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# Schaff Trend Cycle

### Description

This study calculates and displays a Schaff Trend Cycle of the data specified by the **Input Data** Input.

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) be the value of the **Input Data** at Index \(t\). Let the Inputs **Length**, **Short Cycle Length**, **Long Cycle Length**, and **Multiplier** be denoted as \(n\), \(n_S\), \(n_L\), and \(v\), respectively. We begin by comuting the MACD, denoted as \(MACD_t(X,n_F,n_S)\).

**Note**: The default Moving Average Type used in calculating the MACD is the Exponential Moving Average, but depending on the setting of the Input **Moving Average Type**, this could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

For this study, we use variants of the Fast %K and Fast %D that differ slightly from those found in KD - Fast. We define these functions for a generic random variable below.

For a random variable \(Z\), the versions of Fast %K and Fast %D used in this study are denoted as \(Fast\% K_t(Z,n)\) and \(Fast\% D_t(Z,n,v)\), and they are computed as follows.

\(Fast\% K_t(Z,n)\) is computed using the Moving Maximum and Moving Minimum functions.

\(\displaystyle{Fast\% K_t(Z,n) = \left\{ \begin{matrix} \frac{Z_t - \min_t(Z,n)}{\max_t(Z,n) - \min_t(Z,n)} & \max_t(Z,n) - \min_t(Z,n) > 0 \\ \frac{Z_{t - 1} - \min_{t - 1}(Z,n)}{\max_{t - 1}(Z,n) - \min_{t - 1}(Z,n)} & \max_t(Z,n) - \min_t(Z,n) \leq 0 \end{matrix}\right .} \)\(Fast\% D_t(Z,n,v)\) is computed using a unique averaging procedure.

\(\displaystyle{Fast\% D_t(Z,n,v) = \left\{ \begin{matrix} Fast\% K_t(Z,n) & t = 0 \\ Fast\% D_{t - 1}(Z,n,v) + v \cdot (Z_t - Fast\% D_{t - 1}(Z,n,v)) & t > 0 \end{matrix}\right .}\)We use these functions to compute Fast %K and Fast %D for the MACD. That is, we compute Fast %K and Fast %D with \(Z = MACD(X,n_S,n_L)\), as shown below.

\(Fast\% K_t(MACD(X,n_S,n_L),n)\)\(Fast\% D_t(MACD(X,n_S,n_L),n,v)\)

Next we compute Fast %K of \(Fast\% D_t(MACD(X,n_S,n_L),n,v)\), as shown below.

\(Fast\% K_t(Fast\% D_t(MACD(X,n_S,n_L),n,v),n)\)Finally, we compute the **Schaff Trend Cycle**, which is denoted as \(STC_t(X,n_S,n_L,n,v)\). This function is the Fast %D of \(Fast\% D_t(MACD(X,n_S,n_L),n,v)\), as shown below.

#### 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, 27th February, 2023.