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Register a new userOn Some Processes and Distributions in a Collective Model of Investors Behavior
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This article considers a model for alternative processes for securities prices and compares this model with actual return data of several securities. The distributions of returns that appear in the model can be Gaussian as well as non-Gaussian; in particular they may have two peaks. We consider a discrete Markov chain model. This model in some aspects is similar to well-known Ising model describing ferromagnetics. Namely we consider a set of N investors, each of whom has either bullish or bearish opinion, denoted by plus or minus respectively. At every time step each of N investors can change his/her sign. The probability of a plus becoming a minus and the probability of a minus becoming a plus depends only on the bullish sentiment described as the number of bullish investors among the total of N investors. The number of bullish investors then forms a Markov chain whose transition matrix is calculated explicitly. The transition matrix of that chain is ergodic and any initial distribution of bullish investors converges to stationary. Stationary distributions of bullish investors in this Markov chain model are similar to continuous distributions of the theory of social imitation of Callen and Shapero. Distributions obtained this way can represent 3 types of market behavior: one-peaked distribution that is close to Gaussian, transition market (flattening of the top), and two-peaked distribution.
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We calculate a measure of statistical complexity from the global dynamics of electroencephalographic (EEG) signals from healthy subjects and epileptic patients, and are able to stablish a criterion to characterize the collective behavior in both groups of individuals. It is found that the collective dynamics of EEG signals possess relative higher values of complexity for healthy subjects in comparison to that for epileptic patients. To interpret these results, we propose a model of a network of coupled chaotic maps where we calculate the complexity as a function of a parameter and relate this measure with the emergence of nontrivial collective behavior in the system. Our results show that the presence of nontrivial collective behavior is associated to high values of complexity; thus suggesting that similar dynamical collective process may take place in the human brain. Our findings also suggest that epilepsy is a degenerative illness related to the loss of complexity in the brain.
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