# Technical Studies Reference

### Double Stochastic

This study calculates and displays a Double Stochastic for the Price Data.

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 respective values at Index $$t$$. Let $$n$$ and $$n_{MA}$$ be the Length and Moving Average Length Inputs, respectively. We then execute the first of two Fast %K and Fast %D calculations, which we denote as $$Fast\%K_t^{(1)}(n)$$ and $$Fast\%D_t^{(1)}(n,n_{MA})$$, respectively. It should be noted that our calculation of Fast% K is slightly different here than it is in the study KD - Fast. See that study for an explanation of the notation used in the following formulas.

$$Fast\%K_t^{(1)}(n) = \displaystyle{Fast\%K_t(H,L,C,n) = \left\{ \begin{matrix} 100\cdot\frac{C_t - \min_t(L,n)}{\max_t(H,n) - \min_t(L,n)} & \max_t(H,n) - \min_t(L,n) \neq 0 \\ 100\cdot\left(C_t - \min_t(L,n)\right) & \max_t(H,n) - \min_t(L,n) = 0 \end{matrix}\right .}$$

$$Fast\%D_t^{(1)}(n,n_{MA}) = Fast\%D_t(H,L,C,n,n_{MA}) = SMA_t\left(Fast\%K(H,L,C,n),n_{MA}\right)$$

Next we execute the second of two Fast %K and Fast %D calculations, which we denote as $$Fast\%K_t^{(2)}(n,n_MA)$$ and $$Fast\%D_t^{(2)}(n,n_{MA})$$, respectively.

$$Fast\%K_t^{(2)}(n,n_{MA}) = Fast\%K_t\left(Fast\%D^{(1)}(n,n_{MA}), Fast\%D^{(1)}(n,n_{MA}), Fast\%D^{(1)}(n,n_{MA}), n\right)$$

$$Fast\%D_t^{(2)}(n,n_{MA}) = SMA_t\left(Fast\%K^{(2)}(n,n_{MA}), n_{MA}\right)$$

Note: Depending on the setting of the Input Moving Average Type, the Simple Moving Averages in the above Fast %D formulas could be replaced with Exponential Moving Averages, Linear Regression Moving Averages, Weighted Moving Averages, Wilders Moving Averages, Simple Moving Averages - Skip Zeros, or Smoothed Moving Averages.

The value of the Double Stochastic at Index $$t$$ is denoted as $$DS_t(n,n_{MA})$$, and it is equal to the value of this second Fast %D calculation.

$$DS_t(n,n_{MA}) = Fast\%D_t^{(2)}(n,n_{MA})$$

The Subgraph of the Double Stochastic is calculated and displayed for $$t \geq 0$$.

This study also displays horizontal lines at levels determined by the Line 1 and Line 2 Inputs.