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### Moving Average - Zero Lag Exponential

This study calculates and displays a Zero Lag Exponential Moving Average of the data specified by the **Input Data** Input. This indicator was created by John Ehlers and Ric Way.

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 Input **Zero Lag EMA Length** be denoted as \(n\). The Lag \(L(n)\) in the data is computed as follows.

For an explanation of the ceiling function (\(\left\lceil{\space\space}\right\rceil\)), refer to our description here.

The de-lagged data \(\Xi_t(X,n)\) is computed as follows.

\(\Xi_t(X,n) = \left\{ \begin{matrix} 2X_t - X_0 & 0 \leq t < L(n) \\ 2X_t - X_{L(n)} & t \geq L(n) \end{matrix}\right .\)In the above notation, \(\Xi\) is the capital Greek leter "Xi".

We denote the **Moving Average - Zero Lag Exponential** at Index \(t\) for the given Inputs as \(ZLEMA_t(X,n)\), and we compute it for \(t \geq 0\) in terms of an Exponential Moving Average as follows.

**Note**: \(ZLEMA_t(X,n)\) is computed for \(t \geq 0\), but it is only displayed for \(t \geq n - 1\).

If \(L(n) = 0\), then \(ZLEMA_t(X,n)\) becomes identical to \(EMA_t(X,n)\).

#### 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**.

*Last modified Friday, 19th June, 2020.