# Technical Studies Reference

### Instantaneous Trendline

This study calculates and displays an Instantaneous Trendline and Trigger Line for the data given by the Input Data Input. This study is an ACSIL implementation of the ITrend Indicator given in Figure 3.4 of the book Cybernetic Analysis for Stocks and Futures by John Ehlers.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_t$$ be the value of $$X$$ at Index $$t). Let the Length Input be denoted as \(n$$.

We compute a Smoothing Factor $$\alpha(n)$$ (Greek letter alpha) as follows.

$$\displaystyle{\alpha(n) = \frac{2}{n + 1}}$$

We will suppress the functional dependence of $$\alpha$$ on $$n$$ in the following formulas.

The Instantaneous Trendline and the Trigger Line at Index $$t$$ are denoted as $$IT_t(X,n)$$ and $$Trig^{(IT)}_t(X,n)$$, respectively. Both of these are displayed as Subgraphs, and we compute them as follows.

$$\displaystyle{IT_t(X,n) = \left\{ \begin{matrix} \frac{1}{4}(X_t + 2X_{t - 1} + X_{t - 2}) & t < 6 \\ \left(\alpha - \frac{\alpha^2}{4}\right)X_t + \frac{\alpha^2}{2}X_{t - 1} - \left(\alpha - \frac{3\alpha^2}{4}\right)X_{t - 2} + 2(1 - \alpha) IT_{t - 1} - (1 - \alpha)^2 IT_{t - 2}(X,n) & t \geq 6 \end{matrix}\right .}$$

$$Trig^{(IT)}_t(X,n) = 2IT_t(X,n) - IT_{t - 2}(X,n)$$