Covariance matrix of measurement errors

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. 

Most techniques for process data reconciliation start with the assumption that the measurement errors are random variables obeying a known statistical distribution, and that the covariance matrix of measurement errors is given. In Chapter 10 direct and indirect approaches for estimating the variances of measurement errors are discussed, as well as a robust strategy for dealing with the presence of outliers in the data set. 

V covariance matrix of measurement error estimates W covariance matrix of d... 

Only a few publications in the literature have dealt with this problem. Almasy and Mah (1984) presented a method for estimating the covariance matrix of measured errors by using the constraint residuals calculated from available process data. Darouach et al. (1989) and Keller et al. (1992) have extended this approach to deal with correlated measurements. Chen et al. (1997) extended the procedure further, developing a robust strategy for covariance estimation, which is insensitive to the presence of outliers in the data set. 

The covariance matrix of measurement errors is a very useful statistical property. Indirect methods can deal with unsteady sampling data, but unfortunately they are very sensitive to outliers and the presence of one or two outliers can cause misleading results. This drawback can be eliminated by using robust approaches via M-estimators. The performance of the robust covariance estimator is better than that of the indirect methods when outliers are present in the data set. 

Here, m j (subvector of m ) represents the measured (integrated) mass flowrates, y = m , the unmeasured ones, and the states of the inventories at time. The vector s. i is formally constant and the measured value of x thus x is composed of the actually measured value of (say, = (nifk) X and of x = s - k., where sj is the actually measured value of the constant represents the preceding estimate of 8,., . The covariance matrix of measurement errors in x used in the reconciliation is F (11.2.1). 

In the given case, the rows la, 4a, and columns la, 4a, 6b, 7a, 8bde are absent. We can further delete the rows 2 (equations concerning nonrequired unn asured quantities only), and the column 8 (vertical band 3) that is zero in the rows lb. Let us further suppose that the covariance matrix of measurement errors is diagonal. Then also the row 4b and the columns 9 and 10 (representing the bands 7b and 8c) can be deleted. In order to determine all the required unmeasured quantities, it is necessary to measure in addition the quantities (mass flowrates) and, while the variables nig and need no more be measured. The reduced model (matrices A and B ) according to Fig. 12-5 is shown in Fig. 12-9. 

The positive definite character of a matrix can be assumed by hypothesis, due to its physical nature. Let us for example have N physical variables Xj, the values of which are measured. The t-th measurement error is the difference e = x] - Xj where x is the measured value, x, the true value. The statistical theory of measurement is discussed in another part of the book (see Chapter 9). Here, let us suppose that having a large set of measurements, the average error equals zero, and that the covariance matrix of measurement errors, say F of elements, can be approximated by the averages... 

E((e-Co )(e-eo ) ), covariance matrix of measurement errors, symmetric positive definite, / x 7 if not otherwise specified given a priori by the statistical model of measurement submatrices of F (11.2.1), covariance matrices of measurement errors in the components of s, and mj, , respectively... 

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