  State aggregation is a popular model reduction method rooted in optimal
control. It reduces the complexity of engineering systems by mapping the
system's states into a small number of meta-states. The choice of aggregation
map often depends on the data analysts' knowledge and is largely ad hoc. In
this paper, we propose a tractable algorithm that estimates the probabilistic
aggregation map from the system's trajectory. We adopt a soft-aggregation
model, where each meta-state has a signature raw state, called an anchor state.
This model includes several common state aggregation models as special cases.
Our proposed method is a simple two-step algorithm: The first step is spectral
decomposition of empirical transition matrix, and the second step conducts a
linear transformation of singular vectors to find their approximate convex
hull. It outputs the aggregation distributions and disaggregation distributions
for each meta-state in explicit forms, which are not obtainable by classical
spectral methods. On the theoretical side, we prove sharp error bounds for
estimating the aggregation and disaggregation distributions and for identifying
anchor states. The analysis relies on a new entry-wise deviation bound for
singular vectors of the empirical transition matrix of a Markov process, which
is of independent interest and cannot be deduced from existing literature. The
application of our method to Manhattan traffic data successfully generates a
data-driven state aggregation map with nice interpretations.
