Simple linear regression: Difference between revisions

Revision as of 14:34, 5 August 2008

Revision as of 18:01, 8 November 2008

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== Estimating the regression line ==

The parameters of the linear regression linemodel, <math>Y Y_i = a + bXbX_i + \varepsilon_i </math>, can be estimated using the method of [[ordinary least squares]]. This method finds the line that minimizes the sum of the squares of the regression residualserrors, <math> \sum_{i = 1}^Nn \hat{\varepsilon}_varepsilon_{i}^2 </math>. The residual is the difference between the observed value and the predicted value: <math> \hat{\varepsilon} _{i} = y_{i} - \hat{y}_{i} </math>

The minimization problem can be solved using calculus, producing the following formulas for the estimates of the regression parameters:

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Ordinary least squares produces the following features:

# The line goes through the point <math> (\bar{Xx},\bar{Yy}) </math>.

# The sum of the residuals is equal to zero.

# The linear combination of the residuals in which the coefficients are the ''x''-values is equal to zero.