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Publication Number: FHWAHRT08035 Date: March 2008 
Relative to the least squares error associated with linear regression, assuming that y_{1} = ax_{1} + b, then the error (r_{i}) and the variance (r_{i}^{2}) at a point can be expressed as:
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The total variance over all points n is:
Setting the derivatives of the variance with respect to the coefficients a and b to zero gives:
and yields two equations in the two unknown coefficients a and b:
Which expresses the definition of linear regression. In matrix form, where there are a number (i) independent variables x_{i} associated with observations y_{j} (dependent variable) that form a matrix of independent variables, x_{i,j} can be expressed as:
Where:
= vector of j observations  
X  = matrix of x_{i,j} 
= vector of unknown coefficients  
= vector of regression errors 
Solving for :
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Where the second part of the above expression represents the residual regression error. Formulating this on the basis of partial derivatives:
Differentiating with respect to the vector of unknown coefficients and setting to zero:
Rearranging and solving for :
Where again the second part of the above expression represents the residual regression error. Drawing the analogy to the system identification method (SID):
Where () is the matrix of model predictions. Rearranging:
Where:
[F]  = which is a rectangular sensitivity matrix (k x n); k = number of coefficients a 
{ß}  = which is the matrix of change in the model coefficient (n x 1) 
= the matrix of change in the model prediction or the residual error (k x 1) 
Therefore:
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This yields a solution for the changes in the model coefficients based on the residual error in the model prediction.
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