# Technical Studies Reference

### Polarized Fractal Efficiency

This study calculates and displays a polarized fractal efficiency for the data specified by the Input Data Input, as well as a moving average of the polarized fractal efficiency.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be the value of the Input Data at Index $$t$$. Let the Period Input be denoted as $$n$$. Then we denote the Polarized Fractal Effiency at Index $$t$$ for the given Inuts as $$PFE_t(X,n)$$, and we compute it for $$t \geq n$$ as follows.

$$\displaystyle{PFE_t(X,n) = \left\{ \begin{matrix} \frac{\sqrt{(X_t - X_{t - n})^2 + n^2}}{\sum_{i = t - n + 2}^t \sqrt{(X_i - X_{i - 1})^2 +1}} & X_t \geq X_{t - 1} \\ -\frac{\sqrt{(X_t - X_{t - n})^2 + n^2}}{\sum_{i = t - n + 2}^t \sqrt{(X_i - X_{i - 1})^2 +1}} & X_t < X_{t - 1} \end{matrix}\right .}$$

Let the Smoothing Period Input be denoted as $$n_S$$. We denote the Smoothed Polarized Fractal Efficiency at Index $$t$$ for the given Inputs as $$PFE^{(S)}(X,n,n_S)$$, and we compute it for $$t \geq n$$ in terms of an Exponential Moving Average as follows.

$$PFE^{(S)}(X,n,n_S) = EMA_t(PFE(X,n),n_S)$$

Note: Depending on the setting of the Input Moving Average Type, the Exponential Moving Average in the above calculation could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.