Thesis title: Change Points Detection for Spherical Functional Autoregressive Processes
This dissertation aims to deepen the understanding of dynamic processes by investigating the occurrence of sudden shifts in their structure.
To this end, we introduce the Spherical Change Point Autoregressive model, able to incorporate the change point framework into the Spherical Functional Autoregression (SPHAR) model by introducing non-stationarity and consequently relaxing one of its main assumptions.
The first part of the thesis provides some background on change point analysis and spherical random processes. Having introduced all the necessary tools, the Spherical Autoregressive Change Point model is introduced, giving details on its structure and its observations. Two change point detection techniques are then introduced.
The main contribution of the thesis is the development of a Lasso-penalized detection method capable of retroactively identifying the presence of change points within a series of observations, without prior knowledge of the number or location of the true breaks. We prove theoretical guarantees on the estimation results of this technique.
In addition, this thesis proposes a change point detection method for the At-Most-One-Changepoint scenario through a likelihood-based approach. The procedure involves defining a risk function whose optimization is equivalent to maximize a suitable likelihood function.
To conclude this dissertation, the performance of the proposed techniques are evaluated on different settings through an extensive simulation study.