Time Series Analyses: Time Domain, Spectral Analysis and Cpestral Analysis

• Grant number: 25/02138-7
• Support Opportunities: Research Grants - Visiting Researcher Grant - Brazil
• Start date: May 01, 2025
• End date: April 30, 2026
• Field of knowledge: Physical Sciences and Mathematics - Probability and Statistics - Statistics
• Principal Investigator: Aluísio de Souza Pinheiro
• Grantee: Aluísio de Souza Pinheiro
• Visiting researcher: Valderio Anselmo Reisen
• Visiting researcher institution: Universidade Federal do Espírito Santo (UFES). Centro de Ciências Exatas (CCE), Brazil
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil
• Associated research grant: 23/02538-0 - Time series, wavelets, high dimensional data and applications, AP.TEM

Abstract

We consider here heteroskedastic time series models, specifically from the family $GARCH(p,q)$, both on time domain and well as on the frequency domain. For the former we estimate the $(p+q)$-dimensional parameter through $M$-quantiles. On the latter, we have two situations. First we employ the discrete Fourier transform (DFT) to estimate the spectrum. Further, we discuss the cpestrum, defined as the inverse Fourier transform for the log spectral density. Two main cpestrum class of estimators based on the smoothed empirical periodogram are studied in this project. The first is defined through the $M$-quantile periodogram. The second is based on the thresholded empirical DWT. The estimators' statistical properties will be developed in this project. Moreover, scalability and on-line feasibility issues will be discussed here. For these high- and ultra-high dimensional properties, several approximate DFT and DWT methods will be employed, such as the fast DFT (FFT) and fast DWT (DFWT), among others. Simulation studies will be used to: compare the proposed methods to the state-of-the-art; and to illustrate the feasibility of such methods for high- and ultra-high dimension data. The proposed methodologies will be employed in three relevant real data problems: Air quality real-time monitoring; Cronic Respiratory Diseases; and Early Asymptomatic Degenerative Diseases Detection. (AU)