Hilbert Space Valued Signal Plus Noise Models: Analysis of Structural Breaks under High Dimensionality and Temporal Dependence

This thesis is concerned with change point analysis for time series, i.e. with detection of structural breaks in time-ordered, random data. This long-standing research field regained popularity over the last few years and is still undergoing, as statistical analysis in general, a transformation to high-dimensional problems. We focus on the fundamental »change in the mean« problem and provide extensions of the classical non-parametric Darling-Erdős-type cumulative sum (CUSUM) testing and estimation theory within highdimensional Hilbert space settings. In the first part we contribute to (long run) principal component based testing methods for Hilbert space valued time series under a rather broad (abrupt, epidemic, gradual, multiple) change setting and under dependence. For the dependence structure we consider either traditional m-dependence assumptions or more recently developed m-approximability conditions which cover, e.g., MA, AR and ARCH models. We derive Gumbel and Brownian bridge type approximations of the distribution of the test statistic under the null hypothesis of no change and consistency conditions under the alternative. A new formulation of the test statistic using projections on subspaces allows us to simplify the standard proof techniques and to weaken common assumptions on the covariance structure. Furthermore, we propose to adjust the principal components by an implicit estimation of a (possible) change direction. This approach adds flexibility to projection based methods, weakens typical technical conditions and provides better consistency properties under the alternative. In the second part we contribute to estimation methods for common changes in the means of panels of Hilbert space valued time series. We analyze weighted CUSUM estimates within a recently proposed »high-dimensional low sample size (HDLSS)« framework, where the sample size is fixed but the number of panels increases. We derive sharp conditions on »pointwise asymptotic accuracy« or »uniform asymptotic accuracy« of those estimates in terms of the weighting function. Particularly, we prove that a covariance-based correction of Darling-Erdős-type CUSUM estimates is required to guarantee uniform asymptotic accuracy under moderate dependence conditions within panels and that these conditions are fulfilled, e.g., by any MA(1) time series. As a counterexample we show that for AR(1) time series, close to the non-stationary case, the dependence is too strong and uniform asymptotic accuracy cannot be ensured. Finally, we conduct simulations to demonstrate that our results are practically applicable and that our methodological suggestions are advantageous.

Functional analysis of Arabidopsis thaliana lesion mimic mutant cfs1 in ESCRT complex-related protein trafficking

The Endosomal Sorting Complex Required for Transport (ESCRT)-complex is composed of four complexes, ESCRT-0-III. They sequentially act on a late endosome to sort mono-ubiquitinated transmembrane proteins into the intralumenal vesicle, forming of a multivesicular body(MVB) that is delivered to vacuole for degradation. In Arabidopsis thaliana, the loss of an ESCRT-I component, elch displays a cytokinesis defect; while a dominant negative expression of an ESCRT-III component results in cell death due to vacuolar loss. In this work, the function of a plant-specific ELCH-interactor, CELL DEATH RELATED FYVE/SYLF DOMAIN CONTAINING 1 (CFS1) and its influences on the ESCRT-complex function are investigated. CFS1 is a phosphatidylinositol-3-phosphate- and actin-binding protein. The cfs1 mutants mimic lesions in the first eldest leaf that propagate to the next eldest one. Genetic analyses have demonstrated that cell death in cfs1 does not require a functional ESCRT-I component; nevertheless, the loss of CFS1 alleviates elchcytokinesis defect, suggesting its influence on the ESCRT-I function. Further analyses reveal that cfs1 accumulates autophagosomes throughout its lifespan due to a decrease in autophagosome degradation, suggesting that as the plant ages, the cumulated autophagosomes falsely trigger effectors-triggered immunity that executes cell death in cfs1. As the ESCRT-complex has been demonstrated to be involved in the delivery of autophagosomes to vacuole and CFS1 homolog, CFS2 reportedly interacts with ATG8, it can be postulated from the results of this work that CFS1 alone or together with CFS2 function in sequestering mature autophagosomes onto MVBs. At the MVBs, the ESCRT-complex then mediates the fusion of autophagosome and MVB for subsequent delivery to vacuole.