Local Bootstrap for Incomplete Stationary Time Series in the Presence of Atypical Observations: An Application to Problems in the Air Quality Area

Name: Carlo Corrêa Solci
Type: PhD thesis
Publication date: 23/09/2022

Namesort descending Role
Valdério Anselmo Reisen Advisor *

Examining board:

Namesort descending Role
Elisa Valentim Goulart Internal Examiner *
Felipe Elorrieta López External Examiner *
Glaura da Conceição Franco External Examiner *
Neyval Costa Reis Jr. Internal Examiner *
Pascal Bondon External Examiner *
Paulo Jorge Canas Rodrigues Co advisor *
Valdério Anselmo Reisen Advisor *

Summary: Studies about air pollution typically involve measurements and analysis of pollutants, such
as PM10 (particulate matter with diameter lower than 10 µm), SO2 (sulfur dioxide) and
others. These data typically have important features like serial correlation, seasonality,
missing observations and the presence of peaks that despite not being atypical observations
(outliers) because of their high frequency of occurrence, can be modeled as such owing to
the effect that they have on the series. All these features demand special attention during
data analysis and complicate the obtainment of confidence intervals for the parameters
of stationary time series models through asymptotic theory. With this motivation, this
study proposed bootstrap methodologies in the frequency domain for weakly stationary
time series in the presence of missing observations and/or of contamination by additive
outliers. The suggested methodologies are based on the local bootstrap of Paparoditis
& Politis (1999), with the robustness being achieved by the substitution of the classical
periodogram with the M-periodogram of Reisen, L´evy-Leduc & Taqqu (2017) and when
there is presence of missing observations the original time series is replaced by its amplitude modulated version proposed by Parzen (1963). In this context, the efficiency of
the proposed bootstrap methodologies in estimating confidence intervals of parameters of
models for weakly stationary time series was verified through Monte Carlo studies under
different scenarios, including: additive outliers contamination and presence of missing observations. For comparison purposes, in some cases it was also considered the bootstrap
methodology of Paparoditis & Politis (1999), as well as the parameter estimates without
the bootstrap via the classical and robust versions of the methodologies of Whittle (1953)
and of Dunsmuir & Robinson (1981). The practical purpose in air pollution is to evaluate
if the confidence intervals of the parameters obtained by the robust methodologies present
a reduction in the effect of left shift that the classical intervals have due to the memory
loss caused by the additive outliers, in addition to the possibility of calculating these intervals without using imputation techniques to obtain a complete time series. The proposed
bootstrap methodologies were applied to calculate confidence intervals of parameters of
adjustment of the autoregressive (AR) model, and in some cases also of the seasonal autoregressive (SAR) model, to MP10 data of stations of the air quality monitoring network
of the Greater Vit´oria Region - ES.

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