Using just the rules, we can claim that we are appljing Fuzzy logic to our trading; however, the state of the art mdicates that fiizzj logic is still hanging onto the coattails of simple probability. When it lets go, it is likely to be the most significant breakthrough in maiket -is.

FRACTALS AND CHAOS

Another new area that has captured the interest of maiket analjsis is chaos. Chaos thecrj is a way to describe the behavior of nonlinear sjstems, those that cannot be described by a straight line. Chaotic sjstems are not necessarily without any form or method, as the expression is commonly used, but those phenomena that are not a simple variation of a linear relationship. Some examples of this behavior will be given later in this section.

one method of measuring chaotic sjstems is with various geometric shapes. This effort has resulted in an area of mathematics now calledfiactal geometry; its approach shikes a true note about how the real world of numbers actually works. All of us have been taught Euclidean geometryin school; itistheworldof straight lines and clean edges in which we can measure the length of a line or the area of a rectangle very easily, in the real world, however, there are no straight lines; if you look closely enough, they all have ragged edges and all may be described as chaotic.

Fractal Dimension

in fractal geometry we find that there is a way of representing the irregularity of numbers, and the formations seen in nature. We first must accept the notion that there are no whole numbers in nature, that real-world objects are more likely to be described as fractional, or having a fractal dimension. The classic example of this is the algebra of coastline dimen

9 An excellent discussion of this topic can be found in Edgar Peters, Chaos and order in the Capital (John Wiley & Sons, New Yoik, 1991). The coastline example is originally credited to Benoit Mandelbrot,

sion. We will see that the questions "How long is the coastline?" and "How far did prices move?" are very similar

The answer to both these questions is "That depends on how it is measured.-Consider the problem of measuring the coastline of Australia using a laige wall map. if we take a 12inch ruler, we might find that the coastline is about 10 feet (perhaps 10,000 miles according to the scale). Had we taken a slightly smaller ruler we would have been more accurate and perhaps have found the coastline to be 11,000 miles; even a smaller ruler would have followed the contours better and found 12,000 miles of coast. As the ruler gets smaller the coastline appears to get longer. If we had an infinitely smaller ruler, the coast would be infinitely long. There is really no correct answer to the question, "How long is the coastline?

Fractal dimension is the degree of roughness or irregularity of a shncture or sjstem. In many chaotic sjstems, there is a constant fractal dimension: that is. the interval used for measuring will have a predictable impact on the resulting values in a manner similar to a normal disfribution. Therefore, using a 12-Inch ruler, we measure the coastline and get 10.000 miles:

24-inch ruler 5,000-mile coastline

12-inch ruler 10,000-mile coastline

6-inch ruler 20,000-mile coastline

Note that the large 24-inch ruler retums a value that is actually smaller than what we believe is a reasonable answer. This is because, when you place a long ruler from one point to another on the map, it cuts across part of the land mass.

Using Fractal Efficiency

In Chapter 17 (Adtive Techniques"), there is a discussion of Kaufrnans efficiency ratio. This ratio is formed by dividing the net change in price movement over n periods by the sum of all component moves, taken as positive numbers, over the same n periods, if the ratio approaches the value 1.0, then the movement is smooth (not chaotic); if the ratio approaches 0, then there is great inefficiency or chaos. This same measurement has more recently been called fractal efficiency.

Fractal efficiency measures the amount of chaotic movement in prices; this can also be considered similar to market noise, in Chter 17, Kaufinan related this to trending and nontrending pattems, when the ratio approadied 1.0 and 0, respectively. While eadi market has its unique underlying level of noise, the measurement of fractal efficiency is consistent over aU markets. Markets may varj in volatility, although their chaotic behavior is technically the same; therefore, the characteristics of one market may be compared with others by matching both the fractal efficiency and volatility.

Interpreting fractal efficiency as noise allows other trading rules to develop. For exanple, a market with less noise should be entered quickly using a frending sjstem, while it may be best to wait for a better price if the market is classified as having high noise. A noisj market is one that continues to change direction, while an efficient market is smooth. These characteristics are important in choosing trading rules and in turning a theoretical model into a profitable frading sjstem.

Chaotic Pattems and Market Behavior

Chaotic pattems are easj to imagine in the behavior of prices, but very difficult to measure. There would be no problem in predicting price direction if every participant reacted in the same way to the same event, much the way a single planet would smoothly orbit a single sun. In the real world nothing is quite as simple. Consider the pattem of prices represented as planets that are affected equally by two events, E 1 and E2. We will get a wobbly pattem whenever the planet passes across the midpoint where one attrador is sfronger

than the other, shown in Figure 20-7a. At point a, the object is most allected by the nearest atfractor, E 1, but as il circles it becomes closer to E2 and fries to form an orbit around it. The possible pattems are too complex and they varj based on the distance between E 1 and E2 and the size of E2 compared with E1. If attractor E2 is much larger than E1, there will simply be a distortion in the orbit around E2; if E 1 and E2 are the same size, objects a and b will switch orbits, forming figure eights.

As complex as these patterns might get, they are simple when compared with reality Each day brings events of various importance into the market, acting as attractors. Eadi attractor has an initial importance that loses value over time. To make matters worse, we cannot predict when a new attractor, or news event, will appear. This makes the chaotic pattem very similar to raindrops falling on a pond. Each new drop, equivalent to a new event, hits at an unpredictable time and place, with variable size, and forms circular ripples. These ripples dissipate as they get fiirther away from the point of contact in the same way that the importance of an event fades away over time. The interesting aspect of the raindrop analogj is that, while we cannot predict where the next raindrop will fall, once it has landed we can completely determine its effects-until the next drop hits. This is remarkably similar to the market. Less often there is a significant event, a price shodc that overwhehns the smaller events, the market noise, for a short time.

NEURAL NETWORKS

Another area of analysis that has continued to grow rapidly is neural networks. During the past 5 years there has been a deluge of material on the advantages of using neural net

; "f this section .ne based on PetmK.mfman.cni.nterTra.liiielJci.iw-Hill 1995 pp IM-l-li-

FIGURE 20-7 (a) Equal attractors cause a sjmmefric pattem that switches between E 1 and E2. (b) Very unequal atfradors show only a small dishirbance in a regular orbit.

*-Greater Giier \ sTtracilon atlraction \ lo E2

Greater

Greater \ aWadion atlraction \ to E2 loEl

Size ol ortil Eows reflecme etfect ol eventb

works as a tool for uncovering market relationships and finding better sjstems. Most of what has been said is true. This technique offers unparalleled power for discovering nonlinear relationships between any combination of fundamental information, technical indicators, and price data, its disadvantage is that it is potentially so powerful that, without proper control, it will find relationships that exist only by chance.

Although the idea and words for the computerized neural network are based on the biological functions of the human brain, an artificial neural network is not a model of a brain, nor does it leam in the human sense. It is simply very good at finding patterns, whether they are continuous or appear at different times.

The operation of an artificial neural network can be thought of as a feedback process, similar to Pavlovs approach to training a dog:

1. A bell rings.

2. The dog runs to one of three bowls.

3. If right, the dog gets a treat; if wrong, the dog gets a shock.

4. If framed, stop; if not trained, continue at step 1.

Terminologj of Neural Networks

The terminologj used in the computerized neural network is drawn from the human biological 1 , shown in Figure 20-8. The principal elements are:

Neurons, the cells that compose the brain, process and store information.

Networks, groups of neurons.

Dendrites, receivers of information, passing it directly to the neurons.

A-xons, which come out of the neuron and allow information to pass from one neuron to another.

Synapses, which exist on the path between neurons and may inhibit or enhance the flow of information between neurons. They can be considered selectors.

Readers are referred to E. Michael Azoff, Neural Network Time Series Forecasting of Financial Markets (John Wiley & Sons, 1994), for a thorough treatment of this subject, and Edward Gately, Neural Networks for Financial Forecasting (John Wiley & Sons, 1996), for amore infroductory approach.