Expected First Occurrence Time of Uncertain Future Events in One-Dimensional Linear Systems

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Published Nov 5, 2024
David E. Acuña-Ureta
Diego I. Fuentealba-Secul Marcos E. Orchard

Abstract

The rapid advancement of machine learning algorithms has significantly enhanced tools for monitoring system health, making data-driven approaches predominant in Prognostics and Health Management (PHM). In contrast, model-based approaches have seen modest progress, as they are often constrained by the need for prior knowledge of specific governing equations, limiting their applicability to a wide range of problems. Recently, rigorous theoretical foundations have been established to extend dynamical systems theory by incorporating prognosis of uncertain events. This article leverages this formal framework to introduce and demonstrate a fundamental mathematical result for one-dimensional linear dynamical systems. The presented theorem offers an exact expression for calculating the expected time at which an event will first occur in the future. Unlike typical thresholds, this event is triggered by a hazard zone, defined as an uncertain event likelihood function over the system's state space. Applications of this theorem can be found in implementing real-time prognostic frameworks, where it is crucial to quickly estimate the magnitude of impending failures. Emphasis is placed on minimizing computational burden to facilitate prognostic decision-making.

How to Cite

Acuña-Ureta, D. E., Fuentealba-Secul, D. I., & Orchard, M. E. (2024). Expected First Occurrence Time of Uncertain Future Events in One-Dimensional Linear Systems. Annual Conference of the PHM Society, 16(1). https://doi.org/10.36001/phmconf.2024.v16i1.4116
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Keywords

Failure time, Hazard zone, Prognostics, Failure prognosis

References
Acuna-Ureta, D. E., Orchard, M. E., & Wheeler, P. (2021). Computation of time probability distributions for the occurrence of uncertain future events. Mechanical Systems and Signal Processing, 150.

Agliari, E. (2008). Exact mean first-passage time on the Tgraph. Phys. Rev. E, 77, 011128. Beichelt, F. (2001). A first-passage time problem in Reliability Theory. Stochastics and Quality Control, 16(1), 65-73.

Blake, I., & Lindsey,W. (1973). Level-crossing problems for random processes. IEEE Transactions on Information Theory, 19(3), 295-315.

Bo, L., & Lefebvre, M. (2011). Mean first passage times of two-dimensional processes with jumps. Statistics & Probability Letters, 81(8), 1183-1189.

Benichou, O., Guerin, T., & Voituriez, R. (2015). Mean first passage times in confined media: from Markovian to non-Markovian processes. Journal of Physics A: Mathematical and Theoretical, 48(16), 163001.

Domine, M. (1995). Moments of the first-passage time of a wiener process with drift between two elastic barriers. Journal of Applied Probability, 32(4), 1007-1013.

Dybiec, B., Gudowska-Nowak, E., & Hanggi, P. (2006). Lvy-Brownian motion on finite intervals: Mean first passage time analysis. Phys. Rev. E, 73, 046104.

Fink, O., Wang, Q., Svensen, M., Dersin, P., Lee, W.-J., & Ducoffe, M. (2020). Potential, challenges and future directions for deep learning in prognostics and health management applications. Engineering Applications of Artificial Intelligence, 92, 103678.

Gitterman, M. (2000). Mean first passage time for anomalous diffusion. Phys. Rev. E, 62, 6065-6070.

Gut, A. (1974). On the moments and limit distributions of some first passage times. The Annals of Probability, 2(2), 277-308.

Klein, G. (1952). Mean first-passage times of Brownian motion and related problems. Proc. R. Soc. Lond. A, 211, 431-443.

Kulkarni, V. G., & Tzenova, E. (2002). Mean first passage times in fluid queues. Operations Research Letters, 30(5), 308-318.

Latouche, G., & V., R. (1995). Expected passage times in homogeneous quasi-birth-and-death processes. Communications in Statistics. Stochastic Models, 11(1), 103- 122.

Lee, M.-L. T., & Whitmore, G. A. (2003). First hitting time models for lifetime data. In Advances in survival analysis (Vol. 23, p. 537-543). Elsevier.

Lefebvre, M. (2010). Mean first-passage time to zero for wear processes. Stochastic Models, 26(1), 46-53.

Macdonald, I. G. (1995). Symmetric functions and hall polynomials. Oxford: Clarendon Press.

Mattos, T. G., Mejıa-Monasterio, C., Metzler, R., & Oshanin, G. (2012). First passages in bounded domains: When is the mean first passage time meaningful? Phys. Rev. E, 86, 031143.

Orchard, M. E., & Vachtsevanos, G. J. (2009). A particle filtering approach for on-line fault diagnosis and failure prognosis. Trans. Inst. Meas. Control 2009, 31(3-4), 221-246.

Polizzi, N. F., Therien, M. J., & Beratan, D. N. (2016). Mean first-passage times in biology. Israel Journal of Chemistry, 56(9-10), 816-824.

Redner, S. (2001). A guide to first-passage processes. Cambridge: Cambridge University Press. Robbins, N. B. (1976). Convergence of some expected first passage times. The Annals of Probability, 4(6), 1027- 1029.

Salminen, P. (1988). On the first hitting time and the last exit time for a Brownian motion to/from a moving boundary. Advances in Applied Probability, 20(2), 411–426.

Siegert, A. J. F. (1951). On the first passage time probability problem. Physical Review, 81(4), 617-623.

Talkner, P. (1987). Mean first passage time and the lifetime of a metastable state. Z. Physik B - Condensed Matter, 68, 201–207.

Vachtsevanos, G., & Zahiri, F. (2022). Prognosis: Challenges, precepts, myths and applications. 2022 IEEE Aerospace Conference (AERO), 1-13.

Vanvinckenroye, H., & DenoÅNel, V. (2017). Average firstpassage time of a quasi-Hamiltonian mathieu oscillator with parametric and forcing excitations. Journal of Sound and Vibration, 406, 328-345.

Viete, F. (1646). Opera mathematica (F. van Schouten, Ed.). Leiden.

Wickwire, K. (1979). The expected first-passage time for a sufficient statistic arising in the Poisson disorder problem. Journal of Applied Probability, 16(2), 274-286.

Zanga, A., Ozkirimli, E., & Stella, F. (2022). A survey on causal discovery: Theory and practice. International Journal of Approximate Reasoning, 151, 101-129.

Zhang, Z., Qi, Y., Zhou, S., Xie,W., & Guan, J. (2009). Exact solution for mean first-passage time on a pseudo fractal scale-free web. Phys. Rev. E, 79, 021127.
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Technical Research Papers