# Technical Studies Reference

## T3

This study calculates and displays a T3 Moving Average of the data specified by the Input Data Input. The study was developed by Tim Tillson.

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 $$n$$, and let the Input Multiplier be denoted as $$v$$. Then we denote the value of T3 at Index $$t$$ for the given Inputs as $$T3_t(X,n,v)$$, and we compute it using the following sequence of Exponential Moving Averages for the given Inputs.

$$EMA_t^{(1)}(X,n) = EMA_t(X,n)$$
$$EMA_t^{(2)}(X,n) = EMA_t(EMA(X,n),n)$$
$$EMA_t^{(3)}(X,n) = EMA_t(EMA(EMA(X,n),n),n)$$
$$EMA_t^{(4)}(X,n) = EMA_t(EMA(EMA(EMA(X,n),n),n),n)$$
$$EMA_t^{(5)}(X,n) = EMA_t(EMA(EMA(EMA(EMA(X,n),n),n),n),n)$$
$$EMA_t^{(6)}(X,n) = EMA_t(EMA(EMA(EMA(EMA(EMA(X,n),n),n),n),n),n)$$

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 $$T3_t(X,n,v)$$ in terms of these Exponential Moving Averages for $$t > 0$$ as follows.

$$T3_t(X,n,v) = -v^3EMA_t^{(6)}(X,n) + 3v^2(1+v)EMA_t^{(5)}(X,n) - 3v(1+v)^2EMA_t^{(4)}(X,n) + (1+v)^3EMA_t^{(3)}(X,n)$$