# Technical Studies Reference

### Directional Movement Index

This study calculates and displays Welles Wilder's Plus and Minus Directional Indicators.

Let $$H$$, $$L$$, and $$C$$ be random variables denoting the High, Low, and Close prices, respectively, and let $$H_t$$, $$L_t$$, and $$C_t$$ be their values at Index $$t$$. We denote an Up Move and a Down Move at Index $$t$$ as $$\Delta H_t$$ and $$-\Delta L_t$$, respectively. Initially we have $$\Delta H_0 = -\Delta L_0 = 0$$, and for $$t > 0$$ compute the Up Move and Down Move as follows.

$$\Delta H_t = H_t - H_{t - 1}$$

$$- \Delta L_t = L_{t - 1} - L_t$$

We denote the Positive Directional Movement and Negative Directional Movement at Index $$t$$ as $$DM_t^{(+)}$$ and $$DM_t^{(-)}$$, respectively, and we compute them for $$t \geq 0$$ as follows.

$$\displaystyle{DM_t^{(+)} = \left\{\begin{matrix} \Delta H_t & \Delta H_t > -\Delta L_t \space and \space \Delta H_t > 0 \\ 0 & \Delta H_t \leq -\Delta L_t \space or \space \Delta H_t \leq 0 \end{matrix}\right .}$$

$$\displaystyle{DM_t^{(-)} = \left\{\begin{matrix} -\Delta L_t & -\Delta L_t > \Delta H_t \space and \space -\Delta L_t > 0 \\ 0 & -\Delta L_t \leq \Delta H_t \space or \space -\Delta L_t \leq 0 \end{matrix}\right .}$$

Let the Length Input be denoted as $$n_{DX}$$. We compute the Welles Sum the Postive Directional Movement, Negative Directional Movement, and True Range for $$t \geq 0$$. The values of these Welles sums at Index $$t$$ are respectively denoted as $$WS_t\left(DM^{(+)},n_{DX}\right)$$, $$WS_t\left(DM^{(-)},n_{DX}\right)$$, and $$WS_t(TR,n_{DX})$$.

Finally, we denote the indicators of the Directional Movement Index at Index $$t$$ as $$DI_t^{(+)}(n_{DX})$$ (Positive Directional Indicator) and $$DI_t^{(-)}(n_{DX})$$ (Negative Directional Indicator). These are both initially equal to zero, and we compute them for $$t \geq 0$$ as follows.

$$\displaystyle{DI_t^{(+)}(n_{DX}) = \left\{\begin{matrix} 100\cdot\frac{WS_t\left(DM^{(+)},n_{DX}\right)}{WS_t(TR,n_{DX})} & WS_t(TR,n_{DX}) \neq 0 \\ DI_{t - 1}^{(+)}(n_{DX}) & WS_t(TR,n_{DX}) = 0 \end{matrix}\right .}$$

$$\displaystyle{DI_t^{(-)}(n_{DX}) = \left\{\begin{matrix} 100\cdot\frac{WS_t\left(DM^{(-)},n_{DX}\right)}{WS_t(TR,n_{DX})} & WS_t(TR,n_{DX}) \neq 0 \\ DI_{t - 1}^{(-)}(n_{DX}) & WS_t(TR,n_{DX}) = 0 \end{matrix}\right .}$$

Note: The values of the Directional Movement Index Indicators are displayed for $$t \geq n_{DX}$$.