Least-square constraints errors, linear

The adjustment of measurements to compensate for random errors involves the resolution of a constrained minimization problem, usually one of constrained least squares. Balance equations are included in the constraints these may be linear but are generally nonlinear. The objective function is usually quadratic with respect to the adjustment of measurements, and it has the covariance matrix of measurements errors as weights. Thus, this matrix is essential in the obtaining of reliable process knowledge. Some efforts have been made to estimate it from measurements (Almasy and Mah, 1984 Darouach et al., 1989 Keller et al., 1992 Chen et al., 1997). The difficulty in the estimation of this matrix is associated with the analysis of the serial and cross correlation of the data. 

Recall that, in the absence of gross errors, the measurement and linear constraint models are given by Eqs. (7.1) and (7.4), respectively. Furthermore, the solution of the least square estimation problem of x variables is... 

The constrained least-square method is developed in Section 5.3 and a numerical example treated in detail. Efficient specific algorithms taking errors into account have been developed by Provost and Allegre (1979). Literature abounds in alternative methods. Wright and Doherty (1970) use linear programming methods that are fast and offer an easy implementation of linear constraints but the structure of the data is not easily perceived and error assessment inefficiently handled. Principal component analysis (Section 4.4) is more efficient when the end-members are unknown. 

In Eq. 5.15, A, q, g and a are the mxn kernel matrix, the distribution vector, the ACF vector, and the error vector, respectively. The regularizor G is a linear constraint operator that defines the additional constraint to the process. The best possible fit is sacrificed in order to find a reasonable and stable solution. G can be set in various values the identity matrix, the first derivative operator, or the second derivative operator. The coefficient a, called the regularization parameter, allows the process to define the strength of the constraint in the solution. A value of zero leads the algorithm back to a normal non-negative least-squares result. A small value does not introduce any effective constraint and so the result may be unstable and may have little relation to the actual solution. Values that are too large are insensitive to the measured data, resulting... 

---