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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.

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.

Nonparametric bagging clustering methods are studied and compared to identify latent structures from a sequence of dependent categorical data observed along a one-dimensional (discrete) time domain. The frequency of the observed categories is assumed to be generated by a (slowly varying) latent signal, according to latent state-specific probability distributions. The bagging clustering methods use random tessellations (partitions) of the time domain and clustering of the category frequencies of the observed data in the tessellation cells to recover the latent signal, within a bagging framework. New and existing ways of generating the tessellations and clustering are discussed and combined into different bagging clustering methods. Edge tessellations and adaptive tessellations are the new proposed ways of forming partitions. Composite methods are also introduced, that are using (automated) decision rules based on entropy measures to choose among the proposed bagging clustering methods. The performance of all the methods is compared in a simulation study. From the simulation study it can be concluded that local and global entropy measures are powerful tools in improving the recovery of the latent signal, both via the adaptive tessellation strategies (local entropy) and in designing composite methods (global entropy). The composite methods are robust and overall improve performance, in particular the composite method using adaptive (edge) tessellations.

Background: Psychophysiological consequences often persist following musculoskeletal trauma and can result in vastly decreased quality of life. Re-injury anxiety is particularly common among individuals following anterior cruciate ligament (ACL) injury. Existing assessments of re-injury anxiety are, however, restricted to subjective suboptimal questionnaires, which may result in under-reporting and thus poorer injury management. We propose a novel approach to objectively quantify arousal response to movement-related anxiety. A new experimental paradigm was implemented to induce and record a conditioned electrophysiological response to a sudden perturbation, experienced to be potentially injurious.

Objective: To explore the feasibility of detecting anxiety-associated electrocortical response and to evaluate its discriminative ability between asymptomatic individuals and those who had experienced an ACL injury.

Methods: Physically-active asymptomatic persons and individuals post-ACL reconstruction stood blindfolded on a perturbation platform capable of generating high-acceleration translations (1.5 m/s2). Auditory stimuli were repeatedly presented in four-second intervals, as either low- or high-frequency tones. Half of the high-frequency tones were followed 1.5 seconds later by a destabilizing perturbation in one of eight randomized directions. The two tone conditions were thus termed ‘Neutral’ and ‘Anxiety’, as the high-frequency tone was intended to invoke an arousal response in anticipation of a potential perturbation. Event-related potentials (ERP) were computed for nine electrodes by averaging 100 Neutral and 100 Anxiety trials. Significant ERP components were identified using functional data analysis. Paired difference-waves’ amplitudes (Neutral - Anxiety) were compared between groups.

Results: ERP correlates of anxiety were detected for both groups in frontal and central midline locations, with an observable contingent negative variation (CNV) from 500 ms post-stimulus in Anxiety compared with Neutral trials. This ERP component is reflective of a threat-induced arousal response, associated with attention and expectancy of an anxiety-relevant event. Preliminary data indicate no group differences in CNV amplitudes.

Conclusions: Objective evaluation of an arousal response to movement-related anxiety was found to be feasible, resulting in a threat-induced CNV. Further investigation will elucidate the discriminative power of such an approach to differentiate between individuals with high and low re-injury anxiety, as well as potential associations with existing patient-reported outcome measures.

Psychological consequences often persist following musculoskeletal trauma and can result in vastly decreased quality of life. Re-injury anxiety is reported to hinder return to sports and can itself be a precursor for secondary injuries. Existing assessments of re-injury anxiety are restricted to subjective questionnaires, which may result in under-reporting and thus poorer injury management. In the current study, we introduced an experimental approach to objectively quantify movement-related anxiety using a threat-conditioning paradigm. We aimed to explore the feasibility of such an approach among non-injured persons.

Ten physically-active individuals stood blindfolded on a platform capable of generating high-acceleration translations in eight different directions. Consecutive auditory stimuli were presented (four-second intervals), as either high- (conditioned stimulus; CS⁺) or low- (neutral stimulus; CS^–) tones. Half of the CS⁺ trials were followed by a perturbation in a pseudo-random order. Event-related potentials were computed for nine electrodes by averaging 100 X CS^– and 100 X CS⁺ trials. Significant latencies for CS^– – CS⁺ comparisons were identified using interval-wise testing. Mean-amplitudes for significant intervals were used to detect a channel effect.

Large negative CS+ waveforms were observed from 302-627ms post-stimulus and continuing until the end of the trials, most prominently over frontal and central midline locations (p ≤ 0.025). This effect, inferred as a contingent negative variation wave (CNV), may be reflective of threat-induced arousal response.

Our test paradigm was found to be feasible, with a CNV suggested as a potential biomarker for re-injury anxiety. Further validation is needed, as well as exploring the discriminative power of such an approach between individuals with and without previous injury.

Musculoskeletal trauma often leads to lasting psychological impacts stemming from concerns of future injuries. Often referred to as kinesiophobia or re-injury anxiety, such concerns have been shown to hinder return to physical activity and are believed to increase the risk for secondary injuries. Screening for re-injury anxiety is currently restricted to subjective questionnaires, which are prone to self-report bias. We introduce a novel approach to objectively identify electrocortical activity associated with the threat of destabilising perturbations. We aimed to explore its feasibility among non-injured persons, with potential future implementation for screening of re-injury anxiety. Twenty-three participants stood blindfolded on a translational balance perturbation platform. Consecutive auditory stimuli were provided as low (neutral stimulus [CS^–]) or high (conditioned stimulus [CS⁺]) tones. For the main experimental protocol (Protocol I), half of the high tones were followed by a perturbation in one of eight unpredictable directions. A separate validation protocol (Protocol II) requiring voluntary squatting without perturbations was performed with 12 participants. Event-related potentials (ERP) were computed from electroencephalography recordings and significant time-domain components were detected using an interval-wise testing procedure. High-amplitude early contingent negative variation (CNV) waves were significantly greater for CS⁺ compared with CS– trials in all channels for Protocol I (> 521-800ms), most prominently over frontal and central midline locations (P ≤ 0.001). For Protocol II, shorter frontal ERP components were observed (541-609ms). Our test paradigm revealed electrocortical activation possibly associated with movement-related fear. Exploring the discriminative validity of the paradigm among individuals with and without self-reported re-injury anxiety is warranted.

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.

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.

Three-dimensional human motion analysis provides in-depth understanding in order to optimize sports performance or rehabilitation following disease or injury. Recent developments of statistical methods for functional data allow for novel ways to analyze often complex biomechanical data. Even so, for such methods as well as for traditional well-established statistical methods, the interpretations of the results may be influenced by analysis choices made prior to the analysis. We evaluated the consequences of three such choices when comparing one-leg vertical hop (OLVH) performance in individuals who had ruptured their anterior cruciate ligament (ACL), to that of asymptomatic controls, and also athletes. Kinematic data were analyzed using a statistical approach for functional data, targeting entire curve data. This was done not only for one joint at a time but also for multiple lower limb joints and movement planes simultaneously using a multi-aspect methodology, testing for group differences while also accounting for covariates. We present the results of when an individual representative curve out of three available was either: (1) a mean curve (Mean), (2) a curve from the highest hop (Max), or (3) a curve describing the variability (Var), as a representation of performance stability. We also evaluated choice of sample leg comparison; e.g., ACL-injured leg compared to either the dominant or non-dominant leg of asymptomatic groups. Finally, we explored potential outcome effects of different combinations of included joints. There were slightly more pronounced group differences when using Mean compared to Max, while the specifics of the observed differences depended on the outcome variable. For Var there were less significant group differences. Generally, there were more disparities throughout the hop movement when comparing the injured leg to the dominant leg of controls, resulting in e.g., group differences for trunk and ankle kinematics, for both Mean and Max. When the injured leg was instead compared to the non-dominant leg of controls, there were trunk, hip and knee joint differences. For a more stringent comparison, we suggest considering to compare the injured leg to the non-dominant leg. Finally, the multiple-joint analyses were coherent with the single-joint analyses. The direct effects of analysis choices can be explored interactively by the reader in the Supplementary Material. To summarize, the choices definitively have an impact on the interpretation of a hop test results commonly used in rehabilitation following knee injuries. We therefore strongly recommend well-documented methodological analysis choices with regards to comparisons and representative values of the measures of interests.

Here we present and compare functional and spatiotemporal (Sp.T.) kriging approaches to predict spatial functional random processes, which can also be viewed as Sp.T. random processes. Comparisons are focused on Sp.T. kriging versus ordinary kriging for functional data (OKFD), since more flexible functional kriging approaches like pointwise functional kriging and functional kriging total model coincide with OKFD in several situations. Prediction performance is evaluated via functional cross-validation on simulated data as well as on a Canadian weather data set. The two kriging approaches perform in many cases rather equal for stationary Sp.T. processes. For nonstationary Sp.T. processes, OKFD performs better than Sp.T. kriging. The computational time for OKFD is considerably lower compared to those for the Sp.T. kriging methods.

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.