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# Technical Studies Reference

### 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.

$$\displaystyle{\xi_t(X,n_{HL}) = \frac{10\left(X_t - \min_t(X,n_{HL})\right)}{\max_t(X,n_{HL}) - \min_t(X,n_{HL})}} - 5$$, $$\left(\max_t(X,n_{HL}) - \min_t(X,n_{HL}) \neq 0 \right)$$

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.

$$\overline{\xi}_t(X,n_{HL},n_I) = WMA_t(\xi(X,n_{HL}), n_I)$$

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 $$\pm l$$.