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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.

Alzheimer's disease (AD), as one of the most common neurodegenerative diseases, is characterized by the loss of neuronal dysfunction and death resulting in progressive cognitive impairment. The main histopathological hallmark of AD is the accumulation and deposition of misfolded Aβ peptide as amyloid plaques, however the precise role of Aβ toxicity in the disease pathogenesis is still unclear. Moreover, at early stages of the disease the important clinical features of the disorder, in addition to memory loss, are the disruptions of circadian rhythms and spatial disorientation. In the present work I first studied the role of Aβ toxicity by comparing the findings of genome-wide association studies in sporadic AD with the results of an RNAi screen in a transgenic C. elegans model of Aβ toxicity. The initial finding was that none of the human orthologues of these worm genes are associated with risk for sporadic Alzheimer’s disease, indicating that Aβ toxicity in the worm model may not be equivalent to sporadic AD. Nevertheless, comparing the first degree physical interactors (+1 interactome) of the GWAS and worm screen-derived gene products have uncovered 4 worm genes that have a +1 interactome overlap with the GWAS genes that is larger than one would expect by chance. Three of these genes form a chaperonin complex and the fourth gene codes for actin, a major substrate of the same chaperonin. Next I have evaluated the circadian disruptions in AD by developing a new system to simultaneously monitor the oscillations of the peripheral molecular clock and behavioural rhythms in single Drosophila. Experiments were undertaken on wild- type and Aβ-expressing flies. The results indicate the robustness of the peripheral clock is not correlated with the robustness of the circadian sleep and locomotor behaviours, indicating that the molecular clock does not directly drive behaviour. This is despite period length correlations that indicate that the underlying molecular mechanisms that generate both molecular and behavioural rhythms are the same. Rhythmicity in Aβ-expressing flies is worse than in controls. I further investigated the mechanism of spatial orientation in Drosophila. It was established that in the absence of visual stimuli the flies use self-motion cues to orientate themselves within the tubes and that in a Drosophila model of Aβ toxicity this control function is disrupted.