« Live Perspective Broadcast | Main | Keys to Trading Intraday Breakouts »


Polarized Fractal Efficiency

Date: September 20, 2005 14:00GMT

Expert: Erik Long, President of Tetrahex and Principal of Ceres Capital LLC

- We will discuss this useful indicator
- How it can be applied to all trading markets.

Who is Erik Long?
Erik Long has been active in the study of evolutionary feedback systems since his university days at UCLA and CSU from 1987-1992. After obtaining a Bachelors in Bio-Anthropology and Economics, Erik went on to study evolution of markets with the accomplished academic and entrepreneur Dr. Gerald Fecht and notable mathematician Dr. Viktor Fontaine. Erik founded Tetrahex in 1996 as a forum to discuss evolution, entropy, and later fractals in financial markets. In the process Tetrahex later formed into a corporation, offering it's services to private industry. In 1998 Erik studied for his Post Graduate Diploma in Economics at the University of London. Since that time Erik has developed proprietary trading systems for Altea Trading from 2000-2002. Erik has also consulted in non-linear methodologies for Nexbridge Inc., 2000-2002. Erik is one of the founding members of the System Select Group at Peregrine Financial Group, 2002-2004. Mr. Long became a founding principal of Ceres Capital LLC in 2004. Erik Long has been published in Futures magazine and Stocks & Commodities magazine. Mr. Long has been a sponsored member of the New York Academy of Sciences since 1997 and listed in Who'sWho in America since 2003. Erik is available for presentations on a case by case basis.

Speech Material:

There has been much talk lately about Chaos theory, fractals and financial markets. Although large amounts of information are available about the theory of these subjects, there is little in terms of practical application. Previous research has determined that most chaotic systems produce some form of graphic representation. A common example is the turbulent flow in a stream producing swirls, eddies and vortices. Unfortunately it has also been discovered that standard Euclidean geometry does not accurately represent chaotic events, so a new type of geometry is needed.

Benoit Mandelbrot recognized the problem and decided to formulate a new type of geometry that could solve the problem of describing chaos. In his pursuit, he realized that many of the problems being researched had a common graphic representation, the squiggly line. So he asked himself the question, “What is the dimension of a squiggly line?” His answer was the creation of a new type of geometry known as fractal geometry.

The problem can be represented in figure 1.

Click to enlarge:


A straight line has a dimension of one. A plane surface has a dimension of two. A squiggly line has a dimension of between 1 and 2, depending on how much squiggle is in the line. The dimension of the squiggly line is actually a fraction of a dimension between 1 and 2. This fractional measurement is commonly known as the fractal dimension of the line. More importantly, others discovered that systems with the same fractal dimension had other properties in common. This of course led to the importance of the fractal dimension as a tool in chaos work.

This is all very interesting, but as traders we are mainly concerned with the application of the fractal dimension to price motion. Fractal dimensions can be used to analyze the price action of any stock or commodity. Depending on the application, fractal dimensions can measure how trendy or congested price action is. In order to do this we must have a tool based on the fractal dimension. To develop this tool, we will look at a problem that interested Mandelbrot. The question is, “How long is a coastline?” The surprise answer is an infinite answer. In other words it can be measured over and over again with different scales of measurement.

To present this more clearly, please look at fig. 2

Click to enlarge:


Which depicts an island and two ways to measure the coastline. In this diagram, the coastline is measured with two different rulers. You can clearly see that the longer ruler will not measure the coastline as accurately as the smaller ruler. Because of this, the larger ruler will indicate a shorter length around the coast than the smaller ruler. Progressively shorter rulers will produce progressively greater distance. Theoretically this can be carried on indefinitely.

In an effort to find a relationship that he could quantify, Mandelbrot plotted the data on log-log charts. The outcome was a relationship between the length of the ruler and the length of the coastline that had a constant slope on the log-log chart. This relationship held up for any coastline. Because of his discovery, Mandelbrot was able to determine that the slope of the log-log chart was one minus the fractal dimension.

Please see figure 3 for a graphic example.

Click to enlarge:


Why is all of this important you may ask?
The answer is that price movement creates a graphical representation that is similar to coastlines. Based on this similarity, it only makes sense to measure price movement with the same approach as the coastlines. Simply put, price movement is a bunch of squiggly lines. A very squiggly line is symptomatic of congested price movement. A less squiggly line is characteristic of more “efficient” price movement. If I measure price from A to B and the price is a straight line, the price efficiency is 100%. Prices rarely do this however, and are usually some degree or fraction of a dimension between a 2d plane and a 1d line,

figure 4. Click to enlarge:


If we wish to measure price efficiency, all that we need to do is divide the length of the straight line by the length of the squiggly line. If we add a plus sign when it is moving up and a negative sign when it is moving down, we have polarized fractal efficiency (PFE).

Please refer to figure 5 to see how this is measured on a stock or commodity chart. We can measure directly from C1 to C2 to get distance B, or measure along each closed bar to get distance A. If we take distance B and divide it by distance A, the efficiency would be 36%. The fractal efficiency equation expresses this efficiency using logarithms. In terms of the coastline example, the measure along each closed bar represents the short ruler. The C1 to C2 distance represents the long ruler.

The mathematical equation for computing PFE is derived in figure 6. Each close to close line is treated as the hypotenuse of a triangle, the length of which is computed as the square root of the squares of the sides. Length A is the close to close length added together. Distance B is the hypotenuse of the triangle between the first and last close.

Applying the PFE is simply a matter deciding what lookback length to use in the chart and then working the calculation for PFE. Traders may also use filter techniques to remove noise caused by trends switching directions. As an example, Tetrahex has used wavelets, adaptive moving averages and non-linear regression with various degrees of success. The filtered PFE is easily plotted under a price chart in most common charting packages such as MetaStock or TradeStation. Dr. Al Larson, the inventor of PFE has found that a five-period exponential moving average is also useful for filtering market noise. Figure 7 illustrates his use of the PFE on the OEX index with a 10-day PFE. According to Dr. Larson, this is a reasonable compromise between computational delay, which is half the span, and usability of the indicator information.

Please note that Dr. Larson has discovered that all stock indices have approximately a 43% maximum efficiency going up or down. Other stock or commodities markets have various maximums, but a maximum is always observed. Bear in mind that the maximum is a commonly observed event, but is not a hard/fast rule. Variations are always possible. You will also see in Figure 7 that the PFE generally has a smooth transition from trending up to down or vice versa. It is possible however, for the PFE to bounce or hover around the zero line in the indicator. This zero level signifies a balance between the forces of supply and demand. Whenever the PFE maintains this zero state, the market is in a congestion zone.

Dr. Larson has made two other observations about the PFE. When the efficiency passes the 66% level in the OEX market, it tends to move sharply to 80%. In many cases the time from the sharp rise to the next top or bottom is a constant time period. This implies that maximum price velocity can only last for X amount of time.

The other behavior is a hook toward maximum efficiency right before the end of an efficient period. See point B in figure 7. This hook usually appears too late to be tradable, but may be used as a confirming indicator with other indicators. When using PFE to trade it is also advisable to place a stop at the point just beyond the hook and watch the PFE. If the PFE begins to trend and then starts to lose traction around the zero level, you may want to exit the trade. On the other hand, if it continues through the zero line, stay with the trade until it reaches efficiency on the other side. Stay in the trade until efficiency starts to reverse again.

Please bear in mind that PFE is not a leading indicator like the Fractal Dimension Index discussed in previous Q&A sessions. PFE is a measure of past price activity. It is recommended that this indicator be used with other indicators and/or with trailing stops. I have found personally that trailing stops and an adaptive moving average work well with this indicator in the forex markets. I encourage experimentation with PFE. You will find that it is easy to use and offers valuable information about any tradable market.

For your convenience I have included MetaStock code for creating the PFE indicator.

Erik Long and Tetrahex would like to give special thanks to Dr. Al Larson and Stocks & Commodities magazine for references in this Q&A presentation.

MetaStock code for creating the PFE indicator:

Polarized Fractal Efficiency

Mov(If(C,>,Ref(C,-9),Sqr(Pwr(Roc(C,9,$),2) + Pwr(10,2)) /
Sqr(Pwr(Roc(C,9,$),2) + Pwr(10,2)) /

Click image to enlarge:


Session Transcript:

FXstreet Moderator (Sep 20, 2005 10:09:56 AM)
Thank you for joining today’s Live Q&A Session. We will begin with the pre-registered users’ Questions in approximately 5 minutes. Meanwhile please read the text speech that you will find on the left of this chat window

FXstreet Moderator (Sep 20, 2005 10:10:28 AM)
Thank you for your participation.

FXstreet Moderator (Sep 20, 2005 10:10:38 AM)
Please find the whole speech of our guest in the left box of the chat. Be sure to read the whole speech at once without any interruption, making its reading and comprehension easy. You won’t miss a single thing from our guest host.

FXstreet Moderator (Sep 20, 2005 10:11:22 AM)
Our guest host will first start answering a selection of the pre-registered user’s questions.

FXstreet Moderator (Sep 20, 2005 10:11:35 AM)
When reading the speech please jot down any questions you may have so you are ready to post them live.

FXstreet Moderator (Sep 20, 2005 10:11:57 AM)
Our guest will answer live Questions sent to the FXstreet Moderator during the course of the session.

FXstreet Moderator (Sep 20, 2005 10:12:11 AM)
For those who are unfamiliar with chat sessions like this, please feel free to ask questions at anytime to the moderator.

FXstreet Moderator (Sep 20, 2005 10:12:28 AM)
At the end of the session, the link for the transcript and the PASSWORD will be posted

FXstreet Moderator (Sep 20, 2005 10:13:27 AM)
Today I am delighted to welcome back our good friend and guest speaker

FXstreet Moderator (Sep 20, 2005 10:13:39 AM)
Erik Long President of Tetrahex and Principal of Ceres Capital LLC

Erik (Sep 20, 2005 10:14:16 AM)
Thank you once again for the opportunity to be here everyone.

Erik (Sep 20, 2005 10:15:00 AM)
I look forward to answering any questions that you may have about the Polarized Fractal Efficiency.

FXstreet Moderator (Sep 20, 2005 10:15:14 AM)
We will begin with pre reg questions.

Erik (Sep 20, 2005 10:16:21 AM)
A method on how to trade Polarized Fractal Efficiency - and if you have an indicator that one can gain access to –

Erik (Sep 20, 2005 10:17:04 AM)
There are several ways to trade it, but one basic way is the following: If PFE is near or above the 50% + line and then you see the hook I mentioned in the paper, be prepared to enter a stop at the price at the hook. Enter the trade when the PFE starts to proceed toward the – direction (down). If the trade starts to hang up around the middle area, exit the trade. If on the other hand it continues toward the –50%, then stay in the trade. Hold the trade as long as the PFE stays below or right around –50%. If the PFE starts to turn up again, exit the trade. You can also use two at the same time. For instance one with a 5 period PFE and another with 15. This gives you two different scales to look at. The same technique can be used in the opposite direction. We also have an indicator that we can send to FXStreet guests for MetaStock and TradeStation. The code for a MetaStock version is also in the paper.

Erik (Sep 20, 2005 10:21:06 AM)
would like some insight into how to use PFE during both trending and sideways markets.

Erik (Sep 20, 2005 10:21:24 AM)
My understanding is that if the reading is >50 then the market is likely to reverse,

Erik (Sep 20, 2005 10:21:52 AM)
however it can stay above 50 for long periods during a trend.

Erik (Sep 20, 2005 10:22:08 AM)
So any ideas on how to use the PFE in both types of markets would be great.

Erik (Sep 20, 2005 10:22:36 AM)
No. If the market is trendy, the PFE will be above 50 or below it.

Erik (Sep 20, 2005 10:22:50 AM)
below -50 that is.

Erik (Sep 20, 2005 10:23:01 AM)
or in that general area.

Erik (Sep 20, 2005 10:23:30 AM)
remember that the OEX market tends to max out at 43%

Erik (Sep 20, 2005 10:23:51 AM)
This is because price efficiency is higher (closer to a straight line).

Erik (Sep 20, 2005 10:24:09 AM)
In a sideways market the PFE will gravitate around 0

Erik (Sep 20, 2005 10:24:31 AM)
I would not take any trades in this zone with this indicator, unless

Erik (Sep 20, 2005 10:24:47 AM)
you want to sell the highs and buy the bottoms

Erik (Sep 20, 2005 10:25:00 AM)
in the trading range.

Erik (Sep 20, 2005 10:25:27 AM)
This would be a scalping startegy.

Erik (Sep 20, 2005 10:25:35 AM)
strategy rather.

Erik (Sep 20, 2005 10:26:23 AM)
Also, shorter periods in the indicator will measure quicker responses.

Erik (Sep 20, 2005 10:26:42 AM)
eg a 5 period vs a 50 period

FXstreet Moderator (Sep 20, 2005 10:27:52 AM)
Please post your questions now LIVE and Erik will attempt to answer as many as possible.

Erik (Sep 20, 2005 10:27:59 AM)
Also, if you would like the code for TradeStation, I can send it to any guests.

Erik (Sep 20, 2005 10:28:09 AM)
the code for the PFE that is.

Erik (Sep 20, 2005 10:29:26 AM)
Also remember that the PFE can be filtered with other indicators.

Erik (Sep 20, 2005 10:29:46 AM)
I mention the example of the exponential moving average in the paper.

Erik (Sep 20, 2005 10:30:24 AM)
This is used to filter market noise when the PFE is beginning to change to a different trend or direction.

Erik (Sep 20, 2005 10:30:40 AM)
It helps from making false entries and exits.

Erik (Sep 20, 2005 10:30:55 AM)
We have used wavelets and adaptive moving averages.

Erik (Sep 20, 2005 10:31:17 AM)
The results have been encouraging.

FXstreet Moderator (Sep 20, 2005 10:34:06 AM)
LIVE questions:

haha (Sep 20, 2005 10:34:10 AM)
Chaos theory usually combined with Elliott wave, have you done research on PFE and elliott wave together and see what the results are

Erik (Sep 20, 2005 10:36:04 AM)
Although Elliot waves appear to be fractal in nature, I do not put much stock in them. Mostly because of the subjectivity involved. Although the PFE would be a great way to confirm that you are entering the next wave count. My immediate answer is that will help.

Erik (Sep 20, 2005 10:36:52 AM)
The PFE is very good as a confirming indicator.

orion (Sep 20, 2005 10:37:25 AM)
have you had success using the PFE on 1 minute charts, or what time periods work best?

Erik (Sep 20, 2005 10:39:10 AM)
I have traded 1 minute charts with it, although I believe that 1 minute charts are more difficult because of the market noise. You could help filter this by extending the time period on the one minute until the noise and false entry/exits appear filtered out. You should experiment with the number of periods in the PFE for each different time frame.

Erik (Sep 20, 2005 10:39:41 AM)
excuse me, extend the time period on the PFE.

Erik (Sep 20, 2005 10:39:57 AM)
lookback length

Erik (Sep 20, 2005 10:40:31 AM)
It might also be possible to create an adaptive PFE.

Erik (Sep 20, 2005 10:40:55 AM)
Please let me know if anyone does it.

aaaa (Sep 20, 2005 10:43:26 AM)
What is the usual PFE of Currencies (Forex) vs Stocks? And does the time frame matter? IE PFE of 1 minute vs PFE of 1 month chart?

Erik (Sep 20, 2005 10:45:25 AM)
Oddly enough the time frame does not appear to make much of a difference as long as the periods in the PFE are the same. All indices of stocks tend to max out around 43%. I have seen Forex go much higher, over 60. I think this because the currency markets are more efficient and also trend more frequently and longer.

orion (Sep 20, 2005 10:46:44 AM)
one could filter the PFE, or one could apply the PFE to a filter of the price. do both ways work, or which is better?

Erik (Sep 20, 2005 10:47:43 AM)
I filter the PFE. That is if you are using it as your primary trigger.

Erik (Sep 20, 2005 10:48:07 AM)
You want to smooth out the pfe when the trend is changing.

Erik (Sep 20, 2005 10:49:01 AM)
If you jump in a trade at the first sign of a change, you may get stopped out. Keep in mind that PFE is stilla trailing indicator. It shows you what the market is currently doing.

Erik (Sep 20, 2005 10:49:19 AM)
No forecasting here.

aaaa (Sep 20, 2005 10:50:37 AM)
Erik, so are currency markets more efficient (meaning they are harder to predict?) Or just the opposite? Thanks

Erik (Sep 20, 2005 10:51:39 AM)
Just the opposite. If a market is more efficent, it will not be as likely to move based on random noise in the market.

Erik (Sep 20, 2005 10:51:50 AM)
Or small news blurbs.

Erik (Sep 20, 2005 10:52:41 AM)
I always prefer to trade efficent markets, unless I have a strategy based upon the inefficient market

Erik (Sep 20, 2005 10:53:54 AM)
Scalpers in an open out cry trading pit make their money from an inefficient bid/ask system, generally.

Erik (Sep 20, 2005 10:54:41 AM)
electronic markets are much more efficient and better to trade.

Erik (Sep 20, 2005 10:54:54 AM)
As long as there is liquidity.

crazyfrog (Sep 20, 2005 10:56:19 AM)
what way and how do you filter the PFE

Erik (Sep 20, 2005 10:56:33 AM)
It is fairly easy to do.

Erik (Sep 20, 2005 10:57:17 AM)
Simply take the PFE indicator and insert it into another filter, e.g. exponential moving average.

Erik (Sep 20, 2005 10:57:38 AM)
It should be easy to do in most charting packages.

Erik (Sep 20, 2005 10:58:06 AM)
TS, MS, E-signal, etc...

FXstreet Moderator (Sep 20, 2005 10:58:24 AM)
Please post last questions now:

orion (Sep 20, 2005 10:59:49 AM)
i'm interested in creating an adaptive PFE, and I have the skills to implement it in C++, and i'd be glad to show you the results. i was wondering what exactly you would suggest adapting within the PFE to get me started?

Erik (Sep 20, 2005 11:02:53 AM)
Well, this may take some thought, but I would suggest the number of periods in the PFe need to adapt. What to base the adaptation on is the tricky part. I would think time periods would be something toexperiment with. The main concern should be volatitlity or cycles. A combination of the PFE with the Fractal Dimension Index or a Hurst measurement may prove very promising. Keep in touch.

Erik (Sep 20, 2005 11:03:33 AM)
Remember that the cycles are aperiodic if they exist at all at that time.

FXstreet Moderator (Sep 20, 2005 11:04:58 AM)
Last question:

orion (Sep 20, 2005 11:05:07 AM)
where exactly is the place to put stop entry for hook? is it right after the bend in the hook, or right after the sharp deviation away from the hook?

Erik (Sep 20, 2005 11:06:46 AM)
Right after the bend in the hook. When watching it live, you don't yet know if the PFE is going to continue toward the other side. The stop protects you in case you are wrong. At the sharp deviation you should already be in the trade in that direction.

FXstreet Moderator (Sep 20, 2005 11:08:16 AM)
Thank you very much for that, Erik

FXstreet Moderator (Sep 20, 2005 11:08:36 AM)
If your question was not answered during the course of this chat, please contact Erik

FXstreet Moderator (Sep 20, 2005 11:08:49 AM)
Email: Erik Long

Erik (Sep 20, 2005 11:09:50 AM)
I would like to thank all of you for attending. Please contact me with any questions. I look forward to our next meeting. Have a great day.

FXstreet Moderator (Sep 20, 2005 11:11:26 AM)
FEEDBACK: I cannot stress how important feedback is for the continued success of our Live Q&A sessions. In the past month, and the coming month watch out for new speakers as well as the regular and ever popular speakers. In order for us to continue providing informative and educational Q&A Sessions, feedback is vital. Please feel free to post feedback about today's session, topic ideas, & suggestions. We would appreciate this very much. I will be logged on for a while. claire@fxstreet.com

FXstreet Moderator (Sep 20, 2005 11:12:40 AM)
FXstreet.com is launching its new brand new exciting interactive Premium zone on Oct 3rd:

FXstreet Moderator (Sep 20, 2005 11:13:57 AM)
Right now we are offering limited places to users to take advantage and have a sneak preview of what is to come.

FXstreet Moderator (Sep 20, 2005 11:14:35 AM)
If you would like the opportunity , please email me and I will send you a log in password.claire@fxstreet.com

FXstreet Moderator (Sep 20, 2005 11:14:57 AM)
Fxstreet.com premium zone should be your new first-rate zone get in the forex zone.

FXstreet Moderator (Sep 20, 2005 11:15:13 AM)
Thank you for your participation.

FXstreet Moderator (Sep 20, 2005 11:15:24 AM)
• Transcript request: • Link: http://www.fxstreet.com/chat/transcriptrequest.asp • Password: Long159159 • Once inside the transcript request page, follow the instructions.

FXstreet Moderator (Sep 20, 2005 11:15:53 AM)
See you all Tomorrow Ed ponsi is back our FXeducator.

FXstreet Moderator (Sep 20, 2005 11:16:37 AM)
Also a fantastic new showroom ZONETICK of WR consulting

FXstreet Moderator (Sep 20, 2005 11:17:12 AM)
Please join us Tomorrow for an action packed day.

FXstreet Moderator (Sep 20, 2005 11:17:14 AM)
Thank you



Where are Figure 5, Figure 6, Figure 7 please ?

Maud Gilson

Hi Cris,

I'm sorry but it seems they are not on this transcript anymore. This is an old session with a speaker who no longer collaborates with us so I'm afraid it will be impossible for us to recover these figures.

Our apologies for the inconvenience.

Maud Gilson

A Prop Trader

interesting article. A company called Amcan produced an chart analysis indicator for TradeStation which calculated the PFE of price movement over time, which I studied a few years ago but never quite got my head around. This article explains the fractal concepts very nicely, thank you!

The comments to this entry are closed.


Weekly Schedule


  • Online Trading Academy Logo

    Rockwell Trading Inc Logo

    SpyGlass Logo

    Profiting With Forex