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# Moving Average - Adaptive

This study calculates and displays an Adaptive Moving Average of the data specified by the **Input Data** Input. This moving average was developed by Perry Kaufman. Reference: Stocks & Commodities V13:6: (267): Sidebar: Adaptive Moving Average.

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) be the value of the **Input Data** at Index \(t\). Let the Inputs **Fast Smoothing Constant** and **Slow Smoothing Constant** be denoted as \(c_F\) and \(c_S\), respectively, and let the Input **Length** be denoted as \(n\). We denote the values of the Direction, Volatility, and Smoothing Constant for the given Inputs at Index \(t\) as \(Dir_t(X,n)\), \(Vol_t(X,n)\), and \(c_t(X,n)\), respectively. We compute these for \(t \geq n\) as follows.

\(\displaystyle{Vol_t(X,n) = \left\{ \begin{matrix} 0.000001 & \sum_{i=t-n+1}^t\left|X_i - X_{i-1}\right| = 0 \\ \sum_{i=t-n+1}^t\left|X_i - X_{i-1}\right| & \sum_{i=t-n+1}^t\left|X_i - X_{i-1}\right| \neq 0 \end{matrix}\right .}\)

\(\displaystyle{c_t(X,n) = \left[\left|\frac{Dir_t(X,n)}{Vol_t(X,n)}\right|\left(\frac{2}{c_F + 1} - \frac{2}{c_S + 1}\right) + \frac{2}{c_S + 1}\right]^2}\)

We denote the **Moving Average - Adaptive** at Index \(t\) for the given Inputs as \(AMA_t\left(X,n,c_F,c_S\right)\), and we compute it with the following recursion relation for \(t \geq n\).

For an explanation of the Sigma (\(\Sigma\)) notation for summation, refer to our description here.

#### Inputs

- Input Data
- Length
**Fast Smoothing Constant**: This is the**Length**of a fast-moving Exponential Moving Average. It should be set to a value that is less than that of the Input**Slow Smoothing Constant**to obtain sensible results.**Slow Smoothing Constant**: This is the**Length**of a slow-moving Exponential Moving Average.

#### Spreadsheet

The spreadsheet below contains the formulas for this study in Spreadsheet format. Save this Spreadsheet to the Data Files Folder.

Open it through **File >> Open Spreadsheet**.

*Last modified Monday, 26th September, 2022.