2.4 Multiscale methods

Multiscale methods can be seen as a complementary approach preceding feature extraction from the time series. In this case, each process variable is extended or replaced by different versions of the variable at different scales. For example, with multiscale PCA, wavelets are used to decompose the process variables under scrutiny into multiple scale representations before application of PCA to detect and identify faulty conditions in process operations. In this way, autocorrelation of variables is implicitly accounted for, resulting in a more sensitive method for detecting process anomalies. Multiscale PCA constitutes a promising extension of

series are not independent, while nonstationarity means that the parameters governing a process change over time, for example, the mean, covariance or other higher order statistics. Therefore, in principle at least, these systems cannot be

series data are unlabeled. Where supervised methods are used, features are

the time series.

6

Figure 2.

2. Unsupervised feature extraction

extracted based on their ability to predict some label, such as the future evolution of

In principle, any low-dimensional representation of the time series data would

constitute a feature set, that is, the data in the time series window, X ∈ R<sup>N</sup>x<sup>M</sup> containing N measurements of the M plant variables with time lagged copies of

Broadly speaking, methodologies dealing with dynamic process systems are all aimed at dealing with the issues arising from the time dependence of the data. Essentially, these approaches are based on the analysis of a segment of the time series data, as captured by a fixed or a moving window, as indicated in Figure 2. The time series segment amounts to observation of the process over a time interval, and the window length should be sufficient to capture the dynamics of the systems. Dynamic process monitoring can be as simple as monitoring the mean or the variance of a signal, in which case, a test window as shown in Figure 2 would not be required, and model maintenance would not be an issue. In more complex systems, as could be characterized by large multivariate sets of signals or high-dimensional signals, such as streaming video or hyperspectral data, feature extraction is often model-based. That is, a model derived from the data in the base window is applied to the data in the test window. For example, principal component models can be used for this purpose. Where models are used and the nature of the signals changes as a result of process drift, recalibration of the models need to be done either at regular intervals or episodically, that is, when a change occurs. Some models, such as those based on principal and independent components can be updated recursively, as discussed in more detail in Sections 4 and 5. Alternatively, the model is updated ab initio at regular intervals. Moreover, most feature extraction methods are unsupervised, that is, the time

treated directly by the methods dealing with steady state systems.

Dynamic process monitoring as an extension of steady state approaches.

Time Series Analysis - Data, Methods, and Applications

multivariate statistical process control methods, and several authors have reported successful applications thereof [24–27].

of this attractor is a direct result of the underlying dynamics of the system being observed, and the changes in the topology are usually an indication of a change in the parameters or structure of the system dynamics. Therefore, descriptors of the attractor geometry can serve as sensitive diagnostic variables to monitor abnormal

Process Fault Diagnosis for Continuous Dynamic Systems Over Multivariate Time Series

For process monitoring purposes, the data captured in a moving window are embedded in a phase space, and descriptors such as correlation dimension [49–51], Lyapunov exponents, and information entropy [49] have been proposed to monitor deterministic or potentially chaotic systems. These approaches have not found widespread adoption in the industry yet, since the reliability of the descriptors may

Process circuits or plants lend themselves naturally to representation by networks and process monitoring schemes can exploit this. For example, Cai et al. [52] have essentially considered a lagged trajectory matrix in the form of a complex network, whereby the variables and their lagged versions served as network vertices. The edges of the network were determined by means of kernel canonical correlation analysis (a nonlinear approach to correlation relationships between sets of variables). Features were extracted from the variables based on the dynamic average degree of each vertex in the network. A standard PCA model, as described in Section 1.1 was consequently used to monitor the process. Case studies have indicated that this could yield considerable improvement in the reliability of the

Any given sequence of numbers or time series can be characterized by similarity matrix containing measures of similarity (e.g., Euclidean distances) between all pair-wise points in the time series. A recurrence matrix is generated by binary quantization of the similarity matrix, based on a user specified threshold value. This thresholded matrix can be portrayed graphically as a recurrence plot, amenable to qualitative interpretation. The recurrence matrix, consisting of zeros and ones, can also be used as a basis to extract features that are representative of the dynamic behavior of the time series. This approach is widely referred to as recurrence quantification analysis, and in process engineering, it has mainly been used in the description of electrochemical phenomena and corrosion [53–58], but in principle

More recent extensions of recurrence quantification analysis have been considered by using the unthresholded similarity matrix as a basis for feature extraction. This is also referred to as global, as opposed to (local) recurrence quantification described in Section 2.5.3. The resulting recurrence plot can consequently be treated as an artificial image amenable to analysis by a large variety of algorithms normally

applied to textural images, as discussed in more detail in Section 4.

system behavior.

2.5.2 Complex networks

2.5.1 Phase space attractor descriptors

DOI: http://dx.doi.org/10.5772/intechopen.85456

be compromised by high levels of signal noise.

model to detect process disturbances.

2.5.3 Local recurrence quantification analysis

has general applicability to any dynamic system.

2.5.4 Global recurrence quantification analysis

9
