Fractal Analysis of Time Series

Fractal Analysis of Time Series

Fractals are rough or fragmented geometrical forms that can be disassembled into their component parts, each of which in turn represents - at least approximately- a smaller copy of the form of the whole. An example of a fractal form is the fern frond illustrated on this page. Every partial leaf has almost exactly the same shape as the fern leaf as a whole. And this partial leaf itself consist again of smaller elements, which in turn are smaller copies of the partial leaf.

Other examples of fractal forms are snow flakes or even coastlines. By means of an analysis of a small part of the structure, it is possible to reconstruct the form at large.

Fractal analyses can also be used in the technical analysis of financial instruments or quite generally in the analysis of time series. The field of application includes predictions as well as the investigations of trends.

In the application for predictions, it is assumed that the course of the price curve of a stock, for example, follows fractal laws. Accordingly, it is assumed that it is possible to isolate small forms (patterns) in intraday charts, for example, and project these onto weekly or monthly charts. If the assumption of a fractal organization is correct, one would thus be able to predict with rather high probability what development the larger form (i.e. the weekly or monthly chart) will take.

Although some traders swear by such projections and some forecasting tools on the market build on this process in particular, it is difficult to confirm the results in a statistically significant way.

Another method also borrowed from fractal theory, on the other hand, performs very good, statistically verifiable work in the quantification of the dynamics of movements. Let us consider the following illustration:

It represents two alternative paths from point A to point D. In the first example, the shortest path is taken: a straight line. In the second example, the path is not straight. Although the shortest path is taken from one point to the next, the result of the movement as a whole is not the shortest, most effective path from A to D.

The ascertainment of the efficiency of movements is the subject of analyses of partial fractal efficiency. These investigations are based on the consideration that movements always become less efficient prior to changing direction. An example of this is the trajectory of a stone that is thrown. Immediately following the throw, the stone might still move in a straight line. With decreasing energy, however, it leaves the straight trajectory and begins to descend - slowly at first and then ever faster and steeper. Another example is a 100 meter runner. If the runner, following the final spurt, crosses the finish line at high speed, he cannot immediately turn around and sprint back to the starting line at the same speed. First he must decelerate, reverse direction and then accelerate again.

Financial markets follow the same laws, since the price of an instrument is determined by supply and demand and a change in price is brought about through business transactions that have been concluded. Hence, strong trends cannot reverse abruptly into a counter-movement of comparable strength.

In the illustration above, the area filled with red, which is delimited by three successive data points, can be used for the analysis of the efficiency - of the energy level, so to speak - of the movement. Among the indicators obtained from these efficiency analyses for the purpose of technical trading are the partial fractal efficiency and the polarized fractal efficiency. Provided that relevant sections of the time series are investigated, this variant of fractal analysis offers important information for recognizing and assessing trends as well as early warning signals of critical trend reversals.