A framework for fitting sparse data

Abstract

This paper develops a framework for fitting functions with domains in the Euclidean space, when data are sparse but a slow variation allows for a useful fit. We measure the variation by Lipschitz Bound (LB). Functions which admit smaller LB are considered to vary more slowly. Since most functions in practice are wiggly and do not admit a small LB, we extend this framework by approximating a wiggly function, f, by ones which admit a smaller LB and do not deviate from f by more than a specified Bound Deviation (BD). In fact for any positive LB, one can find such a BD, thus defining a trade-off function (LB-BD function) between the variation measure (LB) and the deviation measure (BD). We show that the LB-BD function satisfies nice properties: it is non-increasing and convex. We also present a method to obtain it using convex optimization. For a function with given LB and BD, we find the optimal fit and present deterministic bounds for the prediction error of various methods. Given the LB-BD function, we discuss picking an appropriate LB-BD pair for fitting and calculating the prediction errors. The developed methods can naturally accommodate an extra assumption of periodicity to obtain better prediction errors. Finally we present the application of this framework to air pollution data with sparse observations over time.