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### Fisher Transform

This study calculates and displays a Fisher Transform of the data specified by the **Price** Input.

Let \(X\) be a random variable denoting the **Price**, and let \(X_t\) be the value of the **Price** at Index \(t\). Let the Input **Period** be denoted as \(n\). Prior to computing the Fisher Transform, we subject \(X\) to two transformations.

For the first transformation, we compute a new variable \(\xi\) (Greek letter xi) in terms of \(X\) and \(n\). Its value at Index \(t\) is denoted as \(\xi_t(X,n)\), and we compute it with the following recursion relation.

For \(t = 0\): \(\xi_0(X,n) = 0\)

For \(t > 0\): \(\xi_t(X,n) = \left\{\begin{matrix} 0.66\displaystyle{\left(\frac{X_t - \min_t(X,n)}{\max_t(X,n) - \min_t(X,n)} - 0.5\right)} + 0.67\xi_{t - 1}(x,n) & \max_t(X,n) - \min_t(X,n) \neq 0 \\ 0 & \max_t(X,n) - \min_t(X,n) = 0 \end{matrix}\right .\)

In the above formula, \(\min_t(X,n)\) and \(\max_t(X,n)\) are, respectively, the minimum and maximum values of the **Price** data over a moving window of **Length** \(n\) which terminates at Index \(t\). That is, \(\min_t(X,n) = \min\{X_{t - n + 1},...,X_t\}\), and \(\max_t(X,n) = \max\{X_{t - n + 1},...,X_t\}\).

For the second transformation, we compute a new variable \(\xi^*\) (xi-star), in which we truncate the value of \(\xi\) when necessary. Its value at Index \(t\) is denoted as \(\xi^*_t(X,n)\), and we compute it as follows.

\(\xi^*_t(X,n) = \displaystyle{\left\{\begin{matrix} -0.999 & \xi_t(X,n) < -0.99 \\ \xi_t(X,n) & -0.99 \leq \xi_t(X,n) \leq 0.99 \\ 0.999 & \xi_t(X,n) > 0.99 \end{matrix}\right .}\)The **Fisher Transform** at Index \(t\) for the given Inputs is denoted as \(FT_t(X,n)\), and we compute it for \(t \geq 0\) with the following recursion relation.

For \(t = 0\): \(FT_0(X,n) = 0\)

For \(t > 0\): \(FT_t(X,n) = 0.5\displaystyle{\left(\ln\left(\frac{1 + \xi^*_t(X,n)}{1 - \xi^*_t(X,n)}\right) + FT_{t - 1}(X,n)\right)}\)

This study also calculates and displays an **Offset Fisher Transform**, which is just the delayed function \(FT_{t - 1}(X,n)\). The calculation of this function starts at \(t = 1\).

#### 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, 29th July, 2021.