# Technical Studies Reference

### Description

This study calculates and displays an Adaptive 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 10.1 and 10.2 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 Median Phase Change Length Inputs be denoted as $$n_{CC}$$ and $$n_{Med}$$, 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.

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

Second, we compute the Cyber Cycle $$CC_t(X,n_{CC})$$.

Third, we compute the Dominant Cycle Period for $$CC_t(X,n_{CC})$$. Explicitly, we compute $$T^{(DC)}_t(CC(X,n_{CC}),n_{MED})$$. This is where the adaptive nature of the study comes from. The period is not static, but rather it changes with changing market conditions.

Finally, we denote the Adaptive Cyber Cycle as $$CC^{(A)}_t(X,n_{CC},n_{Med})$$, and we compute it as follows.

Define $$\alpha^{(1)}_t(X,n_{CC},n_{Med}) = 2/(T^{(DC)}_t(CC(X,n_{CC}),n_{Med}) + 1)$$. We will suppress the parameters of $$\alpha^{(1)}$$ in the following formula.

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

This study also displays a Center Line and the following Trigger Line.

$$Trig^{(ACC)}_t(X,n_{CC},n_{Med}) = CC^{(A)}_{t - 1}(C,n_{CC},n_{Med})$$