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Catastrophic events, though rare, do occur and when they occur, they
have devastating effects. The study of the critical
dynamics in complex systems is always interesting yet challenging.
First, we present a brief overview of the random matrix
theory and correlated Wishart ensemble. Then, we choose financial
market
as an example of a complex system, and do the
analysis of the S&P 500 (USA) stock market based on the evolution of
cross-correlation structure patterns. We identify “market
states” as clusters of similar correlation structures, which occur
more frequently than by pure chance (randomness). Power
mapping method from the random matrix theory is used to suppress the
noise on correlation patterns, and an adaptation of the
intra-cluster distance method is used to obtain the optimum number of
market states, and also identify the “precursors” to the
crashes. The dynamics of the transitions between the states are
interesting. Further, the resulting “emerging spectrum” of
eigenvalues near zero, have intriguing properties: (i) the shape of the
emerging spectrum reflects the market instability, (ii) the
smallest eigenvalue is able to statistically distinguish the nature of
a
market crash or crisis. We finally investigate whether the
smallest eigenvalue is able to predict a high market correlation, which
is a signature of a crash. |