The Least Squares Method is a fundamental technique in statistics and data analysis. It helps us find the best-fitting line through a set of data points. The method works by minimizing the sum of squared differences between the observed data points and the predicted values from our line.
The mathematical foundation of least squares involves minimizing the sum of squared errors, or SSE. For each data point, we calculate the difference between the observed y value and the predicted y value from our model. We square these differences and sum them up. Our goal is to find the parameters m and b that minimize this sum.
To find the optimal parameters, we use calculus. We take partial derivatives of the sum of squared errors with respect to both m and b, then set these derivatives equal to zero. This gives us the normal equations, which we can solve to find the best-fit line parameters.
The least squares method provides closed-form solutions for the slope and intercept. The slope formula involves sums of the data points, and the intercept can be calculated once we have the slope. Using our example data, we get m equals 1.5 and b equals 0.5, giving us the line y equals 1.5x plus 0.5.
To summarize what we have learned: The least squares method is a powerful optimization technique that finds the best-fit line by minimizing squared errors. It uses calculus to derive closed-form solutions and serves as the foundation for linear regression and many machine learning algorithms.