AOI probability functions

At each time point, AOI probabilities satisfy

$ p_k(t)\ge 0, \qquad \sum_{k=1}^{K}p_k(t)=1. $

That constraint means the channels are not ordinary unconstrained Euclidean variables.

fit_compositional_fpca() verifies the simplex, applies explicit zero replacement, transforms to additive log-ratio coordinates, fits MFPCA, and reconstructs valid probabilities.

fit = fit_compositional_fpca(
    aoi_probabilities,
    reference_dimension=3,
    epsilon=1e-8,
    n_components=0.95,
)

Report how probabilities were built, the reference AOI, zero-replacement epsilon, retained components, and interpretation on the original probability scale.

Mathematical contract

After explicit zero replacement and renormalization, the additive log-ratio coordinate relative to reference AOI \(r\) is

\[ z_k(t) = \log\frac{p_k^\epsilon(t)}{p_r^\epsilon(t)}, \qquad k\ne r. \]

The inverse transformation normalizes \(q_r(t)=1\) and \(q_k(t)=\exp\{z_k(t)\}\):

\[ p_k(t) = \frac{q_k(t)} {\sum_{\ell=1}^{K}q_\ell(t)}. \]

See the mathematical reference.