The thesis consists of two coherent projects. The first project presents the results of evaluating salinity tolerance in barley using growth curve analysis where different growth trajectories are observed within barley families. The study of salinity tolerance in plants is crucial to understanding plant growth and productivity. Because fully-automated smarthouses with conveyor systems allow non-destructive and high-throughput phenotyping of large number of plants, it is now possible to apply advanced statistical tools to analyze daily measurements and to study salinity tolerance. To compare different growth patterns of barley variates, we use functional data analysis techniques to analyze the daily projected shoot areas. In particular, we apply the curve registration method to align all the curves from the same barley family in order to summarize the family-wise features. We also illustrate how to use statistical modeling to account for spatial variation in microclimate in smarthouses and for temporal variation across runs, which is crucial for identifying traits of the barley variates. In our analysis, we show that the concentrations of sodium and potassium in leaves are negatively correlated, and their interactions are associated with the degree of salinity tolerance. The second project studies change-points detection methods in extremes when multiple time series data are available. Motived by the scientific question of whether the chances to experience extreme weather are different in different seasons of a year, we develop a change-points detection model to study changes in extremes or in the tail of a distribution. Most of existing models identify seasons from multiple yearly time series assuming a season or a change-point location remains exactly the same across years. In this work, we propose a random effect model that allows the change-point to vary from year to year, following a given distribution. Both parametric and nonparametric methods are developed for detecting single and multiple change-points, and their performance is compared by simulation studies. The proposed method is illustrated using sea surface temperature data and the tail distributions before and after the change-point from two models, with and without random effects are compared.

Functional data analysis (FDA) is an active field of statistics, in which the primary subjects in the study are curves. My dissertation consists of two innovative applications of functional data analysis in biology. The data that motivated the research broadened the scope of FDA and demanded new methodology. I develop new nonparametric methods to make various estimations, and I focus on developing large sample theories for the proposed estimators. The first project is motivated from a colon carcinogenesis study, the goal of which is to study the function of a protein (p27) in colon cancer development. In this study, a number of colonic crypts (units) were sampled from each rat (subject) at random locations along the colon, and then repeated measurements on the protein expression level were made on each cell (subunit) within the selected crypts. In this problem, measurements within each crypt can be viewed as a function, since the measurements can be indexed by the cell locations. The functions from the same subject are spatially correlated along the colon, and my goal is to estimate this correlation function using nonparametric methods. We use this data set as an motivation and propose a kernel estimator of the correlation function in a more general framework. We develop a pointwise asymptotic normal distribution for the proposed estimator when the number of subjects is fixed and the number of units within each subject goes to infinity. Based on the asymptotic theory, we propose a weighted block bootstrapping method for making inferences about the correlation function, where the weights account for the inhomogeneity of the distribution of the unit locations. Simulation studies are also provided to illustrate the numerical performance of the proposed method. My second project is on a lipoprotein profile data, where the goal is to use lipoprotein profile curves to predict the cholesterol level in human blood. Again, motivated by the data, we consider a more general problem: the functional linear models (Ramsay and Silverman, 1997) with functional predictor and scalar response. There is literature developing different methods for this model; however, there is little theory to support the methods. Therefore, we focus more on the theoretical properties of this model. There are other contemporary theoretical work on methods based on Principal Component Regression. Our work is different in the sense that we base our method on roughness penalty approach and consider a more realistic scenario that the functional predictor is observed only on discrete points. To reduce the difficulty of the theoretical derivations, we restrict the functions with a periodic boundary condition and develop an asymptotic convergence rate for this problem in Chapter III. A more general result based on splines is a future research topic that I give some discussion in Chapter IV.

The objective of this dissertation is to develop a suitable statistical methodology for functional data analysis. Modern advanced technology allows researchers to collect samples as functional which means the ideal unit of samples is a curve. We consider each functional observation as the resulting of a digitized recoding or a realization from a stochastic process. Traditional statistical methodologies often fail to be applied to this functional data set due to the high dimensionality. Functional hypothesis testing is the main focus of my dissertation. We suggested a testing procedure to determine the significance of two curves with order restriction. This work was motivated by a case study involving high-dimensional and high-frequency tidal volume traces from the New York State Psychiatric Institute at Columbia University. The overall goal of the study was to create a model of the clinical panic attack, as it occurs in panic disorder (PD), in normal human subjects. We proposed a new dimension reduction technique by non-negative basis matrix factorization (NBMF) and adapted a one-degree of freedom test in the context of multivariate analysis. This is important because other dimension techniques, such as principle component analysis (PCA), cannot be applied in this context due to the order restriction. Another area that we investigated was the estimation of functions with constrained restrictions such as convexification and/or monotonicity, together with the development of computationally efficient algorithms to solve the constrained least square problem. This study, too, has potential for applications in various fields. For example, in economics the cost function of a perfectly competitive firm must be increasing and convex, and the utility function of an economic agent must be increasing and concave. We propose an estimation method for a monotone convex function that consists of two sequential shape modification stages: (i) monotone regression via solving a constrained least square problem and (ii) convexification of the monotone regression estimate via solving an associated constrained uniform approximation problem.

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Spectral density function (SDF) plays a critical role in spatio-temporal data analysis, where the data are analyzed in the frequency domain. Although many methods have been proposed for SDF estimation, real-world applications in many research fields, such as neuroscience and environmental science, call for better methodologies. In this thesis, we focus on the spectral density functions for time series and spatial data, develop new estimation algorithms, and use the estimators as features for clustering and classification purposes. The first topic is motivated by clustering electroencephalogram (EEG) data in the spectral domain. To identify synchronized brain regions that share similar oscillations and waveforms, we develop two robust clustering methods based on the functional data ranking of the estimated SDFs. The two proposed clustering methods use different dissimilarity measures and their performance is examined by simulation studies in which two types of contaminations are included to show the robustness. We apply the methods to two sets of resting-state EEG data collected from a male college student. Then, we propose an efficient collective estimation algorithm for a group of SDFs. We use two sets of basis functions to represent the SDFs for dimension reduction, and then, the scores (the coefficients of the basis) estimated by maximizing the penalized Whittle likelihood are used for clustering the SDFs in a much lower dimension. For spatial data, an additional penalty is applied to the likelihood to encourage the spatial homogeneity of the clusters. The proposed methods are applied to cluster the EEG data and the soil moisture data. Finally, we propose a parametric estimation method for the quantile spectrum. We approximate the quantile spectrum by the ordinary spectral density of an AR process at each quantile level. The AR coefficients are estimated by solving Yule- Walker equations using the Levinson algorithm. Numerical results from simulation studies show that the proposed method outperforms other conventional smoothing techniques. We build a convolutional neural network (CNN) to classify the estimated quantile spectra of the earthquake data in Oklahoma and achieve a 99.25% accuracy on testing sets, which is 1.25% higher than using ordinary periodograms.

Doctor of Philosophy / Department of Statistics / Suzanne Dubnicka / Christopher Vahl / Photoplethysmograph sensors use a light-based technology to sense the rate of blood flow as controlled by the heart’s pumping action. This allows for a graphical display of a patient’s pulse wave form and the description of its key features. A person’s pulse wave has been proposed as a tool in a wide variety of applications. For example, it could be used to diagnose the cause of coldness felt in the extremities or to measure stress levels while performing certain tasks. It could also be applied to quantify the risk of heart disease in the general population. In the present work, we explore its use for identity authentication. First, we visualize the pulse waves from individual patients using functional boxplots which assess the overall behavior and identify unusual observations. Functional boxplots are also shown to be helpful in preprocessing the data by shifting individual pulse waves to a proper starting point. We then employ functional analysis of variance (FANOVA) and permutation tests to demonstrate that the identities of a group of subjects could be differentiated and compared by their pulse wave forms. One of the primary tasks of the project is to confirm the identity of a person, i.e., we must decide if a given person is whom they claim to be. We used an equivalence test to determine whether the pulse wave of the person under verification and the actual person were close enough to be considered equivalent. A nonparametric bootstrap functional equivalence test was applied to evaluate equivalence by constructing point-wise confidence intervals for the metric of identity assurance. We also proposed new testing procedures, including the way of building the equivalence hypothesis and test statistics, determination of evaluation range and equivalence bands, to authenticate the identity.

A test for independence and identical distribution of functional observations is proposed in this thesis. To reduce dimension, curves are projected on the most important functional principal components. Then a test statistic based on lagged cross--covariances of the resulting vectors is constructed. We show that this dimension reduction step introduces asymptotically negligible terms, i.e. the projections behave asymptotically as iid vector--valued observations. A complete asymptotic theory based on correlations of random matrices, functional principal component expansions, and Hilbert space techniques is developed. The test statistic has chi-square asymptotic null distribution. Two inferential tests for error correlation in the functional linear model are put forward. To construct them, finite dimensional residuals are computed in two different ways, and then their autocorrelations are suitably defined. From these autocorrelation matrices, two quadratic forms are constructed whose limiting distributions are chi--squared with known numbers of degrees of freedom (different for the two forms). A test for detecting a change point in the mean of functional observations is developed. The null distribution of the test statistic is asymptotically pivotal with a well-known asymptotic distribution. A comprehensive asymptotic theory for the estimation of a change--point in the mean function of functional observations is developed. The procedures developed in this thesis can be readily computed using the R package fda. All theoretical insights obtained in this thesis are confirmed by simulations and illustrated by real life-data examples.

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.

The basic philosophy of Functional Data Analysis (FDA) is to think of the observed data functions as elements of a possibly infinite-dimensional function space. Most of the current research topics on FDA focus on advancing theoretical tools and extending existing multivariate techniques to accommodate the infinite-dimensional nature of data. This dissertation reports contributions on both fronts, where a unifying inverse regression theory for both the multivariate setting (Li 1991) and functional data from a Reproducing Kernel Hilbert Space (RKHS) prospective is developed. We proposed a functional multiple-index model which models a real response variable as a function of a few predictor variables called indices. These indices are random elements of the Hilbert space spanned by a second order stochastic process and they constitute the so-called Effective Dimensional Reduction Space (EDRS). To conduct inference on the EDRS, we discovered a fundamental result which reveals the geometrical association between the EDRS and the RKHS of the process. Two inverse regression procedures, a “slicing” approach and a kernel approach, were introduced to estimate the counterpart of the EDRS in the RKHS. Further the estimate of the EDRS was achieved via the transformation from the RKHS to the original Hilbert space. To construct an asymptotic theory, we introduced an isometric mapping from the empirical RKHS to the theoretical RKHS, which can be used to measure the distance between the estimator and the target. Some general computational issues of FDA were discussed, which led to the smoothed versions of the functional inverse regression methods. Simulation studies were performed to evaluate the performance of the inference procedures and applications to biological and chemometrical data analysis were illustrated.

Functional data analysis (FDA) is a demonstrably effective, practical, and powerful method of data analysis, yet it remains virtually unheard of outside of academic circles and has almost no exposure to industry. FDA adds to the milieu of statistical methods by treating functions of one or more independent variables as data objects, analogous to the way in which discrete points are the data objects we are familiar with in conventional statistics. The first step in functional analysis is to “functionalize” the data, or convert discrete points into a system represented most times by continuous functions. Choosing the type of functions to use is data-dependent and often straightforward – for example, Fourier series lend themselves well to periodic systems, while splines offer great flexibility in approximating more irregular trends, such as chemical spectra. This work explores the question of how B-splines can be rapidly and reliably used to denoised infrared chemical spectra, a difficult problem not only because of the many parameters involved in generating a spline fit, but also due to the disparate nature of spectra in terms of shape and noise intensity. Automated selection of spline parameters is required to support high-throughput analysis, and the heteroscedastic nature of such spectra presents challenges for existing techniques. The heuristic knot placement algorithm of Li et al. (2005) for 1D object contours is extended to spectral fitting by optimizing the denoising step for a range of spectral types and signal/noise ratios, using the following criteria: robustness to types of spectra and noise conditions, parsimony of knots, low computational demand, and ease of implementation in high-throughput settings. Pareto-optimal filter configurations are determined using simulated data from factorial experimental designs. The improved heuristic algorithm uses wavelet transforms and provides improved performance in robustness, parsimony of knots and the quality of functional regression models used to correlate real spectral data with chemical composition. In practical applications, functional principal component regression models yielded similar or significantly improved results when compared with their discrete partial least squares counterparts. / Thesis (Master, Chemical Engineering) -- Queen's University, 2012-10-01 20:18:31.119