Least Squares Fitting of Data

This is the usual introduction to least squares fit by a line when the data represents measurements where the y–component is assumed to be functionally dependent on the x–component. Given a set of samples {(xi, yi)}i=1, determine A and B so that the line y = Ax + B best fits the samples in the sense that the sum of the squared errors between the yi and the line values Axi + B is minimized. Note that the error is measured only in the y–direction. Define E(A,B) = ∑m i=1[(Axi +B)− yi]. This function is nonnegative and its graph is a paraboloid whose vertex occurs when the gradient satistfies ∇E = (0, 0). This leads to a system of two linear equations in A and B which can be easily solved. Precisely,