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

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 $$t = 1$$.