We investigate various methods for testing whether two groups of curves are statistically significantly different, with the motivation to apply the techniques to the analysis of data arising from designed experiments. We propose a set of tests based on pairwise differences between individual curves. Our objective is to compare the power and robustness of a variety of tests, including a collection of permutation tests, a test based on the functional principal components scores, the adaptive Neyman test and the functional F test. We illustrate the application of these tests in the context of a designed 2^4 factorial experiment with a case study using data provided by NASA. We apply the methods for comparing curves to this factorial data by dividing the data into two groups by each effect (A, B, . . . , ABCD) in turn. We carry out a large simulation study investigating the power of the tests in detecting contamination, location, and shift effects on unimodal and monotone curves. We conclude that the permutation test using the mean of the pairwise differences in L1 norm has the best overall power performance and is a robust test statistic applicable in a wide variety of situations. The advantage of using a permutation test is that it is an exact, distribution-free test that performs well overall when applied to functional data. This test may be extended to more than two groups by constructing test statistics based on averages of pairwise differences between curves from the different groups and, as such, is an important building-block for larger experiments and more complex designs.

We investigate various methods for testing whether two groups of curves are statistically significantly different, with the motivation to apply the techniques to the analysis of data arising from designed experiments. We propose a set of tests based on pairwise differences between individual curves. Our objective is to compare the power and robustness of a variety of tests, including a collection of permutation tests, a test based on the functional principal components scores, the adaptive Neyman test and the functional F test. We illustrate the application of these tests in the context of a designed 2^4 factorial experiment with a case study using data provided by NASA. We apply the methods for comparing curves to this factorial data by dividing the data into two groups by each effect (A, B, . . . , ABCD) in turn. We carry out a large simulation study investigating the power of the tests in detecting contamination, location, and shift effects on unimodal and monotone curves. We conclude that the permutation test using the mean of the pairwise differences in L1 norm has the best overall power performance and is a robust test statistic applicable in a wide variety of situations. The advantage of using a permutation test is that it is an exact, distribution-free test that performs well overall when applied to functional data. This test may be extended to more than two groups by constructing test statistics based on averages of pairwise differences between curves from the different groups and, as such, is an important building-block for larger experiments and more complex designs.

This research investigates the spectral reflectance characteristics of soil and vegetation using multi-angular and single view hyperspectral data. The question of the thesis is ???How much information can be obtained from multi-angular hyperspectral remote sensing in comparison with single view angle hyperspectral remote sensing of soil and vegetation???? This question is addressed by analysing multi-angular and single view angle hyperspectral remote sensing using data from the field, airborne and space borne hyperspectral sensors. Spectral mapping, spectral indices and Functional Data Analysis (FDA) are used to analyse the data. Spectral mapping has been successfully used to distinguish features of soil and cotton with hyperspectral data. Traditionally, spectral mapping is based on collecting endmembers of pure pixels and using these as training areas for supervised classification. There are, however, limitations in the use of these algorithms when applied to multi-angular images, as the reflectance of a single ground unit will differ at each angle. Classifications using six-class endmembers identified using single angle imagery were assessed using multi-angular Compact High Resolution Imaging Spectrometer (CHRIS) imagery, as well as a set of vegetation indices. The results showed no significant difference between the angles. Low nutrient content in the soil produced lower vegetation index values, and more nutrients increased the index values. This research introduces FDA as an image processing tool for multi-angular hyperspectral imagery of soil and cotton, using basis functions for functional principal component analysis (fPCA) and functional linear modelling. FDA has advantages over conventional statistical analysis because it does not assume the errors in the data are independent and uncorrelated. Investigations showed that B-splines with 20-basis functions was the best fit for multi-angular soil spectra collected using the spectroradiometer and the satellite mounted CHRIS. Cotton spectra collected from greenhouse plants using a spectrodiometer needed 30-basis functions to fit the model, while 20-basis functions were sufficient for cotton spectra extracted from CHRIS. Functional principal component analysis (fPCA) of multi-angular soil spectra show the first fPCA explained a minimum of 92.5% of the variance of field soil spectra for different azimuth and zenith angles and 93.2% from CHRIS for the same target. For cotton, more than 93.6% of greenhouse trial and 70.6% from the CHRIS data were explained by the first fPCA. Conventional analysis of multi-angular hyperspectral data showed significant differences exist between soil spectra acquired at different azimuth and zenith angles. Forward scan direction of zenith angle provides higher spectral reflectance than backward direction. However, most multi-angular hyperspectral data analysed as functional data show no significant difference from nadir, except for small parts of the wavelength of cotton spectra using CHRIS. There is also no significant difference for soil spectra analysed as functional data collected from the field, although there was some difference for soil spectra extracted from CHRIS. Overall, the results indicate that multi-angular hyperspectral data provides only a very small amount of additional information when used for conventional analyses.

abstract: This paper investigates a relatively new analysis method for longitudinal data in the framework of functional data analysis. This approach treats longitudinal data as so-called sparse functional data. The first section of the paper introduces functional data and the general ideas of functional data analysis. The second section discusses the analysis of longitudinal data in the context of functional data analysis, while considering the unique characteristics of longitudinal data such, in particular sparseness and missing data. The third section introduces functional mixed-effects models that can handle these unique characteristics of sparseness and missingness. The next section discusses a preliminary simulation study conducted to examine the performance of a functional mixed-effects model under various conditions. An extended simulation study was carried out to evaluate the estimation accuracy of a functional mixed-effects model. Specifically, the accuracy of the estimated trajectories was examined under various conditions including different types of missing data and varying levels of sparseness. / Dissertation/Thesis / Masters Thesis Psychology 2016

Clustering is an important analytical technique that has proven to affect human life positively through its application in cancer research, market segmentation, city planning etc. In this time of growing technological systems, mixed data has seen another face of longitudinal, directional and functional attributes which is worth paying attention to and analyzing. Previous research works on clustering relied largely on the inverse weight technique and B-spline in smoothing data and assessing the performance of various clustering algorithms. In 1971, Gower proposed a method of clustering for mixed variable types which has been extended to include functional and directional variables by Hendrickson (2014). In this study, we will do a comparative analysis of the performance of the hierarchical clustering mechanism using a simulated Functional data with mixed structure. We will adopt the Fourier basis smoothing procedure and use the Rand index (Rand 1971) and adjusted Rand index for the comparison of the various clustering algorithms.

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As a branch of statistics, functional data analysis (FDA) studies observations regarded as curves, surfaces, or other objects evolving over a continuum. Although one has seen a flourishing of methods and theories on FDA, two issues are observed. Firstly, the functional data are sampled from common time grids; secondly, methods developed only for univariate functional data are challenging to be applied to multivariate functional data. After exploring model-based fitting for regularly observed multivariate functional data, we explore new visualization tools, clustering, and multivariate functional depths for irregularly observed (sparse) multivariate functional data. The four main chapters that comprise the dissertation are organized as follows. First, median polish for functional multivariate analysis of variance (FMANOVA) is proposed with the implementation of multivariate functional depths in Chapter 2. Numerical studies and environmental datasets are considered to illustrate the robustness of median polish. Second, the sparse functional boxplot and the intensity sparse functional boxplot, as practical exploratory tools that make visualization possible for both complete and sparse functional data, are introduced in Chapter 3. These visualization tools depict sparseness characteristics in the proportion of sparseness and relative intensity of fitted sparse points inside the central region, respectively. Third, a robust distance-based robust two-layer partition (RTLP) clustering of sparse multivariate functional data is introduced in Chapter 4. The RTLP clustering is based on our proposed elastic time distance (ETD) specifically for sparse multivariate functional data. Lastly, the multivariate functional integrated depth and the multivariate functional extremal depth based on multivariate depths are proposed in Chapter 5. Global and local formulas for each depth are explored, with theoretical properties being proved and the finite sample depth estimation for irregularly observed multivariate functional data being investigated. In addition, the simplified sparse functional boxplot and simplified intensity sparse functional boxplot for visualization without data reconstruction are introduced. Together, these four extensions to multivariate functional data make them more general and of applicational interest in exploratory multivariate functional data analysis.

Patients with a history of lateral ankle sprain (LAS) may experience different levels of mechanical and/or sensorimotor deficits following their injuries. Although various factors, such as structural damage, sensorimotor adaptation, perceived instability, swelling and/or pain, can develop and perpetuate the condition of chronic ankle instability (CAI), most previous CAI research on biomechanics has considered all patients with CAI as a homogeneous group. Recent research has clustered patients with CAI into six distinct movement patterns during a maximal jump-landing/cutting task. This approach could motivate clinicians to develop appropriate rehabilitation programs for each patient with CAI depending on their movement patterns. However, evaluating patients with CAI for the quality of movement and sensorimotor deficits using a 3D motion capture system and a force plate is not easily accessible in clinical settings. PURPOSE: (i) to identify subgroups within the CAI population based on their movement patterns using inertial measurement unit (IMU) devices and (ii) to characterize each subgroup's functional movement patterns during maximal jump-landing/cutting relative to the uninjured controls. METHODS: A total of 100 patients with CAI (height = 1.76 ± 0.1 m, mass = 74.0 ± 14.9 kg) were assessed according to the Foot and Ankle Ability Measure (FAAM) (ADL: 84.3 ± 7.6%, Sport: 63.6 ± 8.6%) and the Ankle Instability Instrument (AII) (6.7 ± 1.2) and were fit into clusters based on their movement strategy during the maximal jump-landing/cutting task. A total of 21 uninjured controls (height = 1.74 ± 0.1 m, mass = 70.7 ± 13.4 kg) were compared with each cluster. Seven IMU sensors were placed on the base of the lumbar spine, lower and upper legs, and feet and participants performed 5 trials of the maximal jump-landing/cutting test. Joint kinematics in the lower extremity were collected during the task using IMU sensors. Data were reduced to functional curves; kinematic data from the sagittal and frontal planes were reduced to a single representative curve for each plane. Then, representative curves were clustered using a Bayesian clustering technique. Functional analyses of variance were used to identify between-group differences for outcome measures and describe specific movement characteristics of each subgroup. Pairwise comparison functions as well as 95% confidence interval (CI) bands were plotted to determine specific differences. If 95% CI bands did not cross the zero line, we considered the difference significant. RESULTS: Four distinct clusters were identified from the sagittal- and frontal-plane kinematic data. Specific movement patterns in each cluster compared to either uninjured controls or rest of patients with CAI were also identified. CONCLUSION: The IMUs were able to distinguish 4 clusters within the CAI population based on distinct movement patterns during a maximal jump-landing/cutting task. Thus, IMUs can be effective measuring devices to distinguish and characterize multiple distinct movement patterns without relying on a traditional 3D motion capture system. Clinicians should consider utilizing IMU devices to measure and evaluate specific movement patterns in the CAI population during multiplanar demanding tests before developing appropriate treatment interventions in clinical settings.