Tplot=Tplot,
BurnIn=10 * OneYear)
land_sites = np.arange(Nx // 2)
ocean_sites = np.arange(Nx // 2, Nx)
jj = land_sites
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (1, ), jj)
HMM = dpr.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
# It can be seen that Miyoshi's "Global RMSE" is just the average of the land and ocean RMSE's,
# which explains why this differs so much from DAPPER's (conventionally defined) global RMSE.
import dapper as dpr
from dapper.mods.Lorenz96 import dstep_dx, step
from dapper.tools.localization import nd_Id_localization
t = dpr.Chronology(0.05, dtObs=0.4, T=4**5, BurnIn=20)
Nx = 40
Dyn = {'M': Nx, 'model': step, 'linear': dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(M=Nx, C=0.001)
jj = 1 + np.arange(0, Nx, 2)
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.5
Obs['localizer'] = nd_Id_localization((Nx, ), (2, ), jj)
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
####################
# Suggested tuning
####################
# Compare to Table 1 and 3 from frei2013bridging. Note:
# - N is too large to be very interesting.
# - We obtain better EnKF scores than they report,
# and use inflation and sqrt updating,
# and don't really need localization.
# from dapper.mods.Lorenz96.frei2013bridging import HMM # rmse.a
# xps += EnKF_N(N=400,rot=1) # 0.80
# xps += LETKF( N=400,rot=True,infl=1.01,loc_rad=10/1.82) # 0.79 # short xp. only
# xps += Var3D() # 2.42 # short xp. only
Nx = 100
# def step(x,t,dt):
# return np.roll(x,1,axis=x.ndim-1)
Fm = Fmat(Nx, -1, 1, tseq.dt)
def step(x, t, dt):
assert dt == tseq.dt
return x @ Fm.T
Dyn = {'M': Nx, 'model': step, 'linear': lambda x, t, dt: Fm, 'noise': 0}
X0 = dpr.GaussRV(mu=np.zeros(Nx),
C=homogeneous_1D_cov(Nx, Nx / 8, kind='Gauss'))
Ny = 4
jj = dpr.linspace_int(Nx, Ny)
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.01
HMM = dpr.HiddenMarkovModel(Dyn, Obs, tseq, X0, LP=LPs(jj))
####################
# Suggested tuning
####################
# xps += EnKF('PertObs',N=16 ,infl=1.02)
# xps += EnKF('Sqrt' ,N=16 ,infl=1.0)
# Settings not taken from anywhere
import dapper as dpr
from dapper.mods.LotkaVolterra import step, dstep_dx, x0, LP_setup, Tplot
# dt has been chosen after noting that
# using dt up to 0.7 does not change the chaotic properties much,
# as adjudged with eye-ball and Lyapunov measures.
t = dpr.Chronology(0.5, dtObs=10, T=1000, BurnIn=Tplot, Tplot=Tplot)
Nx = len(x0)
Dyn = {'M': Nx, 'model': step, 'linear': dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(mu=x0, C=0.01**2)
jj = [1, 3]
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.04**2
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0, LP=LP_setup(jj))
####################
# Suggested tuning
####################
# Not carefully tuned:
# xps += EnKF_N(N=6)
# xps += ExtKF(infl=1.02)
Dyn = {
'M': LUV.M,
'model': dpr.with_rk4(LUV.dxdt, autonom=True),
'noise': 0,
'linear': LUV.dstep_dx,
}
X0 = dpr.GaussRV(mu=LUV.x0, C=0.01)
R = 1.0
jj = np.arange(nU)
Obs = dpr.partial_Id_Obs(LUV.M, jj)
Obs['noise'] = R
other = {'name': utils.rel2mods(__file__)+'_full'}
HMM_full = dpr.HiddenMarkovModel(Dyn, Obs, t, X0, **other)
################
# Truncated
################
# Just change dt from 005 to 05
t = dpr.Chronology(dt=0.05, dtObs=0.05, T=4**3, BurnIn=6)
Dyn = {
'M': nU,
'model': dpr.with_rk4(LUV.dxdt_parameterized),
'noise': 0,
}
Dyn = {
'M': LUV.M,
'model': dpr.with_rk4(LUV.dxdt, autonom=True),
'noise': 0,
'linear': LUV.dstep_dx,
}
X0 = dpr.GaussRV(mu=LUV.x0, C=0.01)
R = 0.1
jj = np.arange(nU)
Obs = dpr.partial_Id_Obs(LUV.M, jj)
Obs['noise'] = R
other = {'name': utils.rel2mods(__file__) + '_full'}
HMM_full = dpr.HiddenMarkovModel(Dyn, Obs, t, X0, LP=LUV.LPs(jj), **other)
################
# Truncated
################
# Just change dt from 005 to 05
t = dpr.Chronology(dt=0.05, dtObs=0.05, T=4**3, BurnIn=6)
Dyn = {
'M': nU,
'model': dpr.with_rk4(LUV.dxdt_parameterized),
'noise': 0,
}
X0 = dpr.GaussRV(mu=LUV.x0[:nU], C=0.01)
'M': Ny,
'model': hmod,
'noise': dpr.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 = dpr.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).
# - The boundary cells are all fixed at 0 by BCs,
periodic=True,
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 = dpr.HiddenMarkovModel(
Dyn,
Obs,
t,
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','std']:
# stat = getattr(xp.stats,lbl).f[HMM.t.maskObs_BI]
# plt.plot(sqrt(np.mean(stat**2, axis=0)),label=lbl)
#
