Statistical Process Techniques on Water Toxicity Data

Transcription

1 Statistical Process Techniques on Water Toxicity Data Eleni Smeti, Demetrios Koronakis, Leonidas Kousouris Abstract -- Statistical Process Control (SPC) uses statistical techniques to improve the quality of a process reducing its variability. The main tools of SPC are the control charts. The basic idea of control charts is to test the hypothesis that there are only common causes of variability versus the alternative that there are special causes. Control charts are designed and evaluated under the assumption that the observations from the process are independent and identically distributed (iid) normal. However, the independence assumption is often violated in practice. Autocorrelation may be present in many chemical procedures, and may have a significant effect on the properties of the control charts. Thus, traditional SPC charts are inappropriate for monitoring process quality. In this study, we present methods for process control that deal with autocorrelated data and mainly a more sophisticated method based on time series ARIMA models (Alwan Roberts method). We apply the typical SPC tetechniques and the timeseries method on toxicity data of water for human consumption from treated water tanks. This application shows the serious effects of autocorrelation when typical SPC control charts are applied on autocorrelated observations. Index terms-- Statistical process control; Control charts; Autocorrelation; Times series models; Autocorrelated process control I. INTRODUCTION All systems and processes exhibit variability. Statistical Process Control (SPC) uses statistical methods to improve the quality of a process. This can be achieved by the systematically reduction of the variability. There are two reasons of variability: common causes and special causes. Common causes concern in the natural variability that always exists in every process and there is no way to avoid it. Special causes are not an inherent characteristic of the process and therefore they can be identified and eliminated. Statistical Process Control aims at the separation of the two types of variability. The strategy for eliminating special causes of variation is summarized in: i) The usage of early warning indicators ii) the search for cause of trouble wherever there is an indication that a special cause has occurred (what was different on that occasion) and iii) the elimination of the particular temporary or local problem. Water Quality Control Division, Water Supply & Sewerage Corporation of Athens (E.YD.A.P.). esmeti@eydap.gr The main tools of SPC are the control charts. The basic idea of control charts is to test the hypothesis that there are only common causes versus the alternative that there are special causes of variability. In the former case, the process is in a state of statistical control, and in the latter, the process goes out of statistical control. Typical control charts (Shewhart, CUSUM, EWMA) are designed and evaluated under the assumption that the observations from the process are independent and identically distributed (iid). However, the independence assumption is often violated in practice. Autocorrelation may be present in many processes, and may have a significant effect on the properties of the control charts. When autocorrelation is present, there are problems of noticing special causes that do not exist and not detecting special causes that truly exist, implying a high probability of false positives and / or false negatives. Thus, typical SPC charts are inappropriate for monitoring process quality. In this study, we present methods for process control that deal with autocorrelated data and mainly a more sophisticated method based on time series ARIMA models (Alwan Roberts method). Autocorrelation in the data is usual in many analytical systems. A study is also presented concerning the toxicity in water for human consumption. II. SPC CONTROL CHARTS FOR VARIABLES The most important types of control charts for variables (i.e. quality characteristics which are measurable and can be expressed on a numerical scale) are: Shewhart control charts, the Exponentially Weighted Moving Average (EWMA) charts and the CUSUM charts. All types of control charts have some common features. The center line corresponds to average performance whereas the control limits (the other two lines) correspond to the expected range of variation based on the process. If all the points plot between the two control limits and do not exhibit any identifiable pattern the process is said to be in statistical control. The available data concern in daily samples with size n=. Therefore, the three types of control charts, which were referred above, regard to individual observations. The first and simplest type of control charts is Shewhart chart. X control chart for individuals (I Chart) is used for the control of the process mean. In this chart, the measurement data are plotted according to the time order. The center line is set in the mean of the process or in its estimation and the control limits are usually in a distance of

2 three standard deviations from both sides of the mean. The standard deviation of the process is often, practically, unknown and is estimated from the moving range as M R / d or from the sample standard deviation of the data as s / c 4, where d can be read from special tables for n = (it is exactly,8), and c 4 is a constant which is related to the number n of the individual observations and approximates the value, as n increases. These charts are appropriate for fast detection of large shifts in the process, while they are insensible to small shifts. For the increase of the effectiveness of the Shewhart charts there have been recommended criteria which indicate that a process shift or trend has begun. These criteria are known as run rules and are based on runs of consecutive points increasing/ decreasing or oscillating above and below the center line. Exponentially Weighted Moving Average charts (EWMA) and CUSUM charts are better at detecting small changes in the process mean than Shewhart charts. The EWMA statistic is defined as z i = ëx i + (-ë)z i- and it is a weighted average of all past and current observations with weights that decrease geometrically. The constant ë can take values in the interval <ë. The most common choices for ë are in the interval [.,.]. The starting value z is the process target value ì or the average of preliminary data or, if it is not available, the sample mean. If the observations x i are independent random variables with variance ó, then the variance of z i is: σ λ = σ λ [ ( λ) ] i z i In case that the variance ó is unknown it is estimated from preliminary data or otherwise it is estimated from the moving range or from the sample standard deviation of the data (just as in Shewhart control chart). Therefore, the EWMA control chart, is constructed by plotting z i versus the time order i. The center line for the EWMA control chart is set in the mean of the process or in its estimation and the control limits are the usual limits on the Shewhart chart (3ó) or wider / narrower. CUSUM charts, are also used as alternative to the Shewhart charts for detecting small shifts in the process. According to Champ and Woodall [4], the general opinion is that the Cusum detects the small divergences faster than the Shewhart chart, even though the additional run rules have been used in the last mentioned. For each observation two cumulative sums S Hi and S Li are calculated for the detection of positive and negative shifts of the mean respectively. The cumulative sums are named superior and inferior Cusums respectively. It is: S Hi = max[, i ( X + K) + S Hi- ] and S Li = max[, ( X - K) - i + S Li- ] The initial prices of these sums are zero (S H = S L = ). The price of K is selected to be the half of the mean change that is wished to be detected, usually.ó, and H= hó (with h usually 4 or ) is the decision interval. If either S Hi or S Li exceeds the decision interval H, then the process is considered as out of control. As in the previous types of control charts so in the CUSUM charts, the standard deviation ó is estimated via the moving range or via the sample standard deviation. The moving-range control chart ÌR, where ÌR i = x i+ -x i and x i is the observation at time i, may be used to control variability although some authors (see e.g. Sullivan and Woodall [4]) have pointed out that this chart cannot really contribute enough to identification of a shift in process variability. III. A TYPICAL METHOD OF COMPARING CONTROL CHARTS A typical method of comparing control charts is based on their average run length (ARL) (see Woodall [8]), which is the average number of samples needed for a control chart to signal an out of control situation. The in-control ARL (ARL ) concerns in the expected number of samples that should be received in order to point out an erroneous signal (false alarm) that the process is in an out of control state while it is actually in control. The out-of-control ARL (ARL ) concerns in the expected number of samples that should be received in order to be detected a real change in the process mean. It is obvious that we want the probability of false alarms to be as small as possible while at the same time we want any change in the process mean to become perceptible as soon as possible. In other words we want big prices for ARL and small prices for ARL. The Shewhart chart for individuals gives much bigger ARL prices, for all changes in the process, compared to EWMA and CUSUM charts except if there are very big changes in the process mean (e.g. three standard deviations), in which it acts faster than the others. Narrower limits than three standard deviations in the Shewhart chart for individual observations, do not improve its faculty in the detection of small changes, because the in-control ARL (ARL ) is decreased dramatically giving as a result many false out-of-control signals. The additional run rules on the Shewhart chart for individuals, with 3-sigma limits, have similar effect in ARL. IV. ASSUMPTIONS FOR THE TYPICAL SPC CONTROL CHARTS The assumptions for the typical SPC control charts are that the individual observations are independent and are (approximately) normally distributed. The violation of the normality assumption is of a small importance only in the case of EWMA, which is not sensitive in the lack of normality, because its statistic is a weighted average of all past and current observations. The violation of the independence influences a lot the typical SPC control charts. However, this assumption is not even approximately satisfied in some manufacturing processes, such as chemical processes where consecutive measurements on process or product characteristics are often highly correlated (see Montgomery [9]). In case that the autocorrelation is small, for practical but also statistical reasons, the traditional control charts can be applied to the

3 data with an adjustment at their parameters to count for the autocorrelation. However, the typical SPC control charts are inadequate in case of moderate to high autocorrelation, because they lead to a lot of false positive alarms that the process is out of control while they can cover real positive signals for out of control situation. V. ALWAN & ROBERTS APPROACH FOR AUTOCORRELATED DATA The main approach for autocorrelated data, was proposed by Alwan and Roberts [] and uses time series modeling to help the detection of the existence of systematic variation and to gain a more accurate sorting out of special causes. The basic steps of Alwan and Roberts approach are: Find an appropriate ARIMA time-series model that describes the data well and then use two charts: i) The Common Cause Chart CCC (a time plot of fitted values) and ii) Special Cause Chart SCC (a typical control chart on the residuals of the time - series model). Box Jenkins methodology (see Box et al. [3]) is followed to find an appropriate ARIMA(p,d,q) model. This methodology involves the following steps: o Specification of the number of differences (d), of the original series that is needed to produce stationarity. o Identification of the orders p and q for the autoregressive and moving average operators using the Autocorrelation and Partial Autocorrelation Functions. o Estimation of the parameters of the possible models. o Model selection using information criteria such as MSE, AICC, BIC. o Model diagnostic checking: are the residuals white noise (uncorrelated and normally distributed with stationary variance)? Some of the major reasons that the Alwan and Roberts approach is appealing, as Wardell et al. [] presents them, include the following: o It takes advantage of the fact that the process is correlated to make forecasts of future quality. o The special cause chart is based on the assumption that the residuals are random, so all of the assumptions of traditional SPC are met and hence any of the traditional tools for SPC can be used, including run rules, cumulative sum (CUSUM) charts etc. o Similarly, the special-cause chart can be used to detect any assignable cause, including changes in the structure of the time series. o The methodology used to obtain the charts is straightforward and does not require a great deal of sophistication on the part of the user, especially with the availability of user-friendly software packages to fit time series models. o Unlike other methods dealing with correlated data that have been limited to AR() or MA() time-series models, this method can be applied to any type of time-series models. o The method is often more effective in detecting shifts in the process mean than other more traditional control charts when the underlying process is ARMA(,) (see Wardell et al. []). VI. THE DISADVANTAGES OF ALWAN & ROBERTS APPROACH The main disadvantage of Alwan & Roberts is the application of a time series model which takes quite a lot of time. Besides that, the size of the sample must be quite large in order to obtain an appropriate model that describes the data with good parameter estimates (see Lu and Reynolds [8]). Another point of criticism this method is that it is based on residuals and not on the original data. Therefore the typical control charts on the residuals might not have the same properties as the typical charts on the original observations. VII. OTHER APPROACHES DEALING WITH AUTOCORRELATED DATA In order to overcome the difficulty of the estimation of an explicit time series model for every variable that interests us, approximate methods for autocorrelated data have been proposed. One of them was proposed by Montgomery and Mastragelo [] and it is based on EWMA chart. This method uses the ability of EWMA to forecast the level of process at the next time period, as its statistic z i is a forecast of the value of the process mean at the next time period (i+). Then, the typical control charts are applied in the sequence of one-step-ahead prediction errors, which is independently and identically distributed with mean zero. However, this method is considered approximately suitable for processes which follow the first order integrated moving average model IMA (). In the processes of which the EWMA is not the most suitable model for forecast, it will not explain satisfactorily the autocorrelation and this can influence the statistical performance of the control charts that are based on the prediction errors (see Tseng and Adams []). Another approach for autocorrelated data is based on the unweighted batch means (UBM) and it was proposed by Runger and Willemain []. The UBM control chart is based on the ascertainment that the averages of sufficiently large consecutive and equal size batches of sequential observations are distributed approximately normally and independently for any underlying time series model. Therefore, the typical control charts of SPC can be applied to the batches averages. It is a model-free approach as it does not depend on identifying and estimating a time series model. The basic disadvantage of this method is that in order to achieve the independence and normality of the batch means, the size of batches is often required to be very large.

4 VIII. ANALYTICAL TOXICOLOGICAL CONTROL OF TREATED WATER THE AVAILABLE DATA Water Quality Control Division of Water Supply and Sewerage Company of Athens, according to / guide from the European Council, has been starting register systematically the toxicity of treated water at Athens treatment plants using a bioluminescence biotest, that is based on the correlation between toxicity of the water sample and its effects on the light intensity of marine bacteria Vibrio fischeri (former Photobacterium phosphoreum), measured by the bioluminometer Microtox. The available toxicity data are from Galatsi old treated water tank (variable PDG), Menidi old treated water tank (variable PDM) and Menidi new treated water tank (variable NDM). IX. TYPICAL CONTROL CHARTS APPLIED TO TOXICITY DATA The typical Shewhart charts on original data (Figures, 9, ), using moving ranges ( M R /d ) to estimate standard deviation ó, detect several out-of- control points in every water tank (9 points in Galatsi old treated water tank, 3 in Menidi old tank and 9 in Menidi new tank). On the other hand, the typical Shewhart charts on original data, using the sample standard deviation (s/c 4 ) to estimate standard deviation ó, give only one point out of the control limits in every water tank (Figures,, 8). In the first case there are many false possitive signals and in the second one, it might be some false negatives. In other words the estimator of ó that is based upon moving ranges of size two underestimates ó (see Ryan [3]) giving as a result many false out of control indications. On the other hand, the estimator of ó that is based on the sample standard deviation overestimates ó resulting insensible control charts even though we have large shifts on the process (see Wardell et al. []). Therefore, Shewhart control charts on the original data are inadequate to distinguish common causes from special causes of variation. The inquest of special causes becomes even more complicated if run rules are applied on these control charts (see Alwan []). In both cases, the form of the data in the charts indicates that there exists a serious amount of corellation in the data. Consequently they are inadequate because of the violation of the independence assumption. X. ALWAN & ROBERTS APPROACH ON TOXICITY DATA The Autocorrelation Function (ACF) (Figures 3,, 9) and the Partial Autocorrelation Function (PACF) (Figures 4,, ) indicate that measurements on water toxicity are moderate to highly positively correlated in every treated water tank. The graphs of the ACF and PACF show that the procedure is stationary for PDG and not stationary for PDM and NDM. For the last two variables, stationarity is been achieved after differencing each one of these procedures by one lag. The ACFs and PACFs for PDG and for the new variables that the differencing produces from PDM and NDM are useful identification tools of the orders p and q for the autoregressive and moving average ARIMA operators. The selection of the appropriate ARIMA model is based on AICC and BIC (Schwartz Bayesian Information Criterion). Low AICC or BIC value in comparison to other models is indicative of a good model. The estimated models are ARMA(,) for PDG, ARIMA(,,) for PDM and ARIMA(,,) for NDM. PDG : t =,89X t- + e t,38e t- PDM : t = X t-,4e t- + e t NDM : t = X t-,9e t- + e t Where t is the observation at time t, t- is its previous value, e t are the residuals and e t- are the previous residuals. The Durbin Watson statistic is near and Box Ljung statistic is consistent with the assumption that the residuals are not correlated. The ACF and PACF of the residuals corroborate this. We can therefore conclude that the autocorrelation is eliminated. The residuals are normally distributed and seem to be stationary in variance. We can therefore infer that the residuals can be considered as White Noise and the estimated models are good models for the time series. Now, typical SPC control charts can be applied on the residuals. Figures, 3, show the common cause charts (CCC) for the residuals, which are charts of the fitted values from the ARIMA models for the toxicity in Galatsi old treated water tank, Menidi old tank and Menidi new tank respectively and give a representation of the current and estimated state of the processes. Common Cause Chart is not a control chart because of the fact that it has not any control limits but it basically accounts for the systematic variation that exists in the process. It would be probably useful for the process maintenance between specifically limits if this is necessary. Special cause chart (SCC) is a Shewhart chart for individuals applied on the residuals. Figures, 4, show the SCC charts of the residuals, using moving ranges to estimate the standard deviation, for the treated water tanks. There is point out of the control limits in Galatsi old treated water tank (PDG), 3 in Menidi old water tank and none in Menidi new water tank. Even though we use s to estimate the standard deviation ó there is no substantial difference in these graphs. Therefore, the differences between the two ways to estimate standard deviation, that were noticed, when we applied Shewhart charts on original data, are eliminated. Wardell et al. [] showed that the Shewhart chart on residuals has small sensitivity in the small shifts, when the process is positively autocorrelated and they consider that EWMA or CUSUM on the residuals can give better results in detection small shifts, as in the case of traditional SPC control charts. In order to improve the sensitivity of the

5 Shewhart chart on residuals, Runger et al. [] suggested the use of the CUSUM chart on the residuals. Figures,, 3 present the EWMA charts of the variables PDG, PDM and NDM respectively (Galatsi old tank, Menidi old tank and Menidi new tank). In order to achieve the same in-control ARL (ARL =3,4) as in the case of 3-sigmas Shewhart charts, we use ë=, and,89- sigmas control limits EWMA charts (see Lu and Reynolds []). EWMA charts on the residuals give points out of the control limits for both PDG and PDM and none for NDM. Figures 8,, 4 present the CUSUM charts of the variables PDG, PDM and NDM respectively. These charts are designed using Ê =,ó. Therefore, they are appropriate for detecting shifts of magnitude one standard deviation. The decision interval is Ç = 4,ó in order to achieve the same in-control ARL (ARL =3,4) as in the case of 3-sigmas Shewhart charts (see Hawkins []). The two-sided CUSUM charts on the residuals give points out of the control limits for both PDG and PDM and one for NDM. CUSUM charts show mo re points out of control than the EMWA charts do. These additional out-of-control points were nearby the control limits of the EWMA charts. Therefore, if we compare the CUSUM charts with EWMA charts we can see that CUSUM charts are more effective in detecting small shifts than the EMWA charts. XI. CONCLUSIONS FURTHER RESEARCH The typical SPC control charts are ineffective, because the existence of autocorrelation in data for both of the two examined variables. The Method of Alwan and Roberts uses the common causes chart (CCC) which shows the running level of the process and the special causes chart (SCC) which is a chart of individual observations applied on the residuals. SCC gave out-of-control clues on the PDG, 3 on PDM and none on NDM. For fast localization of small shifts, EWMA and CUSUM applied on the residuals are more suitable. To overrule any pontential concern for the out-of-control indications that were located by the control charts that were used, it must be said that all the variables that concern qualitative characteristics of water and result from its chemical analytical control, are undeniable inside the specifications. Control charts attempt to follow-up of process in order to locate and then eliminate the special causes of variability giving as a result the permanent improvement of the process. As an extension for the analysis of the available data, is proposed the application of Multivariate Statistical process Control. In our analysis, the control charts that were used indicate the size of the shift in every variable independently from the other. For the analytical chemical control of drinking water, measurements are received for many more variables that concern in its qualitative characteristics. All these variables may not be independent. The Multivariate methods use all data simultaneously and examine the behavior of all the variables jointly in relation to the dependence between them. Besides that, in order to take into consideration the autocorrelation in the process, the extension of Alwan and Roberts method in the Multivariate processes can be followed. The additional univariate control charts, of individual observations on each component, help in the revelation of variables that can be more accountable for an out-of-control situation in the Multivariate control chart (see Charnes []). REFERENCES [] Alwan, L.C. (99). Effects of Autocorrelation on Control Chart Performance. Communications in Statistics Theory and Methods, -49 [] Alwan, L.C. and Roberts, H.V. (988). Time-Series Modeling for Statistical Process Control. Journal of Business and Economic Statistics, 8-9 [3] Box, G.E.P., Jenkins, G.M., and Reinsel G.C. (994). Time Series Analysis, Forecasting and Control. Prentice-Hall, Inc., Englewood Cliffs, New Jersey [4] Champ, C.W., and Woodall W.H. (98). Exact Results for Shewhart Control Charts with Supplementary Runs Rules. Technometrics 9 [] Charnes, J.M. (99). Tests for Special Causes with Multivariate Autocorrelated Data. Computers Ops Res., [] Hawkins, D.M. (993a). Cumulative Sum Control Charting: An Underutilized SPC Tool. Quality Engineering [] Lu, C.W. and Reynolds, M.R. Jr (999). Control Charts for Monitoring the Mean and Variance of Autocorrelated Processes. Journal of Quality Technology 3, 9-4 [8] Lu, C.W. and Reynolds, M.R. Jr (999a). EWMA Control Charts for Monitoring the Mean of Autocorrelated Processes. Journal of Quality Technology 3, -88 [9] Montgomery, D.C. (). Introduction to Statistical Quality Control, 4 th ed. John Wiley and Sons, Inc., N. York [] Montgomery, D.C. and Mastrangelo C.M. (99). Some Statistical Process Control Methods for Autocorrelated Data. Journal of Quality Technology 3, 9-93 [] Runger, G.C., Willemain, T.R. (99). Model-Based and Model-Free Approaches for Control of Autocorrelated Processes. Journal of Quality Technology, 83-9 [] Runger, G.C., Willemain, T.R., and Prabhu, S. (99). Average Run Lengths for CUSUM Control Charts Applied to Residuals. Communications in Statistics Theory and Methods 4, 3-83 [3] Ryan, T.P. (989). Statistical Methods for Quality Improvement, John Wiley and Sons, Inc., N. York [4] Sullivan, J.H. and Woodall, W.H. (99a). A Control Chart for Preliminary Analysis of Individual s. Journal of Quality Technology 8, -8

6 [] Tseng, S., and Adams, B.M. (994). Monitoring Autocorrelated Processes with an Exponentially Weighted Moving Average Forecast. Journal of Statistical Computation and Simulation [] Wardell, D.G., Moskowitz, H., and Plante, R.D. (99). Control Charts in the Presence of Data Correlation. Management Science 38, 84- [] Wardell, D.G., Moskowitz, H., and Plante, R.D. (994). Run-Length Distributions of Special-Cause Control Charts for Correlated Processes. Technometrics 3, 3- [8] Woodall, W.H. (98). The Statistical Design of Quality Control Charts. The Statistician 34, - FIGURES I C hart of PD G (ó e s timate d by Av erag e Moving Rang e) I C hart o f PDG ( ó es timated by s ) Ind ivid ual V alu e U CL=, X =-4, LCL=-,4 In div idua l Va lue U CL=34,9 X =-4, LCL=-43, Ob se rva tion Ob se rva tion 8 44 Figure : I chart of PDG (using moving ranges) Figure : I chart of PDG (sample standard deviation) A utocorrelation Function for PDG (with % si gni fi cance li mits for the autoc orrelati ons) Partial Autocorrel ation Func tion for PDG (wi th % signific ance lim its for the partial autoc orrelations),,8,,8 A ut ocor re lat ion,,4,, -, -,4 -, Pa rt ial A u to co rr ela tio n,,4,, -, -,4 -, -,8 -, -,8 -, Figure 3: ACF for PDG Figure 4: PACF for PDG Time S eries Plot of FITS (C CC) I Ch art of R es idu als PDG (S CC ) 3 UC L=,8 FITS - - In div idu al Va lue - - X=-, LC L=-8, In dex Obse rva tion 8 44 Figure : Common Cause Chart of PDG Figure : (SCC) - I chart on the residuals for PDG EWMA Chart of RES (PDG) +,9S L=,9 C USUM Chart of RES (PDG) UC L=44, EW MA - X=-,8 C umulat ive S um - - -,9SL=- 9, - LCL=-44, Sa mple Figure : EWMA chart on the residuals for PDG Figure 8: CUSUM chart on the residuals for PDG

7 I Chart of PDM (ó e stimated by Ave rage Moving Range ) Figure 9 : I chart of PDM (using moving ranges) UCL=,88 X=-4,4 LCL=-, I Chart of PDM (ó e stimated by s) Figure : I chart of PDM (sample standard deviation) UCL=33,8 X=-4,4 LCL=-4,39 Aytocorre lation for PDM (with % significance limits for the autocorrelations) Partial Autocorrelation for PDM (with % significance limits for the partial autocorrelations) A utocorrelation,,8,,4,, -, -,4 -, -,8 -, Figure : ACF for PDM Partial Autocorrelation,,8,,4,, -, -,4 -, -,8 -, Figure : PACF for PDM Time Se rie s Plot of FITS (CCC ) I Chart of R e siduals (SCC) 4 FITS Index UCL=, X=-,8 LCL=-,43 Figure 3 : Common Cause Chart of PDM Figure 4 : (SCC) - I chart on the residuals for PDM EWMA Chart of RES (PDM) CUSUM Chart of RES (PDM) +,9SL=,9 UCL=4,3 EWMA X=-,8 Cumulative Sum ,9SL=-8, LCL=-4,3 Figure : EWMA chart on the residuals for PDM Figure : CUSUM chart on the residuals for PDM

8 I Chart of NDM (ó e stimate d by Moving A ve rage ) I Chart of N DM (ó es timate d by s) UCL=9, X=-4,4 LCL=-8, UCL=3,9 X=-4,4 LCL=-4, Figure : I chart of NDM (using moving ranges) Figure 8: I chart of NDM (sample standard deviation) A utocorrelation,,8,,4,, -, -,4 -, -,8 Autocorrelation Function for NDM (with % significance limits for the autocorrelations) Partial A utocorrelation,,8,,4,, -, -,4 -, Partial Autocorrelation Function for NDM (with % significance limits for the partial autocorrelations) -, -,8 Figure 9: ACF for NDM , Figure : PACF for NDM Time Se rie s Plot of Fits (CCC) I Chart of Residuals NDM (SCC) 3 UCL=8,4 FITSnm X=, Index Figure : Common Cause Chart of NDM - LCL=-, Obs ervation Figure : (SCC) - I chart on the residuals for NDM EWMA Chart of RES (NDM) +,9SL=9,3 CUSUM Chart of RES (NDM) UCL=44, EWMA - X=, Cumulative Sum - -,9SL=-8, Figure 3 : EWMA chart on the residuals for NDM LCL=-44, Figure 4 : CUSUM chart on the residuals for NDM