# Technical Studies Reference

### 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.

$$L(n) = \left\lceil{\frac{n-1}{2}}\right\rceil$$

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.

$$ZLEMA_t(X,n) = EMA_t(\Xi(X,n),n)$$

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 $$EMA_t(X,n)$$.