Dynamic classification using multivariate locally stationary wavelet processes

Abstract

Methods for the supervised classification of signals generally aim to assign a signal to one class for its entire time span. In this paper we present an alternative formulation for multivariate signals where the class membership is permitted to change over time. Our aim therefore changes from classifying the signal as a whole to classifying the signal at each time point to one of a fixed number of known classes. We assume that each class is characterised by a different stationary generating process, the signal as a whole will however be nonstationary due to class switching. To capture this nonstationarity we use the recently proposed Multivariate Locally Stationary Wavelet model. To account for uncertainty in class membership at each time point our goal is not to assign a definite class membership but rather to calculate the probability of a signal belonging to a particular class. Under this framework we prove some asymptotic consistency results. This method is also shown to perform well when applied to both simulated and accelerometer data. In both cases our method is able to place a high probability on the correct class for the majority of time points.

Spectral Dependence

“…Note that Z jk in the multivariate LSW is the analog of dZ(ω) in the Fourierbased stochastic representation. The classification procedure for the LSW model was developed in [153] for univariate time series and in [154] for multivariate time series. Given training data (signals with known group membership), these methods extract the wavelet scale-shift features or projections that separate the different classes of signals.…”

Dynamic detection of anomalous regions within distributed acoustic sensing data streams using locally stationary wavelet time series

“…In contrast, dynamic classification approaches allow for the class assignment of the test signal to vary over time which allows for more flexibility in the classification and covers problems where the underlying nonstationarity is due to class switching. The method that we introduce is an online analogue of the dynamic classification approach of Park et al (2018) which looks to detect subtle changes in the dependence structure of a multivariate signal in a fast and efficient manner.…”

“…Following their work on MvLSW processes, Park et al (2018) introduced an approach to dynamically classify a Multivariate Locally Stationary Wavelet signal X t whose class membership may change over time. The approach assumes that at any time t, the signal X t can belong to one of N c ≥ 2 different classes, where N c is known.…”

“…Before introducing our proposed method in Section 3, we first outline some details of the locally stationary wavelet framework of Park et al (2014) together with the (offline) dynamic classification method introduced in Park et al (2018) that forms the basis of our online approach. We begin with some introductory concepts of wavelets.…”

“…where L(θ|x) is the likelihood and Pr C(k) = c is a prior probability (Park et al 2018). Due to the use of the Fisher-z transform in (4), the likelihood L(θ|x) takes the form of a Gaussian likelihood with mean vector µ (c) and covariance matrix Σ (c) .…”