# Technical Studies Reference

### Heikin-Ashi Smoothed

The Heikin-Ashi Smoothed study is based upon the standard Heikin-Ashi study with additional moving average calculations. We describe these calculations below.

Let $$O$$, $$H$$, $$L$$, and $$C$$ be random variables denoting the Open, High, Low, and Close Prices, respectively, and let their respective values at Index $$t$$ be $$O_t$$, $$H_t$$, $$L_t$$, and $$C_t$$. Let the Inputs Moving Average Period Type 1 and Moving Average Period Type 2 be denoted as $$n_1$$ and $$n_2$$, respectively.

We begin by smoothing the Price Data with Smoothed Moving Averages. We indicate the smoothed Price Data with a superscript $$(S)$$, and we compute them as follows.

$$O_t^{(S)}(n_1) = SMMA_t(O,n_1)$$
$$H_t^{(S)}(n_1) = SMMA_t(H,n_1)$$
$$L_t^{(S)}(n_1) = SMMA_t(L,n_1)$$
$$C_t^{(S)}(n_1) = SMMA_t(C,n_1)$$

Note: Depending on the setting of the Input Moving Average Type 1, the Smoothed Moving Averages in the above formulas could be replaced with Exponential Moving Averages, Linear Regression Moving Averages, Simple Moving Averages, Weighted Moving Averages, Wilders Moving Averages, or Simple Moving Averages - Skip Zeros.

Next we apply the Heikin-Ashi transformation to the smoothed Price Data. We denote the Heikin-Ashi smoothed Price Data with a superscript $$(HAS)$$, and we compute them as follows.

$$\displaystyle{O_t^{(HAS)}(n_1) = \left\{ \begin{matrix} O_0^{(S)}(n_1) & t = 0 \\ \frac{O_{t - 1}^{(HAS)}(n_1) + C_{t - 1}^{(HAS)}(n_1)}{2} & t > 0 \end{matrix}\right .}$$

$$\displaystyle{H_t^{(HAS)}(n_1) = \max\left\{H_t^{(S)}(n_1), O_t^{(HAS)}(n_1)\right\}}$$

$$\displaystyle{L_t^{(HAS)}(n_1) = \min\left\{L_t^{(S)}(n_1), O_t^{(HAS)}(n_1)\right\}}$$

$$\displaystyle{C_t^{(HAS)}(n_1) = \frac{O_t^{(S)}(n_1) + H_t^{(S)}(n_1) + L_t^{(S)}(n_1) + C_t^{(S)}(n_1)}{4}}$$

Finally, we apply a second smoothing with Weighted Moving Averages. We denote the double-smoothed Heikin-Ashi Price Data with a superscript $$(HA2S)$$, and we compute them as follows.

$$O_t^{(HA2S)}(n_1,n_2) = WMA_t\left(O_t^{(HAS)}(n_1),n_2\right)$$
$$H_t^{(HA2S)}(n_1,n_2) = WMA_t\left(H_t^{(HAS)}(n_1),n_2\right)$$
$$L_t^{(HA2S)}(n_1,n_2) = WMA_t\left(L_t^{(HAS)}(n_1),n_2\right)$$
$$C_t^{(HA2S)}(n_1,n_2) = WMA_t\left(C_t^{(HAS)}(n_1),n_2\right)$$

The one exception to these formulas occurs when the Set Close to Current Price for Last Bar Input is set to Yes. In that case, for the last bar in the chart, we have $$C_t^{(HA2S)}(n_1,n_2) = C_t$$.

Note: Depending on the setting of the Input Moving Average Type 2, the Weighted Moving Averages in the above formulas could be replaced with Exponential Moving Averages, Linear Regression Moving Averages, Simple Moving Averages, Wilders Moving Averages, Simple Moving Averages - Skip Zeros, or Smoothed Moving Averages.

If you want to view the Heikin-Ashi Smoothed study as the main price graph and replace the existing chart bars, then open the Study Settings window for the Heikin-Ashi study and enable the Display As Main Price Graph option.