Exploring the 3D Surfaces with Modified Method of Steepest Descent

Abstract

Aim: To prove expediency of the steepest descent method to divide a given cloud of (Y, X1, X2) points into the spatial clusters with purpose to estimate a simple regression model Y = f(Z|X1,X2) at each cluster. Material and Method: The exemplary data sets {Y, X1, X2} were drawn randomly from assumed 3D surface: Y = f(X1,X2), and then a random noise was added to variable Y. A polynomial model Y = f(X1,X2) and a set of models Y = f(Z|X1,X2) were estimated separately, both under Akaike information criterion (AIC), and then compared with respect to their determination coefficients R-square, and the residuals’ distributions. Results: In the artificial data set studied, the both compared methods after several iterations can provide regression models of the quite similar quality. Conclusions: Because the proposed novel method seems to be more robust to outliers, and easier to graphical presentations and to intuitive understanding than the conventional way of building a regression model, the proposed novel method can be recommended to use by non-statisticians, especially in situation when, besides usual moderate noise, the sporadic but influential measurement errors can occur.