The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals made in the results of every single equation.
The most important application is in data fitting. The best fit in the least-squares sense minimizes the sum of squared residuals (a residual being: the difference between an observed value, and the fitted value provided by a model). When the problem has substantial uncertainties in the independent variable (the x variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares.
Least-squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression analysis; it has a closed-form solution. The nonlinear problem is usually solved by iterative refinement; at each iteration the system is approximated by a linear one, and thus the core calculation is similar in both cases.
Polynomial least squares describes the variance in a prediction of the dependent variable as a function of the independent variable and the deviations from the fitted curve.
When the observations come from an exponential family and mild conditions are satisfied, least-squares estimates and maximum-likelihood estimates are identical. The method of least squares can also be derived as a method of moments estimator.
The following discussion is mostly presented in terms of linear functions but the use of least squares is valid and practical for more general families of functions. Also, by iteratively applying local quadratic approximation to the likelihood (through the Fisher information), the least-squares method may be used to fit a generalized linear model.
The least-squares method was officially discovered and published by Adrien-Marie Legendre (1805), though it is usually also co-credited to Carl Friedrich Gauss (1795) who contributed significant theoretical advances to the method and may have previously used it in his work.
I have a question that maybe has to do more with Mathematics, but if you do experimental physics you find it quite often.
Let's assume that we want to measure two quantities x and y that we know that they relate to each other linearly. So we have a set of data points xi and yi...
I would studying the theory of least square and I find that the derivative of the error summation between the predicated line points and the true data is equal zero. Why the first derivative is equal zero?
I have the following data which I would like to model using an exponential function of the form y = A + Becx.
Using wolfram mathematica, solving for these coefficients was computed easily using the findfit function. I was tasked however to implement this using java and have...
The Attempt at a Solution
I used polyfit, but my mean swuare errors are way bigger than they should be- don't see what is wrong with my code! My code is ugly btw, my apologies.