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Non-parametric methods for functional data
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.ORCID iD: 0000-0003-1098-0076
2020 (English)Doctoral thesis, comprehensive summary (Other academic)Alternative title
Icke-parametriska metoder för funktionella data (Swedish)
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 [en]
functional data analysis, testing, clustering, prediction, inference, bagging Voronoi strategy, kriging, dependency
National Category
Probability Theory and Statistics
Research subject
Mathematical Statistics
Identifiers
URN: urn:nbn:se:umu:diva-175594ISBN: 978-91-7855-374-7 (print)ISBN: 978-91-7855-375-4 (electronic)OAI: oai:DiVA.org:umu-175594DiVA, id: diva2:1473189
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
List of papers
1. An inferential framework for domain selection in functional anova
Open this publication in new window or tab >>An inferential framework for domain selection in functional anova
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2014 (English)In: Contributions in infinite-dimensional statistics and related topics / [ed] Bongiorno, E.G., Salinelli, E., Goia, A., Vieu, P, Esculapio , 2014Conference paper, Published paper (Refereed)
Abstract [en]

We present a procedure for performing an ANOVA test on functional data, including pairwise group comparisons. in a Scheff´e-like perspective. The test is based on the Interval Testing Procedure, and it selects intervals where the groups significantly differ. The procedure is applied on the 3D kinematic motion of the knee joint collected during a functional task (one leg hop) performed by three groups of individuals.

Place, publisher, year, edition, pages
Esculapio, 2014
National Category
Probability Theory and Statistics
Research subject
Statistics; Physiotherapy; Mathematical Statistics
Identifiers
urn:nbn:se:umu:diva-108843 (URN)10.15651/978-88-748-8763-7 (DOI)9788874887637 (ISBN)
Conference
IWFOS, Stresa, June 19-21, 2014
Available from: 2015-09-16 Created: 2015-09-16 Last updated: 2020-10-05Bibliographically approved
2. One-leg hop kinematics 20years following anterior cruciate ligament rupture: Data revisited using functional data analysis
Open this publication in new window or tab >>One-leg hop kinematics 20years following anterior cruciate ligament rupture: Data revisited using functional data analysis
Show others...
2015 (English)In: Clinical Biomechanics, ISSN 0268-0033, E-ISSN 1879-1271, Vol. 30, no 10, p. 1153-1161Article in journal (Refereed) Published
Abstract [en]

BACKGROUND: Despite interventions, anterior cruciate ligament ruptures can cause long-term deficits. To assist in identifying and treating deficiencies, 3D-motion analysis is used for objectivizing data. Conventional statistics are commonly employed to analyze kinematics, reducing continuous data series to discrete variables. Conversely, functional data analysis considers the entire data series.

METHODS: Here, we employ functional data analysis to examine and compare the entire time-domain of knee-kinematic curves from one-leg hops between and within three groups. All subjects (n=95) were part of a long-term follow-up study involving anterior cruciate ligament ruptures treated ~20years ago conservatively with physiotherapy only or with reconstructive surgery and physiotherapy, and matched knee-healthy controls.

FINDINGS: Between-group differences (injured leg, treated groups; non-dominant leg, controls) were identified during the take-off and landing phases, and in the sagittal (flexion/extension) rather than coronal (abduction/adduction) and transverse (internal/external) planes. Overall, surgical and control groups demonstrated comparable knee-kinematic curves. However, compared to controls, the physiotherapy-only group exhibited less flexion during the take-off (0-55% of the normalized phase) and landing (44-73%) phase. Between-leg differences were absent in controls and the surgically treated group, but observed during the flight (4-22%, injured leg>flexion) and the landing (57-85%, injured leg<internal rotation) phases in the physiotherapy-only group.

INTERPRETATION: Functional data analysis identified specific functional knee-joint deviations from controls persisting 20years post anterior cruciate ligament rupture, especially when treated conservatively. This approach is suggested as a means for comprehensively analyzing complex movements, adding to previous analyses.

National Category
Physiotherapy
Identifiers
urn:nbn:se:umu:diva-111923 (URN)10.1016/j.clinbiomech.2015.08.010 (DOI)000366790400022 ()26365484 (PubMedID)2-s2.0-84959229856 (Scopus ID)
Available from: 2015-11-26 Created: 2015-11-26 Last updated: 2023-03-23Bibliographically approved
3. Analysis choices provide contrasting conclusions when evaluating jump performance: A multi-aspect inferential method applied to kinematics curves from the one-leg vertical hop in knee-injured and asymptomatic persons
Open this publication in new window or tab >>Analysis choices provide contrasting conclusions when evaluating jump performance: A multi-aspect inferential method applied to kinematics curves from the one-leg vertical hop in knee-injured and asymptomatic persons
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics Physiotherapy
Identifiers
urn:nbn:se:umu:diva-175585 (URN)
Available from: 2020-10-05 Created: 2020-10-05 Last updated: 2020-10-05
4. Multiresolution clustering of dependent functional data with application to climate reconstruction
Open this publication in new window or tab >>Multiresolution clustering of dependent functional data with application to climate reconstruction
2019 (English)In: Stat, E-ISSN 2049-1573, Vol. 8, no 1, article id e240Article in journal (Refereed) Published
Abstract [en]

We propose a new nonparametric clustering method for dependent functional data, the double clustering bagging Voronoi method. It consists of two levels of clustering. Given a spatial lattice of points, a function is observed at each grid point. In the first‐level clustering, features of the functional data are clustered. The second‐level clustering takes dependence into account, by grouping local representatives, built from the resulting first‐level clusters, using a bagging Voronoi strategy. Depending on the distance measure used, features of the functions may be included in the second‐step clustering, making the method flexible and general. Combined with the clustering method, a multiresolution approach is proposed that searches for stable clusters at different spatial scales, aiming to capture latent structures. This provides a powerful and computationally efficient tool to cluster dependent functional data at different spatial scales, here illustrated by a simulation study. The introduced methodology is applied to varved lake sediment data, aiming to reconstruct winter climate regimes in northern Sweden at different time resolutions over the past 6,000 years.

Place, publisher, year, edition, pages
John Wiley & Sons, 2019
Keywords
bagging Voronoi strategy, climate reconstruction, clustering, dependency, functional data
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-164004 (URN)10.1002/sta4.240 (DOI)000506857900010 ()2-s2.0-85081025918 (Scopus ID)
Funder
Swedish Research Council, 340-2013-5203Swedish Research Council, 2016-02763
Available from: 2019-10-11 Created: 2019-10-11 Last updated: 2023-03-24Bibliographically approved
5. Nonparametric clustering methods to identify latent structures from a sequence of dependent categorical data
Open this publication in new window or tab >>Nonparametric clustering methods to identify latent structures from a sequence of dependent categorical data
(English)Manuscript (preprint) (Other academic)
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-175587 (URN)
Available from: 2020-10-05 Created: 2020-10-05 Last updated: 2020-10-05
6. Prediction of spatial functional random processes: comparing functional and spatio-temporal kriging approaches
Open this publication in new window or tab >>Prediction of spatial functional random processes: comparing functional and spatio-temporal kriging approaches
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
Keywords
Functional kriging, Prediction, Spatial functional random processes, Spatio-temporal kriging
National Category
Probability Theory and Statistics
Identifiers
urn:nbn:se:umu:diva-164996 (URN)10.1007/s00477-019-01705-y (DOI)000491084300003 ()2-s2.0-85069204664 (Scopus ID)
Funder
Swedish Research Council, 340-2013-5203
Available from: 2019-11-08 Created: 2019-11-08 Last updated: 2020-10-05Bibliographically approved

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