ABSTRACT
Interpretation of a process signal solely based on its temporal evolution is often risky. Subtle changes in signal characteristics and key transitions may be missed leading to incorrect assessment of process status. In some cases, one can attempt to extract more information from a process signal by transforming it into a domain that might help to accentuate key features of the signal. One such approach is the use of the Fourier transform (FT) to determine the frequency content of a signal. Yet, it would also be interesting to understand if the frequency characteristics of the signal may be changing in time. In the next section (Section 6.1), wavelet transform (WT) will be briefly introduced to show how both frequency and temporal features of a signal can be localized. This will be followed in Section 6.2 by a discussion on signal denoising based on wavelet transforms and a hybrid strategy that can also deal with outliers that are often present in real-world signals. The subsequent sections will introduce methods that help model process signals for later use in monitoring applications. First, in Section 6.3, triangular episodes will be discussed as a means of obtaining a symbolic representation from an otherwise numerical time series data. A more elaborate strategy based on a doubly stochastic model, namely the hidden Markov models (HMMs), will be introduced in Section 6.4.2 and the chapter will conclude in Section 6.5 with the modeling of wavelet coefficients using the HMM paving the way for a trend analysis methodology to be introduced in Chapter 7.
