Implementation of a functional ANOVA (FANOVA) model, based partly on the model in Bayesian functional ANOVA modeling using Gaussian process prior distributions. To implement a FANOVA model, an underlying general framework is defined for modeling functional observations:

$$ Y(t) = X \beta(t),$$

where $$ Y(t) = [y_1(t),\dots,y_m(t)]^T, $$ $$\beta(t) = [\beta_1(t),\dots,\beta_f(t)]^T,$$ $$ X: m \times f$$ for a given time $t$. The design matrix $X$ defines the relation between the functions $\beta$ and observations $y$. In general, the rank of $X$ should match the number of functions $f$. The FANOVA model can then be described by a specific form of $X$ such that

$$ y_{i,j}(t) = \mu(t) + \alpha_i(t) + \beta_j(t) + \alpha\beta_{i,j}(t). $$