## Emergent behaviour in a system of industrial plants detected via manifold learning

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**Published**Nov 13, 2020

**Gueorgui Mihaylov**

**Matteo Spallanzani**

## Abstract

The efficiency behaviour of an industrial plant, part of a huge international structure of plants, is modelled as an emergent phenomenon in a complex adaptive system. The study is based on real in-service data obtained from an industrial production line monitoring system. Models of complex adaptive systems and some modern manifold learning methods are introduced in a unified formalism. The emergent behaviour is efficiently described in this setup.

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Keywords

Spectral Analysis, Spectral Clustering, manifold learning, emergent behaviour, nonlocal features

References

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Singer, A., & Hau-Tieng, W. (2011). Orientability and diffusion map. Applied and Computational Harmonic Analysis, 31(1), 44-58.

Singer, A., & Hau-tieng, W. (2012). Vector diffusion maps and the connection Laplacian. Communications on Pure and Applied Mathematics, 65, 1067–1144.

Singer, A., & Hau-Tieng,W. (2013). Spectral convergence of the connection Laplacian from random samples. preprint available at arXiv:1306.1587.

Sternberg, S. (1963). Lectures on Differential Geometry. New York Chelsea Publishing Company.

Tanenbaum, d. V., J. B., & Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2319–2323.

Xiao, W. (2016). A probabilistic machine learning approach to detect industrial plant faults. IJPHM ISSN2153-2648, 007.

Zhang, Z., & H. Zha, H. (2004). Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J. Sci. Comput., 26, 313-338.

Agricola, F. T., I., & Kassuba, M. (2008). Eigenvalue estimates for Dirac operators with parallel characteristic torsion. Differential Geometry and its Applications, 26, 613–624.

Agricola, I., & Friedrich, T. (1999). Upper bounds for the first eigenvalue of the Dirac operator on surfaces. Journal of Geometry and Physics, 30, 1–22.

Atiyah, M. (1973). Eigenvalues and Riemannian geometry. In Proceedings of the international conference on manifolds and related topics in topology (pp. 5–9).

Baird, P., & Wood, J. (Eds.). (2003). Harmonic morphisms between Riemannian manifolds. Oxford University Press. Belkin, M., & Niyogi, P. (2003). Laplacian eigenmaps for dimensionality reduction and data representation. Neural Computation, 15, 1373–1396.

Belkin, M., & Niyogi, P. (2005). Towards a theoretical foundation for Laplacian-based manifold methods. In Proceedings of the 18th annual conference on learning theory (pp. 486–500).

Berry, T., & Sauer, T. (2015). Local kernels and the geometric structure of data. in press, Applied and Computational Harmonic Analysis.

Bianconi, G., & Barabasi, A. (2001). Competition and multiscaling in evolving networks. Europhysics Letters, 54, 436–442.

Borg, I., & Groenen, P. (1997). Modern multidimensional scaling: theory and applications. Springer.

Bronstein, B. M., A.M., & Kimmel, R. (2006). Generalized multidimensional scaling: a framework for isometryinvariant partial surface matching. Proceedings of the National Academy of Sciences of the United States of America, 103, 1168–1172.

Choo, A. S. W. B. M. J. B. P., B.Y. (2016). Adaptive multiscale prognostics and health management for smart manufacturing systems. IJPHM ISSN2153-2648, 014.

Coifman, R. R., & Lafon, S. (2006). Diffusion maps. Applied and Computational Harmonic Analysis, 21, 5–30.

Demartines, P., & Herault, J. (1997). Curvilinear component analysis: a self-organizing neural network for nonlinear mapping of datasets. IEEE Transactions on Neural Networks, 8, 148–154.

deSilva, V., & Tanenbaum, J. B. (2003). Global versus local methods in nonlinear dimensionality reduction. In Advances in neural information processing systems (pp. 721–728).

Donoho, D. L., & Grimes, C. (2003). Hessian eigenmaps: new locally linear embedding techniques for highdimensional data. Proceedings of the National Academy of Sciences of the United States of America, 100, 5591–5596.

Eells, J. J., & Sampson, J. H. (1964). Harmonic mappings of Riemannian manifolds. American Journal of Mathematics, 86, 109–160.

Esposito, G. (1998). Dirac operators and spectral geometry. Cambridge University Press.

Friedrich, T. (2012). The second Dirac eigenvalue of a nearly parallel G2-manifold. Advances in Applied Clifford Algebras, 22, 301–311.

Gilkey, G. (1995). Invariance theory, the heat equation, and the Atiyah-Singer index theorem. CRC press.

Ilinski, K. (2001). Physics of finance: gauge modelling in non-equilibrium pricing. Wiley.

Jardim, M., & Leao, R. (2008). Survey on eigenvalues of the Dirac operator and geometric structures. International Mathematical Forum, 3, 49–67.

Joyce, D. (2000). Compact manifolds with special holonomy. Oxford Mathematical Monographs.

Mack, G. (2000). Universal dynamics, a unified theory of complex systems. emergence, life and death. preprint available at arXiv:hep-th/0011074.

Nash, J. (1954). C1 isometric imbeddings. Annals of Mathematics. Ng A.Y., M. I., Jordan, & Weiss, Y. (2002). On spectral clustering: analysis and an algorithm. In Advances in neural information processing systems 14 (pp. 849–856). MIT Press.

O’Donovan, B. K. O. D., P. (2016). Adaptive multi-scale prognostics and health management for smart manufacturing systems. IJPHM ISSN2153-2648, 026.

Reuter, W. F., M., & Peinecke, N. (2005). Laplace-spectra as fingerprints for shape matching. In Proceedings of the ACM symposium on solid and physical modeling (pp. 101–106).

Rosenberg, S. (1997). The laplacian on a Riemannian manifold: an introduction to analysis on manifolds. Cambridge University Press.

Sameh, A., & Wisniewski, J. (1982). A trace minimization algorithm for the generalized eigenvalue problem. SIAM Journal on Numerical Analysis, 19, 1243–1259.

Shawe-Taylor, W. C., J. S., Cristianini, N., & Kandola, J. (2005). On the eigenspectrum of the Gram matrix and the generalization error of kernel pca. IEEE Transactions on Information Theory, 51, 2510–2522.

Shi, J., & Malik, J. (1997). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22, 888–905.

Singer, A., & Hau-Tieng, W. (2011). Orientability and diffusion map. Applied and Computational Harmonic Analysis, 31(1), 44-58.

Singer, A., & Hau-tieng, W. (2012). Vector diffusion maps and the connection Laplacian. Communications on Pure and Applied Mathematics, 65, 1067–1144.

Singer, A., & Hau-Tieng,W. (2013). Spectral convergence of the connection Laplacian from random samples. preprint available at arXiv:1306.1587.

Sternberg, S. (1963). Lectures on Differential Geometry. New York Chelsea Publishing Company.

Tanenbaum, d. V., J. B., & Langford, J. C. (2000). A global geometric framework for nonlinear dimensionality reduction. Science, 290, 2319–2323.

Xiao, W. (2016). A probabilistic machine learning approach to detect industrial plant faults. IJPHM ISSN2153-2648, 007.

Zhang, Z., & H. Zha, H. (2004). Principal manifolds and nonlinear dimensionality reduction via tangent space alignment. SIAM J. Sci. Comput., 26, 313-338.

Section

Technical Papers