# Fractal-Wave Process

Figure 8-1: Fractal-Wave process is corresponding to price pattern (1, 5) in the table.

Fractal-Wave process is the representation of the Fractal geometry in the time dimension. Fractal geometry is made from a repeating pattern at many different scales. Simply speaking it is repeating patterns with varying size. Fractal geometry can be a self-similar pattern with the strictly same patterns across at every scale. Or if the pattern loosely matches to the past one, this can be still considered as fractal geometry. We call this as near self-similarity against the strict self-similarity. Many examples of Fractal geometry can be found in nature. Snowflakes, coastlines, Trees are the typical example of the Fractals geometry in space. Fractal-Wave is the fractal geometry generated in time dimension. Just like Fractal Geometry can be described by self-similar patterns. Fractal-Wave can be described by self-similar patterns repeating in time. The concept of Fractal-Wave can be illustrated well by Weierstrass function.

Loosely speaking, Weierstrass function is the cyclic function generated from infinite number of Cosine functions with different amplitude and wavelength. By combining infinite number of Cosine functions, it can generate a complex structure repeating self-similar patterns in different scales. This is a typical synthetic example of Fractal-Wave patterns with strict self-similar patterns. We present this function to help you to understand the properties of the self-similar process. The real world financial market shows the loose fractal geometry. They do not repeat in the identical patterns in shape and in size. The repeating patterns are similar to each other up to certain degree. Since Weierstrss function is the synthetic example for the strict fractal geometry, reader should note that Weierstrss function does not represent the real world financial market.

Figure 8-2: Weierstrass function to give you a feel for the Fractal-Wave process. Note that this is synthetic Fractal-Wave process only and this function does not represent many of real world cases.

In the real world application, Fractal-Wave process appears with the near self-similar patterns most of time. Therefore, detecting them is not easy. The Heart Beat Rate signal is one typical example of the Fractal-Wave process in nature (Figure 8-3 and Figure 8-4). If we zoom in on a subset of time series, we can see the apparent self-similar patterns. In Financial Market, Fractal-Wave process occurs frequently but the market typically shows the near self-similarity too. In terms of pattern shape, the financial market and Heart beat rate signal are different because their underlying pattern generating dynamics inside human organ and crowd behaviour are substantially different.

Figure 8-3: Self-similar process in Heart Beat rate (Goldberger et al, 2000).

Figure 8-4: Continuous Heart Beat time series.

Figure 8-5: Self-similar geometry in EURUSD daily series.

Fractal-Wave process is the basis for many popular trading strategies like Elliott Wave, Harmonic patterns, triangle and wedge patterns. However, these trading strategies were not well connected to existing science in the past. This was one of the important motivation for myself to step up to write and to connect these trading strategies into existing science. Most of time, day traders were referencing that these strategies will work because market repeats themselves. For me, this explanation would be not enough before I make the full time commitment on them.

Therefore, I had to do a lot of research to find the origin of these trading strategies and to see real value behind them. From my research, most likely, the repeating patterns in these strategies come from Fractal-Wave. I was very happy initially until I found that there is big difference between the underlying concepts of these trading strategies and Fractal-wave itself.

To see the difference, you might compare the heart beat signal and EURUSD price series in Figure 8-4 and Figure 8-5. You could also compare the heartbeat signal and S&P 500 series in Figure 8-6. With heart beat signal in Figure 8-4, you might think that it would look similar to Stock or Forex market data but different in some ways. If you felt this way, you are very correct. Here is the main difference.

Heartbeat signal seems to have fixed mean or horizontal patterns as we described in Stationary process in Chapter 5. However, EURUSD price series or S&P 500 series do not show any noticeable horizontal patterns. With human, the heartbeat increases dramatically to get more blood out during intensive activity like sprinting, but they must come back to normal range to maintain our life. Naturally, this makes up the horizontal patterns as we described in Stationary process. On the other hands, the operational range of the financial market does not have to create the horizontal patterns as if heartbeat does. Firstly, financial market is designed to grow or decline with time (Figure 8-6). Secondly, the underlying fundamental drives the rise and fall of the financial market (Figure 8-6) towards unknown destination.

With human, we know that our heartbeat rate must be kept between 60 and 100 times per minute. While you are a living man, this normal range is the destination of your heartbeat rate most of time. It is very predictable that your heartbeat will come back to 80 times per minute once it hits 130 times per minute. In the financial market, the final destination is not known and open. Even if S&P500 hits 3000 points tomorrow, we are not sure if it would even come back to 1000 points next year or ever. Hence, we are talking about similar but very different type of Fractal-Wave. Fractal wave alone can not explain the difference between these two. In the past, many day traders were having misleading picture between fractal wave and these trading strategies because no one or no literature attempted to give clear or easy explanation on them. This was another motivation for me to write this book too.

The open and unknown destination of the financial market is there because the underlying fundamental drives the market towards the price level, at which majority of people think it is most reasonable. In another words, market will move towards the equilibrium price. We have covered this in Equilibrium process in Chapter 8. Financial market is not only made up from Fractal-Wave but with Equilibrium process too. Hence, this is why financial market is different from heartbeat series although both have fractal wave in them. We have just pinned down the important concept of Equilibrium Fractal Wave process. In next two chapters, we will take you to the details of Equilibrium Wave process and Equilibrium Fractal Wave process to give you the complete and systematic view of the trading strategy and financial market.

Figure 8-6: S&P 500 growth last 70 years.

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__Analytical Note__

In the Fractal Geometry, two important quantifiable properties are fractal dimension and self-similarity. Fractal dimension is a measure of how much an N dimensional space is occupied by an object. For example, a straight line have dimension of 1 but a wiggly line has a dimension between 1 and 2. More precisely, their Fractal Dimension can be expressed as Log (N)/Log (r) where N = number of self-similar pieces and r = linear scale factor. Now to illustrate the concept, we will present the four cases from dimension 1 to 2. If we reduce size of straight line by linear scale factor r = 2, then then we get two self-similar piece, N = 2. Therefore, the dimension of the straight line is 1.

Figure 8-7: Iteration of self-similar process for straight line (Dimension = 1).

Likewise, if we remove the middle third of the straight line and replace it with two lines with the same length, then we get the first iteration of the Koch Curve. In this example, for the linear scale factor r = 3, we get the four self-similar pieces. Therefore, the dimension of Koch Curve is 1.26816 = Log (4) / Log (3).

Figure 8-8: Iteration of self-similar process for Koch Curve (Dimension = 1.26816).

Sierpinski Triangle is another good example of Fractal Geometry. If we connect the mid points of the three side inside an equilateral triangle, then we get 3 equilateral triangles inside the first equilateral triangle. Note that we do not count the resulting upside down triangle as a self-similar piece. In this case, the linear scale factor r is 2 since generated triangle have the half of the width and height. The Fractal dimension D = 1.585 = Log (3)/ Log (2).

Figure 8-9: Iteration of self-similar process for Sierpinski Triangle (Dimension = 1.585).

Finally consider a square. For the linear scale factor r = 2, we will get four squares inside the first one. Therefore, our fractal dimension for a square is D = 2 = Log (4) / Log (2). Even we make the linear scale factor r = 3, we still get the same Fractal dimension D = 2 = Log (9) / Log (3). As we have illustrated in the four examples, Fractal Dimension generalize the topological integer dimensions to a fraction. Higher the fractal dimension means that the fractal geometry to occupy more space in the dimension. Therefore, a square have the Fractal Dimension 2 as the smaller square fits larger one without leaving any empty space. If our fractal geometry is kept removing space, then we could get the Fractal Dimension less than 1 too.

Figure 8-10: Iteration of self-similar process for square (Dimension = 2).

If the fractal geometry is known like our four examples above, then calculating Fractal dimension for that fractal geometry is straightforward. However, in many practical applications, the self-similar patterns are not strict and difficult to identify. For example, the rocky coastline have the fractal dimension with near self-similarity. In such a case, we can still calculate the Fractal Dimension using some iterative methods. To calculate the length of wiggly line in the rocky coastline, one can measure the length by counting how many times a measuring stick with a known length can be fitted along the coastline: L(s) = N x s. By plotting the length of the coastline versus the length of the measuring stick on a double logarithmic scale, one obtains the following equations to calculate the Fractal Dimension for the coastline: Log (L (s)) = (1-D) Log (s) + constant. Therefore, the measured slope can be used to calculate the Fractal Dimension. For example, Figure 8-11 shows the plot of the length of the coastline versus the length of the measure stick on double logarithmic scale for different countries. For Great Britain, the Fractal dimension can be calculated as D = 1 – (-0.24) = 1.24. Higher the Fractal Dimensions means greater the jaggedness or non-smoothness of the coastline.

Figure 8-11: Measured slope of Log (L (s)) versus Log (s) for the coastline of the five countries including Australia, South Africa, Germany, Great Britain and Portugal. (Source: Vanderbilt University Webpage, http://www.vanderbilt.edu)

Calculating the Fractal Dimension for financial market data is not easy because it is harder to use such a measuring stick to measure the length of the line. Alternatively one might use Hurst Exponent (H) to calculate Fractal Dimension (D = 2–H) for a rough estimation of Fractal Dimension. Table 8-1 shows Hurst Exponents for several international Stock and Currency pairs. Using the relationship between Hurst Exponent and Fractal dimension, we can find that Fractal Dimension for many stock index lies somewhere between 1.2 and 1.5. Of course, this is very rough estimation only with some margin of error. Considering our four example earlier, we can imagine that the complexity of the financial price series might lies between Koch Curve and Sierpinski Triangle.

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Financial Instrument |
Start Time |
End Time |
FDI for D1 Timeframe |

EURUSD | 2006 7 21 | 2018 2 9 | 1.48 |

GBPUSD | 2006 7 20 | 2018 2 9 | 1.43 |

USDJPY | 2006 7 21 | 2018 2 9 | 1.47 |

AUDUSD | 2006 7 21 | 2018 2 9 | 1.47 |

USDCAD | 2006 7 24 | 2018 2 9 | 1.46 |

NZDUSD | 2006 7 20 | 2018 2 9 | 1.49 |

EURGBP | 2006 7 14 | 2018 2 9 | 1.48 |

USDCHF | 2006 7 20 | 2018 2 9 | 1.45 |

IBM Corporation | 2005 5 19 | 2018 2 9 | 1.46 |

Intel Corporation | 2005 7 8 | 2018 2 9 | 1.45 |

Microsoft Corporation | 2005 6 29 | 2018 2 9 | 1.39 |

Citi Group Inc. | 2005 7 6 | 2018 2 9 | 1.42 |

Walt Disney Company | 2005 6 9 | 2018 2 9 | 1.40 |

General Electric Corporation | 2005 7 13 | 2018 2 9 | 1.44 |

Coca Cola Company | 2005 7 11 | 2018 2 9 | 1.45 |

Boeing Company | 2005 5 20 | 2018 2 9 | 1.36 |

Average | 1.44 | ||

Standard deviation | 0.04 | ||

95% Upper Confidence Interval | 1.51 | ||

95% Lower Confidence Interval | 1.37 |

Table 8-1: Fractal Dimension index for D1 Timeframe

**About this Article**

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