GP Differentiation and Integration — Notebooks

Companion notebooks for the paper:

Differentiation and Integration of Time Series via Gaussian Process Regression for Structural Health Monitoring Applications

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Contents

Notebook	Description
01_Illustrative_Example_1_Differentiation_Duffing.ipynb	GP differentiation: recover velocity and acceleration from noisy displacement — Duffing oscillator (Figure 3)
02_Illustrative_Example_2_Integration_Lorenz.ipynb	GP integration: recover displacement from noisy velocity — Lorenz attractor (Figure 4)
03_Application_WindTurbine_Accel_to_Displacement.ipynb	GP double integration: recover displacement from noisy acceleration — wind turbine SHM application (Figures 6–7)

All notebooks are pre-executed and self-contained — figures and results are visible directly in the browser without running anything.

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Repository structure

All files are in this folder:

notebooks/
├── 01_Illustrative_Example_1_Differentiation_Duffing.ipynb
├── 02_Illustrative_Example_2_Integration_Lorenz.ipynb
├── 03_Application_WindTurbine_Accel_to_Displacement.ipynb
├── data_generation.py           ← ODE solvers and data loaders
├── gp_optimization.py           ← NLL optimisation (Algorithms 1 & 2)
├── KalmanFilter_functions.py    ← Kalman filter + RTS smoother
├── Matern_52_state_space.py     ← Matérn 5/2 state-space matrices
├── plotting_functions.py        ← plotting utilities
└── windturbine2D.txt            ← wind turbine simulation data

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Dependencies

numpy
scipy
matplotlib

Install with:

pip install numpy scipy matplotlib

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Method overview

The core idea is to represent the unknown time series as a Matérn 5/2 Gaussian Process and exploit the fact that its state-space form places the signal and all its derivatives inside a single 3-dimensional latent state:

$$\mathbf{z}(t) = \begin{bmatrix} x(t) \ \dot{x}(t) \ \ddot{x}(t) \end{bmatrix}$$

Observations of any derivative are fused via a Kalman filter, and the RTS smoother recovers the full posterior over all derivatives simultaneously — including uncertainty quantification.

• Differentiation: observe $x(t)$, read off $\dot{x}$ and $\ddot{x}$ from the smoothed state.
• Integration: observe $\dot{x}(t)$, read off $x(t)$ from the smoothed state.
• Double integration: observe $\ddot{x}(t)$, read off $x(t)$ from the smoothed state.