# Technical Studies Reference

### Bill Williams AC

This study calculates and displays the Bill Williams AC study for the Closing Price Data.

Let $$C$$ be a random variable denoting the Closing Price, and let $$C_t$$ denote its value at Index $$t$$. Let the Long Moving Average Length, Short Moving Average Length, and Signal Moving Average Length Inputs be denoted as $$n_L$$, $$n_S$$, and $$n_{Sig}$$, respectively.

We calculate the following sums and averages.

$$\displaystyle{Sum^{(1)}_t(n_L) = \left\{ \begin{matrix} \sum_{i = 50}^t C_i & 50 \leq t < 50 + n_L \\ Sum^{(1)}_{t - 1}(n_L) + C_t - Avg^{(1)}_{t - 1}(n_L) & t \geq 50 + n_L \end{matrix}\right .}$$

$$Avg^{(1)}_t(n_L) = \frac{1}{n_L} Sum^{(1)}_t(n_L)$$

$$\displaystyle{Sum^{(2)}_t(n_S) = \left\{ \begin{matrix} \sum_{i = 50}^t C_i & 50 \leq t < 50 + n_S \\ Sum^{(1)}_{t - 1}(n_S) + C_t - Avg^{(2)}_{t - 1}(n_S) & t \geq 50 + n_S \end{matrix}\right .}$$

$$Avg^{(2)}_t(n_S) = \frac{1}{n_S} Sum^{(1)}_t(n_S)$$

$$Avg^{(3)}_t(n_L,n_S) = Avg^{(2)}_t(n_S) - Avg^{(1)}_t(n_L)$$

$$\displaystyle{Sum^{(4)}_t(n_L,n_S,n_{Sig}) = \left\{ \begin{matrix} \sum_{i = 50}^t Avg^{(3)}_t(n_L,n_S) & 50 \leq t \leq 50 + n_{Sig} - 2 \\ \sum_{i = t - n_{Sig} + 1}^t Avg^{(3)}_t(n_L,n_S) & t > 50 + n_{Sig} - 2 \end{matrix}\right .}$$

$$Avg^{(4)}_t(n_L,n_S,n_{Sig}) = \frac{1}{n_{Sig}}Sum^{(4)}_t(n_S,n_L,n_{Sig})$$

Note: The above formulas are used when the Moving Average Type Input is set to Bill Williams Smoothed. Depending on the setting of that Input, the above averages could be replaced with Simple Moving Averages or Exponential Moving Averages.

The Bill Williams AC indicator at Index $$t$$ for the given Inputs is denoted as $$AC_t(n_L,n_S,n_{Sig})$$, and it is computed for $$t \geq 50$$ as follows.

$$AC_t(n_L,n_S,n_{Sig}) = Avg^{(3)}_t(n_L,n_S) - Avg^{(4)}_t(n_L,n_S,n_{Sig})$$

By default, the Subgraph of $$AC_t(n_L,n_S,n_{Sig})$$ is drawn as a dashed plot, and it is colored as follows.

$$AC_t(n_L,n_S,n_{Sig}) > AC_{t - 1}(n_L,n_S,n_{Sig}) \Longrightarrow Green$$

$$AC_t(n_L,n_S,n_{Sig}) = AC_{t - 1}(n_L,n_S,n_{Sig}) \Longrightarrow Grey$$

$$AC_t(n_L,n_S,n_{Sig}) < AC_{t - 1}(n_L,n_S,n_{Sig}) \Longrightarrow Purple$$