Least squares
Least squares is the name of a procedure in mathematics, to construct a function from a number of observed values. The basic idea is to construct the function in such a way that the sum of the difference between the observed value and its data point is minimized. Since the difference may go in either direction, the value of the difference is squared, for each value.
Carl Friedrich Gauss said he developed the method in 1795. He used it to recover the lost asteroid 1 Ceres and published it in 1807. He used ideas from Pierre-Simon Laplace. Adrien-Marie Legendre developed the same method independently, in 1805.
Least Squares Media
The result of fitting a set of data points with a quadratic function
Conic fitting a set of points using least-squares approximation
Lithograph showing a portrait of the German mathematician Carl Friedrich Gauss at the age of 50
The residuals are plotted against corresponding x values. The random fluctuations about 0 indicate a linear model is appropriate.
The residuals are plotted against the corresponding x values. The parabolic shape of the fluctuations about r_i=0 indicates a parabolic model is appropriate.
"Fanning Out" Effect of Heteroscedasticity
Related pages
Further reading
- Lawson, C. L., & Hanson, R. J. (1995). Solving least squares problems (Vol. 15). SIAM.
- Bjorck, A. (1996). Numerical methods for least squares problems (Vol. 51). SIAM.