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Moving-Window Spectral Neural-Network Feedforward Process Control Dennis Ridley and Felipe Llaugel
Abstract—Unlike reactive feedback control, feedforward control is a proactive method by which information about a measurable disturbance is fed, ahead of time, to the manipulated inputs of a process, the output of which is to be controlled, so as to counteract the effect of the disturbance. Discretized observations on the process variable are indexed to form a time series. A time-series model is fitted to the series. The ultrahigh signal-to-noise ratio fitted values are examined by a neural network, for patterns which detect when the future process is expected to become out of control. The neural-network diagnosis forms the basis for corrective action, prior to the process becoming out of control. In principle, this goes beyond SPC to achieve a process which is never actually out of control. Index Terms—Feedforward control, moving window spectral method, neural network, statistical process quality control.
I. INTRODUCTION
P
ROCESS control is an important managerial function for regulating the quality of the products of a manufacturing firm. Quality affects the firm’s image, reputation, liability, productivity, and costs. Productivity and costs are closely related to quality. Poor quality leads to low productivity and high costs due to defective output, product rework, scrap and other related waste, warranty costs, after-sales replacement and repair, other costs associated with transportation, inspection in the field, and discounts used to offset inferior quality. Competitive management will seek to increase profitability through improved quality. The traditional approach to statistical process control (SPC) is to examine the most recent values of the variable which is obtained from the process to be controlled. The variable is plotted on an SPC chart. The SPC chart is comprised of an upper control limit (UCL) and a lower control limit (LCL). The process variable is assumed to be independently and identically distributed (i.i.d.). Whenever the process variable falls outside the control limits, the process is assumed to be out of control. The cause of the out-of-control condition must then be determined and rectified. This is a reactive feedback control procedure, and depends on the process becoming out of control. In this paper, we define proactive direct and indirect feedforward control as follows:
Manuscript received November 9, 1998; revised July 26, 1999. Review of this manuscript was arranged by Department Editor C. Gaimon. D. Ridley is with the Supercomputer Computations Research Institute, Florida State University, Tallahassee, FL 32306 (e-mail:
[email protected]). F. Llaugel is with the Science and Technology Center, Universidad Dominicana O&M, Dominican Republic. Publisher Item Identifier S 0018-9391(00)06637-X.
Definition: Direct feedforward control: a proactive method by which information about a random unpredictable error input is fed, ahead of time, to the manipulated inputs of a process, the output of which is to be controlled, so as to counteract the effect of the random unpredictable error input. Definition: Indirect feedforward control: a proactive method by which information about a systematic predictable drift is fed, ahead of time, to the manipulated inputs of a process, the output of which is to be controlled, so as to counteract the effect of the systematic predictable drift. The purpose of this paper is to present an indirect feedforward control procedure which is designed to prevent the process from ever being out of control. If the assumption of an i.i.d. process variable were true, then clearly this would be impossible since there would be no way to predict an out-of-control condition. It turns out, however, that in actual manufacturing and other processing systems, the process variables are not i.i.d. These systems exhibit systematic serial correlation estimated to be as high as 80% (Alwan and Roberts [3]). On one hand, this invalidates the SPC chart. If the systematic component adds to the magnitude of the random component, there will be an increase in the probability of a type I error (producer risk), of assigning the cause of systematic variation to an out-of-control process, when, in fact, the cause is the inherent serial correlation in the process variable. If the systematic component subtracts from the magnitude of the random component, there will be an increase in the probability of a type II error (consumer risk), of not detecting an out-of-control process. On the other hand, serial correlation in the process variable enables the fitting of a time-series model, from which predictions can be made. Indirect feedforward control requires that out-of-control information, based on systematic process variation, be available ahead of time. This type of information will be contained in the systematic serially correlated fitted values from a time series model. However, it is very important that the fitted values not be biased. Therefore, care must be taken to correctly specify the time-series model. Otherwise, the residuals from the time-series model may also be serially correlated, resulting in no real progress. Alwan and Roberts [2] and Alwan and Radson [1] suggested the use of a time-domain time-series model to isolate i.i.d. residuals from non-i.i.d. fitted values. They then suggested the separate analysis of these two variables on their own SPC charts. That is still a reactive feedback system, not a proactive feedforward system. However, we mention that research because of some aspects of time-domain models which do not adequately represent the process variable that we are likely to encounter. In particular, Alwan and Radson [1] reported difficulties when the process variable contained cycles. They were
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unable to automate the procedure to reliably fit a time-domain model to the data. Chatfield and Prothero [4] also reported problems in fitting time-domain models to data which contain cycles. In this paper, we use the moving-window spectral (MWS) time-series model. It is a frequency-domain model capable of automatic representation of trend and several hidden cycles. Details of the MWS method may be found in Ridley [15], [17]. As in any time-series methodology, nonstationary data must be made stationary by applying suitable transformations, prior to model fitting. This includes the particular difficult case of heteroscedasticity in which the error variance changes systematically with the mean of the time series (Ridley [14], [16], [18]). In this paper, we discuss the application of the MWS model and the neural-network (NN) model to process control, and the possibility which it creates for reducing the need for feedback trim, reduction in process variability, and quality beyond that currently conceptualized by SPC. The MWS model is used to establish the minimum data required to fit a time-series model. Minimum data usage provides for the earliest detection of an out-of-control condition. An NN is trained to recognize unnatural patterns that might occur in the process variable (see Ramirez-Beltran and Llaugel [13]). The fitted values from a time-series model are then fed to the NN. The NN detects the formation of unnatural patterns. This forms the basis for corrective action to be taken. In this way, the process never actually goes out of control. The remainder of this paper is organized as follows. An overview of the integrated moving-window spectral and neural-network (MWSNN) model-based indirect feedforward SPC system is given in Section II. The MWS model is summarized in Section III. The NN model and training patterns are given in Section IV. In Section V, these models are integrated and applied to the case of a steel-making process. The focus of this paper is on the results of the integrated MWSNN model. Key decision-making details of the time-series model-fitting process are given in the Appendix.
II. MODERN MODEL–BASED SPC A. Artificial Intelligence Many approaches have been used in attempting to automate the manufacturing quality surveillance process (Kuo and Mao [9]). These range from the use of simple runs rules to complex and sophisticated expert systems. A comparison of these methods reveals that the use of artificial intelligence in the form of expert systems based on if–then rules or pattern recognizers using an NN has demonstrable advantages. Hwarng and Hubele [8] and Cheng [5], in particular, benchmark the performance of NN models against CUSUM and Shewhart–CUSUM methods. They show the superior capability of the NN to detect early changes in the process mean. However, one of the observations made by Cheng [5] is that the NN performs poorly when the data are characterized by a low signal-to-noise ratio. By definition, SPC data are comprised primarily of much noise (independent measurements) and little signal (subtle correlation structures). In this paper, by using the MWS time-series model to separate
the data into purely indeterministic residuals (100% noise) and purely deterministic fitted values (0% noise), the fitted values which are fed to the NN are, in theory, characterized by an infinite signal-to-noise ratio. That is, the effectiveness of our integrated MWSNN model will not be diminished by a low signal-to-noise ratio. Ramirez-Beltran and Llaugel [13] showed that it is possible to use an NN to detect unnatural patterns in control charts, other than the process mean. They demonstrated that it is possible to train a neural network to perform pattern recognition in a process-independent manner. This latter approach is used here. B. System Integration An overview of the integrated control system is given in Fig. 1. There are two reasons why the process may become out of control: either the inputs to the process are different from what is expected or the conversion process itself develops a component malfunction. Disturbances to the primary inputs are either measurable or unmeasurable (see Fig. 1—bottom left section). An example of a disturbance is a bad batch of raw materials. Unmeasurable disturbances will enter the process undetected, resulting in defective output from the process. Random, but measurable disturbances are fed to a direct feedforward controller immediately as they occur. The controller then manipulates the input so as to compensate for the disturbance. Any remaining deviations from set point are corrected by feedback trim. Dynamic compensation via direct feedforward control and feedback trim maintain the system in control. Direct feedforward control reduces the amount of feedback trim that is required. For further discussion on the application of direct feedforward control, see Fisher Controls [6]. Malfunctions which develop within the conversion process will only appear in the process output (see Fig. 1—top right section). In order to detect these, a short history of output measurements is analyzed by the MWS time-series model. The fitted values from the MWS model are plotted on a common-cause chart, while residuals are plotted on a special-cause chart. These charts provide an opportunity for visual inspection of the process. The residuals are analyzed (see Fig. 1—top left section) to detect any breaches in the special-cause SPC chart, and to alert the system operator accordingly, for corrective action. Special causes are one of kind in nature. There is no systematic process correction that is possible. For example, in the case of a bad batch of raw materials, any remaining defective raw materials should be removed and replaced. The supplier should be told about it. If it occurs more frequently than is acceptable, a different supplier must be used. The fitted values are also fed to the NN. The fitted values are systematic, and therefore predictable. Predictions by the NN are fed to an indirect feedforward controller. The controller then manipulates the input so as to commence gradual compensation of this systematic disturbance, ahead of time. The objective here is to prevent the system from ever becoming out of control. In a fully automated system, human intervention is minimized. However, continuous monitoring by an operator is possible via the dual system of special-cause and common-cause SPC charts.
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Fig. 1.
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Moving-window spectral-model-based feedforward process control.
III. THE MOVING-WINDOW SPECTRAL (MWS) TIME-SERIES MODEL Consider the process variable, represented as the time series , which is observed at time . The time series is assumed to contain trend, periodic (cyclical), and random components. In order to estimate the correlation struc, a moving window of length is defined in the ture of time domain. The moving window is used to generate sequences of data points in the time domain. This creates multiple observations for obtaining least squares estimates of the parameters which describe the behavior of the component cycles over time. The moving-window spectral (MWS) time-series model is the autoregressive process given by
where is the window number, and the , index of the realization of a cycle at frequency . and Likewise,
Complete details on the moving-window spectral method, and how the above model is estimated, may be found in Ridley [15]. The moving-window spectral time-series model is an integrated procedure of the computer program FOURCAST (Ridley [17]) used in this research. IV. THE NEURAL-NETWORK (NN) MODEL
where parameter, coefficient of
lagged
time periods,
an unobservable error term, sequence of i.i.d. normally distributed random variables with mean zero and constant variance . The Fourier transform is used to estimate the spectral density from function for each window
The literature contains many contributions on the use of the NN in statistical process control (e.g., Cheng [5], Pugh [11], [12], Hwarng and Hubele [7], [8]). Many of these deal with the pattern-recognition approach for identifying the causes of malfunction in the process. All of these approaches focus on detecting an assignable cause after its occurrence has become manifest in the control-chart pattern. A large amount of data is typically required. As a result, if the malfunction is detected, it is detected very late in its development. By then, process failure is imminent. In this paper, we demonstrate that it is possible to increase the lead time until a malfunction becomes imminent. This is accomplished by feeding only a few data points of an
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Fig. 2. Simulated patterns.
ultrahigh signal-to-noise ratio pattern to the NN, very early in the development of the malfunction pattern. The neural network was trained to recognize unnatural patterns in the fitted values from the MWS time-series model. One of the advantages of this approach is that the model is process independent. Process information is only needed subsequently, for comparison with the simulated patterns, to produce an NN reaction and to adjust the process. The neural-network training was performed using simulated patterns of process malfunction. The simulated patterns were prepared using a process mean of 0 with a standard deviation of 1. The following patterns were simulated: sudden up and down, shift up and down in the mean, trend up and down, cycle, normal, systematic, and mixed patterns. The patterns were simulated with small changes in mean because large changes may be detected by conventional methods. All of the disturbances
were simulated so as to produce observations within the three sigma control limits. Some samples of the patterns generated are depicted in Fig. 2. The possible reasons associated with each pattern have been studied in the Western Electric Handbook [20]. The equations used to produce the simulated patterns are based on the formula , where is a normally distributed, set equal to mean-zero random noise with constant variance is the pattern disturbance. The pattern disturbance 1, and is a function of time and a perturbation parameter , is the magnitude of variation. It has a mean where of zero and a standard deviation , where is that for the mean of the actual process being controlled. In this way, in the NN model is the counterpart of in the MWS time-series model, and the integrated MWSNN model is seamless.
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TABLE I
The following disturbance patterns will be considered part of the NN model. Sudden Up and Sudden Down:
where
Shift Up and Shift Down:
Trend Up and Trend Down:
Cycle:
where is the period of the cycle. Normal Pattern:
Systematic:
Mixtures:
where is a random factor with values 1 and . The topology used is a three-layer fully connected network with one input layer with 10 nodes, one output layer with four nodes, and one hidden layer with 45 nodes. This topology was found by experimentation. The training set contained 1000 patterns. The training session stopped when the mean-square error dropped to 0.014. The backpropagation algorithm was used with a momentum of 0.7. A test set was constructed with the same
simulation program that was used for the training set. The test set contained 10 000 patterns, 1000 of each type of ten patterns. The patterns are formed with ten simulated observations, the minimum number required to recognize a patten. The use of more observations only serves to delay the time until pattern recognition and implementation. The algorithm used to train the network (backpropagation) and the node transfer function (sigmoid) produces numbers between 0 and 1 in the nodes of the output layer. Given that ten patterns are to be recognized, it is necessary to represent ten different results lying between 0 and 1 in the output layer. This can be accomplished with a minimum of four binary nodes . Each pattern is represented as a code of four numbers as given in Table I. Experimentation demonstrated that pattern recognition was better if 0.1 and 0.9 were used in place of 0 and 1. The value of each output node must match one of the codes in Table I. In order to accomplish this, a threshold value of 0.5 is used. If the output from a node is greater than 0.5, then its value is matched to 0.9; otherwise, it is matched to 0.1. During the training session, the neural network learns to identify each pattern with its associated code. Those codes are the results reported in the output layer. The network is trained until the difference between the codes computed by the network and those required by the patterns is minimized. This is known as supervised learning [19]. The performance of the NN is assessed by the number of patterns per thousand that it classifies correctly. The number of patterns classified correctly and the corresponding small probabilities of types I and II errors are listed in Table I. The training and diagnostic sessions were conducted on a modestly powered 486-based 100 MHz computer with 8 Mbytes of RAM. The simulated patterns were constructed with a computer program written in the C programming language. The source code is available from the authors. The training session (after the final network topology was selected) took approximately 5 h. The real-time process data time-series analysis and generation of the fitted values required approximately 10 s. During the diagnostic phase, each pattern was recognized in less than 1 s. Therefore, the data can be processed quickly and easily on the production floor. V. CASE STUDY In the case study that follows, the controlled variable is a percentage of phosphorus in iron taken from a steel-making SPC
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Fig. 3.
Time-sequence plot of the phosphorus series.
Fig. 4.
Special-cause chart for phosphorus series (time period 2–101).
application given by Podolski [10]. The phosphorus time series is given in Fig. 3. The original time sequence plot reveals no evidence of any breach of the three sigma limits, that is, no indication of an out-of-control condition. A. Time-Series Analysis The integrated MWSNN model analysis proceeds as follows. First, the MWS time-series model is fitted to the phosphorus time series. The details of the fitting process are given in the Appendix. Next, the residuals and fitted values from the model are examined separately, on a special-cause SPC chart and a common-cause SPC chart, respectively. By separating the independent residuals and the systematic fitted values in this way, the i.i.d. assumption can be applied to the special-cause SPC chart with fidelity. The plot of the residuals for the first 100 data points (not shown because it is not interesting) did not reveal any breach of the three sigma limits. The starting time was then advanced to time 2, and the second 100 data points were examined. The results are given in Fig. 4. Inspection of the special-cause chart in Fig. 4 reveals a breach of the three sigma limit in time period 101. This breach went undetected on the original chart. By definition, this breach is due to a one-of-a-kind special cause such as a bad batch of raw materials. The breach is not predictable. It must occur before it can be observed and corrective action taken. When it does occur, the NN may be used to alert the system operator.
B. Neural-Network Analysis We now turn to the main focus of this paper, which is on the noise (residuals)-free fitted values and the common-cause chart (see Fig. 5). This chart is based on the first 100 data points from the phosphorus time sequence. The computer software alogorithm makes an allowance for a differencing transformation, whether or not differencing is actually performed. This uses up the first data point. The optimal MWS window length is 4 (see Appendix), indicating the presence of a four-period periodic component. Also, this uses up four data points (see Section III). Therefore, the fitted values begin in period 6, one time period , and end in period 100. Variations in after period 5 the fitted values may suggest that there is some wear in a component (bearings, measuring instruments, etc.) of the process or some other source of systematic variation (environmental temperature or humidity, power fluctuation, etc.). All components will wear to some extent. Given enough time, all components will wear out. Therefore, in the absence of specific information, we must always monitor the process for component wear. Examination of the patterns in the common-cause chart may indicate the location of wear. Small variations may be used as a tool in preventative maintenance. However, left unattended, small variations will eventually grow into to large variations, which will show up as a breach of the three sigma limits, indicating that wear out is imminent. There is no breach of the three sigma limits in Fig. 5. Let us interpret that to mean that, if component wear is, in fact, oc-
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Fig. 5.
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Common-cause chart for phosphorus series (time period 1–100/fitted period 6–100).
Fig. 6. Recognized patterns.
curring, it is not yet imminent. Still, component wear could be underway. In order to investigate this, the fitted values were fed to the NN. The fitted values were organized into overlapping sets of ten data points to match the required inputs to the NN.
The first set was constructed from fitted values 6–15, the next set from fitted values 7–16, the next set from fitted values 8–17, and so on up to fitted values 17–26. The series was normalized using the mean and standard deviation of the undisturbed process. Fig.
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Common-cause chart for phosphorus series (time period 17–116/fitted period 23–122).
6 (read from left to right then down) contains the results of the NN analysis. This allows us to compare what the NN indicates is evolving in the immediately subsequent time periods. In addition to the periodic component already detected, in 8 out of 12 cases, the NN classifies the process change as “shift up,” with no contradiction from the other four cases. These are abnormal, and therefore out-of-control conditions, requiring investigation and correction. In practice, action will be taken to correct the process at the first ocurrence of an abnormal pattern. In this case, the first abnormal pattern is “shift up.” Once a worn component has been detected, it can be replaced during the next scheduled maintenance period. If the patterns contained in Fig. 5, and detected in Fig. 6, are ignored (no action taken to alter their course) and the MWS time-series model is updated to include successive sets of 100 data points, the three sigma limits are eventually breached. The breach is caused either by a large absolute slope or by a prolonged upward or downward trend, such that either the lower or upper end of the trend extends beyond one of the three sigma limits. It does not matter which end of the trend breaches a three sigma limit. Fig. 7 is a common-cause chart corresponding to the case when the time-series model is based on data points 17–116 and is the first to show a breach of its lower three sigma limit. The breach occurs in period 28. The lower three sigma limit breach occurring on the left of the chart implies that there is an upward trend that has rotated counterclockwise from where it was earlier in Fig. 5. If this breach is indicative that component wear out is imminent, then failure will ultimately occur. This would lead to an unscheduled shutdown of the process. Such a shutdown would be a costly disruption of production. The worn-out component would have to be replaced anyway, albeit during the unfortunate shutdown. Furthermore, if the worn-out component were to be unavailable at the time of the shutdown, additional costly delays would occur. VI. CONCLUDING REMARKS This paper introduced the use of the integrated moving-window spectral (MWS) time-series and artificial-intelligence neural-network (NN) model-based statistical process control. This seamless integrated indirect feedforward (MWSNN) model was applied to a process variable represented as a digitized time sequence of percentage phosphorus in iron used in a steel-making process. The data contained substantial
Fig. 8.
F
ratio versus window length (T ).
Fig. 9.
F
ratio versus number of data points (n).
randomness, some trend, and periodicity in their very subtle correlation structure. The MWS model was used to decompose the process variable into trend, periodic component, and residuals. The integrated procedure was performed by the computer program FOURCAST (Ridley [17]), with a call to the NN. The program automatically provided statistical process-control charts for residuals and for fitted values. These charts, marked with one, two, and three sigma limits, provided visual inspection capability and human backup to the NN for difficult cases where the automatic system may fail. The residuals were analyzed to detect breaches of the three sigma limits of the special-cause chart. The system detected special-cause disturbance effects which went undetected in the traditional SPC chart of original data. The ultrahigh signal-to-noise ratio fitted values were fed to the NN. The integrated MWSNN
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Fig. 10
F
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ratio for various window lengths (T ) and number of data points (n).
model identified systematic disturbances early in their development. This provided the basis for an integrated indirect feedforward control system to take advance corrective action. The indirect feedforward control system achieved its objective of providing the information necessary to prevent the process from ever becoming out of control. A suggestion for further research is to incorporate an assignable cause interpreter expert system (ES) into the artificial-intelligence system, with the diagnostic capability to identify and locate specific components of the production system which may be failing (see Kuo and Mao [9] for a system overview). The component list could include people, equipment, and/or process. Therefore, such an interpreter would, of necessity, have to be production plant specific. The output from the NN would be fed to an ES which is trained on the particular plant.
APPENDIX This Appendix provides a detailed description of the steps completed in fitting the MWS time-series model to the phosphorus data. The computer program FOURCAST (see Ridley [17]) was used to fit the MWS model to the phosphorus data. The program provides clear and automatic diagnostic aids to identify the optimal time-series model. These include time-series plots and histograms of the process data; spectral decomposition for identifying trend, periodicity (seasonal and other values; ratios; critical and racycles), and residuals; tios; and time-series plots and histograms of fitted values and residuals. In that way, the optimal model is easily determined by a process-control specialist. Once selected, the optimal model provides stable and consistent results, as the model is updated to reflect new data, without the need to update the structure of the model. In order to determine the optimal window length, models with different window lengths were fitted to the data, so as to discover the maximum ratio, in the time domain where the
original data were observed. The automatic optimization option in FOURCAST was set to perform these computations. Fig. 8 shows the results for window lengths ranging from 3 to 80. These results indicate an optimal window length of 4. Next, we consider the number of data points to be used in building the model. It is better to use fewers data where possible, not more data. This is particularly true in the case of detecting common-cause effects. In order to detect special effects that cause a breach of the three sigma limits of the residuals SPC chart, a pattern of randomness must first be established. This period is then followed immediately by a breach of the three sigma limits. In order to detect common effects that cause a breach of the three sigma limits of the fitted-values SPC chart, a pattern of correlated structure must first be established. Therefore, the fewer the number of data points required, the sooner this system will provide a warning of common-cause effects. Setting the window length to 4, the number of data points was varied from 10 to 100. The results are given in Fig. 9. These results show that the ratio is not significant below 80 data ratio rises sharply, until points. Above 80 data points, the about 98 data points is reached, before leveling off. The ratio is reviewed for all combinations of window length and number of data points in Fig. 10. Once again, it appears that the optimal model will use about 98 data points and a window length of 4. Using round numbers, then, the optimal MWS model was set to 100 data points and a window length of 4. The final discrete time fitted model is
fitted residuals
ACKNOWLEDGMENT The authors would like to thank Prof. L. Alwan for providing the data for the case study, the Departmental Editor (Models and
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Methodologies), and three anonymous referees for suggestions that improved the presentation of the paper. REFERENCES [1] L. C. Alwan and D. Radson, “Implementation issues of time-series based statistical process control,” Prod. Oper. Manage., vol. 4, no. 3, pp. 263–276, 1995. [2] L. C. Alwan and H. V. Roberts, “Time-series modeling for statistical process control,” J. Bus. Econ. Statist., vol. 6, no. 1, pp. 83–95, 1988. [3] L. C. Alwan and H. V. Roberts, “The problem of misplaced control limits (with discussions),” Appl. Statist., vol. 44, no. 3, pp. 269–278, 1995. [4] C. Chatfield and D. L. Prothero, “Box-Jenkins seasonal forecasting: Problems in a case-study (and discussion),” J. Roy. Statist. Soc., ser. A, vol. 136, pp. 295–336, 1973. [5] C. Cheng, “A multi-layer neural network model for detecting changes in process mean,” Comput. Ind. Eng., vol. 28, no. 1, pp. 51–61, 1995. [6] Fisher Controls, Feedforward Control (Student Guide). Marshalltown, IA: Fisher Controls Educ. Services, 1988. [7] H. B. Hwarng and N. F. Hubele, “Back-propagation pattern recognizers for x control chart: Methodology and performance,” Comput. Ind. Eng., vol. 24, pp. 219–235, 1993a. , “x control chart pattern identification through efficient off-line [8] neural network training,” IIE Trans., vol. 25, pp. 27–40, 1993b. [9] T. Kuo and W. Mao, “Real-time on-line quality control systems,” Int. J. Ind. Eng., vol. 2, no. 4, pp. 271–278, 1995. [10] G. Podolski, “Standard deviation: Root mean square versus range conversions,” Qual. Eng., vol. 2, no. 2, pp. 155–161, 1989. [11] G. A. Pugh, “Synthetic neural networks for process control,” Comput. Ind. Eng., vol. 17, pp. 24–26, 1989. , “A comparison of neural networks to SPC charts,” Comput. Ind. [12] Eng., vol. 21, pp. 253–255, 1991. [13] N. D. Ramirez-Beltran and F. Llaugel, “The ratio control chart and pattern recognition,” in Proc. ASQC 48th Ann. Qual. Contr., , 1994, p. 583. [14] A. D. Ridley, “Variance stabilization: A direct yet robust method,” Rev. Bus., vol. 15, pp. 28–30, 1993. [15] , “The Global Univariate Moving Window Spectral Method,” Supercomput. Computations Res. Inst., Florida State Univ., Tallahassee, FL,
[email protected], FSU-SCRI-94-17, 1994a. [16] , “A model-free power transformation to homoscedasticity,” Int. J. Prod. Econ., vol. 36, pp. 191–202, 1994b. [17] , FOURCAST—Multivariate Spectral Time Series Analysis and Forecasting(MWS) [Online]. Available: http://www.fourcast.net/, EMC, Box 12518, Tallahassee, FL 32 317–2518, USA, 1998.
, “Optimal antithetic weights for lognormal time series forecasting,” Comput. Oper. Res., vol. 26, pp. 189–209, 1999. [19] D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning internal representations by error propagation,” in Parallel Distributed Processing, D. E. Rumelhart and J. L. McClelland, Eds. Cambridge, MA: M.I.T. Press, 1986, vol. 1. [20] B. Small-Western Electric Co.. (1958) Statistical Quality Control Handbook. Western Electric, AT&T Customer Information Center, Indianapolis, IN. [Online]. Available: www.lucent.com/work/family/docs/small.html/
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Dennis Ridley was born in Jamaica. He studied electrical engineering in England and Trinidad,where he received the M.Sc. degree in engineering power systems from the University of the West Indies. He received the Ph.D. degree in engineering management from Clemson University, Clemson, SC. He is an Engineering Management/Operations Research/Statistics Professor at Florida A&M University, and Faculty Associate of the Supercomputer Computations Research Institute, Florida State University. Previous engagements include the International Atomic Energy Agency, in Vienna, Austria; Howard, George Mason and Clemson Universities; and Jamaica Public Service Company. He also serves as a Consultant on forecasting. Dr. Ridley is a member of the Institute for Operations Research and the Management Sciences (INFORMS), the International Institute of Forecasters, the Institute of Business Forecasting, the Production and Operations Management Society, and the American Statistical Association.
Felipe Llaugel was born in the Dominican Republic. He received the B.S. (industrial engineering) and M.B.A. degrees from Instituto Tecnologico de Santo Domingo, and the M.S. degree in economics from Universidad Catolica Madre y Maestra, Dominican Republic. He received the M.Eng. degree in management systems engineering from the University of Puerto Rico. Currently, he is Director of the Science and Technology Center, Universidad Dominicana O&M, Dominican Republic. Mr. Llaugel is a member of the Institute of Industrial Engineers and the American Society for Quality (ASQ).
