Models for describing cluster change over time Suppose I have a set of observations encoded as a matrix $X$ whose entries $X_{it} \in \mathcal{C} $ correspond to cluster labels of experimental unit $i$ with repeated measures at time $t$, where $\mathcal{C}$ denotes a finite set of nominal cluster indicators. For clarity, here nominal is meant to emphasize that particular values of $x_{it}$ have no inherent value, and are only meaningful given other $x_{it}$s values.
What models are suitable for describing the change in clusters over time?

To get the brainstorm-ball rolling, I'll start with a naive proposal.
A challenge in modelling cluster change over time is the inherently nominal values of cluster indicators. One way to work around this roadblock is to define random variables whose values correspond to the discrepancy between contiguous measures of partitions. In other words, define new random variables $Y_t=D(X_{\cdot t}, X_{\cdot t-1})$ for some sensible map $D$ measuring the discrepancy or similarity of two partitions. Then the problem simplifies to modelling a set of  univariate time series data. For example, $D$ could be the rand index.
More generally, the idea is to define some sensible test function that maps partitions to real numbers, e.g., one could imagine computing the discrepancy of each partition versus some constant partition instead of computing the discrepancy between contiguous partitions. For this particular example, the interpretation can become fuzzy, as the range of values is limited, i.e., $D$ is not injective. The challenge now becomes finding a sensible test function, which likely depends on the data at hand.
