Lorenz63 SGD fit to data

In [1]:

# Imports
import sys
import jax

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 jax.config.update("jax_enable_x64", True) import jax.numpy as jnp import jax.random as jr import matplotlib.pyplot as plt

In [2]:

# 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,
)

# 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, ) # Set default config path config_path = "../configs/"

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.

In [3]:

# 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"

In [4]:

# 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 } }

In [5]:

# 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¶

In [6]:

# 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

In [7]:

# 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¶

Learnable model definition¶

In [8]:

# 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"}

In [9]:

# 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: SGD-based optimization of log-likelihood, computed via Extended Kalman Filter (EKF)¶

Extended Kalman Filter (EKF)

In [10]:

# 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"

In [11]:

# 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={} )

SGD for log-likelihood maximization

In [12]:

# Load SGD fitting configuration

# Fit configuration file
fit_config_file = "fitting/sgd_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 SGD fitting configuration # Fit configuration file fit_config_file = "fitting/sgd_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()

In [13]:

# Run SGD for parameter estimation

if "sgd" in optim_configs:
    # SGD configuration
    sgd_config = config["sgd"]
    # SGD MAP estimation via the cd-dynamax learnable_model's fit_sgd method
    print("\nStarting SGD fitting...")
    fit_results = learnable_model.fit_sgd(
        initial_params=learnable_params,
        props=learnable_props,
        emissions=emissions,
        t_emissions=t_emissions,
        filter_hyperparams=ekf_hyperparams,
        inputs=None,
        optimizer=eval(sgd_config.get("optimizer", "optax.adam(1e-3)")),
        batch_size=sgd_config.getint("batch_size", 1),
        num_epochs=sgd_config.getint("num_epochs", 50),
        shuffle=sgd_config.getboolean("shuffle", False),
        return_param_history=sgd_config.getboolean("return_param_history", False),
        return_grad_history=sgd_config.getboolean("return_grad_history", False),
        key=jr.PRNGKey(sgd_config.getint("key", 0)),
    )

# Run SGD for parameter estimation if "sgd" in optim_configs: # SGD configuration sgd_config = config["sgd"] # SGD MAP estimation via the cd-dynamax learnable_model's fit_sgd method print("\nStarting SGD fitting...") fit_results = learnable_model.fit_sgd( initial_params=learnable_params, props=learnable_props, emissions=emissions, t_emissions=t_emissions, filter_hyperparams=ekf_hyperparams, inputs=None, optimizer=eval(sgd_config.get("optimizer", "optax.adam(1e-3)")), batch_size=sgd_config.getint("batch_size", 1), num_epochs=sgd_config.getint("num_epochs", 50), shuffle=sgd_config.getboolean("shuffle", False), return_param_history=sgd_config.getboolean("return_param_history", False), return_grad_history=sgd_config.getboolean("return_grad_history", False), key=jr.PRNGKey(sgd_config.getint("key", 0)), )

Starting SGD fitting...

Organize results for easier plotting

In [14]:

# First element of tuple is the fitted parameters
sgd_fitted_params = fit_results[0]
# Note that fit_sgd returns normalized negative log-likelihoods
# So we multiply it by the number of datapoints in emissions
sgd_loss_history = -fit_results[1] * emissions.size

# Process history outputs if requested
if sgd_config.getboolean("return_param_history", False) and sgd_config.getboolean(
    "return_grad_history", False
):
    sgd_fitted_params_history = fit_results[2]
    sgd_fitted_grad_history = fit_results[3]
elif sgd_config.getboolean("return_param_history", False):
    sgd_fitted_params_history = fit_results[2]
elif sgd_config.getboolean("return_grad_history", False):
    sgd_fitted_grad_history = fit_results[2]

# First element of tuple is the fitted parameters sgd_fitted_params = fit_results[0] # Note that fit_sgd returns normalized negative log-likelihoods # So we multiply it by the number of datapoints in emissions sgd_loss_history = -fit_results[1] * emissions.size # Process history outputs if requested if sgd_config.getboolean("return_param_history", False) and sgd_config.getboolean( "return_grad_history", False ): sgd_fitted_params_history = fit_results[2] sgd_fitted_grad_history = fit_results[3] elif sgd_config.getboolean("return_param_history", False): sgd_fitted_params_history = fit_results[2] elif sgd_config.getboolean("return_grad_history", False): sgd_fitted_grad_history = fit_results[2]

In [15]:

# Plot the marginal log likelihood learning curve
plot_mll_learning_curve(
    true_model=true_model,
    true_params=true_params,
    true_emissions=emissions,
    t_emissions=t_emissions,
    marginal_lls=sgd_loss_history,
    filter_hyperparams=ekf_hyperparams,
    plot_save_path=None,  # Plot here instead of saving
)

# Plot the marginal log likelihood learning curve plot_mll_learning_curve( true_model=true_model, true_params=true_params, true_emissions=emissions, t_emissions=t_emissions, marginal_lls=sgd_loss_history, filter_hyperparams=ekf_hyperparams, plot_save_path=None, # Plot here instead of saving )

In [16]:

# Plot parameter trajectories over SGD iterations ---translating trees to dataframes
plot_param_sequences(
    param_history=learnable_params_to_1d_df(
        sgd_fitted_params_history, learnable_props, has_batch_dim=True
    ),
    true=learnable_params_to_1d_df(true_params, learnable_props),
    init=learnable_params_to_1d_df(learnable_params, learnable_props),
    pointwise_estimate=learnable_params_to_1d_df(sgd_fitted_params, learnable_props),
    burn_in_frac=0.0,
    pairwise_plots=True,
    plot_save_path=None,  # Plot here instead of saving
)

# Plot parameter trajectories over SGD iterations ---translating trees to dataframes plot_param_sequences( param_history=learnable_params_to_1d_df( sgd_fitted_params_history, learnable_props, has_batch_dim=True ), true=learnable_params_to_1d_df(true_params, learnable_props), init=learnable_params_to_1d_df(learnable_params, learnable_props), pointwise_estimate=learnable_params_to_1d_df(sgd_fitted_params, learnable_props), 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.

In [17]:

# 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

In [18]:

# 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

In [19]:

# 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.

In [20]:

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.

In [21]:

# 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¶

In [22]:

print(
    "Running filtering with {filter_name} with Learned 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
(
    test_ekf_filtered,
    test_ekf_forecasted,
    start_idx_filter,
    stop_idx_filter,
    start_idx_forecast,
    stop_idx_forecast,
) = filter_and_forecast(
    model_params=sgd_fitted_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 Learned 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 ( test_ekf_filtered, test_ekf_forecasted, start_idx_filter, stop_idx_filter, start_idx_forecast, stop_idx_forecast, ) = filter_and_forecast( model_params=sgd_fitted_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 Learned model from T=45.0 up to T=50.0 and forecasting up to T=52.0.

In [23]:

# 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_ekf_filtered),
        "forecasted": tree_to_dict(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(test_ekf_filtered), "forecasted": tree_to_dict(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, )

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 [24]:

# 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 [25]:

# 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

In [26]:

# Long simulation with learned parameters
fitted_long_states, fitted_long_emissions = learnable_model.sample(
    params=sgd_fitted_params,
    key=next(keys),
    num_timesteps=len(long_t_emissions),
    t_emissions=long_t_emissions,
    transition_type="path",
)

# Long simulation with learned parameters fitted_long_states, fitted_long_emissions = learnable_model.sample( params=sgd_fitted_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

Chaos dynamics plotting¶

In [27]:

# 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,
    fitted_long_states,
    t=long_t_emissions.squeeze(),
    drift_true=true_params.dynamics.drift.f,
    drift_learn=sgd_fitted_params.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, fitted_long_states, t=long_t_emissions.squeeze(), drift_true=true_params.dynamics.drift.f, drift_learn=sgd_fitted_params.dynamics.drift.f, burnin_frac=0.5, )

Estimated Largest Lyapunov Exponent:
  True   : 1.084
  Learned: 0.995

Estimated Correlation Dimension D2:
  True    : 1.583
  Learned : 1.589
Estimated Correlation Dimensions: True D2=1.583, Learned D2=1.589