In mathematics, the moments of a function are measures that quantify the shape of the function. This function can take various forms, depending on the context of the application. For example, in Physics, if a function represents mass density, the first and second moments are referred to as the center of mass and the moment of inertia of a rigid body, respectively. In statistics, if the function represents a probability distribution, then the first, second, and third moments are interpreted as the expected value, variance, and skewness, respectively.

In climate science, if the function represents the spatial distributions of a climate variable, the moments provide quantitative insights into the spatial patterns of the variable. In particular, the second and third moments can be interpreted as indicators of spatial inhomogeneity and spatial asymmetry of the climate state, which is an important topic in climate change research.

In this presentation, we, based on the concept of 'moments', introduce a general statistical framework for studying the spatial inhomogeneity and spatial asymmetry of climate change. This framework can be applied to (1) quantify the spatial inhomogeneity and asymmetry of the climate state, (2) estimate the long-term variability climate inhomogeneity and asymmetry under climate change, and (3) identify the contributors of these long-term changes. The introduced framework has been successfully applied to research on the asymmetrical nature of global warming and spatial inhomogeneity of global precipitation trends. Given its simplicity and generality, this mathematical framework is adaptable to any variables of interest and/or combinations of variables.