Fractal Dimension Index

» Posted by on Mar 3, 2013 in References | 0 comments

by Erik T. Long

Welcome to the world of Chaos. What is Chaos? Chaos is lightening, weather patterns, earthquakes, and financial markets. In short, Chaos is a nonlinear dynamic system that appears to be random when in effect is a higher form of order.

Complexity is the point where the transition to Chaos takes place and is really what we are interested in when studying markets. Order and randomness exist simultaneously, allowing a degree of predictability. Both social and natural systems including private, governmental, and financial institutions fall within this category. People design these complex systems only to find that they take on a life of their own. Each of these networks is sustained by complex feedback loops, re-entering the system at unpredictable points in their cycles. Because of this little understood abstraction, people go about their business bewildered by the very systems they helped to create.

Chaos is the realm of the nonlinear and therefore important to us as investors. Because financial markets are chaotic, we should use nonlinear tools to forecast market dynamics. Linear tools are inefficient because they are the equivalent to describing objects in our world with Euclidean geometry.

What is needed are tools that identify the hidden order in the apparently random process of the markets. The first step toward that goal is determining what underlies a nonlinear, chaotic system.

Almost all chaotic systems have a quantifying measurement known as a fractal dimension. A fractal is an object in which individual parts are similar to the whole. The fractal dimension is a non-integer dimension that describes how an object takes up space. The manifestations of our world are not simple Euclidean objects with perfect symmetry. On the contrary, objects in our space (and the systems that create them) are infinitely complex. If you examine any object with a microscope, more detail is revealed as the scale changes. In addition to various levels of detail, most objects in nature demonstrate self-similarity. Self-similarity is the organizing principles of fractals. Because of this, fractals will maintain their same dimension regardless of the scale used.

Regularity in irregularity is important as each time frame in a market will have a similar fractal pattern. This also tells us that markets are natural phenomena rather than mechanical processes. The significance of this is high since most that use technical analysis lose money consistently. Why ? These traders are applying linear mathematical principles to a nonlinear natural system. When we apply fractal geometry to the markets, we use the right tool to forecast price movement more accurately. These are the principles that form the groundwork for Fractal Finance.

One tool that allows you to trade on the cutting edge of Chaos and Fractal theory is the Fractal Dimension Index (FDI). This specialized indicator identifies the Fractal Dimension of the market by using re-scaled range analysis and an estimated Hurst exponent. FDI uses all available data on the time/price chart to determine the volatility or trendiness of a given market.

FDI is the same type of tool used by Mandelbrot, Hurst, and Peters in their examination of time series analysis. With FDI you can determine the persistence or anti-persistence of any equity or commodity that you display in your graphing program. A persistent time series will result in a chart that is less jagged, resembling a straight line and subject to fewer reversals. An anti-persistent time series will result in a chart that is more jagged and prone to more reversals. Essentially, FDI will tell you whether a market is a random, independent system or a system with bias.

The history of the FDI begins with a British damn builder and hydrologist H.E. Hurst (1900-1978). He worked on the Nile River Dam Project in the early 20th century. Hurst searched for patterns in the Nile delta as an effort to solve a hydrology problem.

The problem involved the storage capacity of the damn reservoir. This is important to hydrologists because if the damn is too high, resources are wasted. If the damn is too low, water will overflow. To solve this problem, Hurst considered the relationship between annual rainfall, the extremes of high/low water, and the reservoir’s level.

Most hydrologists assumed that water inflow was a random process with no underlying order. Hurst came to a very different conclusion after studying almost a millennium of Nile overflows. He found that large overflows tend to be followed by more large overflows. There appeared to be cycles, but their lengths were non-periodic. Standard statistical analysis revealed no patterns between observations.

Hurst developed his own analytical method to explain the non-periodic cycles. To identify a non-random process, he tested the Nile using Einstein’s work on Brownian motion. Brownian motion is a widely accepted model for a random walk. Einstein found that the distance a random particle travels increases with the square root of time used to measure it. This is called the T to the one-half rule and is commonly used in finance and economics.

Hurst divided the Nile data into segments and examined the logarithmic range and scale of each segment in comparison to the number of total segments. The process is called re-scaled range analysis. The range is re-scaled because it has a zero mean and is expressed in terms of local standard deviation.

The re-scaled range value scales with an increase in the time increment, by a power-law value equal to H. H is referred to as the Hurst exponent. Using re-scaled range analysis, Hurst showed that the water overflows tended to repeat, meaning that the natural overflows were partially predictable.

The famous mathematician Mandelbrot, used the Hurst exponent to experiment with time series found in nature including cotton prices. By virtue of his experiments, Mandelbrot developed a method to measure irregular natural objects. He named the measurement the Fractal Dimension. FDI is based on the work of Mandelbrot and Hurst.

To use the FDI for trading, you need to learn how to compute it first. The formula is not terribly complex and can be programmed into a variety of trading platforms including TradeStation, MetaStock and CQG. To begin, you must compute the Hurst exponent. This can be arrived at with the following formula.

The Fractal Dimension in mathematical formula

Xt,nN= i Epsilon u=1 (Eu – MN)

Where Xt,N = Cumulative deviation over N periods Eu = influx in year u MN = average Eu over N periods

The range then becomes the difference between the maximum and minimum levels above

R = Max(Xt,n) – Min(Xt,n)

Where R = range of X Max(X) = maximum value of X Min(X) = minimum value of X

In order to compare different types of time series, Hurst divided this range by the standard deviation of the original observations. This “rescaled range” should increase with time. Hurst formulated the following relationship:

R/S = (a*N)^H

Where R/S = rescaled range N = number of observations a = a constant H = Hurst exponent

H should equal .5 if the series is a random walk. In other words, the range of cumulative deviations should increase with the square root of time. A deviation from .5 proves that each observation carries a memory of all the events that precede it.

The impact of the present on the future can be expressed as a correlation:


Where C = correlation measure H = Hurst exponent

We can estimate the Hurst exponent by:

Log(R/S) = H * log(N) + log(a)

We can also estimate the value of H from a single R/S value by:

H = log(R/S)/log(n/2)

Where n = number of observations. This is what we do with Fractal Finance.

The equation assumes that variable a above is equal to 0.50

The fractal dimension is simply D = 2 – H

For the markets, I use logarithmic returns instead of percentage changes in prices:

St = ln(Pt/P(t-1))

Where St = logarithmic return a time t Pt = Price at time t

This is because the range used in R/S analysis is the cumulative deviation from the average, and logarithmic returns sum to cumulative returns while percentage changes do not

Once you have computed the Hurst exponent, you must derive the fractal dimension of the time series. This is easily accomplished with the formula D = 2-H. This number is the fraction of a dimension between 1 and 2 that your price data represents. In computing the FDI, we use logarithmic returns as mentioned in the above formula. Because logarithmic returns sum to cumulative returns, most analyst agree that this is more appropriate for financial analysis. Although you may substitute price data for logarithmic returns, this is not recommended.

When experimenting with FDI, remember that a Hurst estimate is used. This may produce strange results if not enough data is included. How much data is needed is speculative at best and still debated among chaoticians and analysts. J. Feder believes any data with less than 2,500 observations is questionable.

Another postulate is the length of time necessary for each data period (periodicity). Some researchers believe that smaller time periods such as daily data are subject to more noise from random information. If this is the case, FDI is less accurate with finer slices of sequential data. The individual length of each period will come into play as noise is filtered.

The FDI is useful for traders because it determines the amount of market volatility. The easiest way to use this indicator is to understand that a value of 1.5 suggests that the market is acting in a completely random fashion. As the market deviates from 1.5, the opportunity for profit earning is increased in proportion to the amount of deviation.

The entire scale is based on a range of one to two suggesting extreme linearity to extreme volatility. An example of this scale is its use in geography. If you examine an island and plot the fractal dimension, you will be able to determine how jagged the edges of the island are for a particular measurement scale. An island with a 1.7 fractal dimension is highly jagged with many peaks and troughs on the periphery. An island with a fractal dimension of 1.3 is much more linear, approaching a single dimension or a straight line. If you examine this island on a map, the coastline will be straighter.

Applying FDI to the market is similar to studying an island on a map. The price plot is analogous to the periphery of the island. The FDI indicator then determines how close the price plot is to two dimensions (a plane) or one dimension (a line).

Because the price plot will never be one extreme or the other, we need to measure the “fraction” of the dimension. That is why we call the FDI number a fractal dimension. The further away this dimension is from 1.5, the more confident we can be that the market is not random.

When a market is not random it is more predictable. In the case of the FDI a fractal dimension closer to two may provide substantial opportunities because of the high volatility and changes in market movement. An FDI closer to one signals a trending market that is moving in one direction.

With the FDI, you may now trade on the cutting edge of fractal analysis and Chaos theory. If you want to apply the FDI as easily as possible, simply follow the above instructions. This is just one market application of this powerful tool. Experimentation is encouraged.

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