Obust methods in multivariate time series
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Data
2019-08-22
Autores
Cotta, Higor Henrique Aranda
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Universidade Federal do Espírito Santo
Resumo
This manuscript proposes new robust estimation me thods for the autocovariance and autocorrelation ma trices functions of stationary multivariates time se ries that may have random additives outliers. These functions play an important role in the identification and estimation of time series model parameters. Ran dom additive outliers can impact the level of one or more components of the multivariate vector. This in creases the overall variability of the series, which has an impact on the periodogram matrix and leads to a decrease in the values of the autocorrelation ma trix function. We first propose new estimators of the autocovariance and of autocorrelation matrices func tions constructed using a spectral approach conside ring the periodogram matrix periodogram which is the natural estimator of the spectral density matrix. As in the case of the classic autocovariance and autocor relation matrices functions estimators, these estima tors are affected by aberrant observations. Thus, any identification or estimation procedure using them is di rectly affected, which leads to erroneous conclusions. To mitigate this problem, we propose the use of robust statistical techniques to create estimators resistant to aberrant random observations. As a first step, we propose new estimators of auto covariance and autocorrelation functions of univariate time series. The time and frequency domains are lin ked by the relationship between the autocovariance function and the spectral density. As the periodogram is sensitive to aberrant data, we get a robust esti mator by replacing it with the M-periodogram. The M-periodogram is obtained by replacing the Fourier coefficients related to periodogram calculated by the standard least squares regression with the ones cal culated by the M-robust regression. The asymptotic properties of estimators are established. Their perfor mances are studied by means of numerical simula tions for different sample sizes and different scena rios of contamination. The empirical results indicate that the proposed methods provide close values of those obtained by the classical autocorrelation func tion when the data is not contaminated and it is re sistant to different contamination scenarios. Thus, th estimators proposed in this thesis are alternative me thods that can be used for time series with or without outliers. The estimators obtained for univariate time series are then extended to the case of multivariate series. This extension is simplified by the fact that the calculation of the cross-periodogram only involves the Fourier co efficients of each component from the univariate se ries. Again, the duality relationship between time and frequency domains is considered via the link between the autocovariance matrix function and the spectral density matrix stationary multivariate time series. The M-periodogram matrix is a robust periodogram matrix alternative to build robust estimators of the autoco variance and autocorrelation matrices functions. The asymptotic properties are studied and numerical ex periments are performed. As an example of an appli cation with real data, we use the proposed functions to adjust an autoregressive model by the Yule-Walker method to Pollution data collected in the Vit´ oria re gion Brazil (particles smaller than 10 micrometers in diameter, PM10). Finally, the robust estimation of the number of fac tors in large factorial models is considered in order to reduce the dimensionality. It is well known that the values random additive outliers affect the covariance and correlation matrices and the techniques that de pend on the calculation of their eigenvalues and ei genvectors, such as the analysis principal compo nents and the factor analysis, are affected. Thus, in the presence of outliers, the information criteria pro posed by Bai & Ng (2002) tend to overestimate the number of factors. To alleviate this problem, we pro poseto replace the standard covariance matrix with the robust covariance matrix proposed in this manus cript. Our Monte Carlo simulations show that, in the absence of contamination, the standard and robust methods are equivalent. In the presence of outliers, the number of estimated factors increases with the non-robust methods while it remains the same using robust methods. As an application with real data, we study pollutant concentrations PM10 measured in the ˆ Ile-de-France region of France.
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Séries temporais multivariadas , Robustez , Observações discrepantes , Domínio do tempo , Domínio da frequência , Multivariate time series , Robustness , Outliers , Time domain , Frequency domain