MATLAB Implementation of Least Squares Fitting with Code Examples

Least squares method is a fundamental approach in data fitting that finds optimal parameters by minimizing the sum of squared errors. Implementing least squares fitting in MATLAB typically involves the following key steps: Determining the Fitting Model First, define the mathematical form of the fitting function, such as linear functions (y = ax + b), polynomial functions (y = ax² + bx + c), or other nonlinear models. The choice of model depends on the data characteristics and the underlying relationship you want to capture. Constructing Matrix Equations The core of least squares method involves formulating a linear system of equations that can be solved through matrix operations. For linear models, this can be expressed as Aθ = y, where A is the design matrix containing the independent variable values, and θ is the parameter vector to be estimated. In MATLAB, you typically construct the design matrix by organizing your input data according to the chosen model structure. Parameter Solving MATLAB provides multiple approaches for solving least squares problems: - The backslash operator (\) performs efficient matrix left division using QR decomposition - polyfit function specifically handles polynomial fitting with automatic degree selection - lsqcurvefit function from the Optimization Toolbox handles nonlinear fitting problems with constraint support - For custom implementations, you can use pinv (pseudoinverse) or explicit matrix operations like (A'*A)\A'*y Fitting Quality Evaluation Assess the fitting quality by calculating error metrics such as residual sum of squares (RSS), mean squared error (MSE), or R-squared values. Visual validation through plotting the fitted curve against original data points provides intuitive feedback on the model's performance. For simple linear fitting, MATLAB's polyfit function offers a straightforward solution where you simply provide the data points (x,y) and polynomial degree, and it returns the optimal coefficients. For complex or custom models, you may need to manually construct the design matrix and use mldivide (the \ operator) which automatically selects the most efficient numerical method based on matrix properties.