# Stochastic Processes and Multi-Resolution Analysis: A Trigonometric Moment Problem Approach and an Analysis of the Expenditure Trends for Diabetic Patients

Spring 8-2023

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

## Department

Computational and Data Sciences

Daniel Alpay

Cyril Rakovski

Didy Danioko

## Abstract

This dissertation is divided into two distinct parts. The main theme of the first part is to study stochastic processes (and related signal processing questions) using tools in wavelet analysis, functional analysis (we use in particular the trigonometric moment problem), the theory of realization of rational functions, and reproducing kernel Hilbert spaces. A novel form of multiresolution analysis is formulated in the discrete case that is used to study some stochastic processes. Using the trigonometric moment problem, we associate with a vector-valued wide-sense stationary process a multiresolution of a new kind. The notion of realization of rational functions was used to study stochastic processes and related signals. When the data is rational (meaning that the generating function of the covariance coefficients is rational) we used realization theory and the Kalman-Yakubovich-Popov (KYP) lemma to study this multiresolution. The idea of trigonometric moment problems in both the general and rational case were discussed. In the second part, we studied the trends of the average expenditure of diabetic patients over time. Four important breakdown costs were used to assess the total average expenditure of patients over time. The breakdown are average hospitalization cost, average management cost, average surgery cost, and average drug cost. The result indicated that the average hospitalization cost is higher among all the breakdowns for each year followed by the average management cost. The least average expenditure for each year is the average drug cost. The total average expenditure trend is expected to increase yearly. However, because of an intervention such as government policy that alleviates the cost of treatment, the average expenditure trend of patients changes sharply. We modeled the change in the total average expenditure using the piecewise linear regression model.

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#### DOI

https://doi.org/10.36837/chapman.000499