# Technical Studies Reference

This study calculates and displays a Smoothed Adaptive Momentum for the data given by the Input Data Input. This study is an ACSIL implementation of the Indicator given in Figures 12.1 and 12.2 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, Median Phase Change Length, and 2-Pole Super Smoother Length Inputs be denoted as $$n_{CC}$$, $$n_{Med}$$, and $$n_{SSF}$$, respectively.

We first smooth the data using a Four Bar Symmetrical Finite Impulse Response Filter. The value of the smoothed data at Index $$t$$ is denoted as $$X^{(S)}_t$$, and we compute it as follows.

$$\displaystyle{X^{(S)}_t = \frac{1}{6}(X_t + 2X_{t - 1} + 2X_{t - 2} + X_{t - 3})}$$

Second, we compute the Cyber Cycle $$CC_t(X,n_{CC})$$.

Third, we compute the Dominant Cycle Period for $$CC_t(X,n_{CC})$$. Explicitly, we compute $$T^{(DC)}_t(CC(X,n_{CC}),n_{Med})$$. This is where the adaptive nature of the study comes from. The period is not static, but rather it changes with changing market conditions.

Fourth, we define a Value, denoted as $$Val_t(X, n_{CC}, n_{Med})$$, and we compute it as follows.

$$\displaystyle{Val_t(X, n_{CC}, n_{Med}) = X_t - X_{\lfloor T^{(DC)}(CC(X,n_{CC}),n_{Med}) - 1 \rfloor}}$$

For an explanation of the floor function ($$\left\lfloor{\space\space}\right\rfloor$$), refer to our description here.

Finally, we denote the Smoothed Adaptive Momentum as $$SAM_t(X,n_{CC},n_{Med})$$, and we compute it using a 3-Pole Super Smoother Filter as follows.

$$SAM_t(X,n_{CC},n_{Med}) = SSF^{(3)}_t(Val_t(X, n_{CC}, n_{Med}), n_{SSF})$$

The study also displays a Center Line at zero.