Testing mean densities with an application to climate change in Vietnam

Given samples of density functions on an interval (a, b) of R, categorized according to a factor variable, we aim to test the equality of their mean functions both overall and across the groups defined by the factor. While the Functional Analysis of Variance (FANOVA) methodology is well-established for functional data, its adaptation to density functions (DANOVA) is necessary due to their inherent constraints of positivity and unit integral. To accommodate these constraints, we naturally use Bayes spaces methodology by mapping the densities using the centered log-ratio transformation into the L^2_0 (a, b) space where we can use FANOVA techniques. Many traditional contrasts in FANOVA rely on squared differences and can be reinterpreted as squared distances between Bayes perturbations within the densities space. We illustrate our methodology on a dataset comprising daily maximum temperatures across Vietnamese provinces between 1987 and 2016. Within the context of climate change, we first investigate the existence of a non-zero temporal trend of the densities of daily maximum temperature over Vietnam and then examine whether there is any regional effect on these trends. Finally, we explore odds ratio based interpretations allowing to describe the trends more locally.