Kernel Koopman Kalman Filter (KKF)

A Python library for non-linear state estimation using Kernel-based Extended Dynamic Mode Decomposition (kEDMD) and Koopman operator theory.

What is KKF?

KKF is a sophisticated filtering library that combines:

• Koopman Operator Theory: Lifts nonlinear dynamics to a linear setting in feature space
• Kernel Methods: Uses kernel functions for feature space approximation
• Kalman Filtering: Applies optimal filtering in the lifted feature space
• Extended Dynamic Mode Decomposition: Learns accurate linear approximations

This enables accurate state estimation for highly nonlinear dynamical systems where traditional Kalman filters struggle.

Features

✨ Core Capabilities

• Kernel Extended Dynamic Mode Decomposition (kEDMD)
• Non-linear Kalman Filter implementation
• Support for arbitrary dynamical systems
• Multiple kernel function options (RBF, Matérn, etc.)
• Robust state estimation with uncertainty quantification
• Covariance propagation in both state and feature spaces

📊 High-Quality Implementation

• Type hints throughout the codebase
• Comprehensive error handling
• Numerical stability optimizations
• Efficient NumPy/SciPy operations

🧪 Well-Tested

• 95+ unit and integration tests
• Edge case coverage
• Multi-platform CI/CD
• Coverage reporting

📚 Comprehensive Documentation

• Detailed docstrings
• 3 complete working examples
• Parameter optimization guide
• API documentation

Installation

Current Release

pip install kkf

Development Version

git clone https://github.com/diegoolguinw/kkf.git
cd kkf
pip install -e ".[dev]"

Optional Dependencies

# For visualization examples
pip install kkf[viz]

# For development and testing
pip install kkf[dev]

# For building documentation
pip install kkf[docs]

# Install everything
pip install kkf[all]

Quick Start

Here's a complete example of using KKF to estimate states in a SIR (Susceptible-Infected-Recovered) epidemiological model (see also examples/02_sir_epidemic_model.py):

import numpy as np
from scipy import stats
from sklearn.gaussian_process.kernels import Matern

from KKF import systems, koopman, filter

# Define SIR dynamics
beta, gamma = 0.12, 0.04

def f(x):  # State transition
    return x + np.array([
        -beta * x[0] * x[1],
        beta * x[0] * x[1] - gamma * x[1],
        gamma * x[1]
    ])

def g(x):  # Observation function (infected population)
    return np.array([x[1]])

# Setup system
nx, ny, n_features = 3, 1, 50
X_dist = stats.dirichlet(alpha=np.ones(nx))
dyn_dist = stats.multivariate_normal(mean=np.zeros(nx), cov=1e-5 * np.eye(nx))
obs_dist = stats.multivariate_normal(mean=np.zeros(ny), cov=1e-3 * np.eye(ny))

system = DynamicalSystem(nx, ny, f, g, X_dist, dyn_dist, obs_dist, discrete_time=True)

# Generate synthetic observations
np.random.seed(42)
n_timesteps = 100
y_obs = np.random.randn(n_timesteps, ny) * 0.1 + 0.1

# Apply Koopman Kalman Filter
kernel = Matern(length_scale=1.0, nu=0.5)
koopman_op = KoopmanOperator(kernel, system)

x0_prior = np.array([0.8, 0.15, 0.05])
initial_prior = stats.multivariate_normal(mean=x0_prior, cov=0.1 * np.eye(nx))

solution = apply_koopman_kalman_filter(
    koopman_operator=koopman_op,
    observations=y_obs,
    initial_distribution=initial_prior,
    n_features=n_features,
    optimize=False,
    noise_samples=100
)

# Access results
print(f"State estimates shape: {solution.x_plus.shape}")  # (n_timesteps, nx)
print(f"State covariances shape: {solution.Px_plus.shape}")  # (n_timesteps, nx, nx)
print(f"Mean estimation error: {solution.get_estimation_error().mean():.6f}")

Examples

The library includes three comprehensive examples:

1. Basic Linear System (examples/01_basic_linear_system.py)

Introduces core concepts with a simple linear system:

• System definition
• Synthetic data generation
• Filter application
• Confidence interval visualization

Run: python examples/01_basic_linear_system.py

2. SIR Epidemic Model (examples/02_sir_epidemic_model.py)

Real-world application to disease modeling:

• Nonlinear dynamics
• Partial state observation
• Population dynamics estimation
• Epidemic curve prediction

Run: python examples/02_sir_epidemic_model.py

3. Parameter Exploration (examples/03_advanced_parameter_exploration.py)

Advanced hyperparameter tuning:

• Multiple kernel comparison
• Feature dimension optimization
• Performance analysis
• Systematic search strategy

Run: python examples/03_advanced_parameter_exploration.py

For more details, see examples/README.md

Documentation

API Reference

# Core classes
from KKF import DynamicalSystem, KoopmanOperator, KoopmanKalmanFilterSolution
from KKF.filter import apply_koopman_kalman_filter

# Utility functions
from KKF import (
    compute_initial_covariance,
    compute_dynamics_covariance,
    compute_observation_covariance
)

Theory

The KKF algorithm works in three main steps:

1. Lifting: Maps nonlinear states to a higher-dimensional feature space using kernel methods
2. Approximation: Constructs a linear Koopman operator approximation via kEDMD
3. Filtering: Applies Kalman filtering in the lifted space, then maps back to original coordinates

Key Parameters

Parameter	Description	Default
n_features	Dimension of lifted feature space	50
kernel	Kernel function (RBF, Matérn, etc.)	RBF(1.0)
optimize	Whether to optimize kernel parameters	False
noise_samples	Monte Carlo samples for covariance	100

System Requirements

• Python 3.8 or higher
• NumPy ≥ 1.20.0
• SciPy ≥ 1.7.0
• scikit-learn ≥ 1.0.0
• Matplotlib ≥ 3.5.0 (optional, for visualization)

Advanced Usage

Custom Kernel Functions

from sklearn.gaussian_process.kernels import (
    RBF, Matern, ExpSineSquared, DotProduct, ConstantKernel
)

# Different kernel choices
kernel1 = RBF(length_scale=1.0)
kernel2 = Matern(length_scale=1.0, nu=2.5)
kernel3 = ExpSineSquared(length_scale=1.0, periodicity=1.0)
kernel4 = ConstantKernel(constant_value=1.0) * Matern(nu=1.5)

Manual Kernel Optimization

solution = apply_koopman_kalman_filter(
    koopman_operator=koopman_op,
    observations=y_obs,
    initial_distribution=initial_prior,
    n_features=50,
    optimize=True,           # Enable kernel optimization
    n_restarts_optimizer=20  # Number of optimization restarts
)

Extracting Uncertainty Estimates

# Posterior state covariance
Px_plus = solution.Px_plus  # (n_timesteps, nx, nx)

# Compute confidence intervals
std_devs = np.sqrt(np.diagonal(Px_plus, axis1=1, axis2=2))
ci_lower = solution.x_plus - 1.96 * std_devs
ci_upper = solution.x_plus + 1.96 * std_devs

Testing

Run the comprehensive test suite:

# All tests
pytest tests/

# Verbose output
pytest tests/ -v

# Specific test file
pytest tests/test_core.py

# Skip slow tests
pytest tests/ -m "not slow"

# With coverage report
pytest tests/ --cov=KKF --cov-report=html

Contributing

We welcome contributions! See CONTRIBUTING.md for guidelines on how to:

• Report bugs
• Suggest features
• Submit pull requests
• Improve documentation

Code of Conduct

Please note that this project is released with a Contributor Code of Conduct. By participating you agree to abide by its terms.

Citation

If you use KKF in your research, please cite:

@software{kkf2024,
  title={KKF: Kernel Koopman Kalman Filter},
  author={Olguin-Wende, Diego},
  year={2024},
  url={https://github.com/diegoolguinw/kkf}
}

License

This project is licensed under the MIT License - see LICENSE file for details.

Changelog

See CHANGELOG.md for version history and release notes.

Support & Contact

• Issues & Bug Reports: GitHub Issues
• Discussions: GitHub Discussions
• Email: dolguin@dim.uchile.cl