It would be interesting to test this method on a stock market index offering a continuity throughout the twentieth century and compare this series and its decomposition with the historical events which impacted this financial index and the economy. The French index does not allow probably this experiment, indeed, it would be necessary to study the continuity between the French index before and after 1988, since in 1988 a new index has been created (i.e. Cac40). The Ramsey and Zhang work (using the Mallat and Zhang algorithm) is based on the S&P500 index of the United States which seems to have a better structural continuity. If one evokes the growth rate of the index, it is possible to observe that the very dense signal is traversed by fast oscillations with localized bursts of very large amplitude. This type of behavior favors the use of analysis tools such as the waveform dictionaries, i.e. the time-frequency atoms dictionaries. The Dirac Delta function have a particular interest in this signal decomposition method to understand these types of behavior.
6.3.1 The Dirac Function Would Allow to Distinguish Isolated and Intense Explosions: Internal Shocks and External Shocks
J.B. Ramsey and Z. Zhang distinguish for a signal between "isolated impulses (i.e. isolated bursts or explosions)" and "intense impulses" (i.e. explosions of high intensity energy). The first one being able to be represented by Dirac delta functions which indeed involve all the frequencies of the spectrum, and the second one involving almost all the frequencies but relatively less at the very high and very low frequencies. The distinction enunciated above comes from the observation of stock market crises. At the observation, the stock exchange crises are periods of high intensity energy which appear "without apparent warning", such that Ramsey and Zhang explain it, and go up very quickly and die out equally abruptly. Nevertheless, these bursts are not the result of isolated impulses represented by Dirac delta functions. In fact, there is many cases of isolated impulses in a stock exchange index but their occurrence is separated in general from intense bursts.
An interesting lighting is that the crisis of 1929 and that of 1987 do not show, by means of their decomposition, significant Dirac delta functions, even if the two periods of crises are the center of narrow zones very localized in time with many energy bursts. This indicates that the major crises of the stock exchange and even the smaller crises are not the result of isolated impulses. Thus, by means of the observation, the intense bursts cannot be associated with isolated impulses, or external shocks. Ramsey and Zhang explain that the intense bursts of high intensity energy which are prominent in the stock-exchange times series seem to be internal to the systems. Thus, according to their own terms, it is possible "reasonably to infer from these results that bursts of intense activity are not the result of isolated, unanticipated, external shocks, but more likely the result of the operation of a dynamical system with some form of intermittency in dynamics". The Ramsey and Zhang analysis conclude that the data of financial markets do not follow a random walk, even if their structure is very complex. This observation is based on the fact that the number of "waveforms" necessary to decompose a financial series is smaller than to decompose a random series. This result is stronger when the period of the financial index observed is rather stable but is traversed by some very strong very localized bursts. That means that when these strong energy explosions occur without possibility to anticipate them by means of the period which precedes and when the following period shows also a strong stability. Nevertheless, these bursts are not the result of isolated impulses which would be represented by Dirac Delta functions. There exist however many cases of isolated impulses, but their occurrence generally are separated from the intense energy explosions that we mentioned previously. The occurrence of Dirac delta functions would represent impulses or shocks of the system from external sources, whereas on the other hand the bursts of high intensity energy, which are preeminent, seem to be internal to the system. These explosions contain high energy at most frequencies, although relatively less at very low and very high frequencies.
Even if this analysis poses many questions, the interesting result shows that the isolated impulses which can be formalized by Dirac Delta functions could be the expression of external shocks which cannot be anticipated. On the other hand the explosions of intense activity are not the result of isolated, unanticipated, external shocks, but according to the own sentence of the authors, "but more likely the result of the operation of a dynamical system with some form of intermittency in dynamics".
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