# Technical Studies Reference

### 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.

$$\displaystyle{\ln(\hat{X}) = r_t(X,n) \cdot T + \ln(k_t(X,n))}$$

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 $$n$$.