  Compositional data find broad application across diverse fields due to their
efficacy in representing proportions or percentages of various components
within a whole. Spatial dependencies often exist in compositional data,
particularly when the data represents different land uses or ecological
variables. Ignoring the spatial autocorrelations in modelling of compositional
data may lead to incorrect estimates of parameters. Hence, it is essential to
incorporate spatial information into the statistical analysis of compositional
data to obtain accurate and reliable results. However, traditional statistical
methods are not directly applicable to compositional data due to the
correlation between its observations, which are constrained to lie on a
simplex. To address this challenge, the Dirichlet distribution is commonly
employed, as its support aligns with the nature of compositional vectors.
Specifically, the R package DirichletReg provides a regression model, termed
Dirichlet regression, tailored for compositional data. However, this model
fails to account for spatial dependencies, thereby restricting its utility in
spatial contexts. In this study, we introduce a novel spatial autoregressive
Dirichlet regression model for compositional data, adeptly integrating spatial
dependencies among observations. We construct a maximum likelihood estimator
for a Dirichlet density function augmented with a spatial lag term. We compare
this spatial autoregressive model with the same model without spatial lag,
where we test both models on synthetic data as well as two real datasets, using
different metrics. By considering the spatial relationships among observations,
our model provides more accurate and reliable results for the analysis of
compositional data. The model is further evaluated against a spatial
multinomial regression model for compositional data, and their relative
effectiveness is discussed.
