# Technical Studies Reference

### Inertia 2

This study calculates and displays the Inertia 2 study for the Price Data.

Let $$C$$ be a random variable denoting the Close Price, and let the Standard Deviation Length, Relative Volatility Index Length, and Linear Regression Length Inputs be denoted as $$n_\sigma$$, $$n_{RVIX}$$, and $$n_{LR}$$, respectively. We denote the Relative Volatility Index Up and Relative Volatility Index Down at Index $$t$$ as $$RVIX^{(U)}_t(n_\sigma)$$ and $$RVIX^{(D)}_t(n_\sigma)$$, respectively, and we compute them in terms of a Standard Deviation for $$t \geq \max\{n_\sigma, n_{RVIX}, n_{LR}\}$$ as follows.

$$\displaystyle{RVIX^{(U)}_t(n_\sigma) = \left\{ \begin{matrix} \sigma_t(C,n_\sigma) & C_t > C_{t - 1} \\ 0 & C_t \leq C_{t - 1} \end{matrix}\right .}$$

$$\displaystyle{RVIX^{(D)}_t(n_\sigma) = \left\{ \begin{matrix} 0 & C_t > C_{t - 1} \\ \sigma_t(C,n_\sigma) & C_t \leq C_{t - 1} \end{matrix}\right .}$$

Next we compute the Smoothed Relative Volatility Index Up and Smoothed Relative Volatility Index Down. The values of these at Index $$t$$ are denoted as $$\overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX})$$ and $$\overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX})$$, respectively. These both have the value $$0$$ for $$t < \max\{n_\sigma, n_{RVIX}, n_{LR}\}$$. We compute them for $$t \geq \max\{n_\sigma, n_{RVIX}, n_{LR}\}$$ as follows.

$$\displaystyle{\overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX}) = \frac{\overline{RVIX}^{(U)}_{t - 1}(n_\sigma,n_{RVIX})\cdot(n_{RVIX} - 1) + RVIX^{(U)}_t(n_\sigma)}{n_{RVIX}}}$$

$$\displaystyle{\overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX}) = \frac{\overline{RVIX}^{(D)}_{t - 1}(n_\sigma,n_{RVIX})\cdot(n_{RVIX} - 1) + RVIX^{(D)}_t(n_\sigma)}{n_{RVIX}}}$$

We denote the Relative Volatility Index at Index $$t$$ as $$RVIX_t(n_\sigma,n_{RVIX})$$, and we compute it for $$t \geq \max\{n_\sigma, n_{RVIX}, n_{LR}\}$$ as follows.

$$\displaystyle{RVIX_t(n_\sigma,n_{RVIX}) = \left\{ \begin{matrix} 100\cdot\frac{\overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX})}{\overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX}) + \overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX})} & \overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX}) + \overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX}) \neq 0 \\ 0 & \overline{RVIX}^{(U)}_t(n_\sigma,n_{RVIX}) + \overline{RVIX}^{(D)}_t(n_\sigma,n_{RVIX}) = 0 \end{matrix}\right .}$$

Finally, we denote Inertia 2 at Index $$t$$ as $$Inertia^{(2)}_t(n_\sigma,n_{RVIX},n_{LR})$$. It is a Moving Linear Regression of the Relative Volatility Index, and we compute it for $$t \geq \max\{n_\sigma,n_{RVIX},n_{LR}\}$$ as follows.

$$Inertia^{(2)}_t(n_\sigma,n_{RVIX},n_{LR}) = MLR_t(RVIX(n_\sigma,n_{RVIX}),n_{LR})$$