Covariance measurement errors

The sequence of the innovation, gain vector, variance-covariance matrix and estimated parameters of the calibration lines is shown in Figs. 41.1-41.4. We can clearly see that after four measurements the innovation is stabilized at the measurement error, which is 0.005 absorbance units. The gain vector decreases monotonously and the estimates of the two parameters stabilize after four measurements. It should be remarked that the design of the measurements fully defines the variance-covariance matrix and the gain vector in eqs. (41.3) and (41.4), as is the case in ordinary regression. Thus, once the design of the experiments is chosen... 

We also use a linearized covariance analysis [34, 36] to evaluate the accuracy of estimates and take the measurement errors to be normally distributed with a zero mean and covariance matrix Assuming that the mathematical model is correct and that our selected partitions can represent the true multiphase flow functions, the mean of the error in the estimates is zero and the parameter covariance matrix of the errors in the parameter estimates is ... 

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. 

Chen, J., Bandoni, A., and Romagnoli, J. A. (1997). Robust estimation of measurement error variance/ covariance from process sampling data. Comput. Chem. Eng. 21, 593-600. 

Accordingly, we have for the estimate of the variables, the measurement errors, and the error estimate covariance... 

To solve the least squares problem for the estimate of the measurement errors we need to invert the covariance matrix <. It is possible to relate to through a simple recursive formula. Let us recall the following matrix inversion lemma (Noble, 1969) ... 

In the previous development it was assumed that only random, normally distributed measurement errors, with zero mean and known covariance, are present in the data. In practice, process data may also contain other types of errors, which are caused by nonrandom events. For instance, instruments may not be adequately compensated, measuring devices may malfunction, or process leaks may be present. These biases are usually referred as gross errors. The presence of gross errors invalidates the statistical basis of data reconciliation procedures. It is also impossible, for example, to prepare an adequate process model on the basis of erroneous measurements or to assess production accounting correctly. In order to avoid these shortcomings we need to check for the presence of gross systematic errors in the measurement data. 

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

We have now completed the preliminary step of obtaining the covariance for the initial estimate as a function of the known covariance matrix of the measurement errors. At time t = t, following the initial estimation, a new observation yi is completed. Now the objective to be minimized by combining the new data with the previous estimate can be written as... 

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. 

This procedure (based on sample variance and covariance) is referred to as the direct method of estimation of the covariance matrix of the measurement errors. As it stands, it makes no use of the inherent information content of the constraint equations, which has proved to be very useful in process data reconciliation. One shortcoming of this approach is that these r samples should be under steady-state operation, in order to meet the independent sampling condition otherwise, the direct method could give incorrect estimates. 

Two alternative procedures have been suggested in the literature to solve the problem and they will be discussed next. Alternative 1 was proposed by Almasy and Mah (1984). They attempt to minimize the sum of the squares of the off-diagonal elements of a measurement error covariance matrix subject to the relation deduced from the... 

The problem of estimating the covariance matrix of the measurement errors may now be formulated as... 

To illustrate the application of the strategy for the diagonal case, we consider the process system taken from Ripps (1965). As indicated in Chapter 5, it consists of a chemical reactor with four streams, two entering and two leaving the process. All the stream flowrates are assumed to be measured, and their true values are x = [0.1739 5.0435 1.2175 4.00]T. The corresponding system matrix, A, and the covariance of the measurement errors, F, are also known and given by... 

General case. If some measurement errors are correlated, the covariance matrix is not diagonal. It is assumed that we know the sensors which are subjected to correlated measurement errors, for example because they share some common elements (e.g., power supplies). There are then s off-diagonal elements of d>,. .., w),... 

As before, all the stream flowrates are assumed to be measured in the process flowsheet presented by Ripps (1965). The corresponding system matrix, A, and the covariance of the measurement errors, (P, are given in Section 10.3. 

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. 

Assuming that the measurement errors follow a certain distribution with covariance P, then r will follow the same distribution with expectation and covariance given by... 

Here the vector e is known from the measurements. Now a postulate that the vector 6 of measurement error is normally distributed is equivalent [10] to stating that the e is also a normal vactor, because by (28) e is approximated by a linear transfomation of a normal vector. Assume E [6] =0, then e will also have zero mean and covariance matrix... 

Suppose that the variables BJ are to be determined by a least-squares fit of the relations, Eq. 16, to the measured values T exp (vector Yexp). Assume that the measurements Yexp are unbiased ( (Yexp) = Ytrue where E() represents the mean or expectation value) and that the measurement errors and their correlations are described by the positive-definite nxn variance-covariance matrix 0Y which can be written as the dyadic P... 

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