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If you look at the formula of the Polarized Fractal Efficiency, you will notice that it is flawed.

This is an example that shows, how easily a false concept can make it into a journal such as Technical Analysis of Stocks and Commodities. The indicator was published in 1994, and neither the author nor the editor were aware of the nonsense that was published. Nonsense is good enough for retail traders, because they are happy, if any line is plotted on their chart, best if it is colored.

But now let us come to the problem. The author developed this formula by measuring the distance of two data points on a chart and using geometry. This caused the real problem. Let us for instance look at the numerator of the indicator, which is

SQRT[(Close[n] - Close[n-1]) *(Close[n] - Close[n-1]) - period * period]

The relative weight of the period is more important, if you use instruments which trade at low values, or if you switch to lower timeframes. The indicator will treat a 1-minute-chart of EURUSD differently than a monthly chart of YM. For the 1-min EURUSD chart the value of the numerator is exclusively determined by the period of the indicator. For the monthly chart of YM the value depends to a higher extent from the close values than from the period.

If you are only a little mathematically minded, you should realize that the concept of the indicator is flawed.

By the way, there is a whole lot of indicators which measure angles on charts, which suffer from the same fallacy. So whenever you hear the word angle in the context of technical analysis, you should be alerted.

I have then searched the internet to get my analysis confirmed by someone else, and I have been successful. William Eckhardt - one of the fathers of the Turtles - has done me that favour and identified the Polarized Fractal Efficiency as one of the black sheep that do not pass the coherency test (C-Test).

The C-Test

The following is an excerpt from an article by William Eckhardt from the blog findmin.blogspot.com:

FRACTAL EFFICIENCY
Sophisticated geometric constructions can have the same problems as simpler angular formulations. For a recent example, Hans Hannula defines a fractal efficiency indicator for a lookback of n days:
[Equation 1][IMG]http://photos1.blogger.com/blogger/5467/539/200/1.jpg[/IMG]
SQRT((Pn-P1)*(Pn-P1)+n)/SQRT((P2-P1)*(P2-P1)+1)+SQRT((P3-P2)*(P3-P2)+1)+...+SQRT((Pn-Pn-1)*(Pn-Pn-1)+1)

where P1,...,Pn are n consecutive closes. This expression is multiplied by -1 if Pn- P1 is negative.

And whatever the angular inclination of the first leg on the original chart, there exists a rescaling of the price axis in which this first leg climbs at more than 45 degrees. Thus, this rule is incoherent. It cannot stand up to the c-test.

Here, price and time variables are combined in such a way as to make it impossible for the indicator to retain the relative importance of price action and time change when both axes are rescaled independent of each other. The trader who prices a market in dollars finds that time components mean much more to that market's fractal efficiency than the trader who prices the same market in cents. This dimensional difference is not insignificant. Consider the following three-day price series: [IMG]http://photos1.blogger.com/blogger/5467/539/200/b.jpg[/IMG]

The first has a fractal efficiency of 1.420..., while the second has a slightly higher one of 1.425.... Now suppose we multiply all these prices by 100, signaling a unit change from dollars to cents. In these units, the first series (5000, 5060, 5080) has a fractal efficiency of 1.000..., but the second (5200, 5250, 5240) has a lower fractal efficiency of 0.667.... Thus, fractal efficiency ranks these price series in an irrational and incoherent manner.
The only way to rid equation 1 of this ranking incoherency is to eliminate the time terms. The modified formula reduces to:
[2][IMG]http://photos1.blogger.com/blogger/5467/539/200/2.jpg[/IMG]

Pn-Pn-1/ABS(P2-P1)+ABS(P3-P2)+...+ABS(Pn-Pn-1)
Simplified and divested of the surrounding incoherencies, the true formula becomes apparent. It is the net change divided by a close-to-close path length.

One More for The Garbage Heap of the History of Technical Analysis

So despite its fancy name, the polarized fractal efficiency is another indicator for the garbage heap of the history of technical analysis. It seems that it has just been created to further contribute to the confusion of retail traders.

The following 17 users say Thank You to Fat Tails for this post:

Thanks Fat Tails for the response. I'd be lying if I said I completely followed your logic. I'm sure it's sound, but the real question is, how does it perform in realtime. Does it completely miss the trend and give seemingly random signals? Does it have reasonable accuracy and allow you to get most of the trend. Does it signal too soon and will get you stopped out before price moves it your direction?

I appreciate your thorough analysis of it, but could you offer an explanation of it's performance that supports the theory?

The design of the indicator is inconsistent. It will give different signals for every instrument and for every timeframe.

If you take a standard indicator, such as the RSI, you will know that above 70 means overbought and 30 means oversold. This indicator here will not use 70/30 for all instruments, but 92/16 for EURUSD 1 min, 84/22 for EURUSD 5 min, 54/44 for ES 1 hour and so on.

The design of the indicator is flawed, so whatever you conclude it will only be vaild for a specific instrument and a specific time frame at a specific given time iin history.

To be clear: This indicator is just rubbish, and your question cannot be answered. I have only cited William Eckardt, because I feared that you would not believe in my analysis and let this indicator go where it belongs: to the trash can.

The following 3 users say Thank You to Fat Tails for this post:

It is also a standard NinjaTrader indicator. But this does not tell us that the indicator is correctly designed.

The commmunity of traders is made up self-taught gurus who make up for an inexhaustible supply of nonsense. All the software packages for sale are then pepped up with that nonsense, because that is what the customer wants. It is a bit similar to synthetic CDOs. They had to be created because there was such a huge demand for the repackaged subprime debt, as investors only were aware of the label awarded by the rating agencies. The only reason that the Polarized Fractal Efficiency is still around is its fancy label, which suggests a scientific background. Those who have no understanding of mathematics are easily impressed.

Let me show you an example. The two charts below show the Polarized Fractal Efficiency with default settings (14,10) on two charts, a 60 min chart of YM 09-11 and a 3-min chart of 6E.

Please have a look at those charts! Would you believe me, if I told you that both charts use the same indicator with the same settings? But that is the truth.

Consider the PFE as being like a chameleon. It becomes a different indicator for each instrument and timeframe.

The following 6 users say Thank You to Fat Tails for this post: