Forecasting of random sequences and Prony decomposition of finance data
Abstract In this paper, we propose an original method for forecasting data based on the Prony decomposition. We interpret the forecasting procedure as a possible prolongation of the fitting function that describes the known optimal trend in terms of the Prony decomposition. We give some arguments justifying the selection of the Prony method for obtaining the fitting function. This function should describe the segment of the random curve with high accuracy, that is supposed to be known and can be continued out from the fitting interval. Naturally, the boundaries of the sequential future interval are limited by the influence of future events that can change the tendencies “stretched out” from the known past, in general. This determination coincides with the definition of a forecasting procedure given by Chatfield. We establish the relationships between modes of the quasi-periodic Prony decomposition with the well-known Kondratiev or Elliot waves (K-, E-waves) that are known in economy for forecasting of long random time-series with trends. Our method adds new features to the conventional procedure and corrects the general conception of K(E)-waves for temporal cycles having different lengths. In fact, it helps us to find additional criteria for identifying the K-waves in the given random sequence analyzed. We propose two independent methods of forecasting: (a) from the segment that starts from the nearest past, and (b) from the curve that is remained as the self-similar curve to the initial one. In both cases we show that the results of such prolongation are similar to each other, which increases the reliability of the novel forecasting procedure. As an example we consider data associated with four basic fund markets.