Linear Fitting and Polynomial Fitting with Code Implementation Examples

Linear fitting and polynomial fitting represent fundamental concepts in numerical calculation and related courses. Linear fitting approximates a set of data points using a straight line, while polynomial fitting employs a polynomial function to model the data distribution. These methods serve practical applications such as predicting unknown data values by fitting existing data points. Linear fitting typically addresses simple linear regression problems, whereas polynomial fitting handles more complex data modeling scenarios. In linear fitting implementation, the least squares method is commonly used to determine the optimal fitting line by minimizing the sum of squared residuals between observed and predicted values. This can be implemented using matrix operations with the normal equation: θ = (XᵀX)⁻¹Xᵀy, where X represents the feature matrix and y the target values. For polynomial fitting, increasing the polynomial degree enhances fitting precision but requires careful consideration to avoid overfitting. The implementation typically involves transforming original features into polynomial features using basis functions, then applying linear regression techniques. Key functions in programming implementations often include polyfit() for coefficient calculation and polyval() for evaluation, with regularization methods like Ridge or Lasso sometimes employed to prevent overfitting in higher-degree polynomials. Both linear and polynomial fitting serve as valuable tools that play significant roles in solving various practical engineering and scientific problems, with implementations available across programming languages like Python (using numpy.polyfit), MATLAB (polyfit function), and R (lm function with polynomial terms).