Identifying economic periods and crisis with the multidimensional scaling

Abstract

This paper applied MDS and Fourier transform to analyze different periods of the business cycle. With such purpose, four important stock market indexes (Dow Jones, Nasdaq, NYSE, S&P500) were studied over time. The analysis under the lens of the Fourier transform showed that the indexes have characteristics similar to those of fractional noise. By the other side, the analysis under the MDS lens identified patterns in the stock markets specific to each economic expansion period. Although the identification of patterns characteristic to each expansion period is interesting to practitioners (even if only in a posteriori fashion), further research should explore the meaning of such regularities and target to find a method to estimate future crisis.

Self-similarity principle: the reduced description of randomness

“…It is necessary to find the fractal dimension, to determine the limits of applicability of the scaling properties and to prove the evidence/absence of log-periodic oscillations [9] that accompany any scaling process in time or space. This same phenomena was discovered in random economical and financial activities (see [10,[13][14][15][16][17][18][19], more references can be found in the review [1], and in papers [2][3][4]). Nowadays research in this field simply supposes, or postulates, the existence of scaling properties of the system studied.…”

“…The modification of the BLR in (9) along with the additional expressions in (14) helps to find the desired values of three important constants, C ( = 1 2 3) (the values of other constants;…”

Fractional dynamics and MDS visualization of earthquake phenomena

“…. , 50 meaning that the seismic events are modelled as Dirac impulses, Mkδ(t − tk), where Mk represents the magnitude, tk is the time of occurrence, parameter t represents time and T is the total time period of study, both expressed in seconds [47]. Hence, each xFE i (t) is a time-domain signal that corresponds to the sequence of all seismic events, registered in every F-E region, over the time of study.…”

Analysis of financial indices by means of the windowed Fourier transform

“…The importance of fractional order mathematical models is that it can be used to make a more accurate description and it can even give a deeper insight into the processes underlying long-range memory behavior [18,20] than classical approaches. The methods and algorithms that have been explored for descrip-tion of physical phenomena become an inspiration for very productive methods used in the analysis of economical data [1,10,21,24,31].…”