Primal path algorithm for compositional data analysis

In modern regression analysis, it is frequently observed that regression predictors consist of the proportions or relative ratios of certain values rather than absolute values. For example, in analyzing air pollution data, the percentages of chemicals in the air are considered relevant predictors to identify the source of a pollutant (Lee et al., 2007). These types of proportional data, typically called compositional data, are widely used in geoscience (Buccianti et al., 2006), microbiology (Montassier et al., 2016), and nutritional biochemistry (Leite, 2016). By the definition of compositional data, all compositional predictors lie on the simplex and are thus linearly dependent. Aitchison and Bacon-shone (1984) proposed a regression model for compositional data as follows.