Measurement instrumentation and selected signal processing techniques for anomalous diffusion analysis

Abstract

Anomalous diffusion phenomena are ubiquitous in many areas, including physics, biology, medicine, climate and also financial markets. The description and analysis of this phenomena are challenging tasks. To fully define the anomalous diffusion and properly understand its nature there is need to combine the skills in advanced sensing techniques, mathematical and statistical knowledge of the corresponding theoretical processes and the experience in physical interpretation of this phenomena. The paper reviews on the measurement instrumentation and signal processing used for detection and description of anomalous diffusion regimes. What is more, estimation of the anomalous diffusion exponent is commented. We present here four methods based on the statistics commonly used in the analyzed problem. Finally the fractional Brownian motion (FBM) is proposed as an analog proper for reliable depiction of properties valid for observed anomalous diffusion phenomena. Simulation studies exhibit the scenario of Hurst exponent estimation in the FBM model when the additive noise of various amplitudes disturbance is added to a simulated trajectory of individual particle. Proposed algorithms have different efficiency depending on the model parameters. This implicates a need to provide more advanced analysis and introduce new methods that could be less sensitive to the measurement noise and/or external forces of noise character. This paper is the starting point for the complete description of the anomalous diffusive processes. It presents the areas for prospective applications of the original sensing techniques but also highlights the needs in a range of statistical and intelligent signal processing suitable for description of the small scale complex systems during the SPT experiment.