Abstract
In the world of connected automated objects, increasingly rich and structured data are collected daily (positions, environmental variables, etc.). In this work, we are interested in the characterization of the variability of the trajectories of one of these objects (robot, drone, or delivery droid for example) along a particular path from irregularly sampled data in time and space. To do so, we model the position of the considered object by a random field indexed in time, whose distribution we try to estimate (for risk analysis for example). This distribution being by construction concentrated on an unknown curve, two phases are proposed for its reconstruction: a phase of identification of this curve, by clustering and polynomial smoothing techniques, then a phase of statistical inference of the random field orthogonal to this curve, by spectral methods and kernel reconstructions. The efficiency of the proposed approach, both in terms of computation time and reconstruction quality, is illustrated on several numerical applications.
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References
Aldenderfer MS, Blashfield RK (1984) Cluster analysis. SAGE Publications, Inc
Donoho DL, Grimes C (2003) Hessian eigenmaps: Locally linear embedding techniques for high-dimensional data. Proc Natl Acad Sci 100(10):5591–5596. https://doi.org/10.1073/pnas.1031596100
Duong T, Cowling A, Koch I, Wand MP (2008) Feature significance for multivariate kernel density estimation. Comput Stat Data Anal 52(9):4225–4242
Ghanem R, Spanos PD (1990) Polynomial Chaos in Stochastic Finite Elements. J Appl Mech 57(1):197–202
Ghanem R, Spanos PD (2003) Stochastic finite elements: a spectral approach, Rev. Dover Publications, New York
Haberstich C, Nouy A, Perrin G (2022) Boosted optimal weighted least-squares. Math Comput
Hastie T, Tibshirani R, Friedman J (2001) The elements of statistical learning. Springer Series in Statistics. Springer, New York, NY, USA
Le Maître O, Knio OM (2010) Spectral methods for uncertainty quantification. Springer, Dordrecht
Maday Y, Nguyen NC, Patera AT, Pau SH (2009) A general multipurpose interpolation procedure: the magic points. Commun Pure Appl Anal 8(1):383–404
Mak S, Joseph VR (2018) Support points. Ann Stat 46:2562–2592
Nürnberger G (1989) Approximation by spline functions. Springer, Heidelberg
Perrin G, Soize C, Duhamel D, Funfschilling C (2012) Identification of polynomial chaos representations in high dimension from a set of realizations. SIAM J Sci Comput 34(6):2917–2945
Perrin G, Soize C, Ouhbi N (2018) Data-driven kernel representations for sampling with an unknown block dependence structure under correlation constraints. J Comput Stat Data Anal 119:139–154
Rokach L, Maimon O (2005). In: Maimon O, Rokach L (eds) Clustering Methods. Springer, Boston, pp 321–352
Santner TJ, Williams BJ, Notz WI (2003) The design and analysis of computer experiments. Springer, New York
Scott DW, Sain SR (2005) Multidimensional density estimation. In: Rao CR, Wegman EJ, Solka JL (eds) Data Mining and Data Visualization, vol 24. Handbook of Statistics. Elsevier, Amsterdam, pp 229–261
Shorack GR, Wellner JA (2009) Empirical processes with applications to statistics. Society for Industrial and Applied Mathematics, Boston, MA, Philadelphia
Silverman BW (1986) Density estimation for statistics and data analysis. In: Monographs on Statistics and Applied Probability, vol. 37, p 120
Soize C (2010) Identification of high-dimension polynomial chaos expansions with random coefficients for non-Gaussian tensor-valued random fields using partial and limited experimental data. Comput Methods Appl Mech Eng 199(33–36):2150–2164
Soize C (2017) Uncertainty quantification. Interdisciplinary applied mathematics, vol 47. Springer, Cham, pp 1–327
Soize C, Ghanem R (2022) Probabilistic learning on manifolds (PLoM) with partition. Int J Numer Meth Eng 123(1):268–290
Teymur O, Gorham J, Riabiz M, Oates CJ (2021) Optimal quantisation of probability measures using maximum mean discrepancy. Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS) 130
van der Maaten L, Hinton G (2008) Visualizing data using t-sne. J Mach Learn Res 9(86):2579–2605
Wand MP, Jones MC (1995) Kernel smoothing. Encycl Stat Behav Sci 60(60):212
Wu J (2012) Cluster analysis and K-means clustering: an introduction. Springer Berlin Heidelberg, Berlin, Heidelberg, pp 1–16
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Perrin, G., Soize, C. Reconstruction of Random Fields Concentrated on an Unknown Curve using Irregularly Sampled Data. Methodol Comput Appl Probab 26, 9 (2024). https://doi.org/10.1007/s11009-024-10079-w
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DOI: https://doi.org/10.1007/s11009-024-10079-w