Process data visualisation with biplots
Introduction
In modern industrial environments, process data provide the basis for the monitoring of product quality, control and optimization. As a result, the collection and storage of large volumes of data have become a routine operation on many plants. In order to exploit the data to get a better understanding of the behaviour of the process, it is important to identify the salient features underlying the data. This reduction in the dimensionality of the problem enables the engineer to summarize the information captured in a large number of variables by a smaller number of latent variables. Principal component analysis is commonly used for this purpose, as are other techniques, such as principal curves, Sammon maps and multidimensional scaling.
In the last few decades, several multivariate statistical methods have been developed to identify correlations between variables and to exploit process information for monitoring and control purposes (Wise et al., 1990, Kourti and MacGregor, 1995, Martin et al., 1996, Martin et al., 1999). The approach outlined in Fig. 1(a) shows the importance of taking correlation between variables into account. The two out-of-control observations can only be detected by exploiting the redundancy between variables, that is by construction of a control ellipse, where the correlation between the two variables is reflected by the aspect ratio of the principal axes of the control ellipse. In the contribution plot in Fig. 1(b) the two variables (X1 and X2) which are associated with the deviations in the observations are identified.
Principal component analysis (PCA) and related techniques are commonly used to construct charts like the one shown in Fig. 1, as discussed by Kresta et al. (1991), Kourti and MacGregor (1995), Nomikos and MacGregor (1995), Kosanovich et al. (1996), Amirthalingam and Lee (1997), Jaeckle and MacGregor (1998), Jia et al. (1998), Seasholtz (1999) and Zhang et al., 1996, Zhang et al., 1997, Zhang et al., 1999 among many others. PCA effectively reduces the observation space, which facilitates process failure and disturbance diagnosis through the capture of correlations between variables and characterizing normal operating conditions (NOC).
In addition to PCA maps and contribution plots, the use of multivariate T2-control charts (e.g. Montgomery (2001); Johnson and Wichern (2002)) is well established for diagnosis of industrial process data. Johnson and Wichern (2002) pointed out that ordinary multivariate T2-control charts can be constructed from principal components extending the usefulness of these graphs in practice. All the above techniques take the relationships among multivariate data into account for producing graphs to display process behaviour relative to NOC. It is clear from Fig. 1 that ignoring relationships among process variables can be highly misleading and thus PCA-based methods are routinely used for monitoring many industrial processes.
However, a major problem remains: Providing information of the individual variables influencing the industrial process together with the multivariate map or chart. Fig. 1(b) shows a contribution plot attempting to provide management with information on the variables responsible for the out-of-control observations in Fig. 1(a). Since the information of the variables is provided in a graph separate from the two-dimensional display of the observations, it is impossible to judge exactly how the out-of-control observations are influenced by the two variables implicated. This paper proposes the employment of biplot methodology for addressing this problem. It will be shown that modern biplot methodology extends PCA maps and T2-charts by providing a single graph for displaying multidimensional observations together with information on all variables concerned. However, the graphical display of the variation in the data is in practice not always enough. What is often needed is to optimally display separation and overlap among different classes or groups in the data. Of major importance to the process engineer in these circumstances is if a single graph can be found that provides information on the role of all the variables in the overlap and separation among the different classes. It will be illustrated that canonical variate analysis (CVA) biplots are ideally suited for this purpose. Moreover, the biplot methodology introduced here allows for non-linear relationships and provides an infrastructure for implementing many novel ideas in monitoring industrial plant processes.
In the next section a brief description of biplot methodology is given. This is followed by two detailed case studies. In the first case study PCA biplots are used to monitor the ore feed to a phosphate flotation plant and to identify optimal plant operating conditions. These different plant operating regimes are than optimally represented in CVA biplots. The second case study illustrates the use of CVA biplots in the hydrolysis of zinc chloride in ammonium chloride solutions. These case studies demonstrate that biplot methodology provides sophisticated, yet easy to understand tools of discovery and not merely ways of displaying a set of multidimensional data.
Section snippets
Biplot methodology
Gabriel (1971) introduced the biplot as a graphical display consisting of a vector for each row and a vector for each column of a matrix of rank two. An element of the matrix is represented by the inner product of the vectors corresponding to its row and column. Since any matrix of rank k > 2 can be approximated by a matrix of rank two, a biplot can be constructed for all matrices by considering its rank two approximation. Such low rank approximations are in practice often not only adequate but
Monitoring of the ore feed to the plant
The recoveries achieved in phosphate flotation (and other) plants depend in a complex way on among other the ore particle sizes, activated state of the milled ore, collector dosages, pulp pH values, etc. For example, Klimpel (1987) has demonstrated that a gradual increase in the dosage of collector increases the recovery and flotation rate only up to a certain level in the flotation of apatite. With increased collector dosage there is a tendency to float other minerals and reduce selectivity,
Case study 2: hydrolysis of zinc chloride in ammonium chloride solutions
Ammonium chloride based hydrometallurgical processes have generated considerable interest in recent years, since aqueous solutions of high concentrations of ammonium chloride are especially appropriate for the treatment of complex raw materials of both oxide and sulphide types (Figueiredo et al., 1993, Limpo et al., 1992). A detailed knowledge of the solubility of metal chlorides as a function of temperature and composition of solution is necessary for process development and in this case study
Discussion and conclusions
Biplots resulting from the modern perspective of Gower and Hand (1996) were introduced in this paper. In particular, the potential of these PCA and CVA biplots to monitor multivariate process systems were demonstrated in two case studies, which focussed on the description of multidimensional variation and the separation of different classes of observations.
In the first case study, the use of a PCA biplot to identify different operating regimes in a phosphate flotation plant was demonstrated.
Acknowledgement
The authors would like to thank the NRF of South Africa for financial supporting this work.
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