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)
from dapper.tools.localization import nd_Id_localization
from dapper.tools.math import Id_Obs
KS = Model(dt=0.5)
Nx = KS.Nx
# nRepeat=10
t = dpr.Chronology(KS.dt,
dkObs=2,
KObs=2 * 10**4,
BurnIn=2 * 10**3,
Tplot=Tplot)
Dyn = {'M': Nx, 'model': KS.step, 'linear': KS.dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(mu=KS.x0, C=0.001)
Obs = Id_Obs(Nx)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx, ), (4, ))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(np.arange(Nx))
####################
# Suggested tuning
####################
# Reproduce (top-right panel) of Fig. 4 of bocquet2019consistency # Expected rmse.a:
# --------------------------------------------------------------------------------
# - FREI, M. & KUNSCH H. R. (2013).
# "Mixture ensemble Kalman filters"
# Comp. Statist. Data Anal. 58, 127–38.
import numpy as np
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,
# 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)
"""Setup parameters for twin experiments."""
import numpy as np
import dapper as dpr
from dapper.mods.Ikeda import step, x0, Tplot, LPs
t = dpr.Chronology(1, dkObs=1, KObs=1000, Tplot=Tplot, BurnIn=4 * Tplot)
Nx = len(x0)
Dyn = {'M': Nx, 'model': step, 'noise': 0}
X0 = dpr.GaussRV(C=.1, mu=x0)
jj = np.arange(Nx) # obs_inds
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = .1 # dpr.GaussRV(C=CovMat(1*eye(Nx)))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(jj)
####################
# Suggested tuning
####################
Nx = len(x0)
Ny = Nx
day = 0.05/6 * 24 # coz dt=0.05 <--> 6h in "model time scale"
t = dpr.Chronology(0.05, dkObs=1, T=200*day, BurnIn=10*day)
Dyn = {
'M': Nx,
'model': step,
'linear': dstep_dx,
'noise': 0
}
# X0 = dpr.GaussRV(C=0.01,M=Nx) # Decreased from Pajonk's C=1.
X0 = dpr.GaussRV(C=0.01, mu=x0)
jj = np.arange(Nx)
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 0.1
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0, LP=LPs(jj))
####################
# Suggested tuning
####################
# xps += ExtKF(infl=2)
# xps += EnKF('Sqrt',N=3,infl=1.01)
# xps += PartFilt(reg=1.0, N=100, NER=0.4) # add reg!
# xps += PartFilt(reg=1.0, N=1000, NER=0.1) # add reg!
################
# Full
################
t = dpr.Chronology(dt=0.005, dtObs=0.05, T=4**3, BurnIn=6)
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
# Localization.
batch_shape = [2, 2] # width (in grid points) of each state batch.
# Increasing the width
# => quicker analysis (but less rel. speed-up by parallelzt., depending on NPROC)
# => worse (increased) rmse (but width 4 is only slightly worse than 1);
# if inflation is applied locally, then rmse might actually improve.
localizer = nd_Id_localization(shape[::-1],
batch_shape[::-1],
obs_inds,
periodic=False)
Obs = {
'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))
####################
from dapper.tools.localization import nd_Id_localization
# Sakov uses K=300000, BurnIn=1000*0.05
t = dpr.Chronology(0.05, dkObs=1, KObs=1000, Tplot=Tplot, BurnIn=2*Tplot)
Nx = 40
x0 = x0(Nx)
Dyn = {
'M': Nx,
'model': step,
'linear': dstep_dx,
'noise': 0
}
X0 = dpr.GaussRV(mu=x0, C=0.001)
jj = np.arange(Nx) # obs_inds
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 1
Obs['localizer'] = nd_Id_localization((Nx,), (2,))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(jj)
####################
# Suggested tuning
####################
"""Reproduce results from Table 1 of Sakov, Oliver, Bertino (2012):
'An Iterative EnKF for Strongly Nonlinear Systems'"""
import numpy as np
import dapper as dpr
from dapper.mods.Lorenz63 import LPs, Tplot, dstep_dx, step, x0
t = dpr.Chronology(0.01, dkObs=25, KObs=1000, Tplot=Tplot, BurnIn=4 * Tplot)
Nx = len(x0)
Dyn = {'M': Nx, 'model': step, 'linear': dstep_dx, 'noise': 0}
X0 = dpr.GaussRV(C=2, mu=x0)
jj = np.arange(Nx) # obs_inds
Obs = dpr.partial_Id_Obs(Nx, jj)
Obs['noise'] = 2 # dpr.GaussRV(C=CovMat(2*eye(Nx)))
HMM = dpr.HiddenMarkovModel(Dyn, Obs, t, X0)
HMM.liveplotters = LPs(jj)
####################
# Suggested tuning
####################
# from dapper.mods.Lorenz63.sakov2012 import HMM # rmse.a:
# xps += Climatology() # 7.6
# xps += OptInterp() # 1.25
# xps += Var3D(xB=0.1) # 1.04
# Furthermore, experiments do not seem to indicate that I can push
# Ny much lower than for the case H = Identity,
# even though the rmse is a lot lower with spectral H.
# Am I missing something?
import numpy as np
import dapper as dpr
from dapper.mods.Lorenz96.sakov2008 import Dyn, Nx, t
# The (Nx-Ny) highest frequency observation modes are left out of H below.
# If Ny>Nx, then H no longer has independent (let alone orthogonal) columns,
# yet more information is gained, since the observations are noisy.
Ny = 12
X0 = dpr.GaussRV(M=Nx, C=0.001)
def make_H(Ny, Nx):
xx = np.linspace(-1, 1, Nx + 1)[1:]
H = np.zeros((Ny, Nx))
H[0] = 1 / np.sqrt(2)
for k in range(-(Ny // 2), (Ny + 1) // 2):
ind = 2 * abs(k) - (k < 0)
H[ind] = np.sin(np.pi * k * xx + np.pi / 4)
H /= np.sqrt(Nx / 2)
return H
H = make_H(Ny, Nx)
# plt.figure(1).gca().matshow(H)
try:
# Load pre-generated
L = np.load(sample_filename)['Left']
except FileNotFoundError:
# First-time use
print('Did not find sample file', sample_filename,
'for experiment initialization. Generating...')
NQ = 20000 # Must have NQ > (2*wnumQ+1)
A = sinusoidal_sample(Nx, wnumQ, NQ)
A = 1 / 10 * center(A)[0] / np.sqrt(NQ)
Q = A.T @ A
U, s, _ = tsvd(Q)
L = U * np.sqrt(s)
np.savez(sample_filename, Left=L)
X0 = dpr.GaussRV(C=dpr.CovMat(np.sqrt(5) * L, 'Left'))
###################
# Forward model #
###################
damp = 0.98
Fm = Fmat(Nx, -1, 1, tseq.dt)
def step(x, t, dt):
assert dt == tseq.dt
return x @ Fm.T
Dyn = {
'M': Nx,
"""Like todter2015 but with Gaussian likelihood.""" import dapper as dpr from dapper.mods.Lorenz96.todter2015 import HMM HMM.Obs.noise = dpr.GaussRV(C=HMM.Obs.noise.C) #################### # Suggested tuning #################### # rmse.a # xps += LETKF(N=40,rot=True,infl=1.04 ,loc_rad=5) # 0.42 # xps += LETKF(N=80,rot=True,infl=1.04 ,loc_rad=5) # 0.42 # xps += LNETF(N=40,rot=True,infl=1.10,Rs=1.9,loc_rad=5) # 0.54 # xps += LNETF(N=80,rot=True,infl=1.06,Rs=1.4,loc_rad=5) # 0.47
def Q(dkObs):
return dpr.GaussRV(M=HMM.Nx, C=0.01 / (dkObs * HMM.t.dt))