Statistical Details for Models with Categorical Predictors
If the Gaussian Process model includes categorical predictors, the Gaussian correlation structure is used for the correlation structure. The elements of the R matrix are defined as follows:
where
K = # of continuous predictors
P = # of categorical predictors
θk = theta parameter for the kth continuous predictor
xik = the value of the kth continuous predictor for subject i
xjk = the value of the kth continuous predictor for subject j
= the correlation between the observed level of predictor p for subject i and the observed level of predictor p for subject j
There is a τ parameter for each combination of levels of a categorical variable, where τij corresponds to the unique combination formed by the observed levels of subject i and subject j. Thus, the covariance element, rij, depends on the combination of levels of the categorical predictors obtained from the ith and jth observations. See Qian et al. (2012).