Lorenz63 MCMC fit to data
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# Imports
import sys
import jax
from jax import vmap
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import jax.random as jr
import matplotlib.pyplot as plt
# Imports
import sys
import jax
from jax import vmap
jax.config.update("jax_enable_x64", True)
import jax.numpy as jnp
import jax.random as jr
import matplotlib.pyplot as plt
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# CD-NLGSSM imports
from cd_dynamax import make_key_sequence
from cd_dynamax.src.utils.experiment_utils import *
from cd_dynamax.src.utils.simulation_utils import filter_and_forecast
from cd_dynamax.src.utils.demo_utils import sample_t_emissions
# Python plotting import
sys.path.append("../scripts")
from filter_forecast_plotting_utils import (
data_plotter,
plot_filter_then_forecast_state_results,
)
from parameter_learning_plotting_utils import (
learnable_params_to_1d_df,
plot_mll_learning_curve,
plot_param_sequences,
plot_param_dist,
)
# Set default config path
config_path = "../configs/"
# CD-NLGSSM imports
from cd_dynamax import make_key_sequence
from cd_dynamax.src.utils.experiment_utils import *
from cd_dynamax.src.utils.simulation_utils import filter_and_forecast
from cd_dynamax.src.utils.demo_utils import sample_t_emissions
# Python plotting import
sys.path.append("../scripts")
from filter_forecast_plotting_utils import (
data_plotter,
plot_filter_then_forecast_state_results,
)
from parameter_learning_plotting_utils import (
learnable_params_to_1d_df,
plot_mll_learning_curve,
plot_param_sequences,
plot_param_dist,
)
# Set default config path
config_path = "../configs/"
/Users/danwaxman/Documents/cd_dynamax/cd_dynamax/src/ssm_temissions.py:785: SyntaxWarning: invalid escape sequence '\s'
$$\theta^\star = \mathrm{argmax}_\theta \; \log p(y_{1:T}, \theta \mid u_{1:T})$$
Define the Lorenz 63 Model¶
We generate data from a Lorenz 63 system, from dynamics with the following stochastic differential equations:
\begin{align*} \frac{d x}{d t} &= a(y-x) + \sigma w_x(t) \\ \frac{d y}{d t} &= x(b-z) - y + \sigma w_y(t) \\ \frac{d z}{d t} &= xy - cz + \sigma w_z(t), \end{align*}
With parameters $a=10, b=28, c=8/3$, the system gives rise to chaotic behavior, and we choose $\sigma=1.0$ for diffusion.
To generate data, we numerically approximate random path solutions to this SDE using Heun's method (i.e. improved Euler), as implemented in Diffrax.
We assume the observation model is \begin{align*} y(t) &= H x(t) + r(t) \\ r(t) &\sim N(0,R), \end{align*} where we choose $R=I$.
Namely, we impose partial observability with H=[1, 0, 0], with noisy observations, sampled at irregular time intervals.
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# Default Lorenz 63 parameter definition in config file
# where only the first component (x_1) is observed
default_lorenz63_config_model = "model/l63_x1_filter"
# Default Lorenz 63 parameter definition in config file
# where only the first component (x_1) is observed
default_lorenz63_config_model = "model/l63_x1_filter"
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# Specify true Lorenz '63 parameters
from cd_dynamax.src.utils.physics_based_models import LearnableLorenz63_Drift
# Define the True Lorenz 63 Model parameters dictionary
true_model_def = {
"initial_values.dynamics_drift": {
"params": LearnableLorenz63_Drift(sigma=10.0, rho=28.0, beta=8.0 / 3.0),
"props": None, # Let us use the default parameter properties
}
}
# Specify true Lorenz '63 parameters
from cd_dynamax.src.utils.physics_based_models import LearnableLorenz63_Drift
# Define the True Lorenz 63 Model parameters dictionary
true_model_def = {
"initial_values.dynamics_drift": {
"params": LearnableLorenz63_Drift(sigma=10.0, rho=28.0, beta=8.0 / 3.0),
"props": None, # Let us use the default parameter properties
}
}
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# Create and initialize the CD-NLGSSM model
true_model, true_params, true_props = create_cddynamax_model_from_config(
config_path=config_path,
true_model_config_file=default_lorenz63_config_model,
overrides=true_model_def,
)
# Create and initialize the CD-NLGSSM model
true_model, true_params, true_props = create_cddynamax_model_from_config(
config_path=config_path,
true_model_config_file=default_lorenz63_config_model,
overrides=true_model_def,
)
Simulate data using true CDNLGSSM model¶
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# Simulate emission times
keys = make_key_sequence(0)
t_emissions = sample_t_emissions(
start=0.0, stop=50.0, dt=0.01, regular=False, key=next(keys)
)
# Simulate data using true parameters (observed at times t_emissions)
# Note that default SDE diffeqsolve settings solve using dfx.Heun() with dfx.ConstantStepSize() and dt0=0.01 step size.
states, emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(t_emissions),
t_emissions=t_emissions,
transition_type="path",
)
# Simulate emission times
keys = make_key_sequence(0)
t_emissions = sample_t_emissions(
start=0.0, stop=50.0, dt=0.01, regular=False, key=next(keys)
)
# Simulate data using true parameters (observed at times t_emissions)
# Note that default SDE diffeqsolve settings solve using dfx.Heun() with dfx.ConstantStepSize() and dt0=0.01 step size.
states, emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(t_emissions),
t_emissions=t_emissions,
transition_type="path",
)
[make_t_emissions] Generated 5000 points from 0.0 to 50.0 with avg step ~0.0100 [make_t_emissions] Smallest time step = 0.000004 Sampling from SDE solver path: this may be an unnecessarily poor approximation if you're simulating from a linear SDE. It is an appropriate choice for non-linear SDEs. CD-NLGSSM Sampling from continuous-discrete non-linear Gaussian SSM path
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# Plot the results in a subplot with 3 rows, with observations overlaid in first emission_dims rows
fig, axes = plt.subplots(true_model.state_dim, 1, figsize=(10, 8), sharex=True)
state_labels = ["x", "y", "z"]
for d in range(true_model.state_dim):
### True data
data_plotter(
ax=axes[d],
t_idx=t_emissions,
states=states[:, d],
state_label=state_labels[d],
state_style={"color": "black", "linestyle": "-", "linewidth": 1.5},
emissions=emissions[:, d] if d < emissions.shape[1] else None,
emission_label="True Observation" if d < emissions.shape[1] else None,
emissions_style={
"color": "orange",
"linestyle": "None",
"marker": "x",
"markersize": 3,
"alpha": 0.5,
},
plot_observations=True and d < emissions.shape[1],
)
axes[-1].set_xlabel("Time")
plt.suptitle("Lorenz '63 System States and Observations")
plt.tight_layout()
plt.show()
# Plot the results in a subplot with 3 rows, with observations overlaid in first emission_dims rows
fig, axes = plt.subplots(true_model.state_dim, 1, figsize=(10, 8), sharex=True)
state_labels = ["x", "y", "z"]
for d in range(true_model.state_dim):
### True data
data_plotter(
ax=axes[d],
t_idx=t_emissions,
states=states[:, d],
state_label=state_labels[d],
state_style={"color": "black", "linestyle": "-", "linewidth": 1.5},
emissions=emissions[:, d] if d < emissions.shape[1] else None,
emission_label="True Observation" if d < emissions.shape[1] else None,
emissions_style={
"color": "orange",
"linestyle": "None",
"marker": "x",
"markersize": 3,
"alpha": 0.5,
},
plot_observations=True and d < emissions.shape[1],
)
axes[-1].set_xlabel("Time")
plt.suptitle("Lorenz '63 System States and Observations")
plt.tight_layout()
plt.show()
Fitting another Lorenz 63 model to the data, using MCMC¶
Learnable model definition¶
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# New Lorenz 63 model definition in config file
# where only the first component (x_1) is observed
learnable_lorenz63_config_model = "model/l63_x1_fit_to_data"
# If desired, we can override the initial parameters here
# We will simply override the prior used to draw the initial drift parameters
learnable_model_def = {"prior.prior_class_file": "prior/l63_mech_drift_hi_info.py"}
# New Lorenz 63 model definition in config file
# where only the first component (x_1) is observed
learnable_lorenz63_config_model = "model/l63_x1_fit_to_data"
# If desired, we can override the initial parameters here
# We will simply override the prior used to draw the initial drift parameters
learnable_model_def = {"prior.prior_class_file": "prior/l63_mech_drift_hi_info.py"}
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# Create and initialize the CD-NLGSSM model
learnable_model, learnable_params, learnable_props = create_cddynamax_model_from_config(
config_path=config_path,
true_model_config_file=learnable_lorenz63_config_model,
overrides=learnable_model_def,
)
# Create and initialize the CD-NLGSSM model
learnable_model, learnable_params, learnable_props = create_cddynamax_model_from_config(
config_path=config_path,
true_model_config_file=learnable_lorenz63_config_model,
overrides=learnable_model_def,
)
Attempting to load prior from: /Users/danwaxman/Documents/cd_dynamax/demos/python/configs/prior/l63_mech_drift_hi_info.py Successfully loaded prior from: /Users/danwaxman/Documents/cd_dynamax/demos/python/configs/prior/l63_mech_drift_hi_info.py Prior instantiated from class: CDNLGSSM_Prior Initializing dynamics_drift with sampled parameters
Fitting: MCMC-based optimization of log-likelihood, computed via Extended Kalman Filter (EKF)¶
Extended Kalman Filter (EKF)
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# Default EKF settings are defined in config file
default_ekf_config = "filter/ekf_StateFirst_EmissionsFirst"
# Default EKF settings are defined in config file
default_ekf_config = "filter/ekf_StateFirst_EmissionsFirst"
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# Figure-out the filtering/smoothing settings from config
ekf_hyperparams, ekf_info = create_cddynamax_filter_from_config(
config_path=config_path, filter_config_file=default_ekf_config, overrides={}
)
# Figure-out the filtering/smoothing settings from config
ekf_hyperparams, ekf_info = create_cddynamax_filter_from_config(
config_path=config_path, filter_config_file=default_ekf_config, overrides={}
)
MCMC for log-likelihood maximization:
In this case, we will resort to NUTS, a flexible MCMC algorithm that leverages gradient information
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# Load MCMC-NUTS fitting configuration
# Fit configuration file
fit_config_file = "fitting/nuts_l63_x1_fit_to_data"
# Full path to the fit config file
fit_config_filepath = os.path.join(config_path, fit_config_file)
# Check if the config file exists
if not os.path.exists(fit_config_filepath):
raise FileNotFoundError(f"Configuration file '{fit_config_filepath}' not found.")
# Read the fitting configuration
from configparser import ConfigParser
config = ConfigParser()
config.read(fit_config_filepath)
# For each optimization method specified in the config
optim_configs = config.sections()
# Load MCMC-NUTS fitting configuration
# Fit configuration file
fit_config_file = "fitting/nuts_l63_x1_fit_to_data"
# Full path to the fit config file
fit_config_filepath = os.path.join(config_path, fit_config_file)
# Check if the config file exists
if not os.path.exists(fit_config_filepath):
raise FileNotFoundError(f"Configuration file '{fit_config_filepath}' not found.")
# Read the fitting configuration
from configparser import ConfigParser
config = ConfigParser()
config.read(fit_config_filepath)
# For each optimization method specified in the config
optim_configs = config.sections()
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# Run MCMC for parameter estimation
if "mcmc" in optim_configs:
# MCMC configuration
mcmc_config = config["mcmc"]
# MCMC MAP estimation via the cd-dynamax learnable_model's fit_mcmc method
print("\nStarting MCMC MAP estimation...")
mcmc_results = learnable_model.fit_mcmc(
initial_params=learnable_params,
props=learnable_props,
emissions=emissions,
t_emissions=t_emissions,
filter_hyperparams=ekf_hyperparams,
inputs=None,
mcmc_algorithm=mcmc_config_to_dict(mcmc_config),
verbose=mcmc_config.getboolean("verbose", True),
key=jr.PRNGKey(mcmc_config.getint("key", 0)),
)
# Run MCMC for parameter estimation
if "mcmc" in optim_configs:
# MCMC configuration
mcmc_config = config["mcmc"]
# MCMC MAP estimation via the cd-dynamax learnable_model's fit_mcmc method
print("\nStarting MCMC MAP estimation...")
mcmc_results = learnable_model.fit_mcmc(
initial_params=learnable_params,
props=learnable_props,
emissions=emissions,
t_emissions=t_emissions,
filter_hyperparams=ekf_hyperparams,
inputs=None,
mcmc_algorithm=mcmc_config_to_dict(mcmc_config),
verbose=mcmc_config.getboolean("verbose", True),
key=jr.PRNGKey(mcmc_config.getint("key", 0)),
)
Starting MCMC MAP estimation... Running window adaptation
100.00% [10/10 00:00<?]
Running HMC inference loop...
Organize results for easier plotting
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# Reformat the results into a dictionary
mcmc_warmup_param_samples = mcmc_results[0]
mcmc_param_samples = mcmc_results[1]
mcmc_warmup_log_probs = mcmc_results[2]
mcmc_log_probs = mcmc_results[3]
# Reformat the results into a dictionary
mcmc_warmup_param_samples = mcmc_results[0]
mcmc_param_samples = mcmc_results[1]
mcmc_warmup_log_probs = mcmc_results[2]
mcmc_log_probs = mcmc_results[3]
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# Plot the marginal log likelihood curve over NUTS warmup
plot_mll_learning_curve(
true_model=true_model,
true_params=true_params,
true_emissions=emissions,
t_emissions=t_emissions,
marginal_lls=mcmc_warmup_log_probs,
filter_hyperparams=ekf_hyperparams,
plot_save_path=None, # Plot here instead of saving
)
# Plot the marginal log likelihood curve over NUTS warmup
plot_mll_learning_curve(
true_model=true_model,
true_params=true_params,
true_emissions=emissions,
t_emissions=t_emissions,
marginal_lls=mcmc_warmup_log_probs,
filter_hyperparams=ekf_hyperparams,
plot_save_path=None, # Plot here instead of saving
)
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# Plot the marginal log likelihood curve over NUTS sampling
plot_mll_learning_curve(
true_model=true_model,
true_params=true_params,
true_emissions=emissions,
t_emissions=t_emissions,
marginal_lls=mcmc_log_probs,
filter_hyperparams=ekf_hyperparams,
plot_save_path=None, # Plot here instead of saving
)
# Plot the marginal log likelihood curve over NUTS sampling
plot_mll_learning_curve(
true_model=true_model,
true_params=true_params,
true_emissions=emissions,
t_emissions=t_emissions,
marginal_lls=mcmc_log_probs,
filter_hyperparams=ekf_hyperparams,
plot_save_path=None, # Plot here instead of saving
)
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# Plot MCMC parameters ---translating trees to dataframes
mcmc_samples_df = learnable_params_to_1d_df(
mcmc_param_samples, learnable_props, has_batch_dim=True
)
true_params_df = learnable_params_to_1d_df(true_params, learnable_props)
init_params_df = learnable_params_to_1d_df(learnable_params, learnable_props)
# Plot MCMC parameters ---translating trees to dataframes
mcmc_samples_df = learnable_params_to_1d_df(
mcmc_param_samples, learnable_props, has_batch_dim=True
)
true_params_df = learnable_params_to_1d_df(true_params, learnable_props)
init_params_df = learnable_params_to_1d_df(learnable_params, learnable_props)
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# Plot MCMC parameter trajectories over MCMC iterations
plot_param_sequences(
param_history=mcmc_samples_df,
true=true_params_df,
init=init_params_df,
pointwise_estimate=mcmc_samples_df.mean(axis=0).to_frame().T,
burn_in_frac=0.0,
pairwise_plots=True,
plot_save_path=None, # Plot here instead of saving
)
# Plot MCMC parameter trajectories over MCMC iterations
plot_param_sequences(
param_history=mcmc_samples_df,
true=true_params_df,
init=init_params_df,
pointwise_estimate=mcmc_samples_df.mean(axis=0).to_frame().T,
burn_in_frac=0.0,
pairwise_plots=True,
plot_save_path=None, # Plot here instead of saving
)
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# Plot parameter distributions from MCMC samples
plot_param_dist(
samples=mcmc_samples_df,
true=true_params_df,
init=init_params_df,
pointwise_estimate=mcmc_samples_df.mean(axis=0).to_frame().T,
burn_in_frac=0.0,
pairwise_plots=True,
plot_save_path=None, # Plot here instead of saving
)
# Plot parameter distributions from MCMC samples
plot_param_dist(
samples=mcmc_samples_df,
true=true_params_df,
init=init_params_df,
pointwise_estimate=mcmc_samples_df.mean(axis=0).to_frame().T,
burn_in_frac=0.0,
pairwise_plots=True,
plot_save_path=None, # Plot here instead of saving
)
Filter then forecast using true and learned models¶
We now generate a new test trajectory from the true model, then do filtering and forecasting (using both true and learned models) to evaluate short-term forecasting performance.
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# Define a set of time indices for filtering and forecasting
test_t_emissions = sample_t_emissions(
start=0.0, stop=100.0, dt=0.01, regular=True, key=next(keys)
)
# Define a set of time indices for filtering and forecasting
test_t_emissions = sample_t_emissions(
start=0.0, stop=100.0, dt=0.01, regular=True, key=next(keys)
)
[make_t_emissions] Generated 10000 points from 0.0 to 100.0 with avg step ~0.0100 [make_t_emissions] Smallest time step = 0.010000
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# Now, generate a new testing trajectory from the true model.
test_states, test_emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(test_t_emissions),
t_emissions=test_t_emissions,
transition_type="path",
)
# Now, generate a new testing trajectory from the true model.
test_states, test_emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(test_t_emissions),
t_emissions=test_t_emissions,
transition_type="path",
)
Sampling from SDE solver path: this may be an unnecessarily poor approximation if you're simulating from a linear SDE. It is an appropriate choice for non-linear SDEs. CD-NLGSSM Sampling from continuous-discrete non-linear Gaussian SSM path
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# Forecasting and filtering on specific range
T0 = 45.0
T_filter_end = 50.0
T_forecast_end = 52.0
# Forecasting and filtering on specific range
T0 = 45.0
T_filter_end = 50.0
T_forecast_end = 52.0
Filtering and Forecasting with the true model¶
Note that the diffusion in the true SDE model (combined with the chaotic sensitivies of the Lorenz drift) makes even high-quality forecasts diverge quickly.
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print(
"Running filtering with {filter_name} with True model from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting
(
true_test_ekf_filtered,
true_test_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = filter_and_forecast(
model_params=true_params,
filter_hyperparams=ekf_hyperparams,
t_emissions=test_t_emissions,
emissions=test_emissions,
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
print(
"Running filtering with {filter_name} with True model from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting
(
true_test_ekf_filtered,
true_test_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = filter_and_forecast(
model_params=true_params,
filter_hyperparams=ekf_hyperparams,
t_emissions=test_t_emissions,
emissions=test_emissions,
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
Running filtering with EKF (1st order) with True model from T=45.0 up to T=50.0 and forecasting up to T=52.0.
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# Plot both filtering and forecasting results
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(true_test_ekf_filtered),
"forecasted": tree_to_dict(true_test_ekf_forecasted),
"start_idx_filter": start_idx_filter,
"stop_idx_filter": stop_idx_filter,
"start_idx_forecast": start_idx_forecast,
"stop_idx_forecast": stop_idx_forecast,
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
# Plot both filtering and forecasting results
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(true_test_ekf_filtered),
"forecasted": tree_to_dict(true_test_ekf_forecasted),
"start_idx_filter": start_idx_filter,
"stop_idx_filter": stop_idx_filter,
"start_idx_forecast": start_idx_forecast,
"stop_idx_forecast": stop_idx_forecast,
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
Filtering and Forecasting with the learned model(s)¶
Note that with MCMC, we have different options to consider:
- To use the mean posterior parameters
- To use all posterior samples as an ensemble, then plot their average
Let us now perform filtering and forecasting with the learned model's MCMC posterior mean
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print(
"Running filtering with {filter_name} with NUTS-learned model's posterior mean from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting, using vma
(
test_nuts_postmean_ekf_filtered,
test_nuts_postmean_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = filter_and_forecast(
model_params=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
),
filter_hyperparams=ekf_hyperparams,
t_emissions=test_t_emissions,
emissions=test_emissions,
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
print(
"Running filtering with {filter_name} with NUTS-learned model's posterior mean from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting, using vma
(
test_nuts_postmean_ekf_filtered,
test_nuts_postmean_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = filter_and_forecast(
model_params=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
),
filter_hyperparams=ekf_hyperparams,
t_emissions=test_t_emissions,
emissions=test_emissions,
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
Running filtering with EKF (1st order) with NUTS-learned model's posterior mean from T=45.0 up to T=50.0 and forecasting up to T=52.0.
In [26]:
Copied!
# Plot both filtering and forecasting results
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(test_nuts_postmean_ekf_filtered),
"forecasted": tree_to_dict(test_nuts_postmean_ekf_forecasted),
"start_idx_filter": start_idx_filter,
"stop_idx_filter": stop_idx_filter,
"start_idx_forecast": start_idx_forecast,
"stop_idx_forecast": stop_idx_forecast,
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
# Plot both filtering and forecasting results
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(test_nuts_postmean_ekf_filtered),
"forecasted": tree_to_dict(test_nuts_postmean_ekf_forecasted),
"start_idx_filter": start_idx_filter,
"stop_idx_filter": stop_idx_filter,
"start_idx_forecast": start_idx_forecast,
"stop_idx_forecast": stop_idx_forecast,
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
Let us now perform filtering and forecasting with every MCMC sample of the learned model
In [27]:
Copied!
print(
"Running filtering with {filter_name} with NUTS-learned model's posterior samples from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting, using vma
(
test_nuts_samples_ekf_filtered,
test_nuts_samples_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = vmap(
filter_and_forecast,
in_axes=(
0,
None,
None,
None,
None,
None,
None,
), # map over axis 0 of mcmc_param_samples
)(
# argument names removed for vmap to work
mcmc_param_samples,
ekf_hyperparams,
test_t_emissions,
test_emissions,
T0,
T_filter_end,
T_forecast_end,
)
print(
"Running filtering with {filter_name} with NUTS-learned model's posterior samples from T={T0} up to T={T_filter_end} and forecasting up to T={T_forecast_end}.".format(
filter_name=ekf_info["name"]
if ekf_info is not None and "name" in ekf_info
else "the specified filter",
T0=T0,
T_filter_end=T_filter_end,
T_forecast_end=T_forecast_end,
)
)
# Perform filtering and forecasting, using vma
(
test_nuts_samples_ekf_filtered,
test_nuts_samples_ekf_forecasted,
start_idx_filter,
stop_idx_filter,
start_idx_forecast,
stop_idx_forecast,
) = vmap(
filter_and_forecast,
in_axes=(
0,
None,
None,
None,
None,
None,
None,
), # map over axis 0 of mcmc_param_samples
)(
# argument names removed for vmap to work
mcmc_param_samples,
ekf_hyperparams,
test_t_emissions,
test_emissions,
T0,
T_filter_end,
T_forecast_end,
)
Running filtering with EKF (1st order) with NUTS-learned model's posterior samples from T=45.0 up to T=50.0 and forecasting up to T=52.0. Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite Warning: matrix is not positive definite
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# Plot filtering and forecasting results, for averaged results over samples
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(
jax.tree.map(lambda x: jnp.mean(x, axis=0), test_nuts_samples_ekf_filtered)
),
"forecasted": tree_to_dict(
jax.tree.map(
lambda x: jnp.mean(x, axis=0), test_nuts_samples_ekf_forecasted
)
),
"start_idx_filter": start_idx_filter[0], # same for all samples
"stop_idx_filter": stop_idx_filter[0], # same for all samples
"start_idx_forecast": start_idx_forecast[0], # same for all samples
"stop_idx_forecast": stop_idx_forecast[0], # same for all samples
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
# Plot filtering and forecasting results, for averaged results over samples
plot_filter_then_forecast_state_results(
data={
"states": test_states,
"emissions": test_emissions,
"t_emissions": test_t_emissions,
},
results={
"filtered": tree_to_dict(
jax.tree.map(lambda x: jnp.mean(x, axis=0), test_nuts_samples_ekf_filtered)
),
"forecasted": tree_to_dict(
jax.tree.map(
lambda x: jnp.mean(x, axis=0), test_nuts_samples_ekf_forecasted
)
),
"start_idx_filter": start_idx_filter[0], # same for all samples
"stop_idx_filter": stop_idx_filter[0], # same for all samples
"start_idx_forecast": start_idx_forecast[0], # same for all samples
"stop_idx_forecast": stop_idx_forecast[0], # same for all samples
},
plot_only_filter_forecast_window=True,
results_file=None, # Plot here, not save
filter_info=ekf_info,
plot_uncertainty=True,
plot_observations=True,
plot_mse=False,
)
Lorenz 63's chaotic behavior: differences in true Vs learned model forecasting¶
Next, we will look at the long-term statistical behavior of the learned vs true models by simulating a long trajectory from each and plotting histograms and others!!
In [29]:
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# Long time emissions for long simulation: training data is too short to see true attractor
long_t_emissions = jnp.arange(start=0.0, stop=1e4, step=0.01).reshape(-1, 1)
# Long time emissions for long simulation: training data is too short to see true attractor
long_t_emissions = jnp.arange(start=0.0, stop=1e4, step=0.01).reshape(-1, 1)
In [30]:
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# Long simulation with true parameters for comparison
true_long_states, true_long_emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(long_t_emissions),
t_emissions=long_t_emissions,
transition_type="path",
)
# Long simulation with true parameters for comparison
true_long_states, true_long_emissions = true_model.sample(
params=true_params,
key=next(keys),
num_timesteps=len(long_t_emissions),
t_emissions=long_t_emissions,
transition_type="path",
)
Sampling from SDE solver path: this may be an unnecessarily poor approximation if you're simulating from a linear SDE. It is an appropriate choice for non-linear SDEs. CD-NLGSSM Sampling from continuous-discrete non-linear Gaussian SSM path
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# Long simulation with posterior mean of parameters
postmean_long_states, postmean_long_emissions = learnable_model.sample(
params=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
),
key=next(keys),
num_timesteps=len(long_t_emissions),
t_emissions=long_t_emissions,
transition_type="path",
)
# Long simulation with posterior mean of parameters
postmean_long_states, postmean_long_emissions = learnable_model.sample(
params=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
),
key=next(keys),
num_timesteps=len(long_t_emissions),
t_emissions=long_t_emissions,
transition_type="path",
)
Sampling from SDE solver path: this may be an unnecessarily poor approximation if you're simulating from a linear SDE. It is an appropriate choice for non-linear SDEs. CD-NLGSSM Sampling from continuous-discrete non-linear Gaussian SSM path
Chaos dynamics plotting¶
In [32]:
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# Let's plot some analyses: Note that we don't expect the learned 2nd and 3rd state variables to match-up at all to the true ones.
from cd_dynamax.src.utils.plotting_chaos_utils import analyze_chaotic_dynamics
analyze_chaotic_dynamics(
true_long_states,
postmean_long_states,
t=long_t_emissions.squeeze(),
drift_true=true_params.dynamics.drift.f,
drift_learn=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
).dynamics.drift.f,
burnin_frac=0.5,
)
# Let's plot some analyses: Note that we don't expect the learned 2nd and 3rd state variables to match-up at all to the true ones.
from cd_dynamax.src.utils.plotting_chaos_utils import analyze_chaotic_dynamics
analyze_chaotic_dynamics(
true_long_states,
postmean_long_states,
t=long_t_emissions.squeeze(),
drift_true=true_params.dynamics.drift.f,
drift_learn=jax.tree.map(
lambda p: jnp.mean(p, axis=0), # Mean over MCMC samples ---axis 0
mcmc_param_samples,
).dynamics.drift.f,
burnin_frac=0.5,
)
Estimated Largest Lyapunov Exponent: True : 1.085 Learned: 1.051
Estimated Correlation Dimension D2: True : 1.576 Learned : 1.597 Estimated Correlation Dimensions: True D2=1.576, Learned D2=1.597
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