Solving Nonlinear Estimation Problems Using Splines

We describe the use of splines for solving nonlinear model estimation problems, in which nonlinear functions with unknown shapes and values are involved, by converting the nonlinear estimation  problems into linear ones at a higher dimensional space. This contrasts with the typical use of the splines [1]–[3] for function interpolation where the functional values at some input points are given and the values corresponding to other input points are sought for via interpolation. The technique described in this column applies to arbitrary nonlinear estimation problems where one or more one-dimensional nonlinear functions are involved and can be extended to cases where higher-dimensional nonlinear functions are used.

The benefit of using the approach described here is obvious. Many real world systems can only be appropriately modeled with nonlinear functions, while the estimation problem is much simpler if only linear functions are involved. It is thus highly desirable if a nonlinear estimation problem can be transformed into a linear estimation problem at a different space. In this column we use the cubic spline (i.e., piecewise third-order polynomials) [1], [2] to illustrate the technique. However, the same approach can be used with other types of spline as illustrated at the end. We demonstrate the applications of the technique in signal processing and pattern recognition with an example.