Real-time interpolation of streaming data

One of the key elements of the 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 those based on finite difference schemas. Two such greedy look-ahead heuristic estimators, based on the calculus of variations (denoted as MinBE and MinAJ2), are formally defined (in closed form), along with the corresponding cubic splines that they generate. They are then comparatively evaluated in a series of numerical experiments involving different types of performance measures. The presented results show that the cubic Hermite splines generated by heuristic MinAJ2 significantly outperformed those that were based on finite difference schemas in terms of all of the 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. Multi-dimensional curves that are 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 the memory (e.g., edge computing devices, embedded time-series databases).