Within the framework of the problem of the best–fit of a set of data through the minimization of the perpendicular offsets (method which dates back to Pearson in 1901), we reconsider the linear fit of a set of data points (either in R^2 or in R^{n+1}, n > 1) and introduce a new estimator of the quality of the fitting. By facing the problem of the best–fit with a second degree polynomial function in R^2, we propose an iterative approach whose underlying ideas in the linear case lead to the solution that is obtained directly. Also in this case an estimator of the quality of fitting is introduced by considering the ratio between the sum of minimized squared distances from the curve and the variance of distances along the curve. It is also shown that the iterative procedure and the computation of the estimator still works in the case of general fitting functions. Some numerical tests are also presented in order to show how the procedure works, and the numerical distributions of the proposed estimator are computed. A Mathematica notebook useful to perform the calculation has been written and is available at the URL http://mat521.unime.it/perpoff/.
A Computational Approach to Least Square Fitting with Perpendicular Offsets
OLIVERI, Francesco;
2012-01-01
Abstract
Within the framework of the problem of the best–fit of a set of data through the minimization of the perpendicular offsets (method which dates back to Pearson in 1901), we reconsider the linear fit of a set of data points (either in R^2 or in R^{n+1}, n > 1) and introduce a new estimator of the quality of the fitting. By facing the problem of the best–fit with a second degree polynomial function in R^2, we propose an iterative approach whose underlying ideas in the linear case lead to the solution that is obtained directly. Also in this case an estimator of the quality of fitting is introduced by considering the ratio between the sum of minimized squared distances from the curve and the variance of distances along the curve. It is also shown that the iterative procedure and the computation of the estimator still works in the case of general fitting functions. Some numerical tests are also presented in order to show how the procedure works, and the numerical distributions of the proposed estimator are computed. A Mathematica notebook useful to perform the calculation has been written and is available at the URL http://mat521.unime.it/perpoff/.Pubblicazioni consigliate
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