Function forecasting::SimpleLinearRegression(xIndepVarValue, yDepVarValue, LRcoeff, VariationComp, yEstimates, eResiduals)

# forecasting::SimpleLinearRegression

The simple linear regression procedure computes the regression line coefficients based on the values of the observations for the independent and the dependent variables. If desired, the values for variation components and the residuals can be returned as well.

## Mathematical Formulation

Using the notation for observations and estimates given in Simple Linear Regression Notation, the estimates of the coefficients of the linear regression line are determined as follows:

$\hat{\beta}_1 = \frac{\sum_{i=1}^{N}(x_i - \bar{x})(y_i - \bar{y})}{\sum_{i=1}^{N}(x_i - \hat{x})^2}$
$\hat{\beta}_0 = \bar{y} - \hat{\beta}_1\bar{x}$

These values provide the minimum in $$\hat{\beta}_0$$, $$\hat{\beta}_1$$ of the function

$F(\hat{\beta}_0,\hat{\beta}_1) = \sum_{i=1}^{N}e_i^2 = \sum_{i=1}^{N}(y_i - \hat{\beta}_0 - \hat{\beta}_1x_i)^2 )$

Therefore, the values $$\hat{\beta}_0$$ and $$\hat{\beta}_1$$ given above are called the least squares estimates of $$\beta_0$$ and $$\beta_1$$. With these coefficients, the regression line (1) is called the least squares regression line. Every least squares regression line has the following two properties:

• It passes through the point $$(\bar{x},\bar{y})$$

• $$\sum_{i=1}^{N} e_i = 0$$

## Function Prototype

In order to provide the variation components and residuals only when needed, there are three flavors of the SimpleLinearRegression procedure provided:

forecasting::SimpleLinearRegression( ! Provides the estimates of the line coefficients, but not the variation components nor the residuals
xIndepVarValue,              ! Input, parameter for independent
yDepVarValue,                ! Input, parameter for dependent
LRcoeff)                     ! Output, parameter for line coefficients

forecasting::SimpleLinearRegressionVC( ! Provides the estimates of the line coefficients and the variation components
xIndepVarValue,                ! Input, parameter for independent
yDepVarValue,                  ! Input, parameter for dependent
LRcoeff,                       ! Output, parameter for line coefficients
VariationComp)                 ! Output, parameter variation components

forecasting::SimpleLinearRegressionVCR( ! Provides the estimates of the line coefficients, the variation components and the residuals
xIndepVarValue,                 ! Input, parameter for independent
yDepVarValue,                   ! Input, parameter for dependent
LRcoeff,                        ! Output, parameter for line coefficients
VariationComp,                  ! Output, parameter variation components
yEstimates,                     ! Output, parameter for estimates
eResiduals)                     ! Output, parameter for residuals


## Arguments

xIndepVarValue

A one dimensional parameter containing the observations for the independent variable

yDepVarValue

A one dimensional parameter containing the observations for the dependent variable

LRcoeff

A one dimensional parameter for storing the coefficients of the regression line

VariationComp

A one dimensional parameter for storing the values of the variation components

yEstimates

A one dimensional parameter for storing the values of the estimates

eResiduals

A one dimensional parameter for storing the values of the residuals

Note

In order to use this function, the Forecasting system library needs to be added to the application.