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### Adaptive RSI Moving Average with Smoothing

This study calculates and displays an adaptive moving average of the Relative Strength Index (RSI), which in turn is a function of the data specified by the **Input Data**. The study allows for optional smoothing of both the **Input Data** and the RSI.

__Smoothed vs Unsmoothed Input Data__Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) be the value of the **Input Data** at chart bar \(t\). If the value of the **Set Price Smoothing** Input is No, then the unsmoothed **Input Data** \(X\) is used in the subsequent calculations. If the value is Yes, then the smoothed **Input Data** is used instead. The smoothing is done with an Exponential Moving Average whose **Length** is given by the **Price Smoothing Period** Input, denoted as \(n_{PS}\).

We define a new random variable \(P(X,n_{PS})\) denoting the Price (smoothed or unsmoothed). The value of this Price at Index \(t\) is given as follows.

\(P_t(X,n_{PS}) =\left\{ \begin{matrix} X_t & Price \space Smoothing = No \\ EMA_t\left(X,n_{PS}\right) & Price \space Smoothing = Yes \end{matrix}\right .\)**Note**: Depending on the setting of the Input **Price Smoothing Average Type**, the Exponential Moving Average in the calculation of the smoothed **Input Data** could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

__Smoothed vs Unsmoothed RSI__The RSI is calculated for \(P(X,n_{PS})\) with **Length** given by the **ARSI Period**, denoted as \(n_{ARSI}\). The type of Moving Average used in the calculation of the RSI is determined by the **ARSI Moving Average Type** Input. If the value of the **Set RSI Smoothing** Input is No, then the RSI at Index \(t\) is given by \(RSI_t\left(P(X,n_{PS}),n_{ARSI}\right)\). If the value is Yes, then the smoothed RSI is used instead. The smoothing is done with an Exponential Moving Average whose **Length** is given by the **RSI Smoothing Period** Input, denoted as \(n_{RSIS}\).

We define a new random variable \(R(X,n_{PS},n_{ARSI},n_{RSIS})\) denoting the RSI (smoothed or unsmoothed). The value of this variable at Index \(t\) is given as follows.

\(R_t(X,n_{PS},n_{ARSI},n_{RSIS}) =\left\{ \begin{matrix} RSI_t\left(P(X,n_{PS}),n_{ARSI}\right) & RSI \space Smoothing = No \\ EMA_t\left(RSI\left(P(X,n_{PS},n_{ARSI}\right),n_{RSIS}\right) & RSI \space Smoothing = Yes \end{matrix}\right .\)**Note**: The calculation of the RSI here differs from that in the RSI study in that \(U(X)\) and \(D(X)\) are not calculated until \(t = n_{ARSI}\).

**Note**: Depending on the setting of the Input **RSI Smoothing Average Type**, the Exponential Moving Average in the calculation of the smoothed **Input Data** could be replaced with a Linear Regression Moving Average, a Simple Moving Average, a Weighted Moving Average, a Wilders Moving Average, a Simple Moving Average - Skip Zeros, or a Smoothed Moving Average.

__Scaling Factor__Let \(SF(X,n_{PS},n_{ARSI},n_{RSIS})\) be a random variable denoting the Scaling Factor, and let \(v\) denote the **ARSI Scale Factor** Input. The value of the Scaling Factor at Index \(t\) as \(SF_t\) is given as follows.

__Adaptive RSI Moving Average__

The **Adaptive RSI Moving Average** at Index \(t\) for the given Inputs is denoted as \(\overline{RSI}_t(X,n_{PS},n_{ARSI},n_{RSIS},v)\), and we compute it for \(t \geq n_{ARSI}\) as follows.

#### Inputs

- ARSI Period
- ARSI Moving Average Type
- Input Data
**Set Price Smoothing**: This Input determines whether or not the**Input Data**is to be smoothed by taking a Moving Average.- Price Smoothing Period
- Price Smoothing Moving Average Type
**Set RSI Smoothing**: This Input determines whether or not the RSI is to be smoothed by taking a Moving Average.- RSI Smoothing Period
- RSI Smoothing Moving Average Type
- ARSI Scale Factor

#### 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 Friday, 08th March, 2019.