The science of chaos means much more than a just new trading approach.

It is an entirely different way of viewing the market, that until the mid-1980s, we didn’t have the computing power needed to deal with on a mathematical basis.

Applying rigorous mathematical thinking to complex forms, chaos theory is the first technique that successfully models turbulent price flows.

One of the tools of the science of chaos is fractal geometry.

It is used to study events that only seem chaotic from the linear mathematics perspective and it has revolutionized research in various fields, including markets.

Linear mathematics has not been impressive when dealing with nonlinear turbulence.

To put it simply, a nonlinear phenomenon happens when the power of an effect is multiplied by the power of the cause.

While there is a direct link between cause and effect in Newtonian world, and all shapes can be reflected with regularity in Euclidian geometry – none of these approaches is very successful in explaining the behavior of markets.

The linear mathematics has been developed by abstracting out the elements deemed unessential to the system, while in reality, these discarded deviations from the norm represent the essential characteristics of the system.

Those are known as fractals now, and with their help we can get a glimpse into the underlying structure of market behavior.

It was discovered that at the border between conflicting forces there is a spontaneous emergence of new kind of organization, rather than the birth of chaos.

This self-organization is found in the exact places that were labeled random and discarded.

The stages that mark the beginning of turbulence can now be forecasted with better precision.

So let’s take a closer look at the typical issue with the linear approach and start to apply this new analysis to trading.

Solved by Mandelbrot, the problem of measuring the length of a coastline, seems silly at first glance, but in reality it raises very serious concerns about viability of traditional measurement for certain objects, and in particular, for financial markets.

If you are to attempt to measure the coastline with the 10-foot long rod, your calculations would have you arrive at a certain number.

Repeating the process with a yardstick will yield much larger measurement.

Using a one foot ruler would give an even bigger number.

The shorter is the measuring stick, the more detail is being captured.

**So what’s it all mean?**

This is an example of an object having an infinite length in a finite space and not being measurable in the Euclidian sense.

A new way to measure such irregular systems was invented by Mandelbrot.

It is called fractional dimension – a degree of irregularity.

As he looked at various problems being studied, he realized that many of these problems had to with determining object’s dimension.

Allow us to explain.

- A line that is straight has a dimension of 1.
- A plane surface has a dimension of two.
- A wavy line has a dimension between one and two, depending on how much it wobbles.
- The fractal geometry allows the line’s dimension to be not an integer like 1 or 2, but also a fraction, hence is the word fractional or fractal dimension.

It is discovered that the fractional dimension would remain constant regardless of degree of magnification when it comes to irregular object.

Turns out all irregularity is actually regular and instead of referring to an occurrence as random we can admit not understanding the structure of said randomness.

In terms of financial markets, this implies that the same pattern formation would be found in different time frames.

What it means is this.

A chart made of 5 minute bars would contain the same fractal patterns as, say, weekly chart, - an effect commonly known as ‘self-similarity’.

Mandelbrot also discovered a similarity between cotton prices and the fractal number of Mississippi River, which held throughout the century of floods, draughts and wars.

The profound meaning of this observation suggests that markets are nonlinear in nature, and just as Euclidian approach is incapable of measuring the coastline, neither it can accurately predict the market behavior.

It would be reasonable to propose that the nature of any pattern that is the result of human interaction in markets should also be fractal.

And the market produced by collective turbulence is, in fact, a nonlinear system.

A fractal is an object with parts that are related to the whole.

Fractals are self-similar.

A tree is one of the easiest fractals to imagine.

Each tree branch, including smaller scale branches, is similar to the entire tree in a mathematical sense.

When it comes to markets, we are interested in both fractal shapes and their space-related self-similarity, as well as, fractal time series which are self-similar in statistical sense.

Since markets are turbulent, nonlinear systems that are consequences of human beings interacting with each other, the behavior of the price is the perfect place to look for fractal structures.

The turbulent processes in nature produce complex, non-random structures that exhibit self-similarity.

But consider this.

- Determining the fractal structure of the market paves the way for understanding the behavior of the system, namely the price movement of any particular market.
- It offers a way of seeing the pattern and predictability where others can only observe the randomness.
- The science of chaos provides us with a new and more efficient paradigm to view the markets offering more accurate and predictable approach to analyzing the current price action.

It gives us a better roadmap to trading by concentrating on the current market behavior, which is a sum of the individual fractal behavior of traders.

Most chaotic phenomena can be described using some form of graphic representation.

Turbulent flow in liquids, for example, produces vortices, swirls and eddies.

However, it was found that Euclidean geometry simply could not help in describing or understanding those issues.

French mathematician Benoit Mandelbrot recognized this problem and solved it by developing fractal geometry.

To understand how such a tool can be developed, it is necessary look at a question that interested Mandelbrot.

The question was, "How long is a coastline?" Surprisingly, the answer was, "As long as you want to make it!", depending on if one uses shorter or longer rulers.

He discovered that for any given coastline, the mathematical relationship existed between the length of the ruler used and the length of the coastline.

But here’s something really interesting.

Market price movements are similar to the coastlines in their appearances.

Conceptually, the answer to a question "How much has price moved?" is "Depends on how we measure it."

Such paradigm shift led to the development of polarized fractal efficiency (PFE) concept.

If one looks at price motion from point A to point B, it can move in a straight line.

That is 100% efficient.

But you know what?

The prices don't usually do this.

They squiggle around, moving with less than 100% efficiency.

This efficiency can be measured by dividing the length of the straight line by the length of the squiggly line.

But measuring the line is the same problem as the coastline, so we need to use fractal dimension.

If we divide the length of the straight line (100% efficient) by the length of the squiggly line, we have a measure of fractal efficiency.

Finally, if we attach a plus sign when the move is down and a minus sign when the move is up, we have polarized fractal efficiency.

The PFE equation expresses the fractal efficiency using logarithms.

The close-to-close spacing represents our short ruler, while the first-to-last close spacing represents our long ruler.

Each close-to-close line is handled as the hypotenuse of a triangle, the length of which is computed as the square root of the squares of the sides.

Polarized fractal efficiency (PFE) is an indicator derived from fractal geometry, the mathematics that describes chaotic systems.

It was developed by Hans Hannula in 1994 with specific intent to quantify the efficiency of price movement and how trendy or choppy the recent past price action has been.

At its core, the polarized fractal efficiency indicator represents a complex variation of a price velocity oscillator, where a slower period change in the closing prices is divided by a faster period average change in price.

- PFE values above zero signal that the trend has been up.
- The higher the PFE, the more efficient the up trend.
- Conversely, the polarized fractal efficiency indicator readings below zero signal that the trend has been down, with lower PFE indicating more efficient downward move.
- The efficiency of a straight-line price drop is also 100%.
- The efficiency of a straight-line price rise is 100%.
- Values near zero signal price congestion, trend-less, choppy, inefficient movement.

To apply the PFE, we need to determine what number of bars to span and then calculate the PFE.

Using a five-period exponential moving average (EMA) for minor smoothing removes noise caused by the sign changes as trends switch directions.

The filtered polarized fractal efficiency indicator is plotted under price action to provide an idea of how efficiently a market is moving.

A balanced compromise between the usability of the indicator information and the computational delay, which is half the span, can be achieved by using 10-day PFE.

Here’s the most important part.

For stock indices, the polarized fractal efficiency indicator testing found the maximum efficiency to be around 43%.

Other financial instruments displayed tendencies to have slightly different maximums, but clear maximum has always been able to be observed.

The transition from bullish trend to bearish is usually smooth.

However, the efficiency may bounce off the zero line or oscillate around it for a time.

This middle region is a congestion point and represents a balance of supply and demand.

One of the behaviors useful to traders exhibited by PFE is that it seems to lock into being efficient.

After passing a certain threshold it tends to jump sharply, suggesting that maximum price velocity can only last for short bursts, just like a short-distance runner who is only capable of sprinting until he runs out of oxygen.

Using the indicator comes with caveat.

The polarized fractal efficiency indicator measures and reflects only what price has been doing in the past.

It does not possess predictive qualities.

The safe way to use it is to trade during the times of maximum efficiency, and to enter by following price action with a trailing stop.

After entering, place a stop and monitor PFE approaching zero.

If it crosses cleanly, the position is considered to be safe to hold until PFE reaches maximum efficiency in the opposite direction.

If price begins to enter into congestion in the vicinity of zero, close the position and wait for a new maximum efficiency entry.

Fractal dimension, the fresh and captivating way of studying markets, offers insight into price behavior in ways not found in other indicators.

As a concept, fractal efficiency has been found to be very useful in trading stocks, commodities and currencies.

This indicator represents one way in which a trader can start incorporating the science of chaos and fractals into their trading.

As you can see, backtesting is quite simple activity in case if you have the right backtesting tools.

The testing of this strategy was arranged in Forex Tester 3 with the historical data that comes along with the program.

To check this (or any other) strategy’s performance you can download Forex Tester 3 for free. In addition, you will receive 16 years of free historical data (easily downloadable straight from the software).

**We would love to hear from you about your experience using polarized fractal efficiency indicator. Please, leave us have your feedback. Any comments of observations would be greatly appreciated.**

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