  Previous research explored various conditions of financial markets based on
the similarity of correlation structures and classified as market states. We
introduce modifications to previous selection criteria for these market states,
mainly due to increased attention to the transition matrix between the states.
Clustering and thus market states are fixed by the optimization of two
parameters -- number of clusters and noise suppression, but in similar
conditions, we give preference to the clustering which avoids large jumps in
the transition matrix. We found statistically significant results applying this
model to the SP 500 and Nikkei 225 markets for the pre-COVID-19 pandemic era
(2006-2019). Retaining the epoch length of 20 trading days but reducing the
shift of the epoch to a single trading day we are led to the concept of a
trajectory of the market in the space of correlation matrices. We may visualize
these states after dimensional scaling to two or three dimensions. This
approach, using dynamics, improves the options of risk assessment, opens the
door to dynamical treatments of markets (e.g. hedging), and shows noise
suppression in a new light.

Statistical Finance