Environmental Modeling and Assessment 6: 77–82, 2001. 2001 Kluwer Academic Publishers. Printed in the Netherlands.

Water treatment control using the joint estimationoutlier detection method

Christine Wright a,∗ and David Booth b

a Department of Management, Western Carolina University, Cullowhee, NC 28723, USAE-mail: cwright@email.wcu.edu

b Administrative Sciences Department, Kent State University, Kent, OH 44242, USA

The loss of contaminated wastewater into the environment by leakage or other means is a serious problem. This problem is essentiallythe same as true of the loss of chemical reagents from a chemical production or purification process. The present article shows how the jointestimation method, an outlier detection method for time series analysis, can be used by a facility manager to deal with these problems.

Keywords: joint estimation, outlier detection, process control, wastewater control

1. Introduction

In many industries, it is important to determine whenthe process is out-of-control, (i.e., when significant adverseprocess changes occur). The idea is to discover these ad-verse process changes while they are still relatively minor,before substandard product or significant pollution is pro-duced. One example of an important chemical process con-trol problem is wastewater treatment. This paper discussesthe use of a process control method for the purpose of moni-toring wastewater data. The objective of the research was todetermine if the out-of-control observations (i.e., abnormalstates) could be detected by JE in the period when they firstoccurred. The process control method reported herein canbe used for any compound for which an analytical chemicaldetection method exists. The method that we consider, JointEstimation (JE), has the potential to be extremely importantto both general pollution control and statistical process con-trol.

2. Background

It has been previously shown that pollution producing sit-uations may be recognized through the detection of statis-tical outliers [1–14]. An outlier is any point that deviatessignificantly from the underlying process model or time se-ries pattern, indicating a change in the process and thus anout-of-control situation with respect to the process model.Points outside of three standard deviations of the targetedprocess mean are usually considered to be outliers. Sucha point can be identified using statistical methods. If suchexist, the process is said to be “out of control” (i.e., thereis a significant adverse process change due to an assignablecause which is a cause that can be identified). Otherwise theprocess is said to be “in control” (i.e., only random variationsof output exist within certain control limits).

∗ Corresponding author.

Traditional statistical process control charts as well asmost of the other methods currently used are based upon theassumption that the observations in the process time seriesare independent and identically distributed (IID) about thetargeted process mean or targeted value at any time t andthat the distribution is normal when the process is in statisti-cal control. Independence implies that there is no particularpattern in the data.

Unfortunately, much of the data used in statistical processcontrol is non-IID [15]. Alwan and Radson [15] also notethat because of the efforts of G.E.P. Box, the chemical in-dustry has recognized for many years that autocorrelation(i.e., relationships across time) exist in their processes. Bax-ley [16], Berthouex et al. [17], Emer et al. [18], Harrisand Ross [19], and Hunter [20] have noted that continuousprocess industries, such as wastewater plants, often have au-tocorrelated process data.

3. Approaches to SPC when standard methods are notappropriate

Process measures over time are often interdependent (i.e.,the observations are autocorrelated). Further, many processtime series exhibit a characteristically repetitive pattern,which can be mathematically modeled by an Autoregres-sive Moving Average [ARMA(p, q)] model. For example,ARMA(1, 1) and other time series models have been empir-ically found in some cases to be appropriate for modeling aprocess time series [21]. Under such conditions, traditionalSPC procedures may be ineffective and inappropriate formonitoring and controlling the process, perhaps even erro-neously indicating an out-of-control situation when the cri-teria of the traditional control chart are applied [15]. In otherwords, they are not as effective as they should be in detect-ing, for example, the escape of pollutants into the environ-ment. Thus, if the process being controlled is one that pro-duces pollutants, these compounds may be introduced into

78 C. Wright, D. Booth / Water treatment control

the environment without the producer’s knowledge. Use oftime series based process control methods, rather than stan-dard statistical process control methods, is appropriate whendata is non-IID or when outlying observations may exist inthe data, such as when the materials are particularly valuableor involve critical safety concerns.

4. Joint estimation method

The method considered is successful in handling theproblems of general statistical process and pollution con-trol [12,14]. Joint Estimation (JE), a time series procedure,developed by Chen and Liu [22] has been applied to otherenvironmental pollution situations [12]. This method is su-perior to the one used earlier by Prasad [23] in that (a) out-liers are obtained iteratively, based on the adjusted residualsand observations, (b) the procedure does not require inter-vention models to be estimated to accommodate the outliers,(c) the identification and location of outliers are based onrobust parameter estimates, (d) the outlier effects are jointlyestimated using multiple regression, and (e) the proceduredifferentiates between and accommodates for four forms ofoutliers: Innovational Outliers (IO), Additive Outliers (AO),Level Shifts (LS), and Temporary Changes (TC). These fourtypes of outliers range between the extremes of a one-timechange (AO), a permanent shift in the level of a process (LS)and two decaying patterns after the initial impact (IO andTC). This method is proprietary; the details of its algorithmsare limited. The JE subroutine is available on the XUTSSoftware from Scientific Computing Associates (SCA), OakBrook, IL. The method is described in appendix. In addition,figure 1 depicts all four types for an ARMA(1, 0) model.

The information provided by the JE method with regardto the location of the outlier can increase the effectivenessof detecting the loss of pollutants into the environment. Thismethod was tested by Prasad et al. and found to be very suc-cessful with nuclear inventory data as well as general SPCdata [2,3]. Wright [14] and Wright et al. [24] show thatthis method can effectively locate outliers in a time serieswith as few as 9 observations where the outlier is the lastobservation in the time series. All outliers are identified asAO when they first occur, this can be seen in figure 1. Fur-thermore, this method has considerably fewer false alarmsthan the Exponentially Weighted Moving Average (EWMA)model [12,14,24].

Figure 1. AO, TC, LS and IO for an ARMA(1, 0) model.

5. Research method

We utilize the joint estimation (JE) outlier detectionmethod of Chen and Liu [22] to detect outliers (i.e., out-of-control observations) in wastewater treatment data. Thisdata consists of 527 daily measurements of 38 different sen-sor readings (variables). These variables are shown in ta-ble 1. The wastewater plant manager identified 13 differentstates of performance; these are shown in table 2 and in-clude such conditions as normal operations, storms, solidsoverload, etc. Of these states, only states 1, 5, 9 and 11 arenormal. The objective of the research was to determine ifthe out-of-control observations (i.e., abnormal states) couldbe detected by JE in the period when they first occurred. Itis of considerable importance to determine that an out-of-control situation exists on the day when it first occurs ratherthan several days later. Clearly the environmental and healthrisks involved necessitate early detection, perhaps even atthe cost of some false alarms.

The joint estimation method is appealing because it per-forms well over a wide variety of both seasonal and non-seasonal ARIMA models. The user must specify the modeltype for the series prior to using the JE routine. This methodis proprietary, the details of its algorithms are limited. TheJE subroutine is available on the XUTS Software from Sci-entific Computing Associates (SCA), Oak Brook, IL. In ad-dition, it is possible to use the JE method as an on-lineprocess control technique through a communication proto-col developed between the on-line data collection unit andthe SCA system. Wright et al. [24] describe the method indetail. A brief summary of the method is included here.

The joint estimation method involves three stages. Thefirst stage obtains maximum likelihood estimates of parame-ters and residuals. Then, outliers are sought and their effectsare removed from the residuals. After all outliers have beendetected, model parameter estimates are revised. In the sec-ond stage, multiple regression is utilized to jointly estimatethe effect of the outliers and model parameters. Then theestimated t-values are compared with the critical value, C.If the t-value of a suspected outlier is smaller than C, theoutlier is deemed not significant. Next, an adjusted seriesis obtained by removing significant outlier effects. Max-imum likelihood estimates of model parameters are foundbased on the adjusted series. In the third stage, outliers aresought based on final parameter estimates found in stagetwo. Residuals are computed using these estimates. Theseresiduals are used as the procedure iterates through the firsttwo stages.

Chen and Liu [22] have shown that the JE method is ex-tremely effective for detecting outliers in autocorrelated timeseries data with a large number of observations. They didnot, however investigate the ability of the method to identifyoutliers when they are the last observation in a time series.Wright et al. [24], through a simulation experiment, showthat the JE method is very effective for identifying outlierswhen they are the last observation in short autocorrelatedtimes series.

C. Wright, D. Booth / Water treatment control 79

Table 1Waste water treatment process variables.

Variable Category Description of variable

1 Influent to plant Flow to plant2 Influent to plant Zinc to plant3 Influent to plant pH to plant4 Influent to plant Biological demand of oxygen to plant5 Influent to plant Chemical demand of oxygen to plant6 Influent to plant Suspended solids to plant7 Influent to plant Volatile suspended solids to plant8 Influent to plant Sediments to plant9 Influent to plant Conductivity to plant

10 Input to primary treatment pH to primary settler11 Input to primary treatment Biological demand of oxygen to primary settler12 Input to primary treatment Suspended solids to primary settler13 Input to primary treatment Volatile suspended solids to primary settler14 Input to primary treatment Sediments to primary settler15 Input to primary treatment Conductivity to primary settler16 Input to secondary treatment pH to secondary settler17 Input to secondary treatment BOD to secondary settler18 Input to secondary treatment Chemical demand of oxygen to secondary settler19 Input to secondary treatment Suspended solids to secondary settler20 Input to secondary treatment Volatile suspended solids to secondary settler21 Input to secondary treatment Sediments to secondary settler22 Input to secondary treatment Conductivity to secondary settler23 Effluent from plant pH24 Effluent from plant Biological demand of oxygen25 Effluent from plant Chemical demand of oxygen26 Effluent from plant Suspended solids27 Effluent from plant Volatile suspended solids28 Effluent from plant Sediments29 Effluent from plant Conductivity30 Performance Input BOD in primary settler31 Performance Input suspended solids to primary settler32 Performance Input sediments to primary settler33 Performance Input BOD to secondary settler34 Performance Input COD to secondary settler35 Global performance Input biological demand of oxygen36 Global performance Input chemical demand of oxygen37 Global performance Input suspended solids38 Global performance Input sediments

Table 2Wastewater treatment process conditions.

Condition Status Condition Number of daysnumber in this state

1 Normal Normal operations 1 2752 Abnormal Secondary settler problems 1 13 Abnormal Secondary settler problems 2 14 Abnormal Secondary settler problems 3 45 Normal Normal operations, above mean 1166 Abnormal Solids overload 1 37 Abnormal Secondary settler problems 4 18 Abnormal Storm conditions 1 19 Normal Normal operations with low influent 69

10 Abnormal Storm conditions 2 111 Normal Normal operations 2 5312 Abnormal Storm conditions 3 113 Abnormal Solids overload 2 1

When using JE, the critical value, C, is specified by theuser. The critical value is compared with the t-value of a sus-pected outlier. The outliers ranged from day 60 to day 467.

This implies that the time series studied ranged from 60 ob-servations to 467 observations. A critical value of C = 4.0was utilized because Chen and Liu [22] recommend C of 3.0

80 C. Wright, D. Booth / Water treatment control

for series between 101 and 200 observations; C > 3.0 forseries greater than 200 observations. The wastewater datacontains 14 outliers. The locations of these outliers are days60, 61, 62, 98, 128, 186, 213, 244, 421, 424, 427, 465, 466and 467.

6. Results

JE is not a multivariate method. Therefore, the concernwas to seek outliers for all 38 sensor readings. If at least oneoutlier was detected within the day’s 38 readings, that daywas determined to be out-of-control. JE was utilized to con-sider each of the 38 sensor readings in such a manner thatdays 60, 61, 62, 98, 128, . . . were the last day in the timeseries under consideration. This replicates the results thatwould be available to the user on the particular days whenout-of-control situations occurred. Table 3 summarizes theresults of the JE method in detecting the outlier when it wasknown to be the last observation in the time series. All ab-normal days were detected except day 467.

The JE method was also tested to determine the falsealarm rate. Of specific concern was whether the methodwould identify the last observation as an outlier when it wasknown to be not an outlier. Thirty days were randomly cho-sen to determine, if they were the last day in the time se-ries, would they be misidentified as outliers? For example,the 51st day is known to be a normal (state 1) day. TheJE method was used to analyze the data up to and includ-ing day 51, where day 51 data was the last observation inthe time series. The JE method reported, as shown in ta-ble 4, that day 51 is not out-of-control (not an outlier). Theresults for all thirty randomly selected days are shown in ta-ble 4 along with their states and the variables determined tobe outliers. The false alarm rate seems a bit high. How-ever, when weighed against the cost of loss of human lifeor environmental damage, the false alarm rate is of less im-portance than the high outlier detection rate of 92.86%. Inaddition, our count of false alarms in table 4 may be overlypessimistic. If, for example, a surge of increased solids en-

Table 3Detection results of JE for days known to be out-of-control.

Day State Number of OOC variablesvariables OOC

60 2 9 24,25,26,28,33,35,36,37,3861 3 4 24,26,37,3862 4 4 25,28,35,3898 7 3 28,32,38

128 6 4 6,8,12,14186 10 3 19,20,32213 8 1 29244 12 5 6,7,12,13,31421 13 3 6,7,13424 6 4 6,8,12,14427 6 2 6,12465 4 7 2,25,26,33,35,36,37466 4 1 29467 5 0 None

tered the system, we might obtain more out of control ob-servations than actually exist. Further, one of the reviewerssuggests that if the false alarm observations are linked in thetreatment process then they may be counted more than once.In either case, our counts for false alarms would be definitelyconservative.

The results reported in table 4 may also be viewed in an-other light. Chen and Liu [22] and Wright et al. [24] showfalse alarm rates of 1% or less for simulated time series.Wright et al. [24] simulated time series and knew with cer-tainty where the outliers were and were not located. Theirfalse alarm rate for ARMA(1, 0) time series ranged from0.79 to 0.10% and for ARMA(0, 1) ranged from 1.27 to0.27%. The data used in this research is ARMA(1, 0) data.Given the results found by Wright et al. [24], one might won-der about the absolute classification of some days as “nor-mal” or in-control. The plant manager of the plant wherethis data originated categorized the wastewater data whenit was recorded. These categories or operating conditionsare, therefore, assigned based on the judgment of the plantmanager (i.e., the 13 conditions mean something to the plantmanager but would mean nothing to other plant managerselsewhere). This leads to the question, given the low falsealarm rates in previous research with this method, is the falsealarm rate as high as reported in table 4, or were many days

Table 4False alarm results of JE for days known to be not out-of-control.

Day State Number of OOCvariables OOC variables

51 1 0 –65 1 0 –71 1 0 –90 1 0 –

104 5 5 6,11,12,17,33117 5 0 –122 5 0 –139 5 0 –144 1 0 –153 1 0 –180 5 1 22209 5 0 –234 9 0 –269 1 1 2285 11 0 –306 11 0 –318 11 1 28331 1 0 –351 5 2 6,21362 11 0 –374 1 0 –380 1 0 –390 1 0 –415 1 0 –422 9 0 –432 1 0 –435 1 0 –497 9 1 35504 5 0 –521 1 0 –

C. Wright, D. Booth / Water treatment control 81

were out-of-control on at least one variable and this condi-tion was not recognized by the plant manager?

Wright et al. [24] considered twenty-two real industrialtime series that were not necessarily normally distributed;the simulated time series were normally distributed (0, 1).Even when the normal distribution was not assumed, falsealarm rates using JE were 8.52% for ARMA(1, 0) and 4.98%for ARMA(0, 1). Thus we might conclude, given that thewastewater data may not be normally distributed, that it isunlikely that this particular data would yield such a largefalse alarm rate when compared with all previous researchusing this method. Because all classifications of “normal”data were made somewhat judgmentally, this leads to theconclusion that this method may help wastewater plant op-erators identify out-of-control days when they first occur andrequire less need to classify days by various judgmental titlesof normal or abnormal.

7. Conclusions

The escape of pollution into the environment by leakageor misidentification presents a serious problem, as it doesfor any chemical production process. Clearly, this methodmay be useful in monitoring and identifying out-of-controlsituations in various environmental data.

The results reported in table 3 indicate that, at least forthis data, some variables were more helpful for detectingoutlying days than others. Table 3 shows that only twenty-one of the thirty-eight variables were useful in determiningwhich days were out-of-control; these variables were 2, 6, 7,8, 12, 13, 14, 19, 20, 24, 25, 26, 28, 29, 32, 33, 35, 36, 37and 38.

In summary, this research has combined manufacturing,chemical, and statistical methods in order to improve ourability to detect pollutants before they become a seriousproblem. Further, as indicated in the introduction, the im-portance of these techniques to decision and policy makerscannot be overemphasized. The requirement of their use bydecision makers and policy makers will allow polluting fa-cilities to be detected more quickly, thus decreasing the totalamount of pollution to the environment.

8. Policy relevance

In addition to making the pollution control methods alongwith the associated safeguards systems more powerful, andthus being able to detect a polluting facility sooner, thismethod has decision making and policy implications. Thereare two special groups for which the results of this researchare targeted. First, appropriate managers in charge of thepotentially polluting facility. These decision makers need toknow as soon as possible that a problem exists so that theycan decide whether a quick fix is possible or whether the fa-cility needs to be shut down for more extensive repairs tostop any potential pollution of the environment. Of course,the sooner such a decision is made the less pollution to the

environment results. The second target group is policy mak-ers. As new methods for pollution detection are developed,policy makers can take advantage of them by requiring theiruse by potential polluters and regulatory agency investiga-tors. Thus, as indicated previously, the amount of pollutioncan be minimized. It should be noted however that policymakers must keep up to date on current methods so that thebest possible pollution detection methods are being requiredfor use.

Acknowledgements

The authors would like to thank David West, Ph.D., EastCarolina University and Paul Mangiameli, Ph.D., Universityof Rhode Island, for their assistance in securing and under-standing the wastewater treatment data.

Appendix: A description of the joint estimation method

A.1. Joint estimation – stage one

The first stage involves estimation of parameters and de-tection of outliers. First, the method obtains the maximumlikelihood estimates of the parameters and the residuals.Then, the procedure searches for an outlier. If it discovers anoutlier, the procedure removes the effect of the outlier fromthe residual. Then the method seeks additional outliers. Af-ter the procedure finds all outliers, it revises the estimates ofthe model parameters. If no outliers are found, the procedurestops.

A.2. Joint estimation – stage two

The procedure jointly estimates the effect of the outliersand the parameters of the model. First, it jointly estimatesthe outlier effects, wj ’s, using multiple regression for j =1, . . . ,m

et =∑

wjπ(B)Lj (B)It (tj ) + at ,

where et is the output variable, wj indicates the magni-tude of the outlier and Lj(B)It (tj ) are the input variables.When the outlier is an IO, Lj (B) = θ(B)/{φ(B)α(B)},Lj (B) = 1 for AO, Lj(B) = 1/(1 − B) for LS, Lj (B) =1/(1 − δB) for TC at t = tj [22]. Lastly, at is a sequence ofrandom errors that are iid and are from a normal distributionwith mean of zero and variance independent of time [25].Next, the procedure computes the estimated t-values of theestimated weights (tj = wj/(std(wj )), j = 1, . . . ,m). Thet-values are compared with the critical value, C. If the t-value of a suspected outlier is less than or equal to C, theoutlier is determined to be not significant and is removedfrom the set of identified outliers. The procedure contin-ues to jointly estimate the weights using multiple regressionand compare the t-values with the critical value until thereare no remaining outliers. Next, the procedure obtains the

82 C. Wright, D. Booth / Water treatment control

adjusted series by removing significant outlier effects. Max-imum likelihood estimates of the model parameters are thenfound based on the adjusted series. This step continues untila specified tolerance level is reached.

A.3. Joint estimation – stage three

The procedure seeks to detect outliers based on final pa-rameter estimates. It computes the residuals using the finalparameter estimates found in the last step of stage two tofilter the original series. These residuals are then used to it-erate through the first two stages. The last iteration of theprocedure ends with the estimated weights from stage two.These weights are final effect estimates of the outliers thatare detected [24].

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