# Technical Studies Reference

This study calculates and displays a Leading Indicator and its Moving Average for the data given by the Input Data Input. This study is an ACSIL implementation of the Indicator given in Figures 16.5 and 16.6 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 1, Length 2, and Moving Average Length Inputs be denoted as \(n_1$$, $$n_2$$, and $$n_{MA}$$, respectively.

We define two smoothing constants, denoted as $$\alpha^{(1)}(n_1)$$ and $$\alpha^{(2)}(n_2)$$, and we compute them as follows.

$$\displaystyle{\alpha^{(1)}(n_1) = \frac{2}{n_1 + 1}}$$
$$\displaystyle{\alpha^{(2)}(n_2) = \frac{2}{n_2 + 1}}$$

Next, we denote a function called the Lead as $$L_t(X,n_1$$, and we compute it as follows.

$$\displaystyle{L_t(X,n_1) = 2X_t + \left(\alpha^{(1)}(n_1) - 2\right)X_{t - 1} + \left(1 - \alpha^{(1)}(n_1)\right)L_{t - 1}(X,n_1)}$$

We now denote the Leading Indicator as $$LI_t(X,n_1,n_2)$$, and we compute it as follows.

$$\displaystyle{LI_t(X,n_1,n_2) = \alpha^{(2)}(n_2)L_t(X,n_1) + \left(1 - \alpha^{(2)}(n_2)\right)L_{t - 1}(X,n_1)}$$

Finally, we denote the Average Leading Indicator as $$\overline{LI}_t(X,n_1,n_2,n_{MA})$$, and we compute it using the Exponential Moving Average as follows.

$$\displaystyle{\overline{LI}_t(X,n_1,n_2,n_{MA}) = EMA_t(LI(X,n_1,n_2),n_{MA})}$$

Note: Depending on the setting of the Input Moving Average Type, the Exponential Moving Average in the above formula could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.