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Prediction of spatial functional random processes: comparing functional and spatio-temporal kriging approaches
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0003-1098-0076
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2019 (English)In: Stochastic environmental research and risk assessment (Print), ISSN 1436-3240, E-ISSN 1436-3259, Vol. 33, no 10, p. 1699-1719Article in journal (Refereed) Published
Abstract [en]

We present and compare functional and spatio-temporal (Sp.T.) kriging approaches to predict spatial functional random processes (which can also be viewed as Sp.T. random processes). Comparisons with respect to computational time and prediction performance via functional cross-validation is evaluated, mainly through a simulation study but also on a real data set. We restrict comparisons to Sp.T. kriging versus ordinary kriging for functional data (OKFD), since the more flexible functional kriging approaches pointwise functional kriging (PWFK) and the functional kriging total model coincide with OKFD in several situations. Here we formulate conditions under which we show that OKFD and PWFK coincide. From the simulation study, it is concluded that the prediction performance of the two kriging approaches in general is rather equal for stationary Sp.T. processes. However, functional kriging tends to perform better for small sample sizes, while Sp.T. kriging works better for large sizes. For non-stationary Sp.T. processes, with a common deterministic time trend and/or time varying variances and dependence structure, OKFD performs better than Sp.T. kriging irrespective of the sample size. For all simulated cases, the computational time for OKFD was considerably lower compared to those for the Sp.T. kriging methods.

Place, publisher, year, edition, pages
Springer, 2019. Vol. 33, no 10, p. 1699-1719
Keywords [en]
Functional kriging, Prediction, Spatial functional random processes, Spatio-temporal kriging
National Category
Probability Theory and Statistics
Identifiers
URN: urn:nbn:se:umu:diva-164996DOI: 10.1007/s00477-019-01705-yISI: 000491084300003Scopus ID: 2-s2.0-85069204664OAI: oai:DiVA.org:umu-164996DiVA, id: diva2:1368678
Funder
Swedish Research Council, 340-2013-5203Available from: 2019-11-08 Created: 2019-11-08 Last updated: 2020-10-05Bibliographically approved
In thesis
1. Non-parametric methods for functional data
Open this publication in new window or tab >>Non-parametric methods for functional data
2020 (English)Doctoral thesis, comprehensive summary (Other academic)
Alternative title[sv]
Icke-parametriska metoder för funktionella data
Abstract [en]

In this thesis we develop and study non-parametric methods within three major areas of functional data analysis: testing, clustering and prediction. The thesis consists of an introduction to the field, a presentation and discussion of the three areas, and six papers.

In Paper I, we develop a procedure for testing for group differences in functional data. In case of significant group differences, the test procedure identifies which of the groups that significantly differ, and also the parts of the domain they do so, while controlling the type I error of falsely rejecting the null hypothesis. In Paper II, the methodology introduced in Paper I is applied to knee kinematic curves from a one-leg hop for distance to test for differences within and between three groups of individuals (with and without knee deficits). It was found that two of the groups differed in their knee kinematics. We also found that the individual kinematic patterns differed between the two legs in one of the groups. In Paper III, we test for group differences in three groups with respect to joint kinematics from a vertical one-leg hop using a novel method that allows accounting for multiple joints at the same time. The aim of Paper III, as one of few within the field of biomechanics, is to illustrate how different choices prior to the analysis can result in different contrasting conclusions. Specifically, we show how the conclusions depend on the choice of type of movement curve, the choice of leg for between-group comparisons and the included joints.

In Paper IV, we present a new non-parametric clustering method for dependent functional data, the double clustering bagging Voronoi method. The objective of the method is to identify latent group structures that slowly vary over domain and give rise to different frequency patterns of functional data object types. The method uses a bagging strategy based on random Voronoi tessellations in which local representatives are formed and clustered. Combined with the clustering method, we also propose a multiresolution approach which allows identification of latent structures at different scales. A simulated dataset is used to illustrate the method's potential in finding stable clusters at different scales. The method is also applied to varved lake sediment data with the aim of reconstructing the climate over the past 6000 years, at different resolutions. In Paper V, we expand and modify the bagging strategy used in Paper IV, by considering different methods of generating the tessellations and clustering the local representatives of the tessellations. We propose new methods for clustering dependent categorical data (e.g., labelled functional data) along a one-dimensional domain, which we also compare in a simulation study. 

In Paper VI, two kriging approaches to predict spatial functional processes are compared, namely functional kriging and spatio-temporal kriging. A simulation study is conducted to compare their prediction performance and computational times. The overall results show that prediction performance is about the same for stationary spatio-temporal processes while functional kriging works better for non-stationary spatio-temporal processes. Furthermore, the computational time for (ordinary) kriging for functional data, was considerably lower than spatio-temporal kriging. Conditions are also formulated under which it is proved that the two functional kriging methods: ordinary kriging for functional data and pointwise functional kriging coincide.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2020. p. 32
Series
Research report in mathematical statistics, ISSN 1653-0829 ; 72/20
Keywords
functional data analysis, testing, clustering, prediction, inference, bagging Voronoi strategy, kriging, dependency
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-175594 (URN)978-91-7855-374-7 (ISBN)978-91-7855-375-4 (ISBN)
Public defence
2020-10-30, Aula Biologica, Biologihuset, Umeå, 10:15 (English)
Opponent
Supervisors
Available from: 2020-10-09 Created: 2020-10-05 Last updated: 2020-10-06Bibliographically approved

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Strandberg, JohanSjöstedt de Luna, Sara

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