Generalized gamma spatial ARMA conditional model for speckled data: theoretical developments and applications

Abstract

This thesis aims to advance the study of conditional spatial ARMA models for SAR image data. First, we propose a conditional spatial ARMA model based on the generalized gamma distribution, which incorporates spatial correlation and accounts for the positive and asymmetric nature of SAR data. We derive the score vector, Fisher information matrix, and sth moments and estimate the two-dimensional generalized gamma ARMA (2D-GGARMA) model parameters through an iterative process. Furthermore, we explicitly establish the relationship between our model and the SAR image formation process. Monte Carlo studies are conducted to analyze the behavior of the conditional maximum likelihood estimators using simulated SAR images. Additionally, we propose a prediction equation and new residuals. To illustrate the practical relevance of our approach, we analyze SAR images of the floods that occurred in Rio Grande do Sul, Brazil, between late November 2023 and early May 2024. We compare our model with other spatial models, such as the Rayleigh-ARMA and Gamma-ARMA spatial models. The results demonstrate that our approach outperforms the competing models in this application. A flood analysis in Rio Grande do Sul is also performed using a proposed classifier based on Mahalanobis distance to detect changes in image texture. The second study focuses on estimating the equivalent number of looks (ENL) in SAR images. Using the proposed spatial model, we introduce two new ENL estimators that explicitly account for the spatial correlation in the data. Monte Carlo simulations compare the proposed estimators’ mean squared errors (MSEs) with those of traditional estimators from the literature across different scenarios for the number of looks and spatial correlation assumptions. The simulation results indicate that, in the presence of spatial correlation, the proposed estimators exhibit lower MSE. The proposed estimators achieve a lower predictive mean squared error in real SAR data applications than conventional estimators. The third study proposes a new spatial Portmanteau test for conditional spatial ARMA models, whether the ARMA model is separable or not, and for any distribution associated with the model. Monte Carlo simulations show good asymptotic properties of the proposed test statistic. In the application to real data, the residuals remain spatially correlated after fitting.