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 Federal Highway Administration > Publications > Research Publications > Infrastructures > Pavements > LTPP Publications > 08035 > LTPP Computed Parameter: Moisture Content
 Publication Number: FHWA-HRT-08-035 Date: March 2008

# LTPP Computed Parameter: Moisture Content

## Appendix B. Characterization of Error in the SID

Relative to the least squares error associated with linear regression, assuming that y1 = ax1 + b, then the error (ri) and the variance (ri2) at a point can be expressed as:

(99)

(100)

(101)

(102)

(103)

(104)

(105)

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:

 = vector of j observations X = matrix of xi,j = vector of unknown coefficients = vector of regression errors

Solving for :

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(115)

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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:

(118)

(119)

This yields a solution for the changes in the model coefficients based on the residual error in the model prediction.

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