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.

This dissertation is composed of four articles describing inference and visualization of periodic sequences. In the first article, a nonparametric method is proposed for estimating the period and values of a periodic sequence when the data are evenly spaced in time. The period is estimated by a "leave-out-one-cycle" version of cross-validation (CV) and complements the periodogram, a widely used tool for period estimation. The CV method is computationally simple and implicitly penalizes multiples of the smallest period, leading to a "virtually" consistent estimator. The second article is the multivariate extension, where we present a CV method of estimating the periods of multiple periodic sequences when data are observed at evenly spaced time points. The basic idea is to borrow information from other correlated sequences to improve estimation of the period of interest. We show that the asymptotic behavior of the bivariate CV is the same as the CV for one sequence, however, for finite samples, the better the periods of the other correlated sequences are estimated, the more substantial improvements can be obtained. The third article proposes an informative exploratory tool, the functional boxplot, for visualizing functional data, as well as its generalization, the enhanced functional boxplot. Based on the center outwards ordering induced by band depth for functional data, the descriptive statistics of a functional boxplot are: the envelope of the 50 percent central region, the median curve and the maximum non-outlying envelope. In addition, outliers can be detected by the 1.5 times the 50 percent central region empirical rule. The last article proposes a simulation-based method to adjust functional boxplots for correlations when visualizing functional and spatio-temporal data, as well as detecting outliers. We start by investigating the relationship between the spatiotemporal dependence and the 1.5 times the 50 percent central region empirical outlier detection rule. Then, we propose to simulate observations without outliers based on a robust estimator of the covariance function of the data. We select the constant factor in the functional boxplot to control the probability of correctly detecting no outliers. Finally, we apply the selected factor to the functional boxplot of the original data.

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