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Least Squares Approximation of ECG Waveforms

This repository contains the implementation and experimental code for the paper
“Least Squares Approximation of ECG Waveforms for Cardiac Signal Analysis”,
developed as part of the Linear Algebra and Geometry course.

The project demonstrates how least squares approximation can be applied to real ECG signals by modeling them as vectors in a finite-dimensional space and projecting them onto polynomial subspaces.

📌 Overview

Electrocardiogram (ECG) signals are time-series measurements of cardiac electrical activity. From a mathematical perspective, each ECG segment can be represented as a vector in ( \mathbb{R}^n ).
This project applies polynomial least squares approximation to ECG segments from the MIT-BIH Arrhythmia Database, with the goal of:

  • illustrating least squares as a projection operation
  • analyzing the effect of polynomial degree on approximation accuracy
  • visualizing approximation errors geometrically
  • extracting simple, interpretable features from approximation results

The implementation focuses on mathematical modeling and analysis, not medical diagnosis.

📂 Repository Structure

.
├── src/
│   ├── ecganalysis.py   # Main implementation script
│
├── figures/
│   ├── ecg_vs_ls_degree6.png     # Example approximation plot
│   ├── rmse_vs_degree.png        # RMSE vs polynomial degree
│
└── README.md

🧠 Methodology Summary

  1. Dataset

    • MIT-BIH Arrhythmia Database (PhysioNet)
    • Record 101, MLII lead
    • Sampling rate: 360 Hz

  2. Preprocessing

    • Baseline removal (mean subtraction)
    • Noise reduction (moving average filter)
    • Z-score normalization
    • Fixed-length segmentation

  3. Mathematical Model

    • ECG segment represented as a vector ( \mathbf{y} \in \mathbb{R}^n )
    • Polynomial basis constructed using a Vandermonde matrix
    • Least squares problem: [ \min_{\mathbf{c}} |\mathbf{y} - A\mathbf{c}|_2^2 ]
    • Solution interpreted as orthogonal projection onto a polynomial subspace

  4. Evaluation

    • Mean Squared Error (MSE)
    • Root Mean Squared Error (RMSE)
    • Residual ( \ell_2 ) norm
    • Visual comparison of original and approximated signals

🧪 Experimental Results

  • Increasing polynomial degree leads to monotonic error reduction
  • Error improvement shows diminishing returns beyond degree 6
  • Low-degree polynomials capture global ECG trends
  • Sharp QRS peaks remain in the residual component due to their high-frequency nature

These results align with the geometric interpretation of least squares approximation.

🛠️ Requirements

Install dependencies using:

pip install -r requirements.txt

Required packages

  • numpy
  • matplotlib
  • wfdb

▶️ Running the Code

Execute the main script:

python src/ecg_ls_approximation.py

The script will:

  • download ECG data from PhysioNet (via WFDB)
  • preprocess the signal
  • compute least squares approximations for multiple polynomial degrees
  • output error metrics
  • generate visualization plots

📊 Output Examples

  • ECG waveform vs polynomial approximation
  • RMSE vs polynomial degree
  • Residual analysis

These outputs are used directly in the experimental results section of the paper.

⚠️ Notes and Limitations

  • This project does not perform medical diagnosis
  • ECG annotations are not used for classification
  • Polynomial approximation is global and may not capture localized sharp features
  • The implementation is intended for educational and analytical purposes

📚 References

  • MIT-BIH Arrhythmia Database, PhysioNet
  • G. Strang, Linear Algebra and Its Applications
  • L. Xie et al., “Computational diagnostic techniques for electrocardiogram signal analysis,” Sensors, 2020

(Full references are provided in the accompanying paper.)

🎓 Academic Context

This project was developed for:

  • Linear Algebra and Geometry
  • Undergraduate (Semester 3)
  • Informatics / Computer Science