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 (r_i) and the variance (r_i²) 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 xi,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.