**8. References**


260 Principal Component Analysis

This chapter introduces online monitoring approaches of batch process to detect fine abnormal at early stage. MICA reveals more nature that occurs abnormal than MPCA. By DTW/OFA, two kinds of synchronization method, more accurate multivariate statistical models are constructed and new batch run is manipulated as much for correct monitoring. GCC method speculates the unknown data of future for MPCA/MICA well when batch process is online. However, in spite of its accuracy, the computation of MICA is more complicated than one of MPCA. It is not suggested to use the methods of synchronization if it is not serious asynchronous among the batch processes, because any method of synchronization consumes a large amount time and memory. Similarly, than other three traditional solutions, GCC needs more time of computation to compare with each other, and huge history model database. None of methods is predominant on the online monitoring of batch processes. The future work may combine the integrative approaches with SDG (Signed Direct Graph) to detect the root cause of the faults (Vedam &

The author wishes to acknowledge the assistance of Miss Lina Bai and Mr. Fuqiang Bian,

A., Kassidas; J. F., MacGregor; P. A., Taylor (1998). Synchronization of Batch Trajectories Using Dynamic Time Warping. *AIChE Journal*, Vol.44, No.4, pp. 864-875. Chang Kyoo, Yoo; Jong-Min, Lee; Peter A., Vanrolleghem; In-Beum, Lee (2004) On-line

*Chemometrics and Intelligent Laboratory Systems*, Vol.71, Issue 2, pp. 151-163. Debashis, Neogi & CoryE., Schlags (1998) Multivariate Statistical Analysis of an Emulsion

Fuqiang, Bian (2008). OFA Synchronize Method and Its Improvement for Batch Process. *Journal of Beijing Union University (Nature Sciences)*, Vol. 22, No.4, pp. 48-53. Fuqiang, Bian; Xiang, Gao; Ming Zhe, Yuan (2009). Monitoring based on improved OFA-

Hiranmayee, Vedam & Venkat, Venkatasubramanian (1999) PCA-SDG based process

Italura, F. (1975) Mimimum Prediction Residential Principle Applied to Speech Recognition.

Junghui, Chen & Hsin-hung, Chen (2006). On-line Batch Process Monitoring Using MHMTbased MPCA. *Chemical Engineering Science*, Vol.61, Issue 10, pp. 3223-3239.

Monitoring of Batch Processes Using Multiway Independent Analysis.

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monitoring and fault diagnosis. *Control Engineering Practice*, Vol.7, No.8, pp. 903-

*IEEE Trans. on Acoustics, Speech and Signal Processing*, Vol.ASSP-23, No.1, pp. 67-72. J. A., Westerhuis; T., Kourti, J. F., MacGregor (1999). Comparing Alternative Approaches for

Multivariate Statistical Analysis of Batch Process Data. *J. Chemometrics*, Vol.13, Issue

who have done some work of the simulation of the chapter.

**6. Conclusion** 

Venkatasubramanian, 1999).

3971-3979.

pp. 209-214.

3-4, pp. 397-413.

917.

**7. Acknowledgment** 

**8. References** 


**0**

**14**

Darko Dimitrov *Freie Universität Berlin*

*Germany*

**Computing and Updating Principal Components**

Efficient computation and updating of the principal components are crucial in many applications including data compression, exploratory data analysis, visualization, image processing, pattern and image recognition, time series prediction, detecting perfect and reflective symmetry, and dimension detection. The thorough overview over PCA's applications can be found for example in the textbooks by Duda et al. (2001) and Jolliffe (2002). Dynamic versions of the PCA applications, i.e., when the point set (population) changes, are of big importance and interest. Efficient solutions of those problems depend heavily on an efficient dynamic computation of the principal components (eigenvectors of the covariance matrix). Dynamic updates of variances in different settings have been studied since the sixties by Chan et al. (1979), Knuth (1998), Pébay (2008), Welford (1962) and West (1979). Pébay (2008)

The principal components of discrete point sets can be strongly influenced by point clusters (Dimitrov, Knauer, Kriegel & Rote (2009)). To avoid the influence of the distribution of the point set, often continuous sets, especially the convex hull of a point set is considered, which lead to so-called continuous PCA. Computing PCA bounding boxes (Gottschalk et al. (1996), Dimitrov, Holst, Knauer & Kriegel (2009)), or retrieval of 3D-objects (Vrani´c et al. (2001)), are

The organization and the main results presented in this chapter are as follows: In Section 2, we present a standard approach of computing principal components of discrete point set in **R***d*. In Section 3, we present closed-form solutions for efficiently updating the principal components of a set of *n* points, when *m* points are added or deleted from the point set. For both operations performed on a discrete point set in **R***d*, we can compute the new principal components in *O*(*m*) time for fixed *d*. This is a significant improvement over the commonly used approach of recomputing the principal components from scratch, which takes *O*(*n* + *m*) time. In Section 4 we consider the computation of the principal components of a dynamic continuous point set. We give closed form-solutions when the point set is a convex polytope **R**3. Solutions for the cases when the point set is the boundary of a convex polytope in **R**<sup>2</sup> or **R**3, or a convex polygon in **R**2, are presented in the appendix. Conclusion and open problems are presented

also investigated the dynamic maintenance of covariance matrices.

typical applications where continuous PCA are of interest.

**1. Introduction**

in Section 5.

**of Discrete and Continuous Point Sets**

