Regression analysis using a blending type spline construction

Abstract

Regression analysis allows us to track the dynamics of change in measured data and to investigate their properties. A sufficiently good model allows us to predict the behavior of dependent variables with higher accuracy, and to propose a more precise data generation hypothesis. By using polynomial approximation for big data sets with complex dependencies we get piecewise smooth functions. One way to obtain a smooth spline representation of an entire data set is to use local curves and to blend them using smooth basis functions. This construction allows the computation of derivatives at any point on the spline. Properties such as tangent, velocity, acceleration, curvature and torsion can be computed, which gives us the opportunity to exploit these data in the subsequent analysis. We can adjust the accuracy of the approximation on the different segments of the data set by choosing a suitable knot vector. This article describes a new method for determining the number and location of the knot-points, based on changes in the Frenet frame. We present a method of implementation using generalized expo-rational B-splines (GERBS) for regression problems (in two and three variables) and we evaluate the accuracy of the model using comparison of the residuals.