# Wilks2005 uses dt=1e-4 with RK4 for the full model,
# and dt=5e-3 with RK2 for the forecast/truncated model.
# As berry2014linear notes, this is possible coz
# "numerical stiffness disappears when fast processes are removed".
################
# Full
################
# tseq = modelling.Chronology(dt=0.001,dto=0.05,T=4**3,BurnIn=6) # allows using rk2
tseq = modelling.Chronology(dt=0.005, dto=0.05, T=4**3,
BurnIn=6) # requires rk4
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 = 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,
import numpy as np
import dapper.mods as modelling
from .extras import LPs, d2x_dtdx, dstep_dx
__pdoc__ = {"demo": False}
# Constants
sig = 10.0
rho = 28.0
beta = 8.0 / 3
@modelling.ens_compatible
def dxdt(x):
"""Evolution equation (coupled ODEs) specifying the dynamics."""
d = np.zeros_like(x)
x, y, z = x
d[0] = sig * (y - x)
d[1] = rho * x - y - x * z
d[2] = x * y - beta * z
return d
step = modelling.with_rk4(dxdt, autonom=True)
Tplot = 4.0
x0 = np.array([1.509, -1.531, 25.46])
import numpy as np
from matplotlib import pyplot as plt
import dapper.mods as modelling
from dapper.mods.LorenzUV.lorenz96 import LUV
# Setup
sd0 = modelling.set_seed(4)
# from dapper.mods.LorenzUV.wilks05 import LUV
nU, J = LUV.nU, LUV.J
dt = 0.005
t0 = np.nan
K = int(10/dt)
step_1 = modelling.with_rk4(LUV.dxdt, autonom=True)
step_K = modelling.with_recursion(step_1, prog=1)
x0 = 0.01*np.random.randn(LUV.M)
x0 = step_K(x0, int(2/dt), t0, dt)[-1] # BurnIn
xx = step_K(x0, K, t0, dt)
# Grab parts of state vector
ii = np.arange(nU+1)
jj = np.arange(nU*J+1)
circU = np.mod(ii, nU)
circV = np.mod(jj, nU*J) + nU
iU = np.hstack([0, 0.5+np.arange(nU), nU])
def Ui(xx):
interp = (xx[0]+xx[-1])/2
# In unitless time (as used here), this means LyapExps ∈ ( −4.87, 1.66 ) .
if mod == "L96":
from dapper.mods.Lorenz96 import step
Nx = 40 # State size (flexible). Usually 36 or 40
T = 1e3 # Length of experiment (unitless time).
dt = 0.1 # Step length
# dt = 0.0083 # Any dt<0.1 yield "almost correct" Lyapunov expos.
x0 = randn(Nx) # Init condition.
eps = 0.0002 # Ens rescaling factor.
N = Nx # Num of perturbations used.
# Lyapunov exponents with F=10: [9.47 9.3 8.72 ..., -33.02 -33.61 -34.79] => n0:64
if mod == "LUV":
from dapper.mods.LorenzUV.lorenz96 import LUV
ii = np.arange(LUV.nU)
step = with_rk4(LUV.dxdt, autonom=True)
Nx = LUV.M
T = 1e2
dt = 0.005
LUV.F = 10
x0 = 0.01*randn(LUV.M)
eps = 0.001
N = 66 # Don't need all Nx for a good approximation of upper spectrum.
# Lyapunov exponents: [8.36, 7.58, 7.20, 6.91, ..., -4.18, -4.22, -4.19] => n0≈164
if mod == "L05":
from dapper.mods.LorenzIII import Model
model = Model()
step = model.step
x0 = model.x0
dt = 0.05/12
# `[x, Force]` as the state variable.
# - enclosing `dxdt` in an outer function which binds the value of the Force
# - creating `dxdt` in an outer "factory" function which determines the
# shape and allocation of the input vector to dxdt.
# - Do the latter using OOP. This would probably be more verbose,
# but even more flexible. In particular, OOP excels at working with multiple
# realisations of models at the same time. However, when using ensemble methods,
# the number of realisations, and the required level of automation
# (input vector/ensemble --> output vector/ensemble) is very high.
# It is not immediately clear if OOP is then more convenient.
# - There are also some intriguing possibilities relating to namedtuples.
# TODO 4: revise the above text.
# Turn dxdt into `step` such that `x_{k+1} = step(x_k, t, dt)`
step = modelling.with_rk4(dxdt_augmented, autonom=True)
# #### HMM
# Define the sequence of the experiment
# See `modelling.Chronology` for more details.
t = modelling.Chronology(
dt=0.05, # Integrational time step
dkObs=1, # Steps of duration dt between obs
KObs=10**3, # Total number of obs in experiment
BurnIn=5, # Omit from averages the period t=0 --> BurnIn
Tplot=7) # Default plot length
# Define dynamical model
Dyn = {
###########################
PRMS = 'Lorenz'
if PRMS == 'Wilks':
from dapper.mods.LorenzUV.wilks05 import LUV
else:
from dapper.mods.LorenzUV.lorenz96 import LUV
nU = LUV.nU
K = 400
dt = 0.005
t0 = np.nan
modelling.set_seed(30) # 3 5 7 13 15 30
x0 = np.random.randn(LUV.M)
true_step = with_rk4(LUV.dxdt, autonom=True)
model_step = with_rk4(LUV.dxdt_trunc, autonom=True)
true_K = with_recursion(true_step, prog=1)
###########################
# Compute truth trajectory
###########################
x0 = true_K(x0, int(2 / dt), t0, dt)[-1] # BurnIn
xx = true_K(x0, K, t0, dt)
###########################
# Compute unresovled scales
###########################
gg = np.zeros((K, nU)) # "Unresolved tendency"
for k, x in enumerate(xx[:-1]):
X = x[:nU]
