def HMMs(stepper="Tay2", resolution="Low", R=1):
"""Define the various HMMs used."""
# Use small version of L96. Has 4 non-stable Lyapunov exponents.
Nx = 10
# Time sequence
# Grudzien'2020 uses the below chronology with Ko=25000, BurnIn=5000.
t = modelling.Chronology(dt=0.005, dto=.1, T=30, Tplot=Tplot, BurnIn=10)
if resolution == "High":
t.dt = 0.001
elif stepper != "Tay2":
t.dt = 0.01
# Dynamical operator
Dyn = {'M': Nx, 'model': steppers(stepper)}
# (Random) initial condition
X0 = modelling.GaussRV(mu=x0(Nx), C=0.001)
# Observation operator
jj = range(Nx) # obs_inds
Obs = modelling.partial_Id_Obs(Nx, jj)
Obs['noise'] = R
return modelling.HiddenMarkovModel(Dyn, Obs, t, X0)
Nx = 80
Dyn = {
'M': Nx,
'model': step,
'noise': 0,
}
X0 = modelling.GaussRV(M=Nx, C=0.001)
jj = np.arange(0, Nx, 2)
Obs = modelling.partial_Id_Obs(Nx, jj)
Obs['localizer'] = nd_Id_localization((Nx,), (1,), jj)
# Obs['noise'] = RVs.LaplaceRV(C=1,M=len(jj))
Obs['noise'] = RVs.LaplaceParallelRV(C=1, M=len(jj))
HMM = modelling.HiddenMarkovModel(Dyn, Obs, t, X0)
####################
# Suggested tuning
####################
# rmse.a
# xps += LETKF(N=20,rot=True,infl=1.04 ,loc_rad=5) # 0.44
# xps += LETKF(N=40,rot=True,infl=1.04 ,loc_rad=5) # 0.44
# xps += LETKF(N=80,rot=True,infl=1.04 ,loc_rad=5) # 0.43
# These scores are quite variable:
# xps += LNETF(N=40,rot=True,infl=1.10,Rs=2.5,loc_rad=5) # 0.57
# xps += LNETF(N=80,rot=True,infl=1.10,Rs=1.6,loc_rad=5) # 0.45
# In addition to standard post-analysis inflation,
# we also find the necessity of tuning the inflation (Rs) for R,
Fm = Fmat(Nx, -1, 1, tseq.dt)
def step(x, t, dt):
assert dt == tseq.dt
return x @ Fm.T
Dyn = {
'M': Nx,
'model': lambda x, t, dt: damp * step(x, t, dt),
'linear': lambda x, t, dt: damp * Fm,
'noise': modelling.GaussRV(C=modelling.CovMat(L, 'Left')),
}
HMM = modelling.HiddenMarkovModel(Dyn, Obs, tseq, X0, LP=LPs(jj))
####################
# Suggested tuning
####################
# Expected rmse.a = 0.3
# xp = EnKF('PertObs',N=30,infl=3.2)
# Note that infl=1 may yield approx optimal rmse, even though then rmv << rmse.
# Why is rmse so INsensitive to inflation, especially for PertObs?
# Reproduce raanes'2015 "extending sqrt method to model noise":
# xp = EnKF('Sqrt',fnoise_treatm='XXX',N=30,infl=1.0),
# where XXX is one of:
# - Stoch
# - Mult-1
'model': modelling.with_rk4(LUV.dxdt, autonom=True),
'noise': 0,
'linear': LUV.dstep_dx,
}
X0 = modelling.GaussRV(mu=LUV.x0, C=0.01)
R = 0.1
jj = np.arange(nU)
Obs = modelling.partial_Id_Obs(LUV.M, jj)
Obs['noise'] = R
other = {'name': rel2mods(__file__) + '_full'}
HMM_full = modelling.HiddenMarkovModel(Dyn,
Obs,
tseq,
X0,
LP=LUV.LPs(jj),
**other)
################
# Truncated
################
# Just change dt from 005 to 05
tseq = modelling.Chronology(dt=0.05, dto=0.05, T=4**3, BurnIn=6)
Dyn = {
'M': nU,
'model': modelling.with_rk4(LUV.dxdt_parameterized),
'noise': 0,
}
'model': hmod,
'noise': modelling.GaussRV(C=4*np.eye(Ny)),
'localizer': localizer,
}
# Moving localization mask for smoothers:
Obs['loc_shift'] = lambda ii, dt: ii # no movement (suboptimal, but easy)
# Jacobian left unspecified coz it's (usually) employed by methods that
# compute full cov, which in this case is too big.
############################
# Other
############################
HMM = modelling.HiddenMarkovModel(Dyn, Obs, t, X0, LP=LP_setup(obs_inds))
####################
# Suggested tuning
####################
# Reproducing Fig 7 from Sakov and Oke "DEnKF" paper from 2008.
# Notes:
# - If N<=25, then typically need to increase the dissipation
# to be almost sure to avoid divergence. See counillon2009.py for example.
# - We have not had the need to increase the dissipation parameter for the EnKF.
# - Our experiments differ from Sakov's in the following minor details:
# - We use a batch width (unsure what Sakov uses).
# - The "EnKF-Matlab" code has a bug: it forgets to take sqrt() of the taper coeffs.
# This is equivalent to: R_actually_used = R_reported / sqrt(2).
Dyn = {
'M': LUV.M,
'model': modelling.with_rk4(LUV.dxdt, autonom=True),
'noise': 0,
'linear': LUV.dstep_dx,
}
X0 = modelling.GaussRV(mu=LUV.x0, C=0.01)
R = 1.0
jj = np.arange(nU)
Obs = modelling.partial_Id_Obs(LUV.M, jj)
Obs['noise'] = R
other = {'name': rel2mods(__file__) + '_full'}
HMM_full = modelling.HiddenMarkovModel(Dyn, Obs, t, X0, **other)
################
# Truncated
################
# Just change dt from 005 to 05
t = modelling.Chronology(dt=0.05, dtObs=0.05, T=4**3, BurnIn=6)
Dyn = {
'M': nU,
'model': modelling.with_rk4(LUV.dxdt_parameterized),
'noise': 0,
}
X0 = modelling.GaussRV(mu=LUV.x0[:nU], C=0.01)
import dapper.mods as modelling
from dapper.mods.LotkaVolterra import step, dstep_dx, x0, LP_setup, Tplot
t = modelling.Chronology(0.5, dto=10, T=1000, BurnIn=Tplot, Tplot=Tplot)
Nx = len(x0)
Dyn = {
'M': Nx,
'model': step,
'linear': dstep_dx,
'noise': 0,
}
X0 = modelling.GaussRV(mu=x0, C=0.01**2)
jj = [1, 3]
Obs = modelling.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.04**2
HMM = modelling.HiddenMarkovModel(Dyn, Obs, t, X0, LP=LP_setup(jj))
####################
# Suggested tuning
####################
# Not carefully tuned:
# xps += EnKF_N(N=6)
# xps += ExtKF(infl=1.02)
jj, zoomy=0.8, dims=[0, Nx], labels=["$x_0$", "Force"]),
),
]
# Labels for sectors of state vector.
# DAPPER will compute diagnostic statistics for the full state vector,
# but also for each sector of it (averaged in space according to the
# methods specified in your .dpr_config.yaml:field_summaries key).
# The name "sector" comes from its typical usage to distinguish
# "ocean" and "land" parts of the state vector.
# Here we use it to get individual statistics of the parameter and state.
parts = dict(state=np.arange(Nx),
param=np.arange(Np)+Nx)
# Wrap-up model specification
HMM = modelling.HiddenMarkovModel(Dyn, Obs, t, sectors=parts, LP=LP)
# #### Treat truth and DA methods differently
# Bocquet et al. do not sample the true parameter value from the
# Bayesian (random) prior / initial cond's (ICs), given to the DA methods.
# Instead it is simply set to 8.
TRUTH = 8
GUESS = 7
# Seeing how far off the intial guess (and its uncertainty, defined below)
# is from the truth, this constitutes a kind of model error.
# It is not a feature required to make this experiment interesting.
# However, our goal here is to reproduce the results of Bocquet et al.,
BurnIn=10 * OneYear)
land_sites = np.arange(Nx // 2)
ocean_sites = np.arange(Nx // 2, Nx)
jj = land_sites
Obs = modelling.partial_Id_Obs(Nx, jj)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (1, ), jj)
HMM = modelling.HiddenMarkovModel(
Dyn,
Obs,
t,
X0,
LP=LPs(jj),
sectors={
'land': land_sites,
'ocean': ocean_sites
},
)
####################
# Suggested tuning
####################
# Reproduce Miyoshi Figure 5 # rmse.a rmse.land.a rmse.ocean.a
# ------------------------------------------------------------------------------
# xps += LETKF(N=10,rot=False,
# infl=sqrt(1.015),loc_rad=3) # 2.1 0.38 2.9
np.arange(Nx)[:, None],
domain=(Nx, ))
batches = np.arange(40)[:, None]
# Define operator
Obs = {
'M': len(H),
'model': lambda E, t: E @ H.T,
'linear': lambda E, t: H,
'noise': 1,
'localizer': localization_setup(lambda t: y2x_dists, batches),
}
HMM = modelling.HiddenMarkovModel(
Dyn,
Obs,
tseq,
X0,
LP=LPs(),
sectors={'land': np.arange(*xtrema(obs_sites)).astype(int)})
####################
# Suggested tuning
####################
# Reproduce Anderson Figure 2
# -----------------------------------------------------------------------------------
# xp = SL_EAKF(N=6, infl=sqrt(1.1), loc_rad=0.2/1.82*40)
# for lbl in ['err', 'spread']:
# stat = getattr(xp.stats,lbl).f[HMM.tseq.masko]
# plt.plot(sqrt(np.mean(stat**2, axis=0)),label=lbl)
#
