Uncertainty Quantification of Power Spectrum and Spectral Moments Estimates Subject to Missing Data
Publication: ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 3, Issue 4
Abstract
In this paper, the challenge of quantifying the uncertainty in stochastic process spectral estimates based on realizations with missing data is addressed. Specifically, relying on relatively relaxed assumptions for the missing data and on a kriging modeling scheme, utilizing fundamental concepts from probability theory, and resorting to a Fourier-based representation of stationary stochastic processes, a closed-form expression for the probability density function (PDF) of the power spectrum value corresponding to a specific frequency is derived. Next, the approach is extended for also determining the PDF of spectral moments estimates. Clearly, this is of significant importance to various reliability assessment methodologies that rely on knowledge of the system response spectral moments for evaluating its survival probability. Further, utilizing a Cholesky decomposition for the PDF-related integrals kept the computational cost at a minimal level. Several numerical examples are included and compared against pertinent Monte Carlo simulations for demonstrating the validity of the approach.
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Acknowledgments
The first author is grateful for the financial support from the China Scholarship Council.
References
Au, S.-K., and Beck, J. (2001). “Estimation of small failure probabilities in high dimensions by subset simulation.” Prob. Eng. Mech., 16(4), 263–277.
Billingsley, P. (2008). Probability and measure, Wiley, New York.
Broersen, P., and Bos, R. (2006). “Time-series analysis if data are randomly missing.” IEEE Trans. Instrum. Meas., 55(1), 79–84.
Bucher, C. (2011). “Simulation methods in structural reliability.” 1st Int. Conf. of Maritime Technology and Engineering (MARTECH), CRC Press, Boca Raton, FL.
Comerford, L., Jensen, H., Mayorga, F., Beer, M., and Kougioumtzoglou, I. (2017). “Compressive sensing with an adaptive wavelet basis for structural system response and reliability analysis under missing data.” Comput. Struct., 182, 26–40.
Comerford, L., Kougioumtzoglou, I. A., and Beer, M. (2014). “Compressive sensing based power spectrum estimation from incomplete records by utilizing an adaptive basis.” 2014 IEEE Symp. on Computational Intelligence for Engineering Solutions, IEEE, New York, 117–124.
Comerford, L., Kougioumtzoglou, I. A., and Beer, M. (2015a). “An artificial neural network approach for stochastic process power spectrum estimation subject to missing data.” Struct. Saf., 52, 150–160.
Comerford, L., Kougioumtzoglou, I. A., and Beer, M. (2015b). “On quantifying the uncertainty of stochastic process power spectrum estimates subject to missing data.” Int. J. Sustainable Mater. Struct. Syst., 2(1–2), 185–206.
Comerford, L., Kougioumtzoglou, I. A., and Beer, M. (2016). “Compressive sensing based stochastic process power spectrum estimation subject to missing data.” Probab. Eng. Mech., 44, 66–76.
Cramer, H., and Leadbetter, R. (1967). Stationary and related stochastic processes: Sample function properties and their applications, Wiley, New York.
Crandall, S. H. (1970). “First-crossing probabilities of the linear oscillator.” J. Sound Vib., 12(3), 285–299.
Eldar, Y., and Kutyniok, G. (2012). Compressed sensing: Theory and apps, Cambridge University Press, Cambridge, U.K.
Gaspar, B., Teixeira, A. P., and Guedes, S. C. (2014). “Assessment of the efficiency of kriging surrogate models for structural reliability analysis.” Probab. Eng. Mech., 37, 24–34.
Golub, G. H., and Van Loan, C. F. (1996). Matrix computations, Johns Hopkins, Baltimore.
Jia, G, and Taflanidis, A. (2013). “Kriging metamodeling for approximation of high-dimensional wave and surge responses in real-time storm/hurricane risk assessment.” Comput. Methods Appl. Mech. Eng., 261–262, 24–38.
Kaymaz, I. (2005). “Application of kriging method to structural reliability problems.” Struct. Saf., 27(2), 133–151.
Kougioumtzoglou, I. A., dos Santos, K. R., and Comerford, L. (2017). “Incomplete data based parameter identification of nonlinear and time-variant oscillators with fractional derivative elements.” Mech. Syst. Signal Process., 94, 279–296.
Kougioumtzoglou, I. A., Kong, F., Spanos, P. D., and Li, J. (2012). “Some observations on wavelets based evolutionary power spectrum estimation.” Proc., Stochastic Mechanics Conf. (SM12), Ustica, Italy, Vol. 3, 37–44.
Kougioumtzoglou, I. A., and Spanos, P. D. (2012). “An analytical wiener path integral technique for non-stationary response determination of nonlinear oscillators.” Probab. Eng. Mech., 28, 125–131.
Lomb, N. (1976). “Least-squares frequency-analysis of unequally spaced data.” Astrophys. Space Sci., 39(2), 447–462.
Lutes, L. D., and Sarkani, S. (2004). Random vibrations: Analysis of structural and mechanical systems, Elsevier, Burlington, MA.
Newland, D. E. (1993). An introduction to random vibrations, spectral and wavelet analysis, Dover Publications, Mineola, NY.
Papoulis, A., and Pillai, S. U. (2002). Probability, random variables and stochastic processes, McGraw-Hill, New York.
Politis, N. P., Giaralis, A., and Spanos, P. D. (2007). “Joint time-frequency representation of simulated earthquake accelerograms via the adaptive chirplet transform.” Proc., Computational Stochastic Mechanics, City Univ., London, 549–557.
Priestley, B. (1982). Spectral analysis and time series, Academic Press, London.
Qian, S. (2002). Introduction to time-frequency and wavelet transforms, Prentice Hall, Upper Saddle River, NJ.
Roberts, D., Lehar, J., and Dreher, J. (1987). “Time-series analysis with clean. I: Derivation of a spectrum.” Astron. J., 93(4), 968–989.
Scargle, J. (1982). “Studies in astronomical time-series analysis. II: Statistical aspects of spectral-analysis of unevenly spaced data.” Astrophys. J., 263(2), 835–853.
Shinozuka, M., and Deodatis, G. (1991). “Simulation of stochastic processes by spectral representation.” Appl. Mech. Rev., 44(4), 191–203.
Spanos, P. D. (1978). “Non-stationary random vibration of a linear structure.” Int. J. Solids Struct., 14(10), 861–867.
Spanos, P. D., Beer, M., and Red-Horse, J. (2007). “Karhunen-Loève expansion of stochastic processes with a modified exponential covariance kernel.” J. Eng. Mech., 773–779.
Spanos, P. D., and Failla, G. (2004). “Evolutionary spectra estimation using wavelets.” J. Eng. Mech., 952–960.
Spanos, P. D., and Kougioumtzoglou, I. A. (2014). “Survival probability determination of nonlinear oscillators subject to evolutionary stochastic excitation.” J. Appl. Mech., 81(5), 051016-1–051016-9.
Stein, M. (1999). Interpolation of spatial data: Some theory for kriging, Springer, New York.
Vanmarcke, E. H. (1972). “Properties of spectral moments with applications to random vibration.” J. Eng. Mech. Div., 98(2), 425–446.
Vanmarcke, E. H. (1975). “On the distribution of the first passage time for normal stationary random process.” J. Appl. Mech., 42(1), 215–220.
Wang, Y., Li, J., and Stoica, P. (2005). Spectral analysis of signals, the mising data case, Morgan & Caypool, San Rafael, CA.
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Published In
ASCE-ASME Journal of Risk and Uncertainty in Engineering Systems, Part A: Civil Engineering
Volume 3 • Issue 4 • December 2017
Copyright
©2017 American Society of Civil Engineers.
History
Received: May 2, 2016
Accepted: Apr 18, 2017
Published online: Jul 22, 2017
Published in print: Dec 1, 2017
Discussion open until: Dec 22, 2017
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