Adaptive Moment Estimation of the Spatial Autoregressive Model Parameters
Michael Oduor Otieno *
Department of Mathematics, Laikipia University, Nyahururu, Kenya.
Adolphus Wagala
Department of Data Science, Dana Farber Cancer Institute, Boston, USA and Department of Biostatistics, Harvard University, Boston, USA.
Gregory Kerich
Department of Mathematics, Physics and Computing, Moi University, Eldoret, Kenya.
*Author to whom correspondence should be addressed.
Abstract
Traditionally, estimation of the spatial autoregressive (SAR) model parameters has been done using the maximum likelihood estimation (MLE) technique. Several variants of the MLE technique, such as quasi-MLE (QMLE) and adjusted-QMLE, among others, have also been used. However, in the presence of large data sets, MLE has been found to be computationally costly due to the matrix inversion involved in maximizing the log-likelihood function. The adaptive moment (ADAM) estimation method is a viable alternative. It eliminates the cost of matrix inversion in the widely used MLE method. Moreover, it eliminates the need to store the whole data set since the gradients can be computed from every data point, or a batch of data points sampled from the whole data set. However, ADAM has been developed and widely used for independent data. This study seeks to design ADAM for use in a spatial data setting. ADAM is found to perform better than the standard MLE in large data setting since it has the least values of Akaike Information Criterion and higher values of the McFadden’s R-squared. It also converges faster than MLE and hence is most appropriate for large data sets.
Keywords: Spatial autocorrelation, spatial autoregressive model, maximum likelihood estimation, ADAM stochastic optimization