Stochastic Interpolation of Sparsely Sampled Time Series via Multi-point Fractional Brownian Bridges

Abstract:

A considerable number of problems in physics, e.g., cosmic ray propagation in multi-scale magnetic fields, suffer from sparse sampling and thus have to rely on suitable interpolation methods. A common problem of these interpolation methods, e.g., polynomial interpolations or kriging, is that they oftentimes are too smooth and cannot reproduce a certain roughness of the data. We present a novel method to interpolate sparsely sampled signals by a stochastic process with a broad range of spatial and/or temporal scales. To this end, we extend the notion of a fractional Brownian bridge, defined as fractional Brownian motion with a given scaling (Hurst) exponent H and with prescribed start and end points, to a bridge process with an arbitrary number of prescribed intermediate and non-equidistant points. Furthermore, we devise an optimization procedure which ultimately finds the optimally interpolating bridge (i.e., the optimal Hurst exponent) of the sparse points. We demonstrate our method at the example of a signal from fluid turbulence in a high Reynolds number flow. Furthermore, we discuss possible extensions of the present work to include the non-self-similar character of the signal and give an outlook on reconstructing multi-dimensional spatial fields, e.g., reconstructions of Eulerian velocity fields from a given set of Lagrangian tracers.