A Julia package implementing Kalman filtering and smoothing for state-space models with quadratic measurement equations, based on the methodology developed in Monfort et. al. (2013, Journal of Econometrics).
The package implements filtering and smoothing for the following state-space model with Gaussian noise:
State equation:
Measurement equation:
Check out the latest documentation for installation, API reference, and usage examples.
- Kalman filtering and smoothing for quadratic state-space models.
- Extends original implementation by allowing for autoregressive measurement equations.
- Gradiant and hessian of negative log-likelihood computed using automatic differentiation using ForwardDiff.jl.
- Visualization tools for filtered and smoothed states.
- Efficient parameter-model conversion for optimization.
- Automatically reparametrizes model parameters to ensure positive-definiteness of covariance matrices in an unconstrained parameter space.
- Benchmarking results show an improvement in speed of 9.7x - 62x faster than R code depending on the dimensionality of the problem.
- Numerically stable implementation.
- TODO: Add support for state-dependent measurement noise.
If you're using QuadraticKalman.jl in your research or projects, I'd love to know! Please consider:
- ⭐ Star the repository to help others discover it
- 🔍 Open an issue sharing your use case or results
- 📝 Submit a PR to add your project to this list
The package is especially valuable for:
- Econometric analysis with non-linear dynamics
- Financial modeling with quadratic effects
- Spatial filtering with interactions.
using Pkg
Pkg.add("QuadraticKalman")using QuadraticKalman, Random, Plots, LinearAlgebra
Random.seed!(2314)
# Define model parameters
N = 2 # Number of states
M = 2 # Number of measurements
# Generate stable state transition parameters
μ = [0.1, 0.2] # N x 1 vector
Φ = [0.5 0.1; 0.1 0.3] # N x N matrix
Σ = [0.6 0.15; 0.15 0.4] # N x N matrix
Ω = cholesky(Σ).L
# Generate measurement parameters
A = [0.0, 0.0] # M x 1 vector
B = [1.0 0.0; 0.0 1.0] # M x N matrix
C = [[0.2 0.1; 0.1 0.0], # M x 1 vector of N x N matrices
[0.0 0.1; 0.1 0.2]]
V = [0.2 0.0; 0.0 0.2] # M x M matrix
D = cholesky(V).L
α = zeros(M, M) # M x M matrix
# Simulate data (see example in Documentation)
# X, Y = ... simulation code ... # X is N x T and Y is M x T
# Create model and run filter/smoother
model = QKModel(N, M, μ, Φ, Ω, A, B, C, D, α)
data = QKData(Y)
results_filter = qkf_filter(data, model)
results_smoother = qkf_smoother(results_filter, model)
# Visualize smoothed states
p = plot(kalman_smoother_truth_plot(X, results_smoother))
savefig(p, "smoother_example.png") # Save plot for README
# Parameter estimation example
params = model_to_params(model)
nll(p) = qkf_negloglik(p, data, N, M)
grad = ForwardDiff.gradient(nll, params)
hess = ForwardDiff.hessian(nll, params)| T | N | M | Julia Median Time | Julia Min Time | R Median Time | R Min Time |
|---|---|---|---|---|---|---|
| 10 | 1 | 1 | 0.13 | 0.12 | 7.74 | 7.13 |
| 100 | 1 | 1 | 1.24 | 1.19 | 84.90 | 80.73 |
| 1000 | 1 | 1 | 13.75 | 12.80 | 851.76 | 837.77 |
| 10 | 2 | 2 | 0.88 | 0.74 | 8.576 | 7.88 |
| 100 | 2 | 2 | 9.08 | 8.12 | 94.19 | 89.32 |
| 1000 | 2 | 2 | 94.49 | 87.56 | 1013.17 | 926.05 |
| 10 | 5 | 5 | 28.94 | 25.14 | 791.03 | 764.57 |
| 100 | 5 | 5 | 324.25 | 318.02 | 8591.90 | 8376.84 |
| 1000 | 5 | 5 | 3267.50 | 3249.10 | 93159.17 | 85904.42 |
Note: Times are measured in milliseconds. Lower values indicate better performance. Benchmarked on 100 runs each. Julia code used BenchmarkTools.jl package where R code uses microbenchmark package. Effort was made to remove error-checking and compilation steps from R code to make comparison as relevant as possible.
If you use this package in your research, please cite the original paper:
@article{monfort2015quadratic,
title = {A Quadratic Kalman Filter},
journal = {Journal of Econometrics},
volume = {187},
number = {1},
pages = {43-56},
year = {2015},
issn = {0304-4076},
doi = {https://doi.org/10.1016/j.jeconom.2015.01.003},
url = {https://www.sciencedirect.com/science/article/pii/S0304407615000123},
author = {Alain Monfort and Jean-Paul Renne and Guillaume Roussellet},
}And please also consider citing this package directly:
@software{leather2025quadratickalman,
author = {Leather, David},
title = {{QuadraticKalman.jl}: A Julia package for quadratic Kalman filtering},
url = {https://github.com/dleather/QuadraticKalman.jl},
version = {0.1.1},
doi = {10.5281/zenodo.14908624},
year = {2025},
}MIT License
Copyright (c) 2025 David Leather
Permission is hereby granted, free of charge, to any person obtaining a copy of this software and associated documentation files (the "Software"), to deal in the Software without restriction, including without limitation the rights to use, copy, modify, merge, publish, distribute, sublicense, and/or sell copies of the Software, and to permit persons to whom the Software is furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in all copies or substantial portions of the Software.
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Contributions are welcome! Please feel free to submit a Pull Request.