Periodic models and variations applied to health problems

• Ano de defesa: 2019
• Autor(a) principal: Prezzoti Filho, Paulo Roberto
• Tipo de documento: Tese
• Tipo de acesso: Acesso aberto
• Idioma: eng
• Instituição de defesa: Universidade Federal do Espírito Santo; BR; Doutorado em Engenharia Ambiental; Centro Tecnológico; UFES; Programa de Pós-Graduação em Engenharia Ambiental
• Palavras-chave em Português: Count time series; periodicity; INAR Models; PINAR Models; ZIP Models; seasonality; Séries temporais de contagem; periodicidade; Modelo INAR; Modelo PINAR; Modelo ZIP; sazonalidade; Engenharia sanitária; 628
• Link de acesso: http://repositorio.ufes.br/handle/10/10930

This manuscript deals with some extensions to time series taking integer values of the autoregressive periodic parametric model established for series taking real values. The models we consider are based on the use of the operator of Steutel and Van Harn (1979) and generalize the stationary integer autoregressive process (INAR) introduced by Al-Osh & Alzaid (1987) to periodically correlated counting series. These generalizations include the introduction of a periodic operator, the taking into account of a more complex autocorrelation structure whose order is higher than one, the appearance of innovations of periodic variances but also at zero inflation by relation to a discrete law given in the family of exponential distributions, as well as the use of explanatory covariates. These extensions greatly enrich the applicability domain of INAR type models. On the theoretical level, we establish mathematical properties of our models such as the existence, the uniqueness, the periodic stationarity of solutions to the equations defining the models. We propose three methods for estimating model parameters, including a method of moments based on Yule-Walker equations (YW), a conditional least squares method, and a quasi-maximum likelihood method (QML) based on the maximization of a Gaussian likelihood. We establish the consistency and asymptotic normality of these estimation procedures. Monte Carlo simulations illustrate their behavior for different finite sample sizes. The models are then adjusted to real data and used for prediction purposes. The first extension of the INAR model that we propose consists of introducing two periodic operators of Steutel and Van Harn, one modeling the partial autocorrelations of order one on each period and the other capturing the periodic seasonality of the data. Through a vector representation of the process, we establish the conditions of existence and uniqueness of a solution periodically correlated to the equations defining the model. In the case where the innovations follow Poisson’s laws, we study the marginal law of the process. As an example of real-world application, we are adjusting this model to daily count data on the number of people who received antibiotics for the treatment of respiratory diseases in the Vit ´ oria region in Brazil. Because respiratory conditions are strongly correlated with air pollution and weather, the correlation pattern of the daily numbers of people receiving antibiotics shows, among other characteristics, weekly periodicity and seasonality. We then extend this model to data with periodic partial autocorrelations of order higher than one. We study the statistical properties of the model, such as mean, variance, marginal and joined distributions. We are adjusting this model to the daily number of people receiving emergency service from the public hospital of the municipality of Vit ´ oria for treatment of asthma. Finally, our last extension deals with the introduction of innovations according to a Poisson law with zero inflation whose parameters vary periodically, and on the addition of covariates explaining the logarithm of the intensity of the Poisson’s law. We establish some statistical properties of the model, and we use the QML method to estimate its parameters. Finally, we apply this modeling to daily data of the number of people who have visited a hospital’s emergency department for respiratory problems, and we use the concentration of a pollutant in the same geographical area as a covariate.