Curve Fitting and Interpolation

In mathematics and statistics, curve fitting and interpolation are two fundamental concepts with distinct purposes. Curve fitting involves constructing a mathematical function or curve that best approximates a set of data points, enabling predictions for unknown values through regression analysis techniques. Common implementations include linear regression (using polyfit in MATLAB or scipy.optimize.curve_fit in Python) and polynomial fitting, where algorithms minimize residuals between observed data and the fitted model. Interpolation, by contrast, estimates unknown values within the range of known data points using methods like linear interpolation, spline interpolation (e.g., MATLAB's spline or scipy.interpolate.interp1d), or polynomial interpolation to fill data gaps. Both techniques find extensive applications across various domains: curve fitting predicts stock price trends through time-series analysis, while interpolation reconstructs missing meteorological measurements using spatial/temporal neighbor data. Understanding the underlying algorithms—such as least-squares optimization for fitting and piecewise polynomial functions for interpolation—is crucial for effective implementation in data science and engineering applications.