PhD Defence Giorgio Micali | Ordinal patterns based time series analysis and applications in privacy

Ordinal patterns based time series analysis and applications in privacy

Giorgio Micali is a PhD student in the Department of Mathematics of Operations Research. (Co)Promotors are dr. R. de Heide and dr. A. Betken from the Faculty of Electrical Engineering, Mathematics and Computer Science.

Time series analysis is a fundamental tool across a wide range of scientific disciplines. The present thesis develops this underlying mathematical theory,
ranging from statistical methodology to secure data treatment. A major part of our work is carried out through the ordinal-pattern representation of time series.

This approach offers several advantages, since ordinal patterns are invariant under strictly increasing marginal transformations of the data, robust to noise,
and sensitive to qualitative properties such as distributional asymmetries. In the domain of time series inference, we focus on two main problems:
change-point detection and hypothesis testing for qualitative properties of time series. To address them, we establish asymptotic results for empirical
ordinal patterns frequencies and related functionals under broad dependence assumptions. In particular, we prove central limit theorems and develop
covariance estimation tools for stationary processes, covering linear processes and more general weak-dependence frameworks formulated through mixing
conditions.

The theory developed for these purposes extends naturally to the statistical analysis of neural networks. More specifically, our proof techniques can also be
used to study convolutional neural networks with time series input, leading to asymptotic normality of their outputs.

If our results are to be applied in practice, the treatment of real world data is unavoidable. Such data treatment is increasingly required to comply with strict
regulations, including privacy constraints. This motivates another perspective on time series analysis, which is also developed in this thesis. In particular,
we introduce privacy-preserving procedures for change-point detection. Under homomorphic encryption, we develop implementations of ordinal-pattern-based CUSUM statistics. In parallel, we study differentially private mechanisms for statistical query release via stochastic convex optimization, yielding synthetic outputs with end-to-end privacy guarantees.