Abstract:
Computerized process control systems and methods for the production of melt polycarbonate include a plurality of sensors for obtaining a plurality of measurements relating to a plurality of predetermined process variables, a preprocessor for preprocessing each of the plurality of measurements for multivariate statistical analysis, an identifier for identifying which of the plurality of predetermined process variables affect each of a plurality of predetermined product variables, a correlator for correlating the plurality of predetermined process variables and the plurality of predetermined product variables, and a model generator for modeling the relationship between the plurality of predetermined process variables and the plurality of predetermined product variables. The plurality of predetermined process variables are analyzed to predict polymer performance and/or to ensure that the value of each of the plurality of predetermined process variables is within a predetermined range.

Description:
BACKGROUND OF INVENTION 
     The present invention is directed to systems and methods for the control of chemical manufacturing processes and, more specifically, to multivariate statistical process analysis systems and methods for the production of melt polycarbonate. 
     Manufacturing process variables (X i ) such as flows, pressures, concentrations, temperatures, and others have traditionally been subject to statistical process control (SPC) strategies. These SPC strategies, introduced in the 1930&#39;s, include a variety of statistical methods designed to maintain process quality and productivity. The statistical methods focus on a single variable X i  at a time, using univariate controls such as Shewhart charts, cumulative sum charts, and exponentially-weighted moving average charts. Such charts are used to monitor the performance of a process, such as a chemical manufacturing process, over time to verify that the process consistently operates within the production specifications of a given product. 
     As the number of monitored variables X i  affecting the behavior of a manufacturing process increases, however, univariate SPC methods become inadequate. The application of these techniques in such situations may result in misleading information being presented to a process operator, leading him or her to take unnecessary or erroneous control actions. 
     An alternative approach is to employ multivariate statistical process analysis (MSPA) methods to extract more relevant information from measured data. MSPA methods provide the staff of a manufacturing plant, for example, with a greater understanding of process performance, allowing them to make sound business decisions. Thus, the application of multivariate methodologies to industrial manufacturing processes has experienced increasing popularity in recent years. For example, MSPA methods have been utilized in emulsion polymerization, low-density continuous polyethylene polymerization, batch polymerization, and pilot-scale penicillin fermentation processes. Similarly, MSPA methods have been utilized to improve the productivity of a titanium dioxide plant, monitor the processing conditions of a nuclear waste storage tank, and control the performance of chromatographic instrumentation. 
     The application of multivariate statistical analysis methods to industrial process data characterized by a large number of correlated chemical process measurements is the area of process chemometrics. The objectives of process chemometrics include the determination of key process variables, the generation of inference models used to forecast and optimize product quality, the detection and diagnosis of faults and potential process abnormalities, and the overall monitoring of chemical processes to ensure production control. Achieving these goals is often difficult with regard to the production of melt polycarbonate, however, as the determination of key process variables may be an inexact and time consuming process, and accurate and reliable inference models may be difficult to generate. 
     SUMMARY OF INVENTION 
     Thus, the present invention is directed to automated multivariate statistical process analysis systems and methods for the production of melt polycarbonate. These systems and methods allow process variables causing abnormal performance to be detected and identified. As a result, a manufacturing plant staff may better understand process performance and make sound business decisions. 
     In one embodiment, a computerized system for the production of melt polycarbonate includes a plurality of sensors for obtaining a plurality of measurements relating to a plurality of predetermined process variables, a preprocessor for preprocessing each of the plurality of measurements for multivariate statistical analysis, an identifier for identifying which of the plurality of predetermined process variables affect each of a plurality of predetermined product variables, a correlator for correlating the plurality of predetermined process variables and the plurality of predetermined product variables, a model generator for modeling the relationship between the plurality of predetermined process variables and the plurality of predetermined product variables, and an analyzer for analyzing the plurality of predetermined process variables to predict polymer performance and/or to ensure that the value of each of the plurality of predetermined process variables is within a predetermined range. 
     In another embodiment, a computerized method for the production of melt polycarbonate includes the steps of obtaining a plurality of measurements relating to a plurality of predetermined process variables, preprocessing each of the plurality of measurements for multivariate statistical analysis, identifying which of the plurality of predetermined process variables affect each of a plurality of predetermined product variables, correlating the plurality of predetermined process variables and the plurality of predetermined product variables, modeling the relationship between the plurality of predetermined process variables and the plurality of predetermined product variables, and analyzing the plurality of predetermined process variables to predict polymer performance and/or to ensure that the value of each of the plurality of predetermined process variables is within a predetermined range. 
    
    
     BRIEF DESCRIPTION OF DRAWINGS 
     FIG. 1 is a flow chart of a computerized multivariate statistical process analysis method for the production of melt polycarbonate; 
     FIG. 2 is a plot of the eigenvalues of process variables, measured using the method of FIG. 1, after autoscaling; 
     FIG. 3 is a graph of the percent of captured variance for each process variable for the first two principal components (PCs) of the principal components analysis (PCA) model of the present invention; 
     FIG. 4 is a loadings plot from the PCA model of FIG. 3; 
     FIG. 5 is a loadings plot for the first PC from the PCA model of FIG. 3; 
     FIG. 6 is a loadings plot for the fourth PC from the PCA model of FIG. 3; 
     FIG. 7 is a plot of the first two scores for the PCA model of FIG. 3; 
     FIG. 8 is a plot of the result of the prediction of pellet intrinsic viscosity (IV) using only process variables; 
     FIG. 9 is a plot of the result of the prediction of Fries concentration using only process variables; 
     FIG. 10 is a Q control chart for the multivariate statistical process analysis of the production of melt polycarbonate resin; 
     FIG. 11 is a T 2  control chart for the multivariate statistical process analysis of the production of melt polycarbonate resin; and 
     FIG. 12 is a functional block diagram of a computerized multivariate statistical process analysis system for the production of melt polycarbonate. 
    
    
     DETAILED DESCRIPTION 
     Polycarbonates are typically prepared from dihydric phenol compounds and carbonic acid derivatives. For example, one important polycarbonate, melt polycarbonate, may be prepared via the melt polymerization of diphenyl carbonate and Bisphenol A (BPA). The reaction is conducted at high temperatures, allowing the starting monomers and product to remain molten while the reactor pressure is staged in order to more effectively remove phenol, the by-product of the polycondensation reaction. 
     During the melt polycarbonate manufacturing process, data may be collected via sensors in order to monitor process performance. Using this collected information, the relative importance of various process variables (X i ) may be determined and an inference model may be developed to predict such outcomes as Fries concentration, pellet intrinsic viscosity (IV), melt polycarbonate grade, and other product parameters. Further, process variables X i  causing abnormal performance may be detected and identified. As a result, a manufacturing plant staff may better understand process performance and make sound business decisions. 
     Referring to FIG. 1, a multivariate statistical process analysis (MSPA) method  10  for the consistent production of melt polycarbonate begins with the collection of sensor data over a predetermined period of time  12 . For example, data from one melt polycarbonate manufacturing plant included information about process variables X i  and product variables Y i  from about 343 polymerization runs. For each polymerization run, process variables X i  and product variables Y i  were recorded once. A list of process variables X i  and product variables Y i  used for multivariate analysis is presented in Table 1. 
     
       
         
               
             
               
               
             
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Process and Product Variables 
               
             
          
           
               
                 Process Variables X i   
                 Product Variables Y i   
               
               
                   
               
             
          
           
               
                 1. 
                 Molar Ratio 
                 1. 
                 Fries Concentration 
               
               
                 2. 
                 Flow Rate 
                 2. 
                 Melt Flow Ratio (MFR) 
               
               
                 3. 
                 Adjusted Molar Ratio 
                 3. 
                 Pellet Intrinsic Viscosity (IV) 
               
               
                 4. 
                 Vacuum in R3 
                 4. 
                 End Cap (EC) 
               
               
                 5. 
                 Temperature in R3 
                 5. 
                 Yellowness Index (YI) 
               
               
                 6. 
                 Torque in R3 
                 6. 
                 Melt Polycarbonate Grade 
               
               
                 7. 
                 Discharge Pressure in R3 
               
               
                 8. 
                 Melt Viscosity in R3 
               
               
                 9. 
                 Vacuum in R4 
               
               
                 10. 
                 Temperature in R4 
               
               
                 11. 
                 Stirring Speed in R4 
               
               
                 12. 
                 Torque in R4 
               
               
                 13. 
                 Discharge Pressure in R4 
               
               
                 14. 
                 Melt Viscosity in R4 
               
               
                 15. 
                 Throughput 
               
               
                 16. 
                 Pellet Speed 
               
               
                   
               
             
          
         
       
     
     In the above table, “R 3 ” and “R 4 ” refer to specific polymerization stages within the physical melt polycarbonate manufacturing system. An exemplary description of such polymerization stages is presented in International Patent Application WO 00/37531. 
     The variables used for multivariate analysis are further described in Table 2. 
     
       
         
               
             
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Description of Variables 
               
             
          
           
               
                 Variable 
                 Units 
                 Description 
               
               
                   
               
               
                 Molar 
                 — 
                 Ratio of moles of DPC to moles of BPA 
               
               
                 Ratio 
               
               
                 Flow 
                 kg/hour 
                 Flow rate of the monomer mix into R1 
               
               
                 Rate 
                   
                 oligomerization reactor 
               
               
                 Adjusted 
                 — 
                 Ratio of moles of DPC (and additional 
               
               
                 Molar 
                   
                 moles of DPC) to moles of BPA 
               
               
                 Ratio 
               
               
                 Vacuum 
                 torr 
                 Measured in headspace of R3 stage using a 
               
               
                 in R3 
                   
                 pressure gauge 
               
               
                 Temperature 
                 degrees C. 
                 Temperature of reaction components in R3 
               
               
                 in R3 
               
               
                 Torque 
                 N*m 
                 Torque on a stirrer in R3 reactor 
               
               
                 in R3 
               
               
                 Discharge 
                 torr 
                 Pressure after the gear pump in the 3rd 
               
               
                 Pressure 
                   
                 reactor stage 
               
               
                 in R3 
               
               
                 Melt 
                 poise 
                 Melt viscosity of reaction components in R3 
               
               
                 Viscosity 
               
               
                 in R3 
               
               
                 Vacuum 
                 torr 
                 Measured in headspace of R4 stage using a 
               
               
                 in R4 
                   
                 pressure gauge 
               
               
                 Temperature 
                 degrees C. 
                 Temperature of reaction components in R4 
               
               
                 in R4 
               
               
                 Stirring 
                 RPM 
                 Speed of a shaft in R4 reactor 
               
               
                 Speed 
               
               
                 in R4 
               
               
                 Torque 
                 — 
                 Torque on a shaft in R4 reactor 
               
               
                 in R4 
               
               
                 Discharge 
                 torr 
                 Pressure in the 4th reactor 
               
               
                 Pressure 
               
               
                 in R4 
               
               
                 Melt 
                 poise 
                 Melt viscosity of reaction components in R4 
               
               
                 Viscosity 
               
               
                 in R4 
               
               
                 Throughput 
                 kg/h 
                 Amount of polycarbonate material per unit 
               
               
                   
                   
                 of time coming fom the last reactor stage 
               
               
                 Pellet 
                 m/s 
                 Speed of a polycarbonate strand entering 
               
               
                 Speed 
                   
                 the pelletizer 
               
               
                 Fries 
                 ppm 
                 Concentration of Fries product measured by LC 
               
               
                 Concen- 
               
               
                 tration 
               
               
                 Melt 
                 g/10 min 
                 Measure of weight in grams extruded 
               
               
                 Flow 
                   
                 througn a capillary for a 10 min test 
               
               
                 Ratio 
               
               
                 (MFR) 
               
               
                 Pellet 
                 dL/g 
                 Measured at 20 degrees C. on a solution of a 
               
               
                 Intrinsic 
                   
                 sample in methylene chlorine 
               
               
                 Viscosity 
               
               
                 (IV) 
               
               
                 End 
                 % 
                 Calculated from measured concentration of 
               
               
                 Cap (EC) 
                   
                 terminal OH groups 
               
               
                 Yellowness 
                 — 
                 Yellowness index of pellets 
               
               
                 Index (YI) 
               
               
                 Melt 
                 — 
                 Based upon Fries concentration, MFR, and EC 
               
               
                 Polycar- 
               
               
                 bonate 
               
               
                 Grade 
               
               
                   
               
             
          
         
       
     
     Prior to multivariate analysis, gathered data may be preprocessed  14 . For example, the process data may be arranged as a matrix of samples and product variables Y i . The product variables Y i  may be used to label data points after applying pattern recognition tools. The process variables X i  and product variables Y i  in the data set may, however, have different physical measurement units. This difference in units between the variables may be eliminated by autoscaling the data  16 . Autoscaling is typically the application of both variance scaling and mean centering. The data may be autoscaled by subtracting the mean and dividing by the standard deviation for each variable: 
     
       
           X   ij   a =( x   ij   −{overscore (x)}   j )/σ j ,  (1) 
       
     
     where X a   ::  is the autoscaled matrix X of size i×j and 
     and σ j  are the mean and standard deviation of the ith column of the original matrix X. Thus, each column of the autoscaled matrix has zero mean and unit standard deviation. 
     Following the data preprocessing step  14 , as part of the determination of key process variables step  18 , information about the relations between the process variables X i  and their effects on the product variables Y i  may be obtained using principal components analysis (PCA) tools  20 . PCA is a multivariate data analysis tool that projects a data set onto a subspace of lower dimensionality. In this reduced space, data is represented with reduced colinearity. PCA achieves this objective by describing the variance of the preprocessed data matrix X a  (Eq. 1) in terms of the weighted sums of the original variables with no significant loss of information. These weighted sums of the original variables are called principal components (PCs). The preprocessed matrix X a  is decomposed into a set of scores T (I×K) and loadings P (J×K), where K is the number of principal components. Preferably, the number of PCs is chosen to be as few as possible to explain important variation in the data set. The matrix X a  is then expressed as a linear combination of orthogonal vectors along the directions of the PCs: 
     
       
           X   a   =t   1   p   1   T   +t   2   p   2   T   + . . . +t   A   p   K   T   +E,   (2) 
       
     
     where t i  and p i  are the score and loading vectors, respectively, E is a residual matrix that represents random error, and T is the transpose of the matrix. 
     To determine the number of principal components to retain in the PCA model, the percent variance captured by the PCA model may be analyzed (see Table 3 below) in combination with a plot of eigenvalues as a function of PCs  21  (see FIG.  2 ). In one example, four PCs captured about 80.8% of the variance in the data. Higher-order PCs had eigenvalues less than unity and did not describe any systematic variance in the system. Thus, four PCs were determined to be adequate for the PCA model. 
     
       
         
               
             
               
               
               
               
               
             
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Percent Variance Captured by PCA Model 
               
             
          
           
               
                   
                 Principal 
                 Eigenvalue 
                 % Variance 
                 % Variance 
               
               
                   
                 Component 
                 of CoV (X) 
                 this PC 
                 Cumulative 
               
               
                   
                   
               
             
          
           
               
                   
                 1 
                 6.85e + 000 
                 42.83 
                 42.83 
               
               
                   
                 2 
                 3.58e + 000 
                 22.37 
                 65.21 
               
               
                   
                 3 
                 1.50e + 000 
                 9.37 
                 74.58 
               
               
                   
                 4 
                 9.96e − 001 
                 6.22 
                 80.80 
               
               
                   
                 5 
                 8.57e − 001 
                 5.36 
                 86.16 
               
               
                   
                 6 
                 5.95e − 001 
                 3.72 
                 89.88 
               
               
                   
                 7 
                 4.71e − 001 
                 2.94 
                 92.82 
               
               
                   
                 8 
                 3.49e − 001 
                 2.18 
                 95.00 
               
               
                   
                 9 
                 2.93e − 001 
                 1.83 
                 96.84 
               
               
                   
                 10  
                 2.46e − 001 
                 1.54 
                 98.37 
               
               
                   
                 11  
                 8.81e − 002 
                 0.55 
                 98.92 
               
               
                   
                 12  
                 5.96e − 002 
                 0.37 
                 99.29 
               
               
                   
                 13  
                 5.12e − 002 
                 0.32 
                 99.61 
               
               
                   
                 14  
                 3.42e − 002 
                 0.21 
                 99.83 
               
               
                   
                 15  
                 1.46e − 002 
                 0.09 
                 99.92 
               
               
                   
                 16  
                 1.29e − 002 
                 0.08 
                 100.00 
               
               
                   
                   
               
             
          
         
       
     
     Information regarding the amount of variance for each process variable X i  captured by individual PCs in the PCA model of this example is presented in FIG.  3 . PC 1   22  describes about 43% of the total variance in the data, but no single process variable X i  provides an exclusive contribution. The five process variables X i  that individually contributed more than about 50% of the captured variance in PC 1   22  were discharge pressure in R 3 , temperature in R 4 , stirring speed in R 4 , discharge pressure in R 4 , and melt viscosity in R 4 . The two process variables X i  that individually contributed more than about 50% of the captured variance in PC 2   24  were flow rate and throughput. 
     Referring to FIG. 4, a loadings plot  26  may also be used as a diagnostic tool for the PCA model and process variables X i . The loadings plot  26  determines which variables are important for describing variation in the data set. When a given process variable X i  contributes a significant variation to the PC, the absolute value of the loading of the variable will be close to unity. When the process variable X i  does not contribute a significant variation to the PC, the absolute value of the loading of the variable will be close to zero. The different signs of the loadings indicate that the process variables inversely contribute to the variance described by the PC. In the above example, discharge pressure in R 4  contributes the most to PC 1 , while vacuum in R 3  contributes the least. Most of the contribution to PC 2  is provided by flow rate and throughput, while vacuum in R 4  and adjusted molar ratio contribute the least to PC 2 . 
     It is also important to note the amount of variation described by a PC when interpreting loadings. A variable with a large loading value contributes significantly to a particular PC. However, the variable may not be truly important if the PC does not describe a large amount of the variation in the data set. FIG. 5 presents the values of loadings of process variables X i  to PC 1   28  and FIG. 6 presents the values of loadings of process variables X i  to PC 4   30 . For example, vacuum in R 4  has a large loading of about −0.5 to PC 4   30  and a relatively small loading of about −0.2 to PC 1   28 . However, PC 1   28  describes about seven times more variation in the data set. 
     Another step in the multivariate statistical process analysis method  10  of the present invention is establishing a correlation between process variables X i    32 . Multivariate methods are capable of detecting changes in the correlation structure of a group of variables that may not be detected by univariate methods. The correlation structure in the data set may be visualized using, for example, a correlation analysis. The correlation between a pair of variables x and y is defined as:                  R     x   ,   y       =       ∑   i   N                       (       x   i     -     x   _       )                       (       y   i     -     y   _       )     /     (       σ   x            σ   y          (     N   -   1     )         )             ,           (   3   )                                
     where R is the correlation coefficient and N is the number of data points. The correlation coefficient R is between −1 and 1 and is independent of the scale of x and y values. For an exact linear relation between x and y, R=1 if increasing x values correspond to increasing y values and R=−1 if increasing x values correspond to decreasing y values. R=0 if the variables are independent. 
     Results of the correlation analysis of process variables X i  may be presented as pseudocolor maps, and may, optionally, be reordered using, for example, a k-nearest neighbor (KNN) cluster analysis. This hierarchical cluster analysis determines the similarity of process variables X i  based on their measured properties. 
     In the example discussed above, two pairs of process variables X i , temperature in R 3  and temperature in R 4  and flow rate and throughput, had correlation coefficients close to 1. Stirring speed in R 4  and both temperature in R 4  and discharge, pressure in R 4  had correlation coefficients close to −1. A further step in the multivariate statistical process analysis method  10  of the present invention is establishing a correlation between process variables X i  and product variables Y i    34 . An immediate benefit of this step  34  may include the ability to forecast product quality based upon measurements involving only process variables X i . As discussed above, during the manufacturing of melt polycarbonate, material quality may be monitored as a function of several product variables X i . 
     Initial analysis of the correlation structure in the combined data set of process variables X i  and product variables Y i  may also be performed using correlation analysis. For example, a pseudocolor correlation map of the process variables X i  and product variables Y i  may be reordered using KNN cluster analysis. In the example discussed above, none of the individual product variables Y i  correlated with process variables X i  with a correlation coefficient of about 1. A strong inverse correlation (R≈−1) was found between the stirring speed in R 4  and pellet IV. 
     A detailed analysis of the correlation between process variables X i  and product variables Y i  may be performed using PCA of the process variables X i  and analyzing a PC scores plots. FIG. 7 shows the plot of the first two scores for PCA  36 . The plot  36  illustrates a general relationship between samples of melt polycarbonate resin manufacture and different process variable X i  conditions. Several possible clusters of data may be assigned on the scores plot  36 . Intuitively, polycarbonate produced under similar process conditions should have similar properties. 
     A more in-depth understanding of the relationships between process variables X i  and product variables Y i  may be obtained given additional knowledge of the data. In particular, the data points on the scores plots may be labeled according to information about product variables Y i . Histogram plots may be used to determine the distribution of product variables Y i  and to assign labels to respective portions of the distribution. For example, the correlation between process variables X i  captured by the first two PCs and product variables Y i  such as melt flow ratio, pellet IV, Fries concentration, and melt polycarbonate grade may be plotted. Analysis of histograms of these product variables Y i  may, for example, indicate that melt flow ratio and pellet IV are more rightly arranged in clusters on the histogram plots than, for example, Fries concentration. This variation in distribution may be explained by more pronounced effects of the process variables X i  on Fries formation. 
     Referring again to FIG. 1, the multivariate statistical process analysis method  10  of the present invention further includes the generation of an inference model for predicting and analyzing the melt polycarbonate production process and its performance  38 . For example, pellet IV and Fries concentration of manufactured polycarbonate resin may be predicted using “virtual analyzers,” analyzing only the information from process variables X i . For this application, a multivariate calibration method such as partial least-squares (PLS) regression may be used. The quality of the developed PLS models for quantitation of pellet IV and Fries concentration may be evaluated using, for example, the root mean squared error of calibration (RMSEC). Performance of the PLS models developed for quantitation of pellet IV and Fries concentration may also be validated using a leave-one-out cross-validation algorithm. The root mean squared error of cross-validation (RMSECV) may be used to estimate the ability of the models to predict pellet IV and Fries concentration. RMSECV is essentially the standard deviation of the predicted values minus laboratory estimated values (i.e. the standard deviation of the test set residuals). A large RMSECV indicates poor correlation with the reference method and/or poor precision. 
     In the example discussed above, the results of the prediction of pellet IV and Fries concentration using only process variables X i  are presented in FIGS. 8 and 9, respectively. For quantitation of pellet IV, a six-factor PLS regression model was found to be adequate. This model accounted for about 85% of the variance in process variables X i  and about 98% of the pellet IV variance. For quantitation of Fries concentration, a four-factor PLS regression model was found to be optimal. It accounted for about 75% of the variance in process variables X i  and about 90% of Fries concentration variance. A summary of the performance of the PLS models for the prediction of pellet IV and Fries concentration from the measured process variables X i  is presented in Table 4. 
     
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 4 
               
             
             
               
                   
               
               
                 Summary of Calibration Model Performances 
               
             
          
           
               
                   
                   
                 Root Mean 
                 Root Mean 
                   
               
               
                   
                 Product 
                 Squared Error 
                 Squared Error 
               
               
                   
                 Variable 
                 of Calibration 
                 of Cross-Validation 
                 R 2   
               
               
                   
                   
               
               
                   
                 Pellet IV 
                 0.00703 dL/g 
                 0.00758 dL/g 
                 0.99 
               
               
                   
                 Fries 
                 0.0146% Fries 
                 0.0152% Fries 
                 0.90 
               
               
                   
                 Concentration 
               
               
                   
                   
               
             
          
         
       
     
     To ensure normal manufacturing plant operation, the quality of collected process variables X i  may be evaluated using statistical tools such as multivariate control charts and multivariate contribution plots, among others. This allows for the detection of faults and the diagnosis of problems in the process variables X i    40 . Multivariate control charts use two statistical indicators of the PCA model, such as Q and T 2  values plotted as a function of manufactured sample. The significant principal components of the PCA model are used to develop the T 2 -chart and the remaining PCs contribute to the Q-chart. The Q residual is the squared prediction error and describes how well the PCA model fits each sample. It is a measure of the amount of variation in each sample not captured by K principal components retained in the model: 
     
       
           Q   i   =e   i   e   i   T   =x   i ( I−P   k   P   k   T ) X   i   T ,  (4) 
       
     
     where e i  is the ith row of E, x i  is the ith sample in X, P k  is the matrix of the k loading vectors retained in the PCA model (where each vector is a column of P k ), and I is the identity matrix of appropriate size (n×n). The Q residual chart monitors the deviation from the PCA model for each sample. The sum of normalized squared scores, known as Hotelling&#39;s T 2  statistic, is a measure of the variation within the PCA model and allows for the identification of statistically anomalous samples. T 2  is defined as: 
     
       
           T   i   2   =t   i λ −1   t   i   T   =x   i   Pλ   −1   P   T   x   i   T ,  (5) 
       
     
     where t i  is the ith row of T k , the matrix of k scores vectors from the PCA model, and λ −1  is the diagonal matrix containing the inverse of the eigenvalues associated with the k eigenvectors (PCs) retained in the model. The T 2 -chart monitors the multivariate distance of a new sample from a target value in the reduced PCA space. 
     Referring to FIGS. 10 and 11, the Q and T 2  control charts may be statistical indicators in the multivariate statistical process control of the production of melt polycarbonate resin  42  and  44 . These charts illustrate samples that exceed some predetermined confidence limit, for example the 95% confidence limit, described by the PCA model. The contribution plots of Q residuals and Hotelling&#39;s T 2  statistic may provide an indication of which process variables X i  cause problems in a given sample. A large Q residual may occur, for example, due to data collection errors or process disturbances. 
     Referring to FIG. 12, a multivariate statistical process analysis (MSPA) system  42  for the consistent production of melt polycarbonate includes a plurality of sensors  44  for collecting manufacturing process data and a computer  46  for determining the relative importance of various process variables X i  and developing an inference model to predict such outcomes as Fries concentration, pellet intrinsic viscosity (IV), and melt polycarbonate grade. The computer  46  may also be used to detect and identify process variables X i  causing abnormal performance on-line. The system  42  of the present invention thus allows a manufacturing plant staff to monitor process performance, better understand it, and make sound business decisions. 
     Structurally, the computer  46  typically includes inputs/outputs, a memory, and a processor for receiving, sending, storing, and processing signals and data to operate, monitor, record, and otherwise functionally control the operation of the system  42 . The computer  46  may include software, hardware, firmware, and other similar components for functionally controlling the operation of the system  42 . The computer  46  may be a single device, or it may be a plurality of devices working in concert. The computer  46  is preferably in communication with all of the other components of the system  42 . The input/output devices may include, for example, a keyboard and a mouse for entering data and instructions into the computer  46 . A video display allows the user or process operator to view what the computer  46  has accomplished. Other output devices may include, for example, a printer, a plotter, a synthesizer, and speakers. The memory typically includes a random-access memory (RAM) and a read-only memory (ROM). The memory may also include other types of memory, such as programmable read-only memory (PROM), erasable programmable read-only memory (EPROM), and electrically erasable programmable read-only memory (EEPROM). The memory also preferably includes an operating system that executes on the processor. The operating system performs basic tasks which include recognizing input from input devices, sending output to output devices, keeping track of files and directories, and controlling various peripheral devices. The memory may also contain one or more databases. The processor accepts data and instructions from the memory and performs various calculations. The processor may include an arithmetic logic unit (ALU), which performs arithmetic and logical operations, and a control unit, which extracts instructions from the memory. Optionally, the computer  46  may also include a modem or other network connection, a mass storage device, and any other suitable peripheral. The above-described computer  46  may take the form of a hand-held digital computer, a personal computer, a workstation, a mainframe computer, and a supercomputer. 
     The computer&#39;s memory preferably contains a number of programs or algorithms for functionally controlling the operation of the system  42 , including a preprocessor  48 , for preprocessing collected process variable X i  and product variable Y i  data, and an identifier  50 , for identifying the process variables X i  of importance in, for example, the melt polycarbonate manufacturing process. The preprocessor  48  may, for example, include an algorithm for scaling each of the measurements related to the process variables X i  and product variables Y i . The identifier  50  may, for example, include an algorithm for performing principal components analysis (PCA). The computer  46  may also contain a correlator  52 , for performing a correlation analysis between. process variables X and process variables X i  and product variables Y i , and a model generator  54 , or generating a “virtual analyzer,” used to understand and predict the performance of the manufacturing process. The model generator  54  may, for example, include an algorithm for performing partial least-squares (PLS) regression. Further, the computer  46  may contain an analyzer  56  for detecting faults and diagnosing problems with monitored process variables X i  that may lead to inferior-polycarbonate product. This may be accomplished through the use of multivariate control charts. Process and product data may be analyzed using multivariate techniques included in, for example, a chemometrics software package, such as PLSToolbox (Version 2.0, Eigenvector Research, Inc., Manson Wash.). This software package may operate, for example, with MATLAB (Version 5.3, The Mathworks, Inc., Natick, Mass.). 
     The present invention has been described with reference to examples and preferred embodiments. Other examples and embodiments may achieve the same results. Variations in and modifications to the present invention will be apparent to those skilled in the art and the following claims are intended to cover all such equivalents.