On Clustering Non-smooth Functional Observations

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Journal Title
Journal ISSN
Volume Title
Perustieteiden korkeakoulu | Master's thesis
Date
2018-12-04
Department
Major/Subject
Applied Mathematics
Mcode
SCI3053
Degree programme
Master’s Programme in Mathematics and Operations Research
Language
en
Pages
56+5
Series
Abstract
Nowadays, we are able to store large amounts of data. This has led to development of methods for analyzing very high dimensional observations. When we consider observations that are not only high dimensional, but infinite dimensional, we are in the setting of Functional data analysis. In functional data analysis, the methods are often based on the assumption that the observations are smooth. However, in some applications, the smoothness assumption is not reasonable. For example, the nature of financial data is often non-smooth. In this thesis, we consider non-smooth functional observations. We introduce a novel method for clustering non-smooth functional data. First, we review basic concepts related to functional data analysis and clustering. After that, we present our new method. The method relies on Holder continuity assumption. We map each functional observation to an index that measures the roughness of the observation. After that, we apply k-means clustering to the obtained indices. We consider theoretical properties of the method under the assumption of fractional Brownian motions and we present several simulated examples to assess its performance in practice. Based on these simulated examples, the method works extremely well in clustering fractional Brownian motions with different Hurst indices.
Description
Supervisor
Ilmonen, Pauliina
Thesis advisor
Viitasaari, Lauri
Keywords
Functional data, clustering, k-means clustering, roughness, Holder continuity, fractional Brownian motions
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