# Technical Studies Reference

### Inertia

This study calculates and displays the Inertia study. The step-by-step calculation method is explained below.

The components of the Relative Vigor Index (RVI) are calculated as follows. We let $$C_i$$, $$O_i$$, $$H_i$$, and $$L_i$$ be the closing, opening, high, and low prices at time index $$i$$, respectively.

Close-Open Average:

$$CloseOpenAvg_i =\frac{(C_i-O_i)+2(C_{i-1}-O_{i-1})+2(C_{i-2}-O_{i-2})+(C_{i-3}-O_{i-3})}{6}$$ $$CloseOpenAvg_i = \frac{1}{3}\sum_{k=1}^3{\frac{(C_{i-k+1}-O_{i-k+1})+(C_{i-k}-O_{i-k})}{2}}$$

The summand is the average of the difference in the closing and opening prices for consecutive values of the index, and $$CloseOpenAvg_i$$ is the average of three of these averages.

High-Low Average: $$HighLowAvg_i = \frac{(H_i-L_i) + 2(H_{i-1}-L_{i-1})+2(H_{i-2}-L_{i-2})+(H_{i-3}-L_{i-3})}{6}$$ $$HighLowAvg_i = \frac{1}{3}\sum_{k=1}^3{\frac{(H_{i-k+1}-L_{i-k+1})+(H_{i-k}-L_{i-k})}{2}}$$

The summand is the average of the difference in the closing and opening prices for consecutive values of the index, and $$HighLowAvg_i$$ is the average of three of these averages.

For the next calculations, let $$d$$ be the starting value of the index, and let $$r$$ be the RVI Length, which is an input. For $$i=d$$, we have the following.

RVI Numerator:

$$RVINum_i = \sum_{k=1}^r{CloseOpenAvg_{k+i-1}}$$

This is the sum of the sequence of $$CloseOpenAvg$$'s starting at $$CloseOpenAvg_i$$ and ending at $$CloseOpenAvg_{r+i-1}$$.

RVI Denominator:

$$RVIDenom_i = \sum_{k=1}^r{HighLowAvg_{k+i-1}}$$

This is the sum of the sequence of $$HighLowAvg$$'s starting at $$HighLowAvg_i$$ and ending at $$HighLowAvg_{r+i-1}$$.

RVI:

From these we calculate the RVI as follows.

$$RVI_i=\left\{ \begin{matrix} \frac{RVINum_i}{RVIDenom_i} & RVIDenom_i \neq 0 \\ 0 & RVIDenom_i=0 \end{matrix}\right .$$

Inertia Calculation:

The inertia of the RVI is precisely the Linear Regression Indicator (LRI) of the RVI. The LRI is described in detail in Moving Linear Regression / Moving Average - Least Squares. We let $$l$$ be the Linear Regression Length, which is an input. The regression statistics are calculated for each $$i$$ where the data points are taken $$l$$ at a time. Explicit formulas are given below.

$$b_i=\frac{l \cdot \sum_{k=1}^l{(k+i-1)RVI_{k+i-1}}-\sum_{k=1}^l{(k+i-1)}\cdot \sum_{k=1}^l{RVI_{k+i-1}}}{l \cdot \sum_{k=1}^l{(k+i-1)^2}-\left(\sum_{k=1}^l{(k+i-1)}\right)^2}$$ $$a_i=\frac{\sum_{k=1}^l{RVI_{k+i-1}}-b_i \cdot \sum_{k=1}^l{(k+i-1)}}{l}$$ $$LRI_i=a_i+lb_i$$