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Technical Studies Reference


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.

\(Dir_t(X,n) = X_t - X_{t-n}\)

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

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

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 \(t \geq n\).

\(\displaystyle{AMA_t\left(X,n,c_F,c_S\right) = \left\{ \begin{matrix} X_{t-1} + c_t(X,n)\cdot\left(X_t - X_{t - 1}\right) & AMA_{t-1}\left(X,n,c_F,c_S\right) = 0 \\ AMA_{t-1}\left(X,n,c_F,c_S\right) + c_t(X,n)\cdot(X_t - AMA_{t-1}\left(X,n,c_F,c_S\right)) & AMA_{t-1}\left(X,n,c_F,c_S\right) \neq 0 \end{matrix}\right.}\)

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

Inputs

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.

Moving_Average_-_Adaptive.93.scss


*Last modified Wednesday, 03rd January, 2018.