Exploring the Higher Reconstruction Step (HRS) in time series.

Author:Filho, Antonio Carlos da Silva

    Long-term dependence can be defined, according to Mandelbrot and Hudson (2006), as the property that certain series have in which the dependence between observations away from each other slowly decreases with the distance between them. Such studies on the long memory processes began with the study by Hurst (1951). This work began with a biologist named Harold E. Hurst, who worked on a project to build an ideal reservoir. The reservoir had to adopt a policy according to the following flow condition: do not let it overflow or without water. By doing this analysis, Hurst realized that the rain in the region did not behave randomly but had a tendency. But many natural phenomena exhibit the characteristic of bias in their distributions over time. Hurst discovered the presence of memory in the behavior of the Nile floods, as evidenced by a parameter proposed by him and known today as the Hurst exponent.

    Mandelbrot and Hudson (2006), trying to study the memory in financial data, enhances the study of Hurst (1951) data for financial time series. The problem then would be like searching for and identifying periods of time during which a series of financial asset returns exhibit a dependence on long-term memory (LIMA, 2011).

    This paper aims to examine a new parameter, the Higher Reconstruction Step (HRS), correlating their values to predictions made (by any method, in principle) in two categories, the "good" and the "bad" ones (the criterion for this separation is another parameter to be analyzed here). The fundamental question here is: the HRS can be an indicator of the quality of the predictions?

    The computation of the HRS was made in windows (with fixed amount of points) which traveled throughout the series until its end. A given window was used for calculating the HRS and simultaneously to make forecasts for some values ahead. The forecasts had errors which have been grouped into two or more categories according to their magnitude. Finally, we study the existence of some correlation between the values of the HRS and the average values of the errors in the various categories. Cajueiro, Tabak and Souza (2005), for example, used windows of 504 data to calculate the Hurst exponent because, following these authors, 504 is a good approximation for two years considering only the days of financial market activity. In this work we varied the sizes of the windows (keeping the size fixed along each scan series) because it could be that for some window size the HRS would not be a good indicator, while it could be so for other sizes.

    Using the number 504 as an example, the HRS values were calculated for blocks [1, ..., 504], [2, ..., 505], until the end of the series. For each block of data the was calculated for the first 503 values and the prediction was made one steps forward, for the 504[degrees] value in the window (the last one) using polynomial regression.


    The motivation for the definition of the HRS comes from the study of Dynamical Systems. A well-studied type of solution...

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