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### Inverse Fisher Transform

This study calculates and both displays an Inverse Fisher Transform (IFT) and a Moving Average of the IFT for 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 \(n_{HL}\) denote the **Highest Value/Lowest Value Length**. We subject \(X_t\) to the following transformation.

In the above formula, \(\xi\) is the Greek letter "xi", \(\max\) is the Moving Maximum function, and \(\min\) is the Moving Minimum function.

Let the **Input Data Mov Avg Length** Input be denoted as \(n_I\). We denote the Weighted Moving Average of \(\xi_t(X,n_{HL})\) as \(\overline{\xi}_t(X,n_{HL},n_I)\), and we compute it as follows.

**Note**: Depending on the setting of the Input **Input Data Mov Avg Type**, the Weighted Moving Average in the above formula could be replaced with an Exponential Moving Average, a Linear Regression Moving Average, a Simple Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

Let the **Output Mov Avg Length** Input be denoted as \(n_O\). We denote the **Inverse Fisher Transform** of \(\overline{\xi}_t(X,n_{HL},n_I)\) at Index \(t\) as \(IFT_t\left(\overline{\xi}(X,n_{HL},n_I)\right)\), and we compute it for \(t \geq n_I + n_O\) as follows.

For \(t = 0\):

\(IFT_0\left(\overline{\xi}(X,n_{HL},n_I)\right) = 0\)For \(t > 0\):

\(\displaystyle{IFT_t\left(\overline{\xi}(X,n_{HL},n_I)\right) = \left\{ \begin{matrix} \frac{\exp\left(2\overline{\xi}_t(X,n_{HL},n_I)\right) - 1}{\exp\left(2\overline{\xi}_t(X,n_{HL},n_I)\right) + 1} & \max_t(X,n_{HL}) - \min_t(X,n_{HL}) \neq 0 \\ IFT_{t - 1}\left(\overline{\xi}(X,n_{HL},n_I)\right) & \max_t(X,n_{HL}) - \min_t(X,n_{HL}) = 0 \end{matrix}\right .}\)We denote the Simple Moving Average of \(\overline{IFT}_t\left(\overline{\xi}(X,n_{HL},n_I)\right)\), and we compute it for \(t \geq n_I + n_O\) as follows.

\(\overline{IFT}_t\left(\overline{\xi}(X,n_{HL},n_I), n_O\right) = SMA_t\left(IFT(\overline{\xi}(X,n_{HL},n_I)), n_O\right)\)**Note**: Depending on the setting of the Input **Output Mov Avg Type**, the Simple Moving Average in the above formula could be replaced with an Exponential Moving Average, a Linear Regression Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

Let the **Line Value** Input be denoted as \(l\). In addition to the graphs of \(IFT_t\left(\overline{\xi}(X,n_{HL},n_I)\right)\) and \(\overline{IFT}_t\left(\overline{\xi}(X,n_{HL},n_I)\right)\), this study displays horizontal lines at levels \(\pm l\).

#### 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 Thursday, 13th June, 2019.