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Spatial Statistics for Distributional Data in Bayes Spaces: From Object-Oriented Kriging to the Analysis of Warping Functions

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Advances in Compositional Data Analysis

Abstract

In the presence of increasingly massive and heterogeneous spatial data, geostatistical modeling of distributional observations plays a key role. Choosing the “right” embedding space for these data is of paramount importance for their statistical processing, to account for their nature and inherent constraints. The Bayes space theory is a natural embedding space for (spatial) distributional data and was successfully applied in varied settings. The aim of this work is to review the state-of-the-art methods for spatial dependence modeling and prediction of distributional data, while shedding light on the strong links between Compositional Data Analysis, Functional Data Analysis, and, more generally, Object-Oriented Data Analysis, in the context of spatial statistics. We propose extensions of these methods to the multivariate setting, and discuss the applicability of the Bayes space approach to the spatial modeling of phase variability in Functional Data Analysis.

This work is dedicated to Vera Pawlowsky-Glahn whose work is far reaching, and inspiring for an entire generation of researchers.

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References

  • J. Aitchison, The statistical analysis of compositional data. J. R. Stat. Soc. Ser. B (Methodol.) 44(2), 139–177 (1982)

    Google Scholar 

  • J. Aitchison, Algebraic Methods in Statistics, chapter Simplicial inference. M. Viana and D. Richards edition (American Mathematical Society, New York, 2001)

    Google Scholar 

  • J.P. Chilès, P. Delfiner, Geostatistics: Modeling Spatial Uncertainty (Wiley, New York, 1999)

    Book  Google Scholar 

  • N. Cressie, Statistics for Spatial Data (Wiley, New York, 1993)

    Book  Google Scholar 

  • P. Delicado, Functional k-sample problem when data are density functions. Comput. Stat. 22, 391–410 (2007)

    Article  MathSciNet  Google Scholar 

  • P. Delicado, Dimensionality reduction when data are density functions. Comput. Stat. Data Anal. 55(1), 401–420 (2011)

    Article  MathSciNet  Google Scholar 

  • J.J. Egozcue, J.L. Díaz-Barrero, V. Pawlowsky-Glahn, Hilbert space of probability density functions based on Aitchison geometry. Acta Math. Sinica, Engl. Ser. 22(4), 1175–1182 (2006)

    Article  MathSciNet  Google Scholar 

  • J.J. Egozcue, V. Pawlowsky-Glahn, Evidence functions: a compositional approach to information. SORT: Stat. Oper. Res. Trans. 42(2), 101–124 (2018)

    Google Scholar 

  • J.J. Egozcue, V. Pawlowsky-Glahn, R. Tolosana-Delgado, M.I. Ortego, K.G. van den Boogaart, Bayes spaces: use of improper distributions and exponential families. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 107(2), 475–486 (2013)

    Google Scholar 

  • J.J. Egozcue, V. Pawlowsky-Glahn, M. Templ, K. Hron, Independence in contingency tables using simplicial geometry. Commun. Stat. Theory Methods 44, 3978–3996 (2015)

    Article  MathSciNet  Google Scholar 

  • O. Grujic, A. Menafoglio, G. Yang, J. Caers, Cokriging for multivariate Hilbert space valued random fields: application to multi-fidelity computer code emulation. Stoch. Env. Res. Risk Assess. 32, 1955–1971 (2018)

    Article  Google Scholar 

  • P.Z. Hadjipantelis, J.A.D. Aston, H.G. Müller, J.P. Evans, Unifying amplitude and phase analysis: a compositional data approach to functional multivariate mixed-effects modeling of Mandarin Chinese. J. Am. Stat. Assoc. 110(510), 545–559 (2015)

    Article  MathSciNet  Google Scholar 

  • C. Happ, F. Scheipl, A.A. Gabriel, S. Greven, A general framework for multivariate functional principal component analysis of amplitude and phase variation. Stat 8(e220), 1–12 (2019)

    MathSciNet  Google Scholar 

  • L. Horváth, P. Kokoszka, Inference for Functional Data with Applications. Springer Series in Statistics (Springer, 2012)

    Google Scholar 

  • K. Hron, A. Menafoglio, M. Templ, K. Hr\(\mathop {\rm u}\limits ^{\circ }\)zová, P. Filzmoser, Simplicial principal component analysis for density functions in Bayes spaces. Comput. Stat. Data Anal. 94, 330–350 (2016)

    Google Scholar 

  • K. Hron, J. Machalová, A. Menafoglio, Bivariate densities in Bayes spaces: orthogonal decomposition and spline representation. Technical report, Politecnico di Milano (2020). MOX-report 83/2020

    Google Scholar 

  • S. Kurtek, A. Srivastava, W. Wu, Signal estimation under random time-warpings and nonlinear signal alignment, in NIPS (2011)

    Google Scholar 

  • J.S. Marron, A.M. Alonso, Overview of object oriented data analysis. Biom. J. 56(5), 732–753 (2014)

    Article  MathSciNet  Google Scholar 

  • J.S. Marron, J.O. Ramsay, L.M. Sangalli, A. Srivastava, Functional data analysis of amplitude and phase variation. Stat. Sci. 30(4), 468–484 (2015)

    Article  MathSciNet  Google Scholar 

  • A. Menafoglio, G. Petris, Kriging for Hilbert-space valued random fields: the operatorial point of view. J. Multivar. Anal. 146, 84–94 (2016). https://doi.org/10.1016/j.jmva.2015.06.012

    Article  MathSciNet  MATH  Google Scholar 

  • A. Menafoglio, P. Secchi, Statistical analysis of complex and spatially dependent data: a review of object oriented spatial statistics. Eur. J. Oper. Res. 258(2), 401–410 (2017)

    Article  MathSciNet  Google Scholar 

  • A. Menafoglio, P. Secchi, O2s2: a new venue for computational geostatistics. Appl. Comput. Geosci. 2, 100007 (2019)

    Article  Google Scholar 

  • A. Menafoglio, P. Secchi, M. Dalla Rosa, A Universal Kriging predictor for spatially dependent functional data of a Hilbert Space. Electron. J. Stat. 7, 2209–2240 (2013)

    Google Scholar 

  • A. Menafoglio, A. Guadagnini, P. Secchi, A Kriging approach based on Aitchison geometry for the characterization of particle-size curves in heterogeneous aquifers. Stoch. Env. Res. Risk Assess. 28(7), 1835–1851 (2014)

    Article  Google Scholar 

  • A. Menafoglio, O. Grujic, J. Caers, Universal kriging of functional data: trace-variography vs cross-variography? Application to forecasting in unconventional shales. Spat. Stat. 15, 39–55 (2016a)

    Article  MathSciNet  Google Scholar 

  • A. Menafoglio, A. Guadagnini, P. Secchi, Stochastic simulation of soil particle-size curves in heterogeneous aquifer systems through a Bayes space approach. Water Resour. Res. 52(8), 5708–5726 (2016b)

    Article  Google Scholar 

  • A. Menafoglio, P. Secchi, A. Guadagnini, A class-kriging predictor for functional compositions with application to particle-size curves in heterogeneous aquifers. Math. Geosci. 48, 463–485 (2016c)

    Article  MathSciNet  Google Scholar 

  • A. Menafoglio, G. Gaetani, P. Secchi, Random domain decompositions for object-oriented kriging over complex domains. Stoch. Env. Res. Risk Assess. 32, 3421–3437 (2018a)

    Article  Google Scholar 

  • A. Menafoglio, M. Grasso, P. Secchi, B.M. Colosimo, Profile monitoring of probability density functions via simplicial functional PCA with application to image data. Technometrics 60(4), 497–510 (2018b)

    Article  MathSciNet  Google Scholar 

  • A. Menafoglio, D. Pigoli, P. Secchi, Kriging riemannian data via random domain decompositions. Technical report, Politecnico di Milano (2018c). MOX-report 64/2018

    Google Scholar 

  • V. Pawlowsky, On spurious spatial covariance between variables of constant sum. Sciences de la Terre série informatique 21, 107–113 (1984)

    Google Scholar 

  • V. Pawlowsky, Cokriging of regionalized compositions. Math. Geol. 21(5), 513–521 (1989)

    Article  Google Scholar 

  • V. Pawlowsky-Glahn, A. Buccianti, Compositional Data Analysis. Theory and Applications (Wiley, 2011)

    Google Scholar 

  • V. Pawlowsky-Glahn, J.J. Egozcue, Geometric approach to statistical analysis in the symplex. Stoch. Env. Res. Risk Assess. 15, 384–398 (2001)

    Article  Google Scholar 

  • V. Pawlowsky-Glahn, J.J. Egozcue, BLU estimators and compositional data. Math. Geol. 34(3), 259–274 (2002)

    Article  MathSciNet  Google Scholar 

  • V. Pawlowsky-Glahn, J.J. Egozcue, R. Tolosana-Delgado, Modelling and Analysis of Compositional Data (Wiley, 2015)

    Google Scholar 

  • A. Pini, A. Stamm, S. Vantini, Hotelling’s T2 in separable Hilbert spaces. J. Multivar. Anal. 167, 284–305 (2018)

    Article  Google Scholar 

  • J. Ramsay, B. Silverman, Functional Data Analysis, 2nd edn. (Springer, New York, 2005)

    Google Scholar 

  • L.M. Sangalli, P. Secchi, S. Vantini, V. Vitelli, K-mean alignment for curve clustering. Comput. Stat. Data Anal. 54(5), 1219–1233 (2010)

    Article  MathSciNet  Google Scholar 

  • A. Srivastava, E.P. Klassen, Functional and Shape Data Analysis (Springer, New York, 2016)

    Book  Google Scholar 

  • A. Srivastava, I. Jermyn, S. Joshi, Riemannian analysis of probability density functions with applications in vision, in IEEE, editor, 2007 IEEE Conference on Computer Vision and Pattern Recognition, vol. 44, pp. 1–8 (2007)

    Google Scholar 

  • R. Talská, A. Menafoglio, J. Machalová, K. Hron, E. Fišerová, Compositional regression with functional response. Comput. Stat. Data Anal. 123, 66–85 (2018)

    Google Scholar 

  • R. Talská, A. Menafoglio, K. Hron, J.J. Egozcue, J. Palarea-Albaladejo, Weighting the domain of probability densities in functional data analysis. Stat (2020). https://doi.org/10.1002/sta4.283

    Article  MathSciNet  Google Scholar 

  • R. Tolosana-Delgado, V. Pawlowsky-Glahn, J.J. Egozcue, Indicator kriging without order relation violations. Math. Geosci. 40(3), 327–347 (2008)

    Article  MathSciNet  Google Scholar 

  • R. Tolosana-Delgado, K.G. van den Boogaart, V. Pawlowsky-Glahn, Geostatistics for Compositions, pp. 73–86, Pawlowsky-Glahn & Buccianti edition (Wiley, 2011)

    Google Scholar 

  • R. Tolosana-Delgado, U. Mueller, K.G. van den Boogaart, Geostatistics for compositional data: an overview. Math. Geosci. 51, 485–526 (2019)

    Article  MathSciNet  Google Scholar 

  • R.D. Tuddenham, M.M. Snyder, Physical growth of California boys and girls from birth to eighteen years. Univ. Calif. Publ. Child Dev. 1, 183–364 (1954)

    Google Scholar 

  • K.G. van den Boogaart, J.J. Egozcue, V. Pawlowsky-Glahn, Bayes Hilbert spaces. Aust. N. Z. J. Stat. 56, 171–194 (2014)

    Article  MathSciNet  Google Scholar 

  • K.G. van den Boogaart, J.J. Egozcue, V. Pawlowsky-Glahn, Bayes linear spaces. SORT 34(2), 201–222 (2010)

    MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. MOX, Department of Mathematics, Politecnico di Milano, Milano, Italy

    Alessandra Menafoglio

Authors
  1. Alessandra Menafoglio
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Correspondence to Alessandra Menafoglio .

Editor information

Editors and Affiliations

  1. Institute of Statistics and Mathematical Methods in Economics, TU Wien, Vienna, Austria

    Peter Filzmoser

  2. Department of Mathematical Analysis and Applications of Mathematics, Palacký University, Olomouc, Czech Republic

    Karel Hron

  3. Department of Computer Science, Applied Mathematics and Statistics, University of Girona, Girona, Spain

    Josep Antoni Martín-Fernández

  4. Biomathematics and Statistics Scotland, Edinburgh, UK

    Javier Palarea-Albaladejo

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Menafoglio, A. (2021). Spatial Statistics for Distributional Data in Bayes Spaces: From Object-Oriented Kriging to the Analysis of Warping Functions. In: Filzmoser, P., Hron, K., Martín-Fernández, J.A., Palarea-Albaladejo, J. (eds) Advances in Compositional Data Analysis. Springer, Cham. https://doi.org/10.1007/978-3-030-71175-7_11

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