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# Moving Linear Regression / Moving Average - Linear Regression

The **Moving Linear Regression** (MLA) and the **Moving Average - Linear Regression** (LRMA) studies calculate and display the value of a linear regression function of the selected **Input Data** (Open, High, Low, Close) over the specified **Length**.

For the calculation method, refer to the **LinearRegressionIndicator_S** function in the /ACS_Source/SCStudyFunctions.cpp file in the folder Sierra_Chart is installed to. We give a detailed mathematical description of this method below.

If you draw a Linear Regression Chart Drawing over the same **Length** that you have set in the study Inputs for this study, then where that drawing ends, it will have the same value as the **Moving Average - Linear Regression** study.

Next we describe the calculation of the Linear Regression Indicator (LRI). Let \(n\) be the Input **Length**. Let \(T\) be the variable measured along the horiztonal axis, let \(X\) be a random variable denoting the **Input Data**, which is measured along the vertical axis. We denote the values of these variables at Index \(i\) as \(T_i = i\) and \(X_i\), respectively , where \(i\) runs from \(1\) to \(n\) in each calculation. We denote the Current Index Value as \(t\). The LRI function computes each of the following sums for \(t \geq n - 1\). These sums are used to calculate the regression statistics.

\(\displaystyle{\left(\sum_{i=1}^n {i}\right)^2 = \frac{n^2(n+1)^2}{4}}\)

\(\displaystyle{\sum_{i=1}^n {i^2} = \frac{n(n+1)(2n+1)}{6}}\)

\(\displaystyle{\sum_{i=t-n+1}^t {X_i}}\)

\(\displaystyle{\sum_{i=t-n+1}^t{T_{i - t + n}X_i} = \sum_{i=t-n+1}^t{(i-t+n)X_i}}\)

For an explanation of the Sigma (\(\Sigma\)) notation for summation, refer to our description here.

Note: The sums over the \(T-\) values (that is, the first three sums) do not move, as the sums over the \(X-\) values (the last two sums) do. This is compensated for by using the **Length** \(n\) in certain places instead of the current value \(t\) of the Index.

These sums are used to compute the regression statistics for \(t \geq n - 1\), as shown below.

**Slope:**

**Intercept:**

The regression model is of the form \(\hat{X} = a_t(X,n) + b_t(X,n)\cdot T\), where \(\hat{X}\) is an estimator of \(X\).

**Linear Regression Indicator:**

The Linear Regression Indicator is the vertical coordinate of the right endpoint of the linear regression trendline of **Length** \(n\). Its value \(LRI_t(X,n)\) at Index \(t\) for the given Inputs is calculated for \(t \geq n - 1\) as \(LRI_t(X,n) = a_t(X,n) + b_t(X,n)\cdot n\).

The **Moving Average - Linear Regression** and **Moving Linear Regression** are both mathematically identical to the Linear Regression Indicator for \(t \geq n - 1\). Their respective values at Index \(t\) for the given Inputs are denoted as \(LRMA_t(X,n)\) and \(MLR_t(X,n)\).

#### 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, 28th September, 2022.