Computers and Chemical Engineering 24 (2000) 2755 – 2764
www.elsevier.com/locate/compchemeng
A new approach for improved identification of measurement bias
Sriram Devanathan a, Derrick K. Rollins b,*, Stephen B. Vardeman c
a
Department of Chemical Engineering, Iowa State Uni6ersity, Ames, IA 50011, USA
Departments of Chemical Engineering and Statistics, Iowa State Uni6ersity, Ames, IA 50011, USA
c
Departments of Statistics and Industrial and Manufacturing Systems Engineering, Iowa State Uni6ersity, Ames, IA 50011, USA
b
Received 27 August 1998; received in revised form 28 August 2000; accepted 28 August 2000
Abstract
This work presents a technique that can completely and accurately identify measurement bias in cases where it is not possible
to use the method of Rollins and Davis (1992, 1993) and where the method of Narasimhan and Mah (1987) fail to perform
accurately. This technique makes use of information contained in the relationship between individual measurements and the
corresponding nodal imbalance. The performance of this method is demonstrated on a problem from the literature that has been
difficult for other methods to handle. In addition, this article discusses how the new technique can be used as a visual monitoring
tool for identifying biased measured variables. © 2000 Elsevier Science Ltd. All rights reserved.
Keywords: Measurement bias; Imbalance correlation strategy; Unbiased estimation technique
1. Introduction
In the chemical industry, measurements collected on
process variables are subject to both large random and
systematic errors (i.e. measurement biases). Often accurate values of process variables are required for the
design of new processes, improvement of existing processes, accurate material accounting, and optimal process control. Hence, it is desirable that measured
process variables be close to their true values (i.e. be
accurate) and also satisfy the physical constraints that
govern the process variables (by the laws of conservation). In general, the mathematical reduction of random
variation of measured process variables is broadly
classified as ‘filtering’ or ‘smoothing’. When estimates
(filtered/smoothed values) are required to satisfy the
physical constraints, the task of obtaining such estimates is called data reconciliation (DR). However, in
the presence of measurement biases, although the estimates may satisfy the physical constraints, they can still
be very inaccurate. Hence, it is important to detect,
identify, and eliminate the affect of biases in order to
obtain accurate estimates of process variables.
* Corresponding author. Tel.: +1-515-2947642; fax: + 1-5152942689.
This article considers issues related to the identification of biases in measured process variables. The past
four decades have witnessed the introduction of various
statistical methods in chemical engineering research
(see, for example, Reilly & Carpani, 1963) for the
purpose of detecting and identifying biases in measured
variables. Mah and Tamhane (1982) introduced the
measurement test (MT), which has grown to be perhaps
the most widely used statistical test in this context.
However, it has been shown that when applied to a
process with multiple measurements this test can have a
high probability of type I errors (incorrect identification
of unbiased variables) and low power (small probability
of correct identification of biased variables) (Heenan &
Serth, 1986). Narasimhan and Mah (1987) proposed a
serial compensation strategy (SCS). In this strategy, one
measurement bias is identified at a time, then estimated,
and mathematically removed, before attempting to
identify another bias. Rollins and Davis (1992) discussed the following undesirable characteristics of SCS,
(1) it can have a large probability of making a wrong
conclusion for measured variables that are unbiased
when at least one variable is biased; and (2) estimates
for measurement biases can be inaccurate.
Rollins and Davis (1992, 1993) developed a new
approach for identifying measurement biases under
0098-1354/00/$ - see front matter © 2000 Elsevier Science Ltd. All rights reserved.
PII: S 0 0 9 8 - 1 3 5 4 ( 0 0 ) 0 0 6 2 6 - 8
2756
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
steady state or pseudo steady state conditions and
linear physical constraints. They called this method the
unbiased estimation technique (UBET). Rollins and
Davis (1992) presented results from a study of this
method for various combinations of two non-zero measurement biases (d). UBET was illustrated on the process represented in Fig. 1, which has seven (7) mass
flow variables and four (4) nodes (interconnecting
units).
As indicated in Rollins and Davis (1992), one of the
limitations of UBET is its inability to pinpoint biased
process variables for certain combinations of ds. Their
work showed that SCS also performs poorly in these
cases. In these situations UBET can narrow down the
location to three variables, but is unable to make a
more specific identification than ‘at least two of the
three variables are biased’.
Phillips and Harrison (1993) presented a gross error
detection and data reconciliation analysis for the context of experimental kinetics. Their work was based
upon the modified iterative measurement test (MIMT)
of Serth and Heenan (1986). Hence, the problems of
high type I error probability and low power associated
with the measurement test (as mentioned earlier) exist
here too.
Tong and Crowe (1996) developed a new strategy for
detection of gross errors using principal component
analysis. The main focus of their work was the development of a method that remained effective when the
assumption of normality was not valid. However, for
certain combinations of biases their method does not
appear to be capable of leading to complete identification (due to confounding of the effects of the multiple
biases). Furthermore, the principal component tests
involve intensive computations in calculating eigenvalues and eigenvectors, which could be a drawback for
some large processes requiring on-line detection.
In an effort to improve identification, this work
presents a new strategy that makes use of the relationship between a nodal imbalance (i.e. a mass or energy
balance residual) and the measured variables involved
in the nodal balance. This technique is computationally
simple and straightforward. In addition, this article will
show that it can perform well in determining the specific locations of the measurement biases. We are
Fig. 1. Recycle process network used in the simulation study taken
from Narasimhan and Mah (1987).
calling this technique the ‘imbalance correlation strategy (ICS)’.
Before presenting ICS, the next section reviews the
relevant measurement and process models. This section
is followed by a description of how ICS works using a
process example. Next, the test statistic is presented.
Finally, we discuss the results of a simulation study to
evaluate the performance of ICS.
2. Mathematical models
This section presents the statistical and physical models for a pseudo steady state process related to the work
of Narasimhan and Mah (1987), Rollins and Davis
(1992, 1993). The notation of this section will be important to the introduction and understanding of the proposed ICS. First, the statistical model (relating the
measured and the true values) can be represented by
yij = mi + dij + lij + oij
(1)
where
lij N(0, s 2li),
(2)
oij N(0, s )
(3)
2
oi
and
E[yij ]= mi + dij
(4)
and is subject to
Am = g
(5)
Æm1 Ç
Ãm Ã
à 2Ã
with m= Ã · Ã
÷Ã
à Ã
Èmp É
(6)
where yij is the measured value of variable i at the jth
time instant; mi is the steady state true value of variable
i; dij is the measurement bias of variable i at the jth
time instant; lij is the true value of the random process
deviation of variable i from mi at the jth time instant;
and oij is the random error of variable i at the jth time
instant. A is a q× p matrix often called the constraint
matrix and in this case (since the constraints are simply
total mass balances taken around each node) the number of constraint equations q is equal to the number of
nodes. Eq. (5) represents the linear mass and energy
conservation constraints and g represents the vector of
process leaks. This article assumes that the oij ’s are
normally distributed with mean 0 and known variance.
Additionally, each variable is assumed to be independent at different values of j (i.e. at different times).
Finally, the o’s are assumed to be independent of the
l’s.
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
Rollins and Davis (1992) showed how nonzero elements of g can be identified for steady state conditions.
Hence, for simplicity we set g= 0. In vector form, for
the jth sampling time, Eq. (1) can be written as
yj = m+ dj +lj + oj
(7)
2757
(i.e. the inclusion of the l’s). We would like to note that
the proposed method is subject to the same conditions
and assumptions as Narasimhan and Mah (1987). For
example, dij in Eq. (1) is not restricted to positive
values.
where
Æy1j Ç
à Ã
Ãy2j Ã
yj = Ã · Ã ,
à Ã
÷ Ã
Èypj É
Æd1j Ç
à Ã
Ãd2j Ã
dj = Ã · Ã ,
à Ã
à · Ã
Èdpj É
Æo1j Ç
Ão Ã
à 2j Ã
oj = Ã · Ã
÷Ã
à Ã
Èopj É
3. The imbalance correlation strategy (ICS)
Æl1j Ç
à Ã
Ãl2j Ã
lj = Ã · Ã ,
à Ã
÷ Ã
Èlpj É
ICS will be illustrated using the seven (7) stream and
four (4) nodal steady state process introduced by
Narasimhan and Mah (1987). The process is shown in
Fig. 1. For the conditions given in the previous section,
the true total mass balance around the four nodes is
(8)
Hence, on the average, yj will deviate from m by dj.
The objective of a detection scheme is to determine if
any of the elements of dj are non-zero. Similarly, the
objective of an identification scheme is to determine
which specific elements of dj are non-zero. The steady
state Global Test (Reilly & Carpani, 1963) for the
conditions represented by Eq. (1) can be used for
detection. This test, as well as nodal identification tests,
are based on a linear transformation of yj (at any time
instant j ) to give the vector of nodal imbalances sj (as
in a total mass balance). The transformed measurement
model is given by Rollins, Cheng and Devanathan
(1996) as
sj = Ayj =Am + Adj + Alj +Aoj.
(9)
Let
m1 + m4 + m6 − m2 = 0
(15)
m2 − m 3 = 0
(16)
m3 − m4 − m5 = 0
(17)
m5 − m6 − m7 = 0.
(18)
The transformation vector at time instant j is specified as
ÆsAj Ç
Ãs Ã
sj = Ayj = Ã Bj Ã
à sCj Ã
ÈsDj É
(19)
where
Æ1
Ã0
A= Ã
Ã0
È0
−1
1
0
0
0
−1
1
0
1
0
−1
0
0
0
−1
1
1
0
0
−1
0 Ç
0 Ã
Ã
0 Ã
−1 É
(20)
with
Alj =tj.
(10)
Substituting for tj in the expression for sj, with g= 0
in Eq. (5), we have
sAj N(d1j + d4j + d6j − d2j, s 2sA),
(21)
sBj N(d2j − d3j, s 2sB),
(22)
sj = Adj + tj +Aoj
sCj N(d3j − d4j − d5j, s 2sC)
(23)
(11)
and
with
tj Nq (0, St )
(12)
where St, characterizes the variability due to physical
process changes. Note that
E[sj ] =Adj,
j=1, …, n
(13)
and
Var(sj )=St +ASAT
(14)
where S is the variance – covariance matrix of oj. The
model given in this section is basically the same model
presented by Narasimhan and Mah (1987) but expanded to clearly show the effect of process variability
sDj N(d5j − d6j − d7j, s 2sD).
(24)
As mentioned before, the key feature of ICS lies in
the recognition of a special relationship between a
nodal imbalance and the measured variables associated
with this node. Table 1 is helpful in demonstrating how
ICS works. Rows in the table correspond to the four
material balances around the four nodes in the recycle
process shown in Fig. 1. Columns correspond to the
seven process variables. In this table, the ‘ × ’s’ indicate
the associations between streams and nodes. For example, variables 1, 2, 4, and 6 are associated with node A,
but variables 3, 5, and 7 are not. Thus, a change in bias
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
2758
Table 1
Relationship between nodes and stream of recycle processa
MBA
MBB
MBC
MBD
y1
y2
×
×
×
y3
y4
y5
×
×
×
×
y6
y7
×
×
×
×
×
a
‘MBi ’ means material balance on node i (where i = A, B, C, or
D). ‘×’ means that the stream yi is associated with the node in the
row.
Fig. 2. Elliptical 0.95 confidence region: no biases in any variables.
The data tends to have one cluster and the sample correlation
coefficient between any sk and any yi will tend to be close to zero.
for a variable will also change its associated nodal
balance(s) but will not change an un-associated nodal
balance(s). Note that, under the assumption of steady
state, while mean shifts will change the level of variables (i.e. the y’s) they will not change the level of
nodal balances (i.e. the s’s), thus leaving the correlation
of the y’s and s’s unaffected.
The idea of correlation to detect bias can be illustrated using an example with a single biased variable
(i.e. where only one d is non-zero). Suppose, for example, that d2j " 0. Then,
E[y2j ]= m2 +d2j
(25)
and
E[yij ]= mi,
i" 2, i = 1, …, 7.
(26)
Furthermore,
E[sAj ]= E[y1j +y4j + y6j −y2j ]
=m1 + m4 +m6 −m2 +d1j +d4j +d6j −d2j = −d2j
(27)
since m1 +m4 + m6 − m2 =0 by Eq. (15). Similarly,
E[sBj ]= E[y2j − y3j ]= d2j
E[sCj ]= E[y3j −y4j − y5j ]= 0,
E[sDj ]= E[y5j − y6j −y7j ]= 0.
(28)
and
(29)
(30)
Comparing Eqs. (25), (27) and (28), we see that when
d2j changes from zero (0), y2j, sAj, and sBj will have the
same shift in their expected values. In other words, a
change in the expected value of a variable will also
cause a corresponding positive or negative change in
the expected value of all mass balances containing the
variable. Thus, when d2j changes from zero (0), the
means of y2j, sAj, and sBj will change. Therefore, it
appears that an evaluation of changes in the correlation
of certain combinations of mass balances and flow rates
could be the basis of an effective method to detect and
identify biased variables.
This work proposes two ways to exploit the effect of
measurement bias on the correlation of nodal material
balances and mass flow rates for measurement bias
identification. The first way is through the use of hypothesis testing for non-zero correlation coefficient. The
second way is by visually monitoring plots of nodal
balances (i.e. skj ) versus measured flow rate variables
(i.e. yij ). A visual change in the correlation would
provide diagnostic information on the occurrence of a
measurement bias. The second method will be discussed
in more details before presenting more details of the
first way. It is worth noting that sk and yi (k=A, B, C,
or D, and i= 1, …, 7) are bivariate normal random
variables. Therefore,
(sk, yi )S − 1
sk
x 22.
yi
(31)
Given that the upper fifth percentile for the x 22 distribution is 5.99, then
p (sk, yi )S − 1
n
sk
B 5.99 = 0.95.
yi
(32)
Note that the term in the brackets describes a region
within a solid ellipse in (sk, yi ) space. Thus, when there
are no biases, a scatter plot of skj versus yij should have
most observations contained within the ellipse as shown
in Fig. 2. With the process in steady state, the plot in
Fig. 3 indicates a change in measurement bias of one or
more streams connected to node k other than yi. Although this plot shows a separation, the linear association between sk and yi is unaffected. In contrast, under
steady state conditions, Fig. 4 indicates that with yi
associated with node k, there is a shift in yi due to bias,
which causes an increase in the linear association of yi
with sk. Yet, if yi changes due to a change in mean (i.e.
mi ), the level of sk would not change and the clusters in
a plot of sk versus yi would move in the direction of the
change along the yi axis but not up or down as we will
show in a plot later. Thus, in this case also, the linear
association between sk and yi would not be affected.
In the next section, we describe a formal statistical
test based on the correlation between sk and yi useful
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
for bias identification. However, one may want to use a
nodal strategy for identification first, (such as the one
described by Rollins et al., 1996). Then either simultaneously or after exhausting the use of the nodal strategy, one can apply ICS.
4. Tests of hypotheses
This section describes the ICS statistical test and
shows the development of the null distribution of the
test statistic. With N1 representing the sampling times
from one period (the reference population) and N2
representing the sampling times from another period
Fig. 3. Elliptical 0.95 confidence region: bias in variable other than yi.
Bias has occurred in a variable associated with the node k, but the
variable is not yi. The two clusters of points represent the data before
the bias occurred and after the bias occurred. The sample correlation
coefficient between sk and yi will tend to be close to zero.
2759
(the test population), we consider the following null
and alternative hypotheses, respectively,
“ H0, dij has averaged d0 in the time space represented
by N1 and N2 versus
“ Ha, dij has averaged d0 in the time space represented
by N1, but has not averaged d0 in the time space
represented by N2.
Note that these hypotheses are written in very general terms. That is, they represent any change in dij,
which also includes a change from zero to a constant
value or non-constant value in the test space. Let r,
given by Eq. (33) below, be the correlation coefficient
of sk and yi defined under H0:
r=corr(sk, yi )=
E[(sk − 0)(yi − mi )]
[Var(sk )Var(yi )]1/2
(33)
where the numerator is the covariance of sk and yi
(when the variances and covariances are unknown,
samples from the reference population may be used to
estimate r. However, under the assumptions of Eq. (1),
including S and St, known, with d0 = 0, r can be
determined theoretically). Note that Eq. (33) is just
simply the definition of the population correlation coefficient (see Devore, 1995) for sk and yi. Also note that
if the biases are not zero in the reference population,
Eq. (33) has to be adjusted to the appropriate means to
include the biases. Let R be the sample correlation
coefficient for the pairs (skj, yij ), j= 1, …, n with
n= N1 + N2. If the measurement bias in the test space is
different than the bias in the reference space, R will
tend to be significantly different than r, the true correlation of sk and yi under H0.
In order to find an approximate null distribution for
R, we will use the ‘Z’ transform which is commonly
known in the statistical literature (see Devore, 1995). If
(X1, Y1), (X2, Y2), …, (Xn, Yn ) is a random sample of
size n, sufficiently large for the central limit theorem to
apply (see Devore, 1995), from a bivariate normal
distribution under H0, with correlation r, and R is the
sample correlation coefficient, let us define Z, z, and n
by
1
1+ R
Z= loge
= arctanh[R]
2
1− R
(34)
1
1+ r
z= loge
= arctanh[r]
2
1− r
(35)
n 2 = (n− 3) − 1
(36)
Then Z is approximately N(z, n ) (see Graybill,
1976). The test statistic
2
Fig. 4. Elliptical 0.95 confidence region: bias in variable yi where yi is
a stream that is connected to node k. The two clusters represent data
before and after the bias occurs. The absolute value of sample
correlation coefficient between sk and yi will not tend to be close to
zero.
Ts =
(Z− z)
= (Z− z)
n −3
n
(37)
is approximately N(0, 1) under H0. Rewriting Eq. (37)
using Eqs. (34) and (35) gives
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
2760
Table 2
SCS and UBET results (Table 5) from Rollins and Davis (1992) with
di = 7, dj = 4, a= 0.05, and S=Ia
i
J
AVTI (SCS)
OPF (SCS)
OPF (UBET)
1
1
1
1
1
1
2
2
2
2
2
3
3
3
3
4
4
4
5
5
6
2
3
4
5
6
7
3
4
5
6
7
4
5
6
7
5
6
7
6
7
7
0.0138
0.0116
0.0188
0.0827
0.8733
1.0924
1.0908
0.9131
0.0836
0.0204
0.0135
0.9137
0.0816
0.0190
0.0128
1.1041
0.5426
0.0527
1.0217
0.0960
0.9198
0.9862
0.9884
0.9812
0.9226
0.1550
0.0000
0.0000
0.1071
0.9276
0.9799
0.9865
0.1074
0.9294
0.9813
0.9872
0.0000
0.0000
0.9473
0.0249
0.9046
0.1063
0.9588
0.9749
0.9900
0.9881
1, 6, 7**
1, 6, 7**
2, 3, 4**
2, 3, 4**
0.9796
0.9658
0.9830
2, 3, 4**
0.9866
0.9567
0.9679
4, 5, 6**
4, 5, 6**
0.9900
4, 5, 6**
0.9726
1, 6, 7**
a
5. Results of simulation studies
**, At least two of the three measurements are biased.
Ts = (arctanh[R]−arctanh[r])
n − 3.
(38)
Therefore, the ICS test is,
reject H0: dij = d0Öj, if and only if Ts ]za, (for cases
where r \ 0) or;
“ reject H0: dij =0Öj, if and only if Ts 5za (for cases
where r B 0)
where a is the significance level of the test and za is the
100(1− a)th percentile of the standard normal distribution. When nodal strategies fail to completely identify
all measurement biases, there is a set of variables
declared to be potentially biased. A table like Table 1
can then be used to select (sk, yi ) pairs to be tested
using the test given above. For example, suppose that
the conclusion after implementing nodal strategies for
the process in Fig. 1 is that at least two of the three
variables 1, 6, and 7 are biased. Then the pairs to be
tested are (sA, y1), (sA, y6), (sD, y6), and (sD, y7). Note
ICS is actually a test that detects a change in correlation structure between the reference set (the set which r
is based on) and the test set (the set R is determined
from). When a change in correlation structure occurs
due to a change in bias, this test is designed to detect it.
This change in bias could be from no bias to either a
negative or positive bias. It could also be from a change
in the level of bias in the reference set.
In addition, a change in the mean value of yi satisfying the model of Eq. (1) (i.e. steady state) would not
change sk. Thus, this change would not change the
correlation structure between yi, and sk giving a false
identification of change of measurement bias. However,
“
the correlation between yi and time would change making this relationship ineffective at distinguishing
changes in means from changes in biases. We illustrate
this idea later when we describe ICS visual monitoring.
The next section presents the simulation study to evaluate ICS performance.
The basic purpose of creating ICS is to obtain accurate identification of biased variables in cases where
other GED methods cannot perform well. The specific
purpose of our simulation study was to determine ICS
performance for the cases presented by Rollins and
Davis (1992) that gave their technique and the technique of Narasimhan and Mah (1987) difficulty. These
cases are shown in Table 2. Note that it contains all the
combinations of two biases for the recycle process in
Fig. 1. The problem combinations are identified with
‘**’. The objective of our simulation study was to show
that ICS could achieve excellent performance for these
problem combinations.
Our evaluation consisted of the same conditions and
assumptions originally used by Narasimhan and Mah
(1987) (i.e. the model is given by Eq. (1) and the values
of relevant parameters are given in Table 2). However,
although we could have obtained r theoretically in this
study, we are going to assume that r (which could be
different for each sk, yi pair) is unknown. Hence, this
study estimated r from data. Note that, by using the
known r, we could run the analysis with the same data
sets as Rollins and Davis (1992), Narasimhan and Mah
(1987). By estimating r for each case we can evaluate
this more difficult situation, which will likely represent
the common application. Thus, for each simulated case,
we will create a reference data set of size N1 = 10 with
d= 0 to estimate r and a test data set of size N2. In this
study, N2 was fixed but N1 varied from 5, 10 and 20 to
also evaluate the reference sample size on performance.
Each simulation consisted of generating data for each
of the process variables for a single combination of
biases and then using ICS to identify the biased variables. In this manner 10 000 trials of simulated data
were run for each result obtained. We used two measures of performance for ICS. The first one, given
below, is a measure of the technique’s ability to correctly identify a particular biased variable i and is called
the power (denoted by Pi, where i is the variable
number):
Pi =
number of nonzero d%i s correctly identified
number of nonerzo d%i s simulated
(39)
The second performance measure is called the average type I error (AVTI) and indicates the technique’s
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
tendency toward misidentification of unbiased variables.
AVTI is defined as
AVTI =
number of zerod%i swrongly identified
total number of simulations
(40)
Thus, for the technique to perform well, one would
want Pi to be high (near one) and AVTI to be low (near
zero). Two levels of a were used in this study, 0.30 and
0.05.
Tables 3–8 show results from the simulation study.
The first two columns in these tables give the locations
of the two biases (i and j ), the third and fourth columns
give the corresponding power values (Pi and Pj ), and the
Table 3
ICS resultsa
Table 6
ICS resultsa
i
j
Pi
Pj
AVTI
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
0.99
1.00
0.99
1.00
1.00
1.00
1.00
1.00
0.99
0.79
0.99
0.71
0.71
0.62
1.00
1.00
0.62
0.0121
0.0094
0.0078
0.0066
0.0071
0.0131
0.0057
0.0057
0.0133
a
N1 =5; N2 =10; a =0.30; di =7; dj =4.
Table 7
ICS resultsa
I
j
Pi
Pj
AVTI
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
0.93
1.00
0.90
0.99
0.99
1.00
1.00
0.95
0.98
0.44
0.97
0.37
0.37
0.29
0.95
0.96
0.29
0.0016
0.0008
0.0009
0.0012
0.0012
0.0016
0.0005
0.0004
0.0025
a
2761
i
j
Pi
Pj
AVTI
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.92
1.00
0.86
0.87
0.79
1.00
1.00
0.79
0.0067
0.0067
0.0041
0.0023
0.0029
0.0083
0.0030
0.0036
0.0084
N1 = 5; N2 =10; a =0.05; di = 7; dj = 4.
a
Table 4
ICS resultsa
N1 =10; N2 =10; a =0.30; di =7; dj =4.
Table 8
ICS resultsa
i
j
Pi
Pj
AVTI
i
j
Pi
Pj
AVTI
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
0.99
1.00
0.99
1.00
1.00
1.00
1.00
1.00
1.00
0.67
1.00
0.59
0.60
0.48
1.00
1.00
0.47
0.0006
0.0007
0.0002
0.0004
0.0004
0.0008
0.0003
0.0004
0.0017
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.96
1.00
0.94
0.94
0.88
1.00
1.00
0.88
0.0023
0.0031
0.0018
0.0028
0.0023
0.0056
0.0023
0.0018
0.0045
a
N1 = 10; N2 =10; a = 0.05; di = 7; dj = 4.
a
Table 5
ICS resultsa
i
j
Pi
Pj
AVTI
1
1
6
2
2
3
4
4
5
6
7
7
3
4
4
5
6
6
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
0.81
1.00
0.75
0.75
0.63
1.00
1.00
0.62
0.0005
0.0002
0.0002
0.0003
0.0001
0.0006
0.0001
0.0002
0.0004
a
N1 = 20; N2 =10; a = 0.05; di = 7; dj = 4.
N1 =20; N2 =10; a =0.30; di =7; dj =4.
fifth column shows AVTI. As mentioned earlier, these
combinations are the cases for which Rollins and Davis
(1992) could not completely identify the biases. However,
as shown, ICS accurately identifies the biases for these
cases. (Tables 3–5 show results for a=0.05). For example, for i= 1 and i =6, Table 3 shows that P1 =1.00,
P6 = 0.98 and AVTI is 0.0016. Going down the columns
in Table 3, for certain combinations the power values are
low (e.g. i= 2, j =3, and Pj = 0.37). However, upon
increasing N1 (from 5 to 10, and then to 20), Tables 4
and 5 show that the power increases significantly. Additionally, AVTI, which is already low in Table 3, decreases
even further in Tables 4 and 5.
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S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
The second set of tables (Tables 6 – 8) are for a=
0.30, as opposed to 0.05 for the previous tables. These
tables show the same trends with an increase in N1.
Additionally, both power and AVTI (which, however,
is still quite low) increase with a, as expected. Notice
that Pi and Pj increase significantly going from Table 6
to Table 7 to Table 8 as N1 increases from 5 to 10 to 20.
Also note that for N1 =20, the power is very high and
the AVTI is low.
Finally, as explained earlier, ICS can also be implemented through on-line visual analysis (i.e. visual monitoring). Consider the simple case of d1 =7.0 and
dj = 0.0 for j "1 illustrated in Fig. 5. Now, recall that
node A has three inlet streams (1, 4, and 6) and one
outlet stream (2). Based on the explanations given
earlier (comparison of Fig. 2 with Fig. 3), the inference
obtained from the plots of sA versus y2, y4, and y6 of
Fig. 5 is that bias has occurred in a variable associated
with node A other than y2, y4, and y6. The only choice
is y1, which is confirmed by the plot of sA versus y1 of
Fig. 5, which shows significant change in the correlation. Thus, the occurrence of a measurement bias can
be detected by on-line plots such as sA versus yi and by
comparing the cluster of sampled observations to the
corresponding elliptical confidence region.
In Fig. 6 we illustrate the insensitive nature of ICS to
be affected by changes in means. In this figure mean
shifts have occurred in y1 and y6 (steady state is maintained) but the correlations of sA and the y’s have not
changed. Thus, the ICS plots support the correct conclusion of no bias when process variables change due to
shifts in means. In contrast, if one tried to base a
change in bias on changes in the time series plots of the
y’s, a shift in the mean of y could likely lead to a false
conclusion of bias as illustrated in Fig. 7. In this figure,
the y1 data used in Fig. 5 (d1 goes from 0 to 20 and
m1 = 100) and Fig. 6 (d1 = 0 and m1 goes from 100 to
120) are plotted against time. Although y1 is inherently
different in these two plots (they have different means
and different biases), its plot is the same in Fig. 7.
Hence, it is not possible to know accurately whether the
change in a process variable is due to a mean shift or a
change in bias from its time series plot alone.
6. Concluding remarks
In this work, we have presented a new approach,
ICS, for the identification of measurement bias in linear
Fig. 5. Plots of sA vs. y2, y4, y6, and y1. Bias has occurred in y1. The opened circles are before the bias entered and the solid circles are after the
bias entered the process. The separation in sA vs. y1 shows the effect of the bias in shifting the mean of sA and y1 (i.e. a change in their correlation
structure). The other plots show only a shift in sA.
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
2763
Fig. 6. Plots of sA vs. y2, y4, y6, and y1 showing a change in the steady state level due to a change in the means of y1 and y6. The opened circles
are at the same conditions as Fig. 5 and the solid circles represent the new steady state from the mean changes. The shifts in the means of y1 and
y6 are detectable but there is no shift in sA for any plot. The indication is that no change in bias has occurred in these streams associated with
node A because their correlation structures are unchanged.
pseudo steady state processes. ICS is easy to implement
and is not computationally intensive. For difficult combinations of measurement biases, this approach was
shown to be capable of accurate identification where
other strategies have failed to perform well. Thus, if a
diagnostic analysis cannot reach an accurate or specific
conclusion about the location of a biased variable, ICS
can be quite useful. In addition, one could also use ICS
as a mathematical or visual on-line monitoring method,
which could aid in the early detection and identification
of biased variables. Visual monitoring would simply
consists of plotting the mass or energy balance residuals
against the associated process variables and looking for
linear trends.
If one has a fixed reference set of data of sufficient
size, ICS can identify, on-line, bias for any variable that
differs from its value in the reference set. However, one
could also set up a moving window mathematical online monitoring scheme with N1 equal to the sample size
of the reference set and N2 equal to the sample size of
the test set as we did in the simulation study. N2 would
contain the most current data and N1 would contain
data from the past, beyond the N2 data. These two sets
do not necessarily have to be close together in time.
The amount of data that one would need for high
accuracy will depend on their process variability, sampling error, and sampling frequency. These values affect
Fig. 7. Plot of y1 vs. time showing a shift in the mean or bias of
stream 1. This is the y1 used in Fig. 5 (a shift in bias of y1) and Fig.
6 (a shift in the mean of y1). The opened circles are before the change
and the solid circles are after the change. This graph shows the
inability of times series plots of process variables to distinguish
between shifts in means and shifts in biases and the superiority of
plots like those in Figs. 5 and 6.
S. De6anathan et al. / Computers and Chemical Engineering 24 (2000) 2755–2764
2764
the window sizes (i.e. N1 and N2). Trade-offs in accuracy may be required to identify frequently changing
biases, which require smaller window sizes. However, in
the common situation, given advancements in computer
and sampling technology, large data sets with small
window sizes should be obtainable. In these cases, a
moving window strategy should work well for ICS. For
off-line analysis, where the window sizes must be fairly
large (biases are assumed to occur slowly) the periods
do not have to be very close in time and one could be
somewhat conservative in their selection. In other situations, engineering judgement and knowledge could be
used to select the periods based on historical considerations or some other diagnostic methodology.
n2
r
s 2i
S
St
tj
variance of Z
true value of correlation coefficient
variance of oij
variance–covariance matrix for oj
variance–covariance matrix for tj
vector representing the effects of process deviation
Acknowledgements
We would like to acknowledge partial support for
this project by the National Science Foundation under
grant CTS-9453534, and Meiyu Shen and Molly McNaughton for helping with the final draft.
7. Notation
A
AVTI
I
N1
N2
Pi
P
p
q
R
sAj
TS
yij
Z
Za
q×p matrix representing process physical constraints
average type I error
identity matrix
sample size taken from the reference
population
sample size taken from the test population
power for variable i
probability
number of process variables
number of process constraint equations
random variable representing the sample correlation coefficient
mass balance on node A at time instant i
test statistic for the ICS hypothesis test
measured value of variable i at the jth time
instant
Fisher’s ‘Z’ transforrn of R
the 100(1−a)th percentile of the standard normal distribution
Greek letters
dij
measurement bias of variable i at the jth time
instant
oij
random error of variable i at the jth time
instant
g
vector of process leaks
lij
true value of process deviation of variable i
from mi at the jth time instant
mi
true value of variable i
.
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