Piecewise Linear Fitting for Two Sets of Discrete Data

When processing two sets of discrete data, piecewise linear fitting serves as a simple yet effective method, particularly suitable for datasets exhibiting nonlinear relationships between data points. The core concept involves dividing the data into multiple segments and applying linear fitting within each interval. This approach preserves local data characteristics while minimizing overall fitting errors.

### Fundamental Approach to Piecewise Linear Fitting Data Segmentation: The first step requires determining breakpoint positions. Common methodologies include manually defined intervals, inflection point detection based on data variation patterns, or optimization algorithms for automatic optimal breakpoint selection (e.g., using dynamic programming or genetic algorithms). Local Linear Fitting: Within each segment, employ least squares regression to compute optimal linear equations, ensuring minimized residuals within the segment. Code implementation typically involves solving normal equations or using built-in functions like numpy.polyfit() for slope and intercept calculation. Result Integration: Concatenate all linear segments to form the final piecewise linear function, ensuring continuity at breakpoints through constraint handling or overlap averaging.

### Visualization of Fitting Results To demonstrate fitting effectiveness, plot original data points and segmented fitting curves on the same graph using distinct colors or line styles. Additionally, calculate segment-specific error metrics (e.g., Mean Squared Error or Mean Absolute Error) to evaluate fitting precision programmatically through error analysis functions.

### Application Scenarios Piecewise linear fitting is ideal for data with clear boundaries or abrupt changes, such as sensor signal processing or economic trend analysis. Compared to high-order polynomial fitting, it reduces overfitting risks while maintaining local variation characteristics. Advanced implementations may incorporate adaptive segmentation strategies or smoothing interpolation methods (e.g., cubic spline integration) for enhanced accuracy and continuity.

For further optimization, consider implementing automatic breakpoint detection algorithms like Jenks natural breaks or implementing cross-validation techniques to determine optimal segment count programmatically.