Approximate bayesian inference for directed acyclic graph autoregressive models
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2022-02-02
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Pontificia Universidad Católica del Perú
Abstract
La prevalencia de enfermedades epidemiológicas recolectadas en áreas geográficamente limitadas, como distritos o provincias, son cruciales para la toma de decisiones en salud pública. Usualmente esta variable respuesta presenta dependencia espacial, es decir, es similar en áreas vecinas, debido a la naturaleza de la enfermedad, clima, nivel económico y cultural, entre otras razones. En este sentido, se proponen modelos espaciales de datos áreas para identificar tendencias y factores asociados a enfermedades epidemiológicas, tomando en cuenta la dependencia espacial entre áreas geográficas. Por lo general, estos modelos ajustan a la dependencia espacial a través de efectos aleatorios derivados a través de grafos. En particular, el modelo autorregresivo de gráfico acíclico dirigido (DAGAR) se basa en un grafo acíclico dirigido y algunos efectos aleatorios \del pasado". Como consecuencia, la matriz de precisión (inversa de la covarianza) del modelo es dispersa. Este modelo tiene una interpretación intuitiva de los parámetros asociados con la dependencia espacial y se puede representar como un modelo gaussiano latente. En este contexto, en esta tesis se propone implementar el modelo DAGAR a través del método de inferencia bayesiano aproximado INLA que es determinista, bastante preciso y eficiente. Dentro de este enfoque, la estimación de datos grandes se puede realizar en segundos o minutos, y permite ajustar los datos con distribución gaussiana o no gaussiana. Finalmente, para mostrar el aporte de esta propuesta, el modelo DAGAR se ajusta a datos reales.
The prevalence of epidemiological diseases collected in geographically limited areas, such as districts or provinces, are crucial for making public health decisions. It is common that this response variable presents spatial dependence, that is, it is similar in neighboring areas, due to the nature of the disease, weather, economy and cultural level, among other reasons. In this sense, spatial models for areal data are proposed to identify trends and factors associated with epidemiological diseases, taking into account the spatial dependence between geographic areas. Usually, these models t the spatial dependence through spatial random e ects built from graphs and conditional distributions. In particular, the directed acyclic graph autoregressive (DAGAR) model is based on a directed acyclic graph and some \past" random e ects. As a consequence, the precision matrix (inverse of the covariance) of the model is sparse. This model has an intuitive interpretation of the parameters associated with spatial dependence and can be represented as a latent Gaussian model. In this context, we propose in this project to implement the DAGAR model throughout the approximate Bayesian inference method INLA which is deterministic, quite accurate and e cient. Within this approach, estimation of large data can be carried out in seconds or minutes, and it allows to t data following a Gaussian distribution or non-Gaussian distributions. Finally, in order to show the contribution of this proposal, the DAGAR model will be tted to real data.
The prevalence of epidemiological diseases collected in geographically limited areas, such as districts or provinces, are crucial for making public health decisions. It is common that this response variable presents spatial dependence, that is, it is similar in neighboring areas, due to the nature of the disease, weather, economy and cultural level, among other reasons. In this sense, spatial models for areal data are proposed to identify trends and factors associated with epidemiological diseases, taking into account the spatial dependence between geographic areas. Usually, these models t the spatial dependence through spatial random e ects built from graphs and conditional distributions. In particular, the directed acyclic graph autoregressive (DAGAR) model is based on a directed acyclic graph and some \past" random e ects. As a consequence, the precision matrix (inverse of the covariance) of the model is sparse. This model has an intuitive interpretation of the parameters associated with spatial dependence and can be represented as a latent Gaussian model. In this context, we propose in this project to implement the DAGAR model throughout the approximate Bayesian inference method INLA which is deterministic, quite accurate and e cient. Within this approach, estimation of large data can be carried out in seconds or minutes, and it allows to t data following a Gaussian distribution or non-Gaussian distributions. Finally, in order to show the contribution of this proposal, the DAGAR model will be tted to real data.
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Estadística bayesiana, Variables latentes
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