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# Adaptive Center of Gravity Oscillator

### Description

This study calculates and displays an Adaptive Center of Gravity Oscillator and Trigger Line for the data given by the **Input Data** Input. This study is an ACSIL implementation of the Indicator given in Figures 10.3 and 10.4 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 \(n_{CG}\) 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 Center of Gravity Oscillator** as \(CG^{(A)}_t(X,n_{CG},n_{Med})\), and we compute it as follows.

Define the Integer Period as \(T^{(Int)}_t(X,n_{CC},n_{Med}) = \left\lfloor{\frac{T^{(DC)}_t(CC(X,n_{CC}),n_{Med})}{2}}\right\rfloor\). We will suppress the parameters of \(T^{(Int)}\) in the following formula.

For an explanation of the floor function (\(\left\lfloor{\space\space}\right\rfloor\)), refer to our description here.

\(\displaystyle{CG^{(A)}_t(X,n_{CC},n_{Med}) = \left\{ \begin{matrix} -\frac{\sum_{j = 0}^{T^{(Int)}_t - 1} (1 + j)X_{t - j}}{\sum_{j = 0}^{T^{(Int)}_t - 1} X_{t - j}} + \frac{1}{2}(T^{(Int)} + 1) & \sum_{j = 0}^{n - 1} X_{t - j} \neq 0 \\ 0 & \sum_{j = 0}^{n - 1} X_{t - j} = 0 \end{matrix}\right .}\)

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

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

*Last modified Sunday, 29th January, 2023.