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### 10.28 Polarized Fractal Efficiency

The Polarized Fractal Efficiency indicator by Hans Hannula shows how efficient, meaning how much like a straight line, the price movement has been over the past N days.

The net distance travelled over the past N days is expressed as a percentage of the total of each day’s distance travelled. Distance is measured in two-dimensions, like a ruler on the plotted graph. Rise (or fall) is expressed as a percentage, and each day counts as 1 unit across. So the formula, on closing prices p1 (today) to pN is

```                   Sign(p1-pN) * Hypot(N-1, Pchg(p1,pN))
PFE = 100 * -------------------------------------------------------
Hypot(1, Pchg(p1,p2)) + ... + Hypot(1, Pchg(p[N-1],pN))

Sign(X) = 1 if X>0, or -1 if X<0

new - old
Pchg(new,old) = 100 * ---------
old

Hypot(x,y) = sqrt (x^2 + y^2)
```

Here “Pchg” is a percentage change up or down from “new” to “old” price, and “Hypot” is the distance (the hypotenuse) for a move of X horizontally and Y vertically. “Sign” means that PFE is positive or negative according to whether the change over the past N days is up or down.

At the extremes of 100 or -100, price movement is at maximum efficiency, with the past N days making a perfectly straight line. An almost straight line is generally very close to 100 too. A midpoint of 0 means there’s been no net change over the past N days.

Hannula looked at price changes over 10 day period (horizontal distance of 9), and this is the default in Chart. A smoothing parameter is provided too; it applies an exponential moving average (see Exponential Moving Average) to the PFE. The default is 5 days smoothing, a value of 0 means no smoothing (to see the raw values).