# zbMATH — the first resource for mathematics

Importance sampling the union of rare events with an application to power systems analysis. (English) Zbl 1416.62088
The authors consider importance sampling for estimating the probability $$\mu$$ of a union of $$J$$ rare events defined by a random variable $$x$$, denoted by $$\mu=\mathrm{Prob.} (\bigcup_{j=1}^J H_j)$$. The method described as ‘at least one rare event’ (ALOE) consists of choosing a rare event at random and then taking samples from the system given that the random event occurs. For each such sample, the total number of events is recorded. Denote by $$\bar\mu$$ the upper bound for $$\sum_{j=1}^J \mathrm{Prob.}(H_j)\geq\mu$$. The ALOE estimate $$\hat\mu$$ is derived and it is shown that $$\operatorname{variance}(\hat\mu)\leq(J+(1/J)-2)\mu^2/4n$$ when the system is sampled $$n$$ times and the coefficient of variation $\operatorname{c.v.}(\hat\mu)\leq \{(J+(1/J)-2)/4n\}^{1/2}.$ The motivation for this work is drawn from power system reliability, when the phase differences between connected nodes have a joint normal distribution and the rare events occur whenever there are large phase differences. The random inputs here are fluctuating demands by users and irregular production as in wind farms. The methodology uses a simpler linear Direct Current (DC) model for an electric grid rather than a complicated Alternating Current (AC) model. Further as stated, normality is assumed to model randomness. Numerical examples are provided to compare ALOE to ‘pmvnorm’ described in R-package ‘mvtnorm’.

##### MSC:
 62D05 Sampling theory, sample surveys 62P30 Applications of statistics in engineering and industry; control charts
##### Software:
AS 24; MATPOWER; mvtnorm; R
Full Text:
##### References:
 [1] Adler, R. J., J. Blanchet, and J. Liu (2008). Efficient simulation for tail probabilities of Gaussian random fields. In, Winter Simulation Conference, 2008, pp. 328–336. IEEE. [2] Adler, R. J., J. H. Blanchet, and J. Liu (2012). Efficient Monte Carlo for high excursions of Gaussian random fields., The Annals of Applied Probability22(3), 1167–1214. · Zbl 1251.60031 [3] Ahn, D. and K.-K. Kim (2018). Efficient simulation for expectations over the union of half-spaces., ACM Transactions on Modeling and Computer Simulation (TOMACS)28(3), 23. [4] Asmussen, S. and P. W. Glynn (2007)., Stochastic simulation: algorithms and analysis, Volume 57. Springer Science & Business Media. · Zbl 1126.65001 [5] Botev, Z. I. and P. L’Ecuyer (2015). Efficient probability estimation and simulation of the truncated multivariate student-t distribution. In, Winter Simulation Conference (WSC), 2015, pp. 380–391. IEEE. [6] Botev, Z. I., M. Mandjes, and A. Ridder (2015). Tail distribution of the maximum of correlated gaussian random variables. In, Proceedings of the 2015 Winter Simulation Conference, pp. 633–642. IEEE Press. [7] Chertkov, M., F. Pan, and M. G. Stepanov (2011). Predicting failures in power grids: The case of static overloads., IEEE Transactions on Smart Grid2(1), 162–172. [8] Chertkov, M., M. Stepanov, F. Pan, and R. Baldick (2011). Exact and efficient algorithm to discover extreme stochastic events in wind generation over transmission power grids. In, 2011 50th IEEE Conference on Decision and Control and European Control Conference, pp. 2174– 2180. [9] Cornuet, J., J.-M. Marin, A. Mira, and C. P. Robert (2012). Adaptive multiple importance sampling., Scandinavian Journal of Statistics39(4), 798–812. · Zbl 1319.62059 [10] Cranley, R. and T. N. L. Patterson (1976). Randomization of number theoretic methods for multiple integration., SIAM Journal of Numerical Analysis13(6), 904–914. · Zbl 0354.65016 [11] Cunningham, S. W. (1969). Algorithm AS 24: From normal integral to deviate., Journal of the Royal Statistical Society. Series C18(3), 290–293. [12] Elvira, V., L. Martino, D. Luengo, and M. F. Bugallo (2015a). Efficient multiple importance sampling estimators., IEEE Signal Processing Letters22(10), 1757–1761. [13] Elvira, V., L. Martino, D. Luengo, and M. F. Bugallo (2015b). Generalized multiple importance sampling., arXiv preprint arXiv:1511.03095. [14] Fliscounakis, S., P. Panciatici, F. Capitanescu, and L. Wehenkel (2013). Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, and corrective actions., IEEE Transactions on Power Systems28(4), 4909–4917. [15] Frigessi, A. and C. Vercellis (1985). An analysis of Monte Carlo algorithms for counting problems., Calcolo22(4), 413–428. · Zbl 0604.68048 [16] Genz, A. (2004). Numerical computation of rectangular bivariate and trivariate normal and t probabilities., Statistics and Computing14(3), 251–260. [17] Genz, A. and F. Bretz (2009)., Computation of Multivariate Normal and $$t$$ Probabilities. Berlin: Springer-Verlag. · Zbl 1204.62088 [18] Genz, A., F. Bretz, T. Miwa, X. Mi, F. Leisch, F. Scheipl, and T. Hothorn (2017)., mvtnorm: Multivariate Normal and t Distributions. R package version 1.0-6. [19] Hesterberg, T. C. (1988)., Advances in importance sampling. Ph. D. thesis, Stanford University. [20] Kahn, H. and A. Marshall (1953). Methods of reducing sample size in Monte Carlo computations., Journal of the Operations Research Society of America1(5), 263–278. [21] Karp, R. M. and M. Luby (1983). Monte-Carlo algorithms for enumeration and reliability problems. In, Foundations of Computer Science, 1983, 24th Annual Symposium on, pp. 56–64. IEEE. · Zbl 0596.90033 [22] Kersulis, J., I. Hiskens, M. Chertkov, S. Backhaus, and D. Bienstock (2015, June). Temperature-based instanton analysis: Identifying vulnerability in transmission networks. In, 2015 IEEE Eindhoven PowerTech, pp. 1–6. [23] Lafortune, E. P. and Y. D. Willems (1993). Bidirectional path tracing. In, Proceedings of CompuGraphics, pp. 95–104. [24] L’Ecuyer, P., M. Mandjes, and B. Tuffin (2009). Importance sampling and rare event simulation. In G. Rubino and B. Tuffin (Eds.), Rare event simulation using Monte Carlo methods, pp. 17–38. Chichester, UK: John Wiley & Sons. · Zbl 1165.62027 [25] Liu, J. S. (2001)., Monte Carlo strategies in scientific computing. New York: Springer. · Zbl 0991.65001 [26] Miwa, T., A. Hayter, and S. Kuriki (2003). The evaluation of general non-centred orthant probabilities., Journal of the Royal Statistical Society, Series B65(1), 223–234. · Zbl 1063.62082 [27] Naiman, D. Q. and C. E. Priebe (2001). Computing scan statistic p values using importance sampling, with applications to genetics and medical image analysis., Journal of Computational and Graphical Statistics10(2), 296–328. [28] Niederreiter, H. (1972). On a number-theoretical integration method., Aequationes Math8(3), 304–311. · Zbl 0252.65023 [29] Owen, A. B. (2013)., Monte Carlo theory, methods and examples. [30] Owen, A. B. and Y. Zhou (2000). Safe and effective importance sampling., Journal of the American Statistical Association95(449), 135–143. · Zbl 0998.65003 [31] Priebe, C. E., D. Q. Naiman, and L. M. Cope (2001). Importance sampling for spatial scan analysis: computing scan statistic p-values for marked point processes., Computational statistics & data analysis35(4), 475–485. · Zbl 1080.62538 [32] R Core Team (2015)., R: A Language and Environment for Statistical Computing. Vienna, Austria: R Foundation for Statistical Computing. [33] Sauer, P. W. and J. P. Christensen (1984). Active linear DC circuit models for power system analysis., Electric machines and power systems9(2–3), 103–112. [34] Shi, J., D. O. Siegmund, and B. Yakir (2007). Importance sampling for estimating p values in linkage analysis., Journal of the American Statistical Association102(479), 929–937. · Zbl 05564421 [35] Stott, B., J. Jardim, and O. Alsaç (2009). DC power flow revisited., IEEE Transactions on Power Systems24(3), 1290–1300. [36] Veach, E. and L. Guibas (1994, June 13–15). Bidirectional estimators for light transport. In, 5th Annual Eurographics Workshop on Rendering, pp. 147–162. [37] Yang, J., F. Alajaji, and G. Takahara (2014). A short survey on bounding the union probability using partial information. Technical report, University of, Toronto. · Zbl 1376.60049 [38] Zimmerman, R. D., C. E. Murillo-Sánchez, and R. J. Thomas (2011). MATPOWER: steady-state operations, planning, and analysis tools for power systems research and education., IEEE Transactions on power systems26(1), 12–19.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.