In this paper, we propose a methodology for construction of confidence interval on mean values with interval data for input variable in uncertainty analysis and design optimization problems. The construction of confidence interval with interval data is known as a combinatorial optimization problem. Finding confidence bounds on the mean with interval data has been generally considered an NP hard problem, because it includes a search among the combinations of multiple values of the variables, including interval endpoints. In this paper, we present efficient algorithms based on continuous optimization to find the confidence interval on mean values with interval data. With numerical experimentation, we show that the proposed confidence bound algorithms are scalable in polynomial time with respect to increasing number of intervals. Several sets of interval data with different numbers of intervals and type of overlap are presented to demonstrate the proposed methods. As against the current practice for the design optimization with interval data that typically implements the constraints on interval variables through the computation of bounds on mean values from the sampled data, the proposed approach of construction of confidence interval enables more complete implementation of design optimization under interval uncertainty. 1. Introduction Uncertainty quantification plays an increasingly important role in assessing the performance, safety, and reliability of complex physical systems, even in the absence of adequate amount of experimental data for many applications. Uncertainty in engineering analysis and design arises from several different sources and must be quantified with accuracy for further analysis. Sources of uncertainty may be divided into two types: aleatory and epistemic. Aleatory uncertainty is irreducible. Examples include phenomena that exhibit natural variation like environmental conditions (temperature, wind speed, etc.). Manufacturing variations due to limited precision in tools and processes also result in this type of uncertainty. In contrast, epistemic uncertainty results from a lack of knowledge about the system, or due to approximations in the system behavior models, or due to limited or subjective (e.g., expert opinion that results in interval data) data; it can be reduced as more information about the system or the variables is obtained. Epistemic uncertainty can be viewed in two ways. It can be defined with reference to a stochastic but poorly known quantity [1] or with reference to a fixed but poorly known physical quantity [2]. The
References
[1]
C. Baudrit and D. Dubois, “Practical representations of incomplete probabilistic knowledge,” Computational Statistics and Data Analysis, vol. 51, no. 1, pp. 86–108, 2006.
[2]
J. C. Helton, J. D. Johnson, and W. L. Oberkampf, “An exploration of alternative approaches to the representation of uncertainty in model predictions,” Reliability Engineering and System Safety, vol. 85, no. 1–3, pp. 39–71, 2004.
[3]
S. Ferson, V. Kreinovich, J. Hajagos, W. Oberkampf, and L. Ginzburg, “Experimental uncertainty estimation and statistics for data having interval uncertainty,” Sandia National Laboratories Technical Report SAND 2007-0939, Albuquerque, NM, USA, 2007.
[4]
X. Du, A. Sudjianto, and B. Huang, “Reliability-based design with the mixture of random and interval variables,” Journal of Mechanical Design, vol. 127, no. 6, pp. 1068–1076, 2005.
[5]
K. Zaman, S. Rangavajhala, M. P. McDonald, and S. Mahadevan, “A probabilistic approach for representation of interval uncertainty,” Reliability Engineering and System Safety, vol. 96, no. 1, pp. 117–130, 2011.
[6]
W. L. Oberkampf, J. C. Helton, C. A. Joslyn, S. F. Wojtkiewicz, and S. Ferson, “Challenge problems: uncertainty in system response given uncertain parameters,” Reliability Engineering & System Safety, vol. 85, no. 1–3, pp. 11–19, 2004.
[7]
S. Ferson and J. G. Hajagos, “Arithmetic with uncertain numbers: rigorous and (often) best possible answers,” Reliability Engineering and System Safety, vol. 85, no. 1–3, pp. 135–152, 2004.
[8]
X. Du, “Unified uncertainty analysis by the first order reliability method,” Journal of Mechanical Design, vol. 130, no. 9, Article ID 091401, 10 pages, 2008.
[9]
J. Guo and X. Du, “Sensitivity analysis with mixture of epistemic and aleatory uncertainties,” AIAA Journal, vol. 45, no. 9, pp. 2337–2349, 2007.
[10]
J. Guo and X. Du, “Reliability sensitivity analysis with random and interval variables,” International Journal for Numerical Methods in Engineering, vol. 78, no. 13, pp. 1585–1617, 2009.
[11]
M. McDonald and S. Mahadevan, “Uncertainty quantification and propagation for multidisciplinary system analysis,” in Proceedings of the 12th AIAA/ISSMO Multidisciplinary Analysis and Optimization Conference (MAO '08), Victoria, Canada, September 2008.
[12]
K. Zaman, M. McDonald, and S. Mahadevan, “Probabilistic framework for uncertainty propagation with both probabilistic and interval variables,” Journal of Mechanical Design, vol. 133, no. 2, Article ID 021010, 14 pages, 2011.
[13]
K. Zaman, M. McDonald, S. Mahadevan, and L. Green, “Robustness-based design optimization under data uncertainty,” Structural and Multidisciplinary Optimization, vol. 44, no. 2, pp. 183–197, 2011.
[14]
K. Zaman and S. Mahadevan, “Robustness-based design optimization of multidisciplinary system under epistemic uncertainty,” AIAA Journal, vol. 51, no. 5, pp. 1021–1031, 2013.
[15]
A. Haldar and S. Mahadevan, Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, New York, NY, USA, 2000.
[16]
X. Zhou and S. Gao, “One-sided confidence intervals for means of positively skewed distributions,” The American Statistician, vol. 54, no. 2, pp. 100–104, 2000.
[17]
N. J. Johnson, “Modified tests and confidence intervals for asymmetrical populations,” Journal of the American Statistical Association, vol. 73, no. 363, pp. 536–544, 1978.
[18]
P. Hall, “On the removal of skewness by transformation,” Journal of the Royal Statistical Society B: Methodological, vol. 54, no. 1, pp. 221–228, 1992.
[19]
X. H. Zhou and P. Dinh, “Nonparametric confidence intervals for the one-and two-sample problems,” Biostatistics, vol. 6, no. 2, pp. 187–200, 2005.
[20]
V. Kreinovich, L. Longpré, P. Patangay, S. Ferson, and L. Ginzburg, “Outlier detection under interval uncertainty: algorithmic solvability and computational complexity,” Reliable Computing, vol. 11, no. 1, pp. 59–76, 2005.
[21]
R. Fletcher, Practical Methods of Optimization, John Wiley & Sons, New York, NY, USA, 2nd edition, 1987.
[22]
E. R. Panier and A. L. Tits, “On combining feasibility, descent and superlinear convergence in inequality constrained optimization,” Mathematical Programming, vol. 59, no. 1–3, pp. 261–276, 1993.
[23]
S. Ferson, C. A. Joslyn, J. C. Helton, W. L. Oberkampf, and K. Sentz, “Summary from the epistemic uncertainty workshop: consensus amid diversity,” Reliability Engineering and System Safety, vol. 85, no. 1–3, pp. 355–369, 2004.
