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### Moving Average - Exponential Regression

The **Moving Average - Exponential Regression** (ERMA) study calculates and displays an exponential regression moving average of the data specified by the **Input Data** Input.

Let \(X\) be a random variable denoting the **Input Data**, and let \(X_t\) denote the value of the **Input Data** at Index \(t\). Let the **Length** Input be denoted as \(n\). Then we denote the **Moving Average - Exponential Regression** at Index \(t\) as \(ERMA_t(X,n)\).

The regression model is of the form \(\hat{X} = k_t(X,n)e^{r_t(X,n) \cdot T}\), where \(\hat{X}\) is an estimator for \(X\). We call \(k_t(X,n)\) the coefficient, and we call \(r_t(X,n)\) the growth constant.

Taking the logarithm of both sides of this model, we obtain the following *linear* regression model for the logarithm of the **Input Data**.

We then compute linear regression statistics for \(\ln(X_t)\). That is, we compute \(b_t(\ln(X),n)\) and \(a_t(\ln(X),n)\).

This yields the following.

- \(b_t(\ln(X),n) = r_t(X,n)\)
- \(a_t(\ln(X),n) = \ln(k_t(X,n))\), or \(k_t(X,n) = e^{a_t(\ln(X),n)}\).

Finally, we compute \(ERMA_t(X,n)\) for \(t \geq n - 1\) as follows.

\(\displaystyle{ERMA_t(X,n) = k_t(X,n)e^{r_t(X,n) \cdot n}}\)The value of \(ERMA_t(X,n)\) is the vertical coordinate of the right endpoint of the exponential regression trendline of **Length** \(n\).

#### 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 Wednesday, 07th July, 2021.