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# Adaptive Cyber Cycle

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.

\(\displaystyle{CC^{(A)}_t(X,n) = \left\{ \begin{matrix} \frac{1}{4}(X_t - 2X_{t - 1} + X_{t - 2}) & t < 6 \\ \left(1 - \frac{\alpha^{(1)}}{2}\right)^2\left(X^{(S)}_t - 2X^{(S)}_{t - 1} + X^{(S)}_{t - 2}\right) + 2(1 - \alpha^{(1)}) CC^{(A)}_{t - 1}(X,n_{CC},n_{Med}) - (1 - \alpha^{(1)})^2 CC^{(A)}_{t - 2}(X,n_{CC},n_{Med}) & t \geq 6 \end{matrix}\right .}\)This study also displays a Center Line and the following Trigger Line.

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

*Last modified Monday, 26th September, 2022.