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FHWA > Engineering > Pavements > Research > LTPP > FHWA-HRT-08-035 |
LTPP Computed Parameter: Moisture ContentAppendix B. Characterization of Error in the SIDRelative to the least squares error associated with linear regression, assuming that y1 = αx1 + b, then the error (ri) and the variance (ri2) at a point can be expressed as:
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 xi associated with observations yj (dependent variable) that form a matrix of independent variables, xi,j can be expressed as: Where:
Solving for
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 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 Where:
Therefore:
This yields a solution for the changes in the model coefficients based on the residual error in the model prediction.
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This page last modified on 04/16/08 |