# Technical Studies Reference

This study calculates and displays a Cumulative Adjusted Value of the data specified by the Input Data Input.

Let $$X$$ be a random variable denoting the Input Data Input, and let $$X_t$$ be the value of the Input Data at Index $$t$$. Let the Input Long Term Smoothing Length be denoted as $$n$$. Then we denote the Adjusted Value for the given Inputs at Index $$t$$ as $$AdjVal_t(X,n)$$, and we compute it for $$t \geq 0$$ using an Exponential Moving Average as follows.

$$AdjVal_t(X,n) = X_t - EMA_t(X,n)$$

Note: The internal calculations performed in the Exponential Moving Average calculation are used in the calculation of $$AdjVal_t(X,n)$$. Consequently, there is no delay of $$n$$ units either in this calculation or in the subsequent calculation of the Cumulative Adjusted Value.

We denote the Cumulative Adjusted Value for the given Inputs at Index $$t$$ as $$AdjCumVal_t(X,n)$$, and we compute it with the following recursion relation.

$$CumAdjVal_0(X,n) = AdjVal_0(X,n)$$

$$CumAdjVal_t(X,n) = CumAdjVal_{t - 1}(X,n) + AdjVal_t(X,n)$$

For some background as to why this study was developed, refer to Market Statistics Calculations Compared to Other Data Services.