(Invited) Diffusion Map Analysis of Multi-Channel Time Series Data

Abstract

Sensor data often come in the form of multichannel time series, for example, ECG, EEG, and structural vibration monitoring to name a few. Conventionally, such multichannel time series data are analysed by using FFT for each channel, and/or by computing inter-channel correlations. Mathematically, however, these signals are better represented by dynamical systems, which are sets of differential equations. The behaviour of a dynamical system may be visualised as a trajectory in the so-called phase space. For example, the state of a weight suspended by a spring can be fully described by the weight’s position x and its velocity v, hence its temporal behaviour creates a trajectory in its phase space (x, v). It works similarly for real systems that produce time series data. From a single channel of time series x(t) it is possible to reconstruct the phase space trajectory using lagged variables x(t), x(t-1), x(t-2),…., x(t-n). It is, however, very difficult to find and analyse the trajectory because the dimensionality of the phase space is very large in general. In this presentation, we will show that a method called the diffusion map can extract the lower dimensional manifold in which the trajectory resides, and reveal the structure of the dynamics. Using some toy models we will demonstrate how our technique can be used to extract various useful information about the nature of the dynamics, including identification of anomalies, especially signs for imminent catastrophic changes such as phase transitions or structural collapse.