Technical Studies Reference


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

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 \(t = 1\).

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.

Fisher_Transform.343.scss


*Last modified Wednesday, 03rd January, 2018.