# Technical Studies Reference

### Moving Average - Smoothed

This study calculates and displays a smoothed moving average of the data specified by the Input Data Input.

Let $$X$$ be a random variable denoting the Input Data, and let $$X_i$$ be the value of the Input Data at Index $$i$$. Let the Inputs Length and Offset be denoted as $$n$$ and $$k$$, respectively. Then we denote the Moving Average - Smoothed at Index $$t$$ for the given Inputs as $$SMMA_t(X,n,k)$$, and we compute it for $$t \geq 0$$ as follows.

For $$t = 0$$: $$SMMA_0(X,n,k) = X_0$$

For $$t > 0$$ the calculation may involve Simple Moving Averages, as shown below.

For $$0 < t \leq k + 1$$:

$$\displaystyle{SMMA_t(X,k,n) = \left\{\begin{matrix} \frac{1}{n}(nX_0 - SMMA_{t - 1}(X,n,k) + X_{t - k}) & SMMA_{t - 1}(X,n,k) \neq 0 \\ MA_{t - k - 1}(X,n) & SMMA_{t - 1}(X,n,k) = 0 \end{matrix}\right .}$$

For $$k + 1 < t < n + k$$:

$$\displaystyle{SMMA_t(X,n,k) = \left\{\begin{matrix} \frac{1}{n}\left((n - t + k + 1)X_0 + \sum_{i = 1}^{t - k - 1}X_i - SMMA_{t - 1}(X,n,k) + X_{t - k}\right) & SMMA_{t - 1}(X,n,k) \neq 0 \\ MA_{t - k - 1}(X,n) & SMMA_{t - 1}(X,n,k) = 0 \end{matrix}\right .}$$

For $$t \geq n + k$$:

$$\displaystyle{SMMA_t(X,n,k) = \left\{\begin{matrix} \frac{1}{n}\left(\sum_{i = t - k - n}^{t - k - 1}X_i - SMMA_{t - 1}(X,n,k) + X_{t - k}\right) & SMMA_{t - 1}(X,n,k) \neq 0 \\ MA_{t - k - 1}(X,n) & SMMA_{t - 1}(X,n,k) = 0 \end{matrix}\right .}$$

For an explanation of the Sigma ($$\Sigma$$) notation for summation, refer to the Wikipedia article Summation.

#### Inputs

• Input Data
• Length
• Offset: This Input specifies the number of chart bars by which the summation index is to be shifted forward.

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