Adaptive parameter choice for one-sided finite difference schemes and its application in diabetes technology

In this paper we discuss the problem of an adaptive parameter choice in onesided finite difference schemes for the numerical differentiation in case when noisy values of the function to be differentiated are available only at the given points. This problem is motivated by diabetes therapy management, where it is important to provide estimations of the future blood glucose trend from current and past measurements. Here we show, how the proposed approach can be used for this purpose and demonstrate some illustrative tests, as well as the results of numerical experiments with simulated clinical data. 1. Problem formulation In this paper we consider the problem of approximation of a derivative y(B) at the boundary point of some interval [b, B] under the condition that at the given points B = tN > tN−1 > · · · > t1 ≥ b (1) only noisy values yδ(tj) of y(tj) are available such that |y(tj)− yδ(tj)| ≤ δ. (2) It should be noted that the problem in such formulation arises in various practical applications, for example, in the diabetes therapy management. More about this application will be given in the forthcoming section. At the same time, it is noteworthy to mention that the problem of approximation of y(B) does not seem to have been discussed so intensively as the problem of the reconstruction of the derivative at given interior point. For