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MAThe least squares method is a technique used to find the best-fitting line for a set of data points. Suppose we are given pairs of values (x1, y1), (x2, y2), and so on, and we want to describe the relationship between x and y using a linear equation of the form y = ax + b. In most real-world situations, the points will not lie perfectly on a single line, so we measure the error at each point as the vertical difference between the actual value yi and the predicted value axi + b. The least squares method chooses the numbers a and b that make the total squared error as small as possible. We square the errors so that negative and positive differences do not cancel each other out and so that larger errors are penalized more heavily.
To find the best values of a and b, we form the sum of all squared errors and treat it as a function of these unknowns. We then minimize this function using calculus, which leads to a system of equations known as the normal equations. Solving this system gives formulas for the slope and intercept in terms of averages and sums computed from the data. Geometrically, the least squares solution can be understood as projecting the observed data onto the space of possible linear models. The method extends naturally to more complicated models, including polynomial regression and multiple regression, and it forms the foundation of many techniques in statistics, economics, and data science.
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