2y', t, self.taper)
for ii in state_batches:
# Localize obs
jj, tapering = obs_taperer(ii)
if len(jj) == 0:
return
Y_jj = YR[:, jj] * np.sqrt(tapering)
dy_jj = yR[jj] * np.sqrt(tapering)
# NETF:
# This "paragraph" is the only difference to the LETKF.
innovs = (dy_jj - Y_jj) / self.Rs
if 'laplace' in str(type(Obs.noise)).lower():
w = laplace_lklhd(innovs)
else: # assume Gaussian
w = reweight(np.ones(self.N), innovs=innovs)
dmu = w @ A[:, ii]
AT = np.sqrt(self.N) * funm_psd(
np.diag(w) - np.outer(w, w), np.sqrt) @ A[:, ii]
E[:, ii] = mu[ii] + dmu + AT
E = post_process(E, self.infl, self.rot)
stats.assess(k, kObs, E=E)
def assimilate(self, HMM, xx, yy):
Dyn, Obs, chrono, X0, stats, N = \
HMM.Dyn, HMM.Obs, HMM.t, HMM.X0, self.stats, self.N
R, KObs, N1 = HMM.Obs.noise.C, HMM.t.KObs, N - 1
Rm12 = R.sym_sqrt_inv
assert Dyn.noise.C == 0, (
"Q>0 not yet supported."
" See Sakov et al 2017: 'An iEnKF with mod. error'")
if self.bundle:
EPS = 1e-4 # Sakov/Boc use T=EPS*eye(N), with EPS=1e-4, but I ...
else:
EPS = 1.0 # ... prefer using T=EPS*T, yielding a conditional cloud shape
# Initial ensemble
E = X0.sample(N)
# Loop over DA windows (DAW).
for kObs in progbar(np.arange(-1, KObs + self.Lag + 1)):
kLag = kObs - self.Lag
DAW = range(max(0, kLag + 1), min(kObs, KObs) + 1)
# Assimilation (if ∃ "not-fully-assimlated" obs).
if 0 <= kObs <= KObs:
# Init iterations.
X0, x0 = center(E) # Decompose ensemble.
w = np.zeros(N) # Control vector for the mean state.
T = np.eye(N) # Anomalies transform matrix.
Tinv = np.eye(N)
# Explicit Tinv [instead of tinv(T)] allows for merging MDA code
# with iEnKS/EnRML code, and flop savings in 'Sqrt' case.
for iteration in np.arange(self.nIter):
# Reconstruct smoothed ensemble.
E = x0 + (w + EPS * T) @ X0
# Forecast.
for kCycle in DAW:
for k, t, dt in chrono.cycle(kCycle): # noqa
E = Dyn(E, t - dt, dt)
# Observe.
Eo = Obs(E, t)
# Undo the bundle scaling of ensemble.
if EPS != 1.0:
E = inflate_ens(E, 1 / EPS)
Eo = inflate_ens(Eo, 1 / EPS)
# Assess forecast stats; store {Xf, T_old} for analysis assessment.
if iteration == 0:
stats.assess(k, kObs, 'f', E=E)
Xf, xf = center(E)
T_old = T
# Prepare analysis.
y = yy[kObs] # Get current obs.
Y, xo = center(Eo) # Get obs {anomalies, mean}.
dy = (y - xo) @ Rm12.T # Transform obs space.
Y = Y @ Rm12.T # Transform obs space.
Y0 = Tinv @ Y # "De-condition" the obs anomalies.
V, s, UT = svd0(Y0) # Decompose Y0.
# Set "cov normlzt fctr" za ("effective ensemble size")
# => pre_infl^2 = (N-1)/za.
if self.xN is None:
za = N1
else:
za = zeta_a(*hyperprior_coeffs(s, N, self.xN), w)
if self.MDA:
# inflation (factor: nIter) of the ObsErrCov.
za *= self.nIter
# Post. cov (approx) of w,
# estimated at current iteration, raised to power.
def Cowp(expo):
return (V * (pad0(s**2, N) + za)**-expo) @ V.T
Cow1 = Cowp(1.0)
if self.MDA: # View update as annealing (progressive assimilation).
Cow1 = Cow1 @ T # apply previous update
dw = dy @ Y.T @ Cow1
if 'PertObs' in self.upd_a: # == "ES-MDA". By Emerick/Reynolds
D = mean0(np.random.randn(*Y.shape)) * np.sqrt(
self.nIter)
T -= (Y + D) @ Y.T @ Cow1
elif 'Sqrt' in self.upd_a: # == "ETKF-ish". By Raanes
T = Cowp(0.5) * np.sqrt(za) @ T
elif 'Order1' in self.upd_a: # == "DEnKF-ish". By Emerick
T -= 0.5 * Y @ Y.T @ Cow1
# Tinv = eye(N) [as initialized] coz MDA does not de-condition.
else: # View update as Gauss-Newton optimzt. of log-posterior.
grad = Y0 @ dy - w * za # Cost function gradient
dw = grad @ Cow1 # Gauss-Newton step
# ETKF-ish". By Bocquet/Sakov.
if 'Sqrt' in self.upd_a:
# Sqrt-transforms
T = Cowp(0.5) * np.sqrt(N1)
Tinv = Cowp(-.5) / np.sqrt(N1)
# Tinv saves time [vs tinv(T)] when Nx<N
# "EnRML". By Oliver/Chen/Raanes/Evensen/Stordal.
elif 'PertObs' in self.upd_a:
D = mean0(np.random.randn(*Y.shape)) \
if iteration == 0 else D
gradT = -(Y + D) @ Y0.T + N1 * (np.eye(N) - T)
T = T + gradT @ Cow1
# Tinv= tinv(T, threshold=N1) # unstable
Tinv = sla.inv(T + 1) # the +1 is for stability.
# "DEnKF-ish". By Raanes.
elif 'Order1' in self.upd_a:
# Included for completeness; does not make much sense.
gradT = -0.5 * Y @ Y0.T + N1 * (np.eye(N) - T)
T = T + gradT @ Cow1
Tinv = tinv(T, threshold=N1)
w += dw
if dw @ dw < self.wtol * N:
break
# Assess (analysis) stats.
# The final_increment is a linearization to
# (i) avoid re-running the model and
# (ii) reproduce EnKF in case nIter==1.
final_increment = (dw + T - T_old) @ Xf
# See docs/snippets/iEnKS_Ea.jpg.
stats.assess(k, kObs, 'a', E=E + final_increment)
stats.iters[kObs] = iteration + 1
if self.xN:
stats.infl[kObs] = np.sqrt(N1 / za)
# Final (smoothed) estimate of E at [kLag].
E = x0 + (w + T) @ X0
E = post_process(E, self.infl, self.rot)
# Slide/shift DAW by propagating smoothed ('s') ensemble from [kLag].
if -1 <= kLag < KObs:
if kLag >= 0:
stats.assess(chrono.kkObs[kLag], kLag, 's', E=E)
for k, t, dt in chrono.cycle(kLag + 1):
stats.assess(k - 1, None, 'u', E=E)
E = Dyn(E, t - dt, dt)
stats.assess(k, KObs, 'us', E=E)
