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# Technical Studies Reference

### Cyber Cycle

This study calculates and displays a Cyber Cycle and Trigger Line for the data given by the Input Data Input. This study is an ACSIL implementation of the Indicator given in Figures 4.2 and 4.6 of the book Cybernetic Analysis for Stocks and Futures by John Ehlers.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be the value of $$X$$ at Index $$t$$. Let the Length and Lag Inputs be denoted as $$n$$ and $$\ell$$, respectively.

We first smooth the data using a Four Bar Symmetrical Finite Impulse Response Filter. The value of the smoothed data at Index $$t$$ is denoted as $$X^{(S)}_t$$, and we compute it as follows.

$$\displaystyle{X^{(S)}_t = \frac{1}{6}(X_t + 2X_{t - 1} + 2X_{t - 2} + X_{t - 3})}$$

We compute two Smoothing Factors $$\alpha(n)$$ and $$\beta(\ell)$$ (Greek letters alpha and beta, respectively) as follows.

$$\displaystyle{\alpha(n) = \frac{2}{n + 1}}$$
$$\displaystyle{\beta(\ell) = \frac{1}{\ell + 1}}$$

We will suppress the functional dependence of $$\alpha$$ and $$\beta$$ on $$n$$ and $$\ell$$, respectively, in the following formulas.

The Cyber Cycle and the Trigger Line at Index $$t$$ are denoted as $$CC_t(X,n)$$ and $$Trig^{(CC)}_t(X,n,\ell)$$, respectively. Both of these are displayed as Subgraphs, and we compute them as follows.

$$\displaystyle{CC_t(X,n) = \left\{ \begin{matrix} \frac{1}{4}(X_t - 2X_{t - 1} + X_{t - 2}) & t < 6 \\ \left(1 - \frac{\alpha}{2}\right)^2\left(X^{(S)}_t - 2X^{(S)}_{t - 1} + X^{(S)}_{t - 2}\right) + 2(1 - \alpha) CC_{t - 1}(X,n) - (1 - \alpha)^2 CC_{t - 2}(X,n) & t \geq 6 \end{matrix}\right .}$$

$$Trig^{(CC)}_t(X,n,\ell) = \beta CC_t(X,n) + (1 - \beta)Trig^{(CC)}_{t - 1}(X,n,\ell)$$