“Reverse Engineering” in Econophysics

Methodology

The proposed hereby methodology for predicting specific characteristics of a system by utilizing past-time data, by exploiting nonlinear dynamics for expanding the time horizon, is briefly aposed (in the form of steps) in the lines that follow:

• Step 1: Few real data for all the variables of the system is collected.

• Step 2: The proper set of differential equations as this is “guessed” by theoretical considerations or by trial and error, is constructed.

• Step 3: Initial values of all the parameters are set; some of them maybe partially known.

• Step 4: The system’s set of differential equations is solved, and the solution is compared to the real data (validation process).

• Step 5: The inverse modeling procedure is applied for all or some of the system’s parameters.

• Step 6: The system’s set of differential equations is solved again with the new fitted parameter values (testing).

• Step 7: Solving the set of differential equations is de facto expanding the time horizon, thus forecasting of the values of one or all of the variables is possible.

This methodology supposes that we have collected some real data for all variables of the specific system. But in case where we have collected one’s variable data only, then we must expand the methodology to include the other unknown data. In this case we can reconstruct the phase space according Taken’s theorem (Abarbanel, Brown, & Kadtke, 1990; Takens, 1980). From the reconstructed timeseries one can choose some points as real data. Then we can repeat the steps all over again.

It is apparent that this methodology is a kind of reverse engineering, exploiting at the same time the ability that deterministic chaotic systems have, to reconstruct the equivalent dynamics by one or more system variables.