Determination of parameters for the formation process of rare events in the economy for their subsequent forecasting

Abstract

  The article presents a method for determining unknown parameters of the process that forms rare events in the economy.The idea behind the study of rare events in economy is to consider these events not just from a statistical point of view, but from the point of view of the processes that form these events, moreover, as well as a process can be an arbitrary algorithm. Such an event generation process will use parameters, which can be static or dynamic. For example, if we consider the consumption process, which forms discrete purchases of an uncontrolled customer, then such parameters can be the maximum stock and a dynamically changing consumption rate. In general, the process can be arbitrary and possess various parameters. The task is to determine these parameters of an unknown process obtaining only a sample of rare events. The idea of the method is to minimize the loss function, which is determined based on the differences between the events generated during the operation of the process and events from the initial sample of observations. Each event, in addition to the time of occurrence, is also characterized by additional information, for example, the purchase volume. We are trying to find out such parameters of such a process that would allow us to get a very similar sample of events. The dynamic parameters of the process are set in the form of cubic splines of a special structure. For an unambiguous determination of each dynamic parameter, a roughness penalty of the corresponding splines is introduced into the objective function. An example of a process and its structure of parameters to be determined is shown. Optimization is performed numerically, based on the Nelder–Mead algorithm, which runs on a grid to determine the global optimum. The process parameters are determined in steps, at the beginning just to get a few events, then the next events. That allows one large optimization task to be divided into a sequence of simple tasks, this significantly reduces the overall complexity. An assumption is described that must be fulfilled for such a technique to be valid. An example of determining unknown parameters is considered on the example of the consumption process. After determining the process parameters, one can proceed to extrapolation of parameters and forecast future events.