# Technical Studies Reference

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# Cycle Period

This study calculates and displays a Cycle Period for the data given by the **Input Data** Input. This study is an ACSIL implementation of the Indicator given in Figures 9.4 and 9.5 of the book *Cybernetic Analysis for Stocks and Futures* by John Ehlers.

## On This Page

### Support Function: Dominant Cycle Period

Before discussing this study, we first give the mathematical details of one of its support functions, namely sc.DominantCyclePeriod().

Let \(X\) be any random variable, and let \(X_t\) be the value of \(X\) at Index \(t\). Let \(n\) denote a **Length**.

The function **sc.DominantCyclePeriod** computes the following quantities.

- The Quadrature Component, denoted as \(Q_t^{(1)}(X,n)\)
- The In-Phase Component, denoted as \(I_t^{(1)}(X,n)\)
- The Raw Phase Change, denoted as \(\Delta\phi_t^{(Raw)}(X,n)\)
- The Phase Change, denoted as \(\Delta\phi_t(X,n)\)
- The Median Phase Change, which is the \(n - \) bar Moving Median of the Phase Change, and is denoted as \(\Delta\phi^{(Med)}_t(X,n)\)
- The Dominant Cycle, denoted as \(DC_t(X,n)\)
- The Instantaneous Period, denoted as \(T_t^{(Inst)}(X,n)\)
- The Dominant Cycle Period, denoted as \(T_t^{(DC)}(X,n)\)

These quantities are computed as follows.

\(Q_t^{(1)}(X,n) = (0.0962X_t + 0.5769X_{t - 2} - 0.5769X_{t - 4} - 0.0962X_{t - 6})\left(0.5 + 0.08*T^{(Inst)}_{t - 1}\right)\)\(I_t^{(1)}(X,n) = X_{t - 3}\)

\(\displaystyle{\Delta\phi_t^{(Raw)}(X,n) = \left\{ \begin{matrix} \frac{\frac{I^{(1)}_t}{Q^{(1)}_t} - \frac{I^{(1)}_{t - 1}}{Q^{(1)}_{t - 1}}}{1 + \frac{I^{(1)}_t(X,n)I^{(1)}_{t - 1}(X,n)}{Q^{(1)}_t(X,n)Q^{(1)}_{t - 1}(X,n)}} & Q^{(1)}_t(X,n) \neq 0 \space and \space Q^{(1)}_{t - 1}(X,n) \neq 0 \\ 0 & otherwise \end{matrix}\right .}\)

\(\displaystyle{\Delta\phi_t(X,n) = \left\{ \begin{matrix} 1.1 & \Delta\phi_t^{(Raw)}(X,n) > 1.1 \\ \Delta\phi_t^{(Raw)}(X,n) & 0.1 \leq \Delta\phi_t^{(Raw)}(X,n) \leq 1.1 \\ 0.1 & \Delta\phi_t^{(Raw)}(X,n) < 0.1 \end{matrix}\right .}\)

\(\Delta\phi^{(Med)}_t(X,n) = MMed_t(\Delta\phi, n)\)

\(\displaystyle{DC_t(X,n) = \left\{ \begin{matrix} \frac{2\pi}{\Delta\phi_t^{(Med)}(X,n)} + 0.5 & \Delta\phi^{(Med)}_t(X,n) \neq 0 \\ 15 & \Delta\phi^{(Med)}_t(X,n) = 0 \end{matrix}\right .}\)

\(T^{(Inst)}_t(X,n) = 0.33DC_t(X,n) + 0.67T^{(Inst)}_{t - 1}(X,n)\)

\(T_t^{(DC)}(X,n) = 0.15T^{(Inst)}_t(X,n) + 0.85T_{t - 1}^{(DC)}(X,n)\)

\(T_t^{(DC)}(X,n)\) is the return value of **sc.DominantCyclePeriod()**.

### Main Study Function: Cycle Period

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) be the value of \(X\) at Index \(t\). Let the **Cyber Cycle 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.

\(\displaystyle{X^{(S)}_t = \frac{1}{6}(X_t + 2X_{t - 1} + 2X_{t - 2} + X_{t - 3})}\)We then compute the Cyber Cycle, \(CC_t(X,n_{CC})\).

Finally, we apply the Dominant Cycle Period function defined above to the Cyber Cycle to obtain \(T_t^{(DC)}(CC(X,n_{CC}),n_{Med})\). This is what is displayed as the Subgraph.

#### Inputs

#### Spreadsheet

The spreadsheet below contains the formulas for this study in Spreadsheet format. Save this Spreadsheet to the Data Files Folder.

Open it through **File >> Open Spreadsheet**.

*Last modified Monday, 26th September, 2022.