Interpolation and Curve Fitting

Interpolation Basics

Given data points, we often want a function to fill gaps or make predictions. Interpolation finds a function passing exactly through every data point.

Given $n+1$ points, there is a unique interpolating polynomial of degree $n$. Spline interpolation joins points with low-degree cubic pieces, matching smoothly.

A single high-degree polynomial can oscillate wildly (Runge's phenomenon). Splines avoid this by using piecewise low-deg curves.

Curve Fitting & Pitfalls

Curve fitting (regression) finds a function of a chosen form coming close to points without hitting them. Least-squares fitting minimizes the sum of squared vertical distances (residuals) from points to the curve.

$$\text{exact data: interpolate} \;;\; \text{noisy data: fit}$$

Common pitfall: interpolating noisy data. Forcing a curve through every point when data have noise makes it chase random jitter (overfitting). Conversely, underfitting misses real features by using a too-simple curve. Match the method to the data's noise level.