Least Squares Linear Regression Algorithm

The least squares linear regression algorithm is a classical regression analysis method used to find the optimal linear fit for data points. This approach determines the optimal slope and intercept of the fitting line by minimizing the sum of squared vertical distances between observed values and the fitted line.

Mathematically, given a set of data points (xi, yi), we aim to find a line y = kx + b that minimizes the sum of squared vertical distances from all data points to this line. By constructing an error function and applying derivative calculus, we can derive computational formulas for the slope and intercept parameters.

Implementing this algorithm in MATLAB is straightforward. Typically, we can utilize built-in functions like polyfit, which accepts x and y coordinates of data points along with the polynomial degree (degree 1 for linear fitting), and returns the fitting coefficients. For custom implementations, one can manually calculate the slope and intercept using matrix operations or analytical solutions to optimize computational efficiency. The key matrix operation involves solving the normal equations: (X'X)β = X'y, where X is the design matrix containing x-values with a column of ones for the intercept term.

The advantages of least squares method lie in its mathematical simplicity and wide applicability, particularly exhibiting optimal statistical properties when errors follow a normal distribution. However, it's important to note its sensitivity to outliers, as significant outliers may adversely affect the fitting results.

For more complex fitting requirements, extended methods like weighted least squares or robust regression can be considered. In practical applications, least squares linear fitting is commonly used for trend analysis, data modeling, and prediction scenarios. Code implementation typically involves preprocessing data, computing covariance matrices, and validating results through residual analysis.