# Technical Studies Reference

This study calculates and displays an Adaptive Relative Vigor Index and Trigger Line for the data given by the Input Data Input. This study is an ACSIL implementation of the Indicator given in Figures 10.7 and 10.8 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 and Median Phase Change Length Inputs be denoted as $$n_{CC}$$ and $$n_{Med}$$, 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 compute two Average Prices at Index $$t$$: The Close-Open Average Price and the High-Low Average Price, denoted as $$\overline{P}_t^{(CO)}$$ and $$\overline{P}_t^{(HL)}$$, respectively.

$$\displaystyle{\overline{P}_t^{(CO)} = \frac{(C_{t - 3} - O_{t - 3}) + 2(C_{t - 2} - O_{t - 2}) + 2(C_{t - 1} - O_{t - 1}) + (C_t - O_t)}{6}}$$

$$\displaystyle{\overline{P}_t^{(HL)} = \frac{(H_{t - 3} - L_{t - 3}) + 2(H_{t - 2} - L_{t - 2}) + 2(H_{t - 1} - L_{t - 1}) + (H_t - L_t)}{6}}$$

Fifth, we define the Adaptive RVI Length, denoted as $$n_{(ARVI)}(n_{(CC)},n_{(Med)})$$, and we compute it as follows.

$$n^{(ARVI)}_t(X,n_{(CC)},n_{(Med)}) = \lfloor (4T^{(DC)}_t(CC(X,n_{CC}),n_{Med}) + 3T^{(DC)}_{t - 1}(CC(X,n_{CC}),n_{Med}) + 2T^{(DC)}_{t - 3}(CC(X,n_{CC}),n_{Med}) + 3T^{(DC)}_{t - 4}(CC(X,n_{CC}),n_{Med}))/20 \rfloor$$

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

Finally, we denote the Adaptive Relative Vigor Index as $$RVI^{(A)}_t(X,n_{CC},n_{Med})$$, and we compute it as follows.

$$\displaystyle{RVI^{(A)}_t(X,n_{CC},n_{Med}) = \frac{\sum_{i = t - n^({ARVI})_t + 1}^t \overline{P}^{(CO)}_i}{\sum_{i = t - n^({ARVI})_t + 1}^t \overline{P}^{(HL)}_i}}$$

This formula is used as long as $$\sum_{i = t - n^({ARVI})_t + 1}^t \overline{P}^{(HL)}_i \neq 0$$. If the sum is $$0$$, then $$RVI^{(A)}_t(X,n_{CC},n_{Med}) = 0$$.

The study also displays a Center Line and the following Trigger Line.

$$Trig^{(ARVI)}_t(X,n_{CC},n_{Med}) = RVI^{(A)}_{t - 1}(C,n_{CC},n_{Med})$$