MATLAB Code for Maximum-Likelihood-Estimator-Based Differentiator (MLEdiff) Using Quadratic and Zero-Order Optimal Splines

This repository contains MATLAB code supporting the results in:

"Derivative Estimation from Coarse, Irregular, Noisy Samples: An MLE–Spline Approach"
by Konstantin E. Avrachenkov and Leonid B. Freidovich, 2025.

📧 Contact: [email protected]

---

📄 Description

This code implements a maximum-likelihood-based numerical differentiation method using optimal smoothing quadratic and zero-order splines. It is designed to estimate the derivative of a signal measured under coarse, non-uniform sampling and corrupted by Gaussian noise.

The implementation reproduces the key results and figures described in the above article, including comparisons with classical differentiators and performance under various noise and sampling conditions.

Last updated: 2025-07-29

---

📦 Files

🔹 Primary Scripts

• ml_quadratic_spline_v16.m — Full implementation of the quadratic-spline-based nd zero-order-spline-based differentiators, including data generation, spline fitting, and comparison with two other methods (generates figures for the paper referenced above).
• ml_zero_spline_v1.m — Implementation of the zero-order spline-based differentiator.
• rec_mle_splines_v2.m — Full implementation of the recursive quadratic-spline-based and zero-order-spline-based differentiators, including data generation, spline fitting, and comparison with ttwo other methods.

🔹 Signal Generation

• make_data.m — Generates synthetic signals with non-uniform sampling and Gaussian noise.
• x_fun.m — Defines the test signal and its analytical derivative.

🔹 Spline-Based Differentiation

• quadratic_spline_step_QC.m — Constructs matrices for quadratic spline smoothing.
• quadratic_spline_step_new.m — Alternative formulation of the quadratic spline method.
• zero_order_spline_step_QC.m — Constructs matrices for zero-order spline smoothing.
• z_from_p.m — Computes all the parameters of derivative estimates using quadratic spline parameters.
• interp_z_from_z_K.m — Interpolates quadratic-spline-based derivative estimates over time.

🔹 Other Differentiation Methods for Compsrision

• levant_step.m — Implements Levant’s first-order super-twisting differentiator (requires upper bound on second derivative), discretized using Euler's method.
• hgo_step.m — Implements a high-gain observer (HGO) for derivative estimation (requires tuning of a single small parameter, uses upper bound on the estimated first derivative for a protactive saturation), discretized using Euler's method.

🔹 Recursive and Update Utilities

• update_quadratic_spline.m — Efficient (optimized for speed in Matlab) recursive update of quadratic spline estimates using matrix identities.
• updateQC.m — Computes updates to second-order spline matrices for recursive estimation (for update of zero-order spline matrices see rec_mle_splines_v2.m).

---

🚀 How to Run

1. Clone or download this repository.
2. Open MATLAB and set the current folder to the repository root.
3. Run the main script: ml_quadratic_spline_v16.m

---

🖋 Citation

If you use this code in your research or publications, please cite either the article, once published, or this depository: https://github.com/fleonid/MLEdiff/

---