2.5.2 Complex networks

multivariate statistical process control methods, and several authors have reported

Bakshi [28, 29] has proposed the use of a nonlinear multiscale principal component analysis methodology for process monitoring and fault detection based on multilevel wavelet decomposition and nonlinear component extraction by the use of input-training neural networks. In this case, wavelets are first used to decompose the data into different scales, after which PCA was applied to the reconstituted time series data. Choi et al. [30] have proposed nonlinear multiscale multivariate monitoring of dynamic processes based on kernel PCA, while Xuemin and Xiaogang [31] have proposed an integrated multiscale approach where kernel PCA is used on measured process signals decomposed with wavelets and have also proposed a

With SSA, the time series is first embedded into a p-dimensional space known as the trajectory matrix. Singular value decomposition is then applied to decompose the trajectory matrix into a sum of elementary matrices [32–34], each of which is

Subsequently, the elementary matrices that contribute to the norm of the original matrix are grouped, with each group giving an approximation of the original matrix. Finally, the smoothed approximations or modes of the time series are recovered by diagonal averaging of the elementary matrices obtained from

decomposing the trajectory matrix. Although SSA is a linear method, it can readily

autoassociative neural networks. Nonetheless, it has not been used widely in statistical process monitoring as yet, although some studies have provided promising

Table 1 gives a summary of multiscale methods that have been considered in

Phase space methods rely on the embedding of the data in a so-called phase space, by the use of delayed vector methods, that is, <sup>y</sup><sup>∈</sup> <sup>R</sup><sup>N</sup>�<sup>1</sup> ! <sup>X</sup> <sup>∈</sup> <sup>R</sup>ð Þ� <sup>N</sup>�mþ<sup>1</sup> <sup>m</sup> <sup>¼</sup> ½ � xð Þt ; xð Þ t � k …xð Þ t � k mð Þ � 1 . Embedding can also be done by the use of principal components or singular value decomposition of <sup>X</sup> <sup>∈</sup> <sup>R</sup>ð Þ� <sup>N</sup>�mþ<sup>1</sup> <sup>m</sup>, where <sup>k</sup> <sup>¼</sup> 1 and m is comparatively large. In the latter case, the scores of the eigenvectors would represent an orbit or attractor with some geometrical structure, depending on the frequencies with which different regions of the phase space are visited. The topology

Methodology Comment References

Different variants have been proposed [2, 36, 48]

[37–47]

Wavelets Variable decomposition with wavelets before building PCA

be extended to nonlinear forms, such as kernel-based SSA or SSA with

process monitoring schemes over the last two decades.

models

Data preprocessing methodologies for multiscale process monitoring.

successful applications thereof [24–27].

Time Series Analysis - Data, Methods, and Applications

similarity factor to identify fault patterns.

2.4.2 Singular spectrum analysis

associated with a process mode.

results [2, 35, 36].

2.5 Phase space methods

Singular spectrum analysis

Table 1.

8

2.4.1 Wavelets

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 model to detect process disturbances.
