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A New Method for Measurement Error Covariance Estimation.

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Dev. Chem. Ens. Mineral Process., 9(IL2),pp. IOI-IO8.2001.
A New Method for Measurement Error
Covariance Estimation
Yuhong Zhao', Zhongwen Gu and Chunhui Zhou
National Laboratory of Industrial Controi Technology, Institute of
Systems Engineering, Zhejiang University, Hangzhou 310027, P.R.
China
A new robust indirect algorithm for measurement error covariance estimation is
proposed in this paper. The residual covariance matrix is estimated using Hampel's
three-part redescending M-estimators. Then measurement error covariance can be
obtained from ihe residual covariance. Unlike the conventional indirect estimations
of measurement error covariance, which are very sensitive to gross errors, credible
results can be achieved either with or without :he presence of external causes.
implementation results show the robustness of the proposed method.
Introduction
Reliable process data play an important role in modern chemical plants for the
purpose of process control, optimization implementation or performance evaluation.
However, the measurements are contaminated inevitably by random errors and gross
errors, whch may result fiom the miscalibration and failure of the measuring
instruments, or fiom leaks of production equipment and pipe h e s , so that the data do
not generally satisfy the process constrains. Thus before the measurements can be
used successfully, data rectification is fiequently required which is receiving more
attention in the chemical e n g i n e g and mineral processing fields.
In the data rectification process, gross errors are detected and eliminated and then
measurements are adjusted to satisfy the material and energy balances. Almost
without exception alI rectification strategies start with the hypothesis that the
measurement errors are random and normally distributed with zero mean and a
known covariance matrix. In practice, the covariance matrix is usually unknom or
known approximately. By far the most commonly used statistical technique for
covariance matrix estimation is the direct method
(la)
LI
I
where X is sample of N repeated n-dimensional measurements X i . The unbiased
'Authorfor correspondence (e-mail: yhzhao@iipc.ziu.edu.cn).
10I
Y.Zhao, Z Gu and C.Zhou
maximum likelihood estimator can be produced only under the assumption of an
independent error distribution with constant covariance. This drawback is eliminated
by the indirect method whch makes use of the covariance matrix of constraint
residuals.
Almasy and Mah [l] gave a method for estimating the covariance matrix of
measurement errors by using the constraint residuals calculated from available
process data. Thls method gives an analytical solution which is very sensitive to the
correlated measurements. Keller et al. [23 presented a method based on the relation
deduced from the statistical properties of material balance consiraints, whch is more
robust with regard to the correlated measurements and more suitable in practical
situations. However these algorithms are sensitive to the presence of gross errors in
the data set because of the use of a conventional method to calculate the covariance
matrix of the residuals. Chen and Romagnoli [3] proposed a robust indirect method
based on Huber M-estimator. This approach is an iterative calculation of covariance
matrix of .the residuals with weights assigned to the observations according to their
&stances from the current estimated location. Since Huber type weights are used in
the method, the dluence by moderately large external causes can not be eliminated
completely.
A new indirect method based on Hampel's three-part redescendmg M-estimators
is proposed using Chen and Romagnoli's approach. In the next section, the
formulation of the indirect method for the error covariance estimation is introduced.
Then the detail of the proposed algorithm is presented and two examples are
simulated to show the performance of the algorithm. Finally, conclusions are drawn
and the further research is indxated.
Problem Formulation
Consider a linear system described by the following equation:
AX',= O
(2)
where A is a m x n full-row rank matrix with m 5 n , X't is the true vector of the ndimensional process variables at time k; and the observation equation is:
X,= X; + e,
(3)
where X , is the vector of measurement at time k and e, is the measurement error
vector at time k. From equations (2) and (3), the residuals R, is given by:
R, = AX,= AX't
+ Ae, = Ae,
(4)
Under the assumption that e, is Gaussian with zero mean and positive definite
"
covariance V ( V = ( v ~ ) "),~then:
E(e,) = 0
102
(5)
A New Methodfor Measurement Error Covariance Estimation
E(ekekT)
=V
(6)
E(Rk)= E(Aek)= AE(e,) = 0
(7)
The covariance matrix of the residuals can be obtained as following:
H = cov(R,) = E(R,R:)
= E(Ae,ekTAT)
= AE(ekekT)AT
= AVA'
(8)
Given A and V , H is uniquely defined by equation (9,
but the converse is not
true. However, since the measurements are normally made by independent
instruments, the measurement errors should be uncorrelated. If instruments share
some common elements such as power supplies, errors can be assumed to be weakly
correlated. The implication is that H is diagonal or diagonally dominant. Hence the
estimation of H can be obtain for specified off-diagonal elements.
Using Kronecker matrix products and vecoperator [l], equation (8) can be
rewritten as:
vec(H) = (A 63 A)vec(V)
(9)
where Kronecker product of matrix A=(a,),"
and B,,, is defined as a
( m * s ) x ( n . t )matrix:
AQDB= [a, . B )
( 10)
where ay .B denotes the product of matrix B by the scalar a,, . vec(A) is defined as:
where A,,i= 1,2;.-,n denotes the ith column of matrix A.
The indirect method uses the residuals covariance matrix H to estimate the
measurement error covariance matrix V. Since V is a symmetrical matrix, let
v, vp4 --- v u f , where v,,,, , vU, ( p < q k < Z ) are offD=[vll vu
diagonal elements, then equation (9) can be rewritten as:
---
vec( H)= GD
where
Lam14 a,&
a,A,
anpAq+awAp*..
a,A,+a,,A,
I03
Y. Zhao, Z.Gu and C. Zhou
and A, ( i = l,...,n ) is the ith column of matrix A. Solving the least squares problem:
min{vec(H) - GDIvec(H)-GOT
the maximum likelihood estimation of D is given by:
D = (GTG)-'Grvec(H)
(14)
Under the assumption of uncorrelated error distribution, simplified result can be
obtained.
Robust Estimation Based on Hampel's Three-part Redescending
M-estimators
Hampel's three-part redescendmg M-estimator belongs to M-estimator or generalized
maximum likelihood estimator, which can be stated as:
2[w* )(X,- W W , - M ) T I / ( N - 1)
(u,
=V
(16)
,=I
where W,, W, are arbitrary weights; V is positive definite matrix, whose Cholesky
decomposition is: V = SS' ; and ui = (KS)-'(X,- M ), K > 0, K is a tuning
parameter varying with different M-estimators and K = 1 here. Note that when
W, = W,= 1 , equation (15) and (16) become the form of conventional MLE. The
iterative procedure can be written as:
where p,q =l,...,n and k notes iterative step. The chosen weight functions are
illustrated in Figure 1.
Figure 1. Illustration of the weightfunctions.
104
A New Method for Measurement Error Covariance Estimation
W,(U) = W I 2 ( 4
(20)
c is the rejection point, over which the rnfluence of the gross errors can be
eliminated completely. The selection of (a,b,c) is discussed in detail in Huang's
work [4]. The iterative algorithm is described as follows:
(1) Compute the residuals:
R=AX
(21)
(2) Compute the covariance matrix of R by iteration:
(2.1) Specify (a,b, c ) and the threshold for convergence and initialize the location
and scale parameters:
Mpo
=median(X,)
p=T,...,n; i=l,..-,N
H,O = g ( x , -M,o)(x, - M ~ o )p , ~q = l,.-,n ; i =I,..., N
(22)
(23)
,=I
Compute the Cholesky decomposition of H o: H o= SSTand let B o = S-' .
(2.2) Iterative calculation:
Compute the Cholesky decomposition of C C = DD' and let:
p = D-'
B = PB"
(2.3) Judge the convergence:
If equation (27) is satisfied, H = B-'(B-')T. Else let M o= M and B o = B , return
(2.2).
105
Y. Zhm, Z Gu and C. Zhou
It is worthwhile mentioning that the positive definiteness of matrix C can be
maintained in the iteration procedure.
Simulation
a = 3, b = 5, c = 12 and e = 6 = lo-' are used in the following simulation.
(1) Diagonal case
A system with two entering and two leaving streams is studied in this section,
whose system matrix is described as [2]:
0.1 0.6
- 0.2 - 0.7
0.1 0.3 -0.6 -0.2
All stream flowrates are assumed to be measured and their true values are:
x-= [0.1739
5.0435 1.2175 4.003'
The variance matrix of the measurement errors is:
V = diag(0.003 1 0.0025 0.0006 0.0392)
A multivariate normal distribute data set with the mean X' and variance V is
generated to simulate the process measurements. The sample size is 1000. The
performances of the conventional indirect method, of the indirect method based on
Huber M-estimator and of the proposed algorithm are investigated in the cases of
with and without gross errors respectively. The computation results are shown in
Table 1.
The three methods give similar results without the presence of gross errors. The
conventional indirect method cannot derive a reliable result due to the presence of
gross errors, while the proposed method and the indmect method based on Huber Mestimator can work well. In the case of large gross errors, the results from the
proposed method are more reliable than that from the indirect method based on Huber
M-estimator.
(2) Non-diagonal case
A network consisting of 7 nodes and 12 streams [2] is given in Figure 2. The
measurement errors of stream sensors 2 and 5, and 6 and 11 are to be correlated. The
system matrix is:
1 - 1 0 0 0 0 - 1 0 0
0 1 - 1 0 0 0 0 - 1 0
0 0 1 - 1 0 0 0 0 0
A= 0 0 0 1 - 1 0 0 0 - 1
0 0 0 0 1 - 1 0 0 0 0 0 0 0 0 0 1 1 0
-0 0 0 0 0 0 0 0 1
106
0
0
0
0
1
0
1
0
0
0
0
0
- 1
0
0
0
0
0
0
0 - 1
8.10
20.00
20.00
6.00
9.00
10.00
10.00
30.00
30.00
20.00
20.00
7.50
15.00
10.00
True
Values
0.0400
0.0029
0.0025
0.0006
0.0409
0.0006
0.0030
0.0026
0.0029
0.0025
0.0006
0.0403
0.0404
p,ppLp
PSpzI
0.0030
0.0030
0.0027
0.0007
0.0404
0.0030
0.0026
0.0006
0.0398
0.0380
0.0029
0.0390
0.003 1
0.0030
0.0024
0.0006
0.0396
30.89
30.19
22.08
18.42
7.29
14.86
10.56
9.92
9.34
9.01
21.30
19.60
5.5 I
9. LO
30.02
30.63
22.50
18.37
7.66
14.80
10.34
9.50
9.24
9.22
19.96
18.78
5.86
9.53
30.80
29.98
21.92
18.36
7.28
14.81
10.43
9.93
9.27
9.03
21.08
19.54
5.46
9.08
9.19
m
15.26
10.60
9.76
9.56
8.71
21.34
19.75
ep?
21.59
18.77
&2!2
31.58
30.07
30.94
22.24
18.46
8.13
14.78
10.33
10.38
9.58
9.15
20.06
18.74
6.33
9.36
30.87
30.37
2 I .63
18.70
7.98
14.76
10.45
9.95
9.35
8.92
21.13
19.50
6.00
8.88
22.33
18.57
19.96
20.05
9.57
47.03
14.64
10.32
10.36
9.51
9.17
20.07
18.64
6.29
9.39
m
1194
71.62
16.84
10.80
9.27
10.48
7.60
21.46
20.32
30.03
33.80
30.82
30.04
21.81
18.38
7.31
14.79
10.43
9.90
9.45
8.95
21.16
19.46
5.49
9.08
Table 2. Comparisons of the results in non-diagonal case.
With gross errors
Without gross errors
X,,= 760, X,,= 360
X,,= 600, X , , = 200
CIE
IEHu
IEHa
CIE
IEHu
IEHa
CIE
IEHu
IEHa
0.0030
0.0026
0.0006
0.0407
Annotation: CIE: conventional indirect estimation; IEHu: indirect estimation based on Huber M-estimator: IEHa: indirect
estimation based on Hampel’s three-part redescending M-estimator;underline: unreliable estimation result.
Off-diagonal
elements
Diagonal
elements
Diagonal
elements
True
Values
Table 1. Comparisons of the results in diagonal case.
With gross errors
Without gross errors
XI, = 1.8126
XI, = 7.2126
CIE
IEHu
IEHa
CIE
IEHu
IEHa
CIE
IEHu
IEHa
s
a
g.
5
9
5’
B
5
2.
2
5
Q
e
9
i
3
E
s
0
3
c”
e
2:
cs
Y. Zhm, 2 Gu and C. Zhou
4
Figure 2. Process network.
The diagonal elements of the covariance matrix are:
&O 30 20 20 7.5 15.00 10.00 10.00 10.00 8.10 20.00 20.001
and the off-diagonal elements are: V,, = V,,2= 6.00, V,.,, = V,,., = 9.00. The study is
similar to the uncorrelated measurement error case. The results from the three
methods are shown in Table 2. Similar conclusions can be drawn from the simulation
results. The indirect method based M-estimator can give perfect results both with and
without gross errors, though the proposed method perfom better than the indirect
method based on Huber M-estimator in the case of large gross errors. However the
conventional indirect estimator gives completely incorrect results for the correlated
coefficients when the corresponding measurements are contaminatedby gross errors.
Conclusions
Estimation of measurement error covariance matrix is very useful in data
rectification. A new robust indirect algorithm based on Hampel's three-part
redescendmg M-estimators is presented. The mfluence of gross errors can either be
elmhated or limited, so that the estimator can give reliable result even in the case of
the presence of gross errors. Two examples demonstrate the robustness of the
algorithm. Since the algorithm is restricted to the linear constraint, a further study will
be focused on the robust estimation subject to nonlinear constraints.
References
Almasy, G.A., and Mah, R.S.H. 1984. Estimation of Measurement Error Variances from h c e s s
Data. Ind. Eng. Chem. Process Des. Dev., 23(4), 779-784.
2. Keller, J.Y.; Zasadzinski, M., and h u a c h , M. 1992. Analytical Estimator of Measurement Error
Variances in Data Reconciliation. Computers chem. Engng., 16(3), 185-188.
3. Chen, J.; Bandoni, A. and Romagnoli, J.A. 1997. Robust Estimation of Measurement Error
VariancdCovariance from Process Sampling Data. Computers chem. Engng., 21(6), 593-600.
4. Huang, Y.C. 1990. Data Exploration and Robust Estimation. (In Chinese) Mapping Ress. Beijing.
1.
Received: 20 June 1999; Accepted after revision: 14 May 2000.
108
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