In this paper, we present a new process monitoring approach for uncertain, or highly noisy systems, which is based on the well known Moving Window Principal Component Analysis (MWPCA) extended to the interval case. We propose to use The Midpoints-Radii PCA (MRPCA) for modelling, which independently exploits two PCAs on the center and radius matrices of the system's sensor interval-valued data. Furthermore, by changing the size and the shift of the window, Both center and radius model parameters are updated on-line; thus deriving a new Moving Window Midpoints-Radii PCA (MWMRPCA) approach. Based on the updated MWMRPCA, an interval SPE statistic and its control limit are calculated and updated through time, and are used for monitoring the state of the process. The performances of the proposed approach is illustrated by an application to the detection of faults on the Tennesse Eastman Process (TEP).
Multivariate statistical approaches based on principal
component analysis (PCA) have been widely applied for
fault diagnosis to improve process quality and
productivity. The key idea of PCA is to extract linear
structure from high-dimensional data by nding new principal
axes. In other words, PCA provides a statistical model
of the system while reducing the dimensionality of the
used dataset. For process monitoring, PCA uses several
statistics to detect changes in the system behaviour. An
abnormal situation will cause the corresponding statistic
to exceed the control limits
A major limitation of PCA-based monitoring is that the
PCA model, once built from the data, is time-invariant,
while most real industrial processes are time-varying. The
time-varying characteristics of industrial processes include
changes in the mean, changes in the variance; and changes
in the correlation structure among variables, and even
changes in the number of signi cant principal components
(PCs). When a static PCA model is used to monitor
time-varying processes, false alarms often occurs, which
signi cantly compromise the reliability of the monitoring
system. Thus, a dynamic variant of PCA, such as the
moving window PCA (MWPCA)
PCA based diagnosis strategies have been developed for the analysis of single-valued variables. But in real life, there are many situations in which the use of these singlevalued variables may cause severe loss of information, and thus compromise the reliability of the approach. In this case, more robust strategy can be achieved by describing the process measurements by interval-valued data. Therefore, An interval-valued PCA model is to be used to handle the new nature of data.
In the last two decades or so, many researchers
investigated the possibility to extend PCA to interval-valued
data.
In this section, we present time-invariant PCA models for single-valued, and interval-valued data. The interval MRPCA model is chosen, among other interval PCA methods, for its high suitability with the on-line application.
In classical PCA, the raw data matrix X 2 <n m, after standardization, i.e. reducing the data to zero mean and unit variance, is decomposed as follows:
X = T P T + E T 2 n `, P 2 m ` and E 2 n m represent, respectively, principal components matrix, loading matrix and residuals matrix. Where ` is the number of principal components (` < m). The Euclidean norm of the residual matrix E must be minimized for a given number of components. This criterion is satis ed when the columns of P are the eigenvectors corresponding to the ` largest eigenvalues of the covariance matrix of X. Thus, PCA can be viewed as a linear mapping from <m to a lower dimensional space <`. The mapping has the form: (1)
Thus, based on classical PCA models of centers and radius matrices, principal components for centers T c and for radius T r are given by: and interval estimates of centers X^ c and radius X^ r are given by:
T c = XcP c
`c T r = XrP r
`r X^ c = XcP`cc P cT
`c
X^ r = XrP`rr P`rrT Where X^ is the estimate of the initial data matrix X. Accordingly, the residual matrix E is the di erence between X and X^ :
E = X ^ X
In the case of interval PCA, the data matrix X 2 Rn m is represented by interval-valued variables, denoted [Xj ],j = 1; : : : ; m. According to the uncertainties or approximation error of the process sensors, and for each time sample k, the measurement takes the form [xj (k); xj (k)], where the bounds of the interval are given by: xj (k) = xjc(k)
xjr(k) xj (k) = xjc(k) + xjr(k) [X] = [[X1][X2]; : : : ; [Xj ]] the global interval data matrix is then constructed as : In a real application, xjc(k) is the center of the interval, which is given by the the estimate single-valued measurement provided by the sensor, and xjr(k), also denoted x(k) is the uncertainty of measurement or the range of the interval, which usually presents the sensor precision given by the manufacturer.
The midpoints-radii PCA (MRPCA) on interval-valued
data, introduced in
Xc 1P c = cP c (8) Xr 1P r = rP r (9) Where c; P c and r; P r are, respectively, the eigenvalues and eigenvectors of the two eigen-decompositions of midpoints and range matrices, and is the covariance matrix given by: =
X0cXc +
X0rXr +
X0cXr + X0rXc (10) Where P is the eigenvectors matrix, or the coe cients of the linear transformation. The projection can be reversed back to <m with: X^ = T P T (2) (3) (4) (5) (6) (7) (11) The interval residual matrix [E] is, as in classical PCA, given by the di erence between [X] and [X^ ]: Thus, according to interval arithmetic, the bounds of the interval residual matrix are given by:
(
Where `c and `c are the number of principal components
for the midpoints model and the radius model respectively.
The interval form for the components [T ] and the estimates
[X^ ] can be obtained from the midpoins-radii
representation through a rotation of coordinates. This rotation is
performed based on matrix A = QP T
In general, the PCA based fault detection scheme uses the squared prediction error (SPE) statistic, which is de ned as:
SP E(k) = kx(k) x^(k)k2 = ke(k)k2 (18)
where e(k) is the residual vector given by the di erence
between the measurement vector and its estimate from
the PCA model. A faulty condition is declared if the SP E
index exceeds its control limit 2 determined statistically
For the interval-valued PCA case, several indices are
proposed as extentions of classical SP E to interval valued
data
SP E(k) = ke(k)k2 = e(k)T e(k)
SP E(k) = ke(k)k2 = e(k)T e(k) Given that e(k) is the interval valued residual computed from the MRPCA model. In the presence of faults, decisions are made when both bounds in Eq. 19 exceed their detection threshold.
Another index introduced in
3
given that [x(k)] = [[x1(k); x1(k)]; : : : ; [xm(k); xm(k)]] is
the interval measurement vector; and that
1
ej2(k) + ej (k)ej (k) + ej2(k)
(21)
The control limits for both statistics, i.e. SP Elim and
SP Elim for SP E index, and ISP Elim for ISP E index,
can be computed from their approximate distribution as
detailed in
ISP Elim = g 2h; Where g is a weighting parameter included to account for the magnitude of the ISP E and h accounts for the degrees of freedom with a signi cance level of 1 , typically selected to be 95% to 99%. The parameters g and h can be estimate as:
b g = ; h =
2a2 2a b Given that a is the estimate mean of ISP E, and b is its estimated variance.
In comparison to classical PCA, a PCA model for intervalvalued data o ers more robustness toward uncertainties of sensor measurements, due to the interval nature of data. The interval-valued model considers that every variation inside the interval is a normal process variation. In other words, the radius X of data is considered as a safe zone, were small magnitude o sets, i.e. uncertainties < X are not detected and are considered as normal process variation, while high magnitude o sets f > X are considered as faults. (22) (23)
In dynamic PCA algorithms, new measurements are used to update the PCA model on-line. The updating scheme should include: mean, covariance, principal components and the con dence limit for the monitoring statistic. Several algorithms for dynamic PCA can be found in the literature. In this paper, we investigate the possibility to extend the Moving Window PCA (MWPCA) algorithm to the interval-valued data case.
In MWPCA, a data window of xed length is moved in real time to update the PCA model once a new normal sample is available. Thus, for a window of size w, the treated data matrix at time sample k is Xk = [x(k w + 1); x(k w + 2); : : : ; x(k)], and at time k + 1 becomes Xk+1 = [x(k w + 2); x(k w + 3); : : : ; x(k + 1)]. MWPCA algorithm then updates various parameters, including : normalization parameters, PCA model parameters, and monitoring statistics control thresholds. The overall strategy for time-varying monitoring using MWPCA is given as follows: (1) Collect training data from the process under NOC.
Normalize the training data using its mean and standard deviation. Then compute covariance matrix and carry out an eigen-decomposition to obtain the PCA model for the process. Choose the number of components. Finally, Determine the control limits for the used monitoring statistic. (2) Obtain a new testing sample and normalize it using scaling parameters from the moving data window. (3) Evaluate the monitoring statistic for the normalized testing sample using the PCA model obtained in Step 1, and check with the corresponding control limit from step 1. If not exceeded, the measurement is considered normal. (4) If the measurement is normal, update the moving window by shifting it by one time sample (add new measurement and delete the oldest measurement). Repeat from Step 2. Else, the measurement is faulty, stop update after three consecutive faulty measurements.
Dynamic variants of PCA, such as MWPCA, were developed in order to take into account the dynamics or variations of the system to be monitored, but which remain less robust due to the e ect of measurements uncertainties. To overcome this problem, we propose in this section a dynamic approach to PCA for interval-valued data for diagnosis purpose. The idea is to exploit the MRPCA model for interval-valued data combined with the well known Mowing window PCA algorithm.
First, let us recall the normalization procedure for
intervalvalued data, each interval-valued measurement is
normalized as follows:
Where Xjc is the centers mean of the j-th interval-valued
variable, and V AR([Xj ]) is the interval variance de ned
as follows
x2j (k) + xj (k)xj (k) + x2j (k) (1) Construct the center (Xc) and radius (Xr) matrices from the initial interval-valued data matrix [X] (2) Obtain the initial values for normalization parameters, i.e. centers mean mc0 and interval variance D0 (3) Compute the initial models parameters (for centers aanndd rPa0rid,utsh):e ceoivgaernivaanlcuees m,eaitgreicnevsectoc0r amnadtricr0e,s tPh0ec rotation matrix A0, and the number of principal components for the two sub-models `c0 and `0; based r on the training data block; (4) Compute interval residuals, and ISP E0 index then obtain the control threshold ISP Elim;0. (5) Determine the initial size for moving window w.
At sample time k, use the previous values of time sample (SkP Eli1m) ;kof1,m`ckc0 a1n, dD`kr0; 1, ck 1, rk 1, Pkc 1, Pkr 1, A0, (1) Compute interval residuals and ISP Ek index, for a new sample [x(k)] after standardizing the interval using mck 1, Dk 1; (2) If ISP Ek > ISP Elim;k 1 go to step 3, else go to step 5; (3) Check if the new sample is an outlier. For instance, if ISP Elim;t s > ISP Elim;k t 1, (t=1,2) (i.e. if three consecutive out-of control signals have been generated), the new sample is not an outlier. Otherwise, it is an outlying sample; (4) If it is an outlier, go to step 5. Otherwise, consider the current condition to be abnormal. Typically, the process condition cannot be determined by examining one sample. In this case, the model is retained without updating and the current sample is stored. Then, if the process condition is proven to be normal subsequently, the model can be updated in a blockwise manner; (5) recompute residuals and ISP Ek; (6) Shift the window and calculate new normalization and model parameters for the new window (update) (7) update the number of principal components to retain for the two sub-models, and SP Elim;k.
The performance of the proposed monitoring scheme, using a MWMRPCA model, is illustrated through an application on a classical example: the Tennessee Eastman Process (TEP), whose scheme is shown in Figure 1. This process is developed by Eastman Chemical Company to provide a simulation of a real industrial process for the testing of control and/or monitoring methods. There are ve major units in TEP simulation (as shown in g 1 a reactor, a separator, a stripper, a condenser, and a compressor. The process has 12 manipulated variables, 22 continuous process measurements, and 19 composition measurements sampled less frequently. which all have Gaussian noise. Corresponding to di erent production rates, there are six modes of process operation. The A D E C
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FI XA XB AN XC LA XD ZY XE E XF R
FI 1 2 3 4
LI 6 FI 7 SC
TI Reactor
Condenser PI
CWS 12 TI
CWS
TI 13 CWR FI CWR
Stripper
LI 8 5
TI 11 FI
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XA PI A XB
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XH A XD NA XE LY XF ZE XG R XH
Product Vapor/Liq.
Separator Stm Cond Fig. 1. Tennessee Eastman Process TEP process was run for 1 hours, and we collected 1000 samples from 22 variables. Considering an uncertainty xi of measurements, of the order of 5% of the measurements for each variable, we construct the new interval-valued data matrix of the process, with xi being the radius of the data intervals. The rst 150 samples were used to construct the initial MWMRPCA model, while the rest of the samples will be processed one by one, by sliding the window each time, in testing phase (online monitoring). The model parameters are updated for each normal sample, where the the number of components is selected according to the Cumulative Percentage Variance criterion (PCV), so that the explained variance represents approximately 95% of the total variance, given as follows: 0 `
P j
B j=1 CP V (`) = 100 B
B m @ P j=1 1 C C
C
j A
(26)
In order to compare the performance of the new
MWMRPCA monitoring strategy, a classical dynamic PCA model
using the sliding window PCA (MWPCA)
An interesting property of the PCA for interval-valued data, applied to the diagnosis, is that the de ned radius of the data acts as a safe-zone. In other words, the PCA model is insensitive to outliers with amplitude smaller than the radius (uncertainty), because considered as a normal variation of the process. As for the outliers with greater amplitude than the radius, they are normally detected as faults.
Indeed, by inspecting the gures 2 and 3, which represent the detection indices SP E and ISP E for the MWMRPCA model, we can clearly notice the absence of the bias x9 representing the uncertainty (present between time 200 and 300), as well as the presence/detection of the fx3 faults simulated from moment 700. This is due to the nature of the interval model, which considers the bias x9 as a normal variation of the process. The MWMRPCA model thus manages to follow the dynamics of the system, to detect faults, while providing more robustness with respect to measurement uncertainties. Note that the SP E index depends on two thresholds, one for each of its bounds, while the ISP E depends only on a single threshold. The latter has better performance with regard to the number of false alarms, and in distinguishing the fault. According to the SP E criterion calculated in the case of the MWPCA model, represented in the gure 4, we can notice that the uncertainty x9 present in the variable x9 has mislead the dynamic MWPCA model. More precisely, the simulated bias x5 is considered as a fault by the MWPCA model thus resulting in a stopping of model update. The impact is a huge rate of false detections from the moment of the presence of uncertainties (from time sample 200). A conventional dynamic PCA monitoring strategy, MWPCA in this case, can be thus insu cient to handle highly noisy systems. Figure 5, represents the SP E index combined with static PCA, which demonstrates its total incapacity in detecting faults in a variant system.
In this work, a new algorithm based on moving window PCA is presented, for modelling and monitoring of dynamic processes. The model used is an interval variant of PCA, called MRPCA, which treats separately the matrices of the centers and the radii extracted from the intervalvalued data matrix. Applied on-line, the proposed algorithm veri es for each sample the presence or absence of faults, using interval statistical indices (SP E and ISP E). After that, the update of the model and the normalization parameters is performed. The presented approach improves not only the robustness toward measurement uncertainties, but also allows to have a con dence zone de ned by the radius of the interval, making it possible to consider any additional amount of information within this zone as a part of the normal operation of the system. Applied to the Tenessee Eastman Process, the proposed algorithm demonstrates good performance over the static PCA model, and the MWPCA model with sliding window.