Date of Award

Winter 2026

Language

English

Embargo Period

1-12-2026

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Epidemiology and Biostatistics

Program

Biostatistics

First Advisor

Edward Valachovic

Second Advisor

Kai Zhang

Third Advisor

Eric J. Rose

Committee Members

Edward Valachovic, Kai Zhang, Eric J. Rose

Keywords

Time Series, Periodically Correlated, Variable Bandpass Periodic Block Bootstrap, Bayesian technique

Subject Categories

Biostatistics

Abstract

Time series with multiple periodically correlated (MPC) components present a complex challenge, with relatively limited prior research. Most existing models are designed for simpler periodically correlated (PC) components and often struggle with over-parameterization, optimization issues, and capturing complex PC patterns within a time series. Frequency separation techniques can help preserve the correlation structure of individual PC components, while Bayesian methods can integrate new and prior information to refine beliefs about these components. This study proposes a two-stage approach that combines frequency separation and Bayesian techniques to forecast PC and MPC time series data. This method aims to demonstrate improved effectiveness in modeling MPC components compared to traditional approaches.

Among the various forecasting methods available, the seasonal autoregressive integrated moving average (SARIMA) model and exponential smoothing technique are regarded as traditional approaches. However, their basic forms are primarily designed for modeling single PC components and often face difficulties in handling MPC patterns. Many studies have attempted to adapt these classical statistical models to better accommodate MPC components. Notable examples include the double seasonal ARIMA model, an extension of the exponential smoothing technique based on the Holt-Winters method, and the hidden Markov model with multiple periods. Despite these developments, existing methods frequently encounter over-parameterization and optimization issues and are often inadequate in effectively capturing complex seasonal patterns in time series data. Because theoretical and practical limits must be established in terms of both frequency separations and Bayesian modelling, along with a practical application that is comprehensive, this purpose was operationalized through three objectives.

The first objective is to investigate the presence and characteristics of MPC—including daily, weekly, annual patterns, and their respective harmonics—in Turkey’s industrial electricity consumption time series using the Variable Bandpass Periodic Block Bootstrap (VBPBB) method. A core focus is to integrate the Kolmogorov-Zurbenko (KZ) filter framework (including its extension, the Kolmogorov-Zurbenko Fourier Transform (KZFT) filter) into the VBPBB workflow. Compare the performance of VBPBB with the Generalized Seasonal Block Bootstrap (GSBB) in terms of confidence interval (CI) size and accuracy. Validate the statistical significance of the identified MPC components by examining their 95% CIs, and quantify the contribution of each significant component to total electricity consumption variability using the coefficient of determination; Demonstrate the practical utility of the VBPBB method in resolving MPC data and preserving the correlation structure of individual periodic components—addressing a key limitation of traditional periodic block bootstrap (PBB) methods that only focus on single frequencies.

The second objective was to propose a two-stage VBPBB-Bayesian integrated method that combines VBPBB and Bayesian techniques with Markov Chain Monte Carlo (MCMC) via the Metropolis–Hastings algorithm to address the challenges of forecasting time series with MPC components. Specifically, it aims to: 1) Use VBPBB to isolate significant single periodic correlated (PC) components (and their harmonics) from MPC time series by filtering frequencies, preserving the correlation structure of each target component while attenuating noise and unrelated signals; 2) Employ Bayesian modeling to estimate the amplitude (the key parameter) of each separated PC component, avoiding over-parameterization associated with modeling individual data points; 3) Validate the method’s effectiveness through simulations (with single and double PC components) and real data application (U.S. monthly milk production);

The third objective was to validate the effectiveness and robustness of the proposed VBPBB-Bayesian model in forecasting MPC time series. Specifically, it aims to test the model’s performance across diverse simulation scenarios (single/double PC components, varying frequency levels (low/medium/high), different noise variances (10/100/1000), and harmonic frequencies); 4) Compare the VBPBB-Bayesian model with a standard Bayesian model (without VBPBB) to demonstrate its superiority in accurate amplitude estimation; 5) Verify the model’s suitability for handling complex MPC patterns by showcasing consistent performance across varied data conditions.

The potential applications of VBPBB and Bayesian analytic methods extend across the entire spectrum of health sciences and human services, as well as span the full range of scientific disciplines and technological fields—their scope is nearly boundless. In light of this, the present Dissertation validated the efficacy of these methods by systematically and comprehensively defining and illustrating both their theoretical constraints and practical limitations.

License

Creative Commons Attribution 4.0 International License
This work is licensed under a Creative Commons Attribution 4.0 International License.

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