Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T08:57:06.081Z Has data issue: false hasContentIssue false
  • English
  • Français

Approximation of Spatio-Temporal Random Processes Using Tensor Decomposition

Published online by Cambridge University Press:  03 June 2015

Debraj Ghosh  and
Anup Suryawanshi
Debraj Ghosh*
Affiliation:
Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India
Anup Suryawanshi*
Affiliation:
Department of Civil Engineering, Indian Institute of Science, Bangalore 560012, India
*
Corresponding author.Email:dghosh@civil.iisc.ernet.in, debrajghosh@gmail.com
Email:suryawanshianup@gmail.com
Get access

Abstract

A new representation of spatio-temporal random processes is proposed in this work. In practical applications, such processes are used to model velocity fields, temperature distributions, response of vibrating systems, to name a few. Finding an efficient representation for any random process leads to encapsulation of information which makes it more convenient for a practical implementations, for instance, in a computational mechanics problem. For a single-parameter process such as spatial or temporal process, the eigenvalue decomposition of the covariance matrix leads to the well-known Karhunen-Loève (KL) decomposition. However, for multiparameter processes such as a spatio-temporal process, the covariance function itself can be defined in multiple ways. Here the process is assumed to be measured at a finite set of spatial locations and a finite number of time instants. Then the spatial covariance matrix at different time instants are considered to define the covariance of the process. This set of square, symmetric, positive semi-definite matrices is then represented as a third-order tensor. A suitable decomposition of this tensor can identify the dominant components of the process, and these components are then used to define a closed-form representation of the process. The procedure is analogous to the KL decomposition for a single-parameter process, however, the decompositions and interpretations vary significantly. The tensor decompositions are successfully applied on (i) a heat conduction problem, (ii) a vibration problem, and (iii) a covariance function taken from the literature that was fitted to model a measured wind velocity data. It is observed that the proposed representation provides an efficient approximation to some processes. Furthermore, a comparison with KL decomposition showed that the proposed method is computationally cheaper than the KL, both in terms of computer memory and execution time.

Keywords

60G1265F9965D15Random processspatio-temporal processtensor decompositionuncertainty quantificationprobabilistic mechanics.
Type
Research Article
Information
Communications in Computational Physics , Volume 16 , Issue 1 , July 2014 , pp. 75 - 95
Copyright
Copyright © Global Science Press Limited 2014

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Ghanem, R. and Spanos, P. D.Stochastic Finite Elements: A Spectral Approach. Revised Edition, Dover Publications, 2003.Google Scholar
[2]Van Trees, H. L.Detection, Estimation and Modulation Theory, Part I. Wiley, 2001.Google Scholar
[3]Xiu, D. and Karniadakis, G. E.A new stochastic approach to transient heat conduction modeling with uncertainty. International Journal of Heat and Mass Transfer, 46:4681–4693, 2003.CrossRefGoogle Scholar
[4]Sang, H. and Huang, J. Z.A full scale approximation of covariance functions for large spatial data sets. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 74(1): 111–132, 2011.Google Scholar
[5]Yang, J., Zhang, D., and Lu, Z.Stochastic analysis of saturated-unsaturated flow in heterogeneous media by combining Karhunen-Loève expansion and perturbation method. Journal of Hydrology, 294: 18–38, 2004.CrossRefGoogle Scholar
[6]Schenk, C. A. and Schueller, G. I.Uncertainty Assessment of Large Finite Element Systems. Springer Berlin/Heidelberg, 2005.Google Scholar
[7]Stein, M. L.Space-time covariance functions. Journal of the American Statistical Association, 100(469): 310–321, 2005.Google Scholar
[8]Gneiting, T.Nonseparable, stationary covariance functions for space-time data. Journal of the American Statistical Association, 97(458): 590–600, 2002.CrossRefGoogle Scholar
[9]Kolda, T. G. and Badder, B. W.Tensor decompositions and applications. SIAM Review, 51(3): 455–500, 2009.Google Scholar
[10]De Lathauwer, L., De Moor, B., and Vandewalle, J.On the best rank-1 and rank-(R1, R2, …, Rn) approximation of higher-order tensors. SIAM Journal on Matrix Analysis and Applications, 21(4): 1324–1342, 2000.CrossRefGoogle Scholar
[11]Hitchcock, F. L.The expression of a tensor or a polyadic as a sum of products. Journal of Mathematical Physics, 6: 164–189, 1927.CrossRefGoogle Scholar
[12]Kruskal, J. B.Three-way arrays: Rank and uniqueness of trilinear decompositions, with application to arithmetic complexity and statistics. Linear Algebra and its Applications, 18: 95–138, 1977.Google Scholar
[13]Carroll, J. D. and Chang, J. J.Analysis of individual differences in multidimensional scaling via an N-way generalization of Eckart-Young decomposition. Psychometrika, 35(3): 283–319, 1970.Google Scholar
[14]Harshman, R. A.Foundations of the PARAFAC procedure: Models and conditions for an explanatory multi-modal factor analysis. UCLA Working Papers in Phonetics, Available at http://publish.uwo.ca/~harshman/wpppfac0.pdf, 16: 1–84, 1970.Google Scholar
[15]Tucker, L. R.Implications of factor analysis of three-way matrices for measurement of change. Problems in Measuring Change, Edited by CW, Harris, University of Wisconsin Press, pages 122–137, 1963.Google Scholar
[16]Tucker, L. R.Some mathematical notes on three-mode factor analysis. Psychometrika, 31(3): 279–311, 1966.Google Scholar
[17]De Lathauwer, L., De Moor, B., and Vandewalle, J.A multilinear singular value decomposition. SIAM Journal on Matrix Analysis and Applications, 21(4): 1258–1278, 2000.Google Scholar
[18]Wikle, C. K.Spatio-temporal methods in climatology. Encyclopedia of Life Support Systems (EOLSS), 2003.Google Scholar
[19]Van Trees, H. L.Detection, Estimation and Modulation Theory, Part IV. Wiley, 2002.Google Scholar
[20]Graham, M. D. and Kevrikedis, I. G.Alternative approaches to the Karhunen-Loève decomposition for model reduction and data analysis. Computers & Chemical Engineering, 20(5): 495–506, 1996.Google Scholar
[21]Bellizzi, S. and Sampaio, R.Poms analysis of randomly vibrating systems obtained from Karhunen-Loève expansion. Journal of Sound and Vibration, 297: 774–793, 2006.Google Scholar
[22]Nott, D. J. and Dunsmuir, W. T. M.Estimation of nonstationary spatial covariance structure. Biometrica, 89(4): 819–829, 2002.Google Scholar
[23]To, C. W. S.Time-dependent variance and covariance of responses of structures to nonstationary random excitations. Journal of Sound and Vibration, 93(1): 135–156, 1984.Google Scholar
[24]Das, S., Ghanem, R., and Finette, S.Polynomial chaos representation of spatio-temporal random fields from experimental measurements. Journal of Computational Physics, 228: 8726–8751, 2009.Google Scholar
[25]Ma, C.Recent developments on the construction of spatio-temporal covariance models. Stochastic Environmental Research and Risk Assessment, 22: 39–47, 2008.Google Scholar
[26]Bruno, F., Guttorp, P., Sampson, P. D., and Cocchi, D.A simple non-separable, non-stationary spatiotemporal model for ozone. Environ Ecol Stat, 16: 515–529, 2009.CrossRefGoogle Scholar
[27]Iaco, S. D., Posa, D., and Myers, D. E.Characteristics of some classes of spaceVtime covariance functions. Journal of Statistical Planning and Inference, 143: 2002–2015, 2013.Google Scholar
[28]Ma, C.Families of spatio-temporal stationary covariance models. Journal of Statistical Planning and Inference, 116: 489–501, 2003.CrossRefGoogle Scholar
[29]Ma, C.Linear combinations of space-time covariance functions and variograms. IEEE Transactions on Signal Processing, 53(3): 857–864, 2005.Google Scholar
[30]Kolovos, A., Christakos, G., Hristopulos, D. T., and Serre, M. L.Methods for generating non- separable spatiotemporal covariance models with potential environmental applications. Advances in Water Resources, 27: 815V830, 2004.Google Scholar
[31]Khoromskij, B. N. and Schwab, C.Tensor-structured Galerkin approximation of parametric and stochastic elliptic PDEs. Research Report No. 2010-04, Swiss Federal Institute of Technology Zurich, 2010.Google Scholar
[32]Bader, B. W. and Kolda, T. G. MATLAB Tensor Toolbox. http://csmr.ca.sandia.gov/~tgkolda/TensorToolbox/.Google Scholar
[33]Haslett, J. and Raftery, A. E.Space-time modelling with long-memory dependence: Assessing Ireland’s wind-power resource. Applied Statistics, 38: 1–50, 1989.CrossRefGoogle Scholar