M-quantile estimation and discriminant analysis for heteroscedastic processes
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Data
2025-07-09
Autores
Patrocinio, Patrick Ferreira
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Universidade Federal do Espírito Santo
Resumo
Based on techniques in the time and frequency domains, this work introduces the M-quantile estimator, which combines the well-known quantile and M-regression functions to address model estimation and discriminant problems, such as classification and dimensionality reduction, in the context of time series with short- and long-memory properties and conditional variance (heteroscedasticity). These phenomena are quite common in real-world problems across many scientific fields, particularly in air quality studies, which motivate this work's main contributions from theoretical and applied perspectives. This thesis is divided into three chapters, and the main contributions are as follows: Chapter 2 explores the M-quantile estimator from a time-domain perspective by introducing the M-quantile Huber loss function to minimise process estimation errors. This approach is an alternative estimation method for time series data, offering advantages over standard estimation methods, such as the conditional least squares estimator, which can be seen as a special case of the proposed approach. Some theoretical issues are discussed, and simulations and applications are provided to support its use in real-world problems. The second and third main contributions are presented in Chapters 3 and 4, respectively. These chapters propose the M-quantile periodogram as an estimator of the spectral function to be used in a discriminant technique constructed based on the cepstral function for short- and long-memory processes with heteroscedastic errors. The asymptotic properties of the M-quantile cepstral discriminant functions are derived. Since the proposed approaches, including the M-quantile periodogram, are relatively novel in the literature, the asymptotic properties of certain sample quantities have been left for future research. Simulations were conducted to evaluate the performance of the M-quantile discriminant function in finite sample sizes. The results reveal interesting findings, particularly regarding the superiority of the M-quantile discriminant function over the cepstral periodogram-based discriminant function for both short- and long-memory processes, with and without additive outliers and non-Gaussian distributions. To demonstrate the usefulness of the proposed methodology in real-world applications, large datasets of PM_10 and PM_2.5 pollutant measurements from more than 200 air quality monitoring stations in France were analysed. The empirical evidence showed promising source classification results in both cases, strongly supporting the use of the M-quantile cepstral sample function in real-world applications such as classification, dimensionality reduction, and cointegration, among others
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M-quantílica , Séries temporais , Valores aberrante , Análise espectral , Memória curta e longa , M-quantile , Time series , Outliers , Spectral classification , Data analysis , Short and long memory