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# Smoothed Adaptive Momentum

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.

The study also displays a Center Line at zero.

#### Inputs

*Last modified Monday, 03rd October, 2022.