Real-time interpolation of streaming data

One of the key elements of real-time $C^1$-continuous cubic spline interpolation of streaming data is an estimator of the first derivative of the interpolated function that is more accurate than the ones based on finite difference schemas. Two such greedy look-ahead heuristic estimators (denoted as MinBE and MinAJ2) based on Calculus of Variations are formally defined (in closed form) together with the corresponding cubic splines they generate, and then comparatively evaluated in a series of numerical experiments involving different types of performance measures. The results presented show that the cubic Hermite splines generated by heuristic MinAJ2 significantly outperformed these based on finite difference schemas in terms of all tested performance measures (including convergence). The proposed approach is quite general. It can be directly applied to streams of univariate functional data like time-series. Multidimensional curves defined parametrically, after splitting, can be handled as well. The streaming character of the algorithm means that it can also be useful in processing data sets that are too large to fit in memory (e.g., edge computing devices, embedded time-series databases).