Characterizing financial crisis by means of the three states random field Ising model

We propose a formula of time-series prediction by means of three states random field Ising model (RFIM). At the economic crisis due to disasters or international disputes, the stock price suddenly drops. The macroscopic phenomena should be explained from the corresponding microscopic view point because there are existing a huge number of active traders behind the crushes. Hence, here we attempt to model the artificial financial market in which each trader $i$ can choose his/her decision among `buying', `selling' or `staying (taking a wait-and-see attitude)', each of which corresponds to a realization of the three state Ising spin, namely, $S_{i}=+1$, -1 and $S_{i}=0$, respectively. The decision making of traders is given by the Gibbs-Boltzmann distribution with the energy function. The energy function contains three distinct terms, namely, the ferromagnetic two-body interaction term (endogenous information), random field term as external information (exogenous news), and chemical potential term which controls the number of traders who are watching the market calmly at the instance. We specify the details of the model system from the past financial market data to determine the conjugate hyper-parameters and draw each parameter flow as a function of time-step. Especially we will examine to what extent one can characterize the crisis by means of a brand-new order parameter --- `turnover' --- which is defined as the number of active traders who post their decisions $S_{i}=1,-1$, instead of $S_{i}=0$.