# Technical Studies Reference

## Moving Average - Triple Exponential

This study calculates and displays a Triple Exponential Moving Average of the data specified by the Input Data Input.

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 Length be denoted as $$n$$. Then we denote the Moving Average - Triple Exponential at Index $$t$$ for the given Inputs as $$TEMA_t(X,n)$$, and we compute it using the following sequence of Exponential Moving Averages for the given Inputs.

$$EMA_t^{(1)}(X,n) = EMA_t(X,n)$$
$$EMA_t^{(2)}(X,n) = EMA_t(EMA(X,n),n)$$
$$EMA_t^{(3)}(X,n) = EMA_t(EMA(EMA(X,n),n),n)$$

In the above relations, $$EMA_t^{(j)}$$ denotes the $$j-$$fold composition of the $$EMA$$ function with itself, and $$EMA(X,n)$$ is a random variable denoting the Exponential Moving Average of Length $$n$$ for the Input Data $$X$$. We compute $$TEMA_t(X,n)$$ in terms of these Exponential Moving Averages for $$t \geq n$$ as follows.

$$TEMA_t(X,n) = 3EMA_t^{(1)}(X,n) - 3EMA_t^{(2)}(X,n) + EMA_t^{(3)}(X,n)$$