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?

Let's denote by $S$ the basic set which contains the elements that are grouped in clusters. So the clusters $C_i$ are subsets of $S$. Now, IIUC, we want the following properties to hold:

*

*Any two different clusters $C_i$ and $C_j$ are disjoint ($C_i\cap C_J = \emptyset, i\ne j$).

*All clusters together cover the whole set $S$ ($\bigcup_{i=1}^K C_i = S$)

In other words, the clustering is a set partition. So the goal is to better describe the space of set partitions; lets call it $\cal P$.
I think the first step would be to find a metric (or some weaker version of a metric) for $\cal P$. Once you have a metric, you can do all kinds of things, e.g. talk about groupings (that would then be "clusters of clusterings").
Since $\cal P$ is discrete and finite, we could maybe then think of this space as a graph, with the metric giving us a similarity value on each edge. We could cut all the edges below a certain similarity, so the graph would not have to be complete.
As far as the notion of a metric is concerned, the author of the question already mentioned the rand index; another possibility is e.g. based on the Fowlkes-Mallows index. You could also use mutual information.
For a similar question, see here.
Then, your time series of clusterings becomes a walk in the graph. You could define transition probabilities derived from the similarity values on the edges and could thus obtain some graph random walk. This would help do predictions.
