Display of multivariate data

Livingstone et al. have employed a particular neural net architecture called a reversible nonlinear dimensionality reduction (ReNDeR) net for a low-dimensional display of multivariate data sets (160). The method makes use of the activity values of the hidden neurons in a trained three-layer feedforward network to produce the low-dimensional display. It was claimed that, in contrast to con-... 

Livingstone, D. J., Hesketh, G., and Clayworth, D. (1991) Novel method for display of multivariate data using neural networks. J. Mol. Graphics 9,115-118. 

Self-organizing maps in conjunction with principal component analysis constitute a powerful approach for display and classification of multivariate data. However, this does not mean that feature selection should not be used to strengthen the classification of the data. Deletion of irrelevant features can improve the reliability of the classification because noisy variables increase the chances of false classification and decrease classification success rates on new data. Furthermore, feature selection can lead to an understanding of the essential features that play an important role in governing the behavior of the system or process under investigation. It can identify those measurements that are informative and those measurements that are uninformative. However, any approach used for feature selection should take into account the existence of redundancies in the data and be multivariate in nature to ensure identification of all relevant features. 

A similar method for the display of multivariate chemical data was proposed by Lin and Chen C1693 and Drack C73, 74D three pairs of reference points in the d-dimensional space are employed and new coordinates are calculated so as to give the same ratio of distances to a pair of reference points. 

Application of multivariate statistics to fatty acid data from the Tyrolean Iceman and other mummies is a mosaic stone in the investigation of this mid-European ancestor, which is still a matter of research (Marota and Rollo 2002 Murphy et al. 2003 Nerlich et al. 2003). The iceman is on public display in the South Tyrol Museum of Archaeology in Bolzano, Italy, stored at —6°C and 98% humidity, the conditions as they probably were during the last thousands of years. 

There are broadly two uses of chemometrics that interest the process chemist. The first of these is simply data display. It is a truism that the human eye is the best analytical tool, and by displaying multivariate data in a way that can be easily assimilated by eye a number of diagnostic assessments can be made of the state of health of a process, or of reasons for its failure [ 153], a process known as MSPC [154—156]. The key concept in MSPC is the acknowledgement that variability in process quality can arise not just by variation in single process parameters such as temperature, but by subtle combinations of process parameters. This source of product variability would be missed by simple control charts for the individual process parameters. This is also the concept behind the use of experimental design during process development in order to identify such variability in the minimum number of experiments. 

One of the keys to multivariate analysis is the ability to reduce the dimensionality of the data so that it can be displayed in two, three or four (time-dependent) dimensional displays. The primary tool for achieving this, principal component analysis (PCA) [158], is the cornerstone of chemometrics as it accomplishes several things ... 

Principal component analysis and Kohonen self-organizing maps allow multivariate data to be displayed as a graph for direct viewing, thereby extending the ability of human pattern recognition to uncover obscure relationships in complex data sets. This enables the scientist or engineer to play an even more interactive role in the data analysis. Clearly, these two techniques can be very useful when an investigator believes that distinct class differences exist in a collection of samples but is not sure about the nature of the classes. 

However, what if we had more than one variable to consider In other words, we have multivariate data. For example, what if we want to identify trends in the properties of a range of organic molecules The variables we might want to consider could be melting point, boiling point, M, solubility in a solvent and vapour pressure. We can, of course, tabulate the data, as before, but this does not allow us to consider any trends in the data. To do this we need to be able to plot the data. However, once we exceed three variables (which we need to be able to plot in three dimensions) it becomes impossible to produce a straightforward plot. It is in this context that chemometrics offers a solution, reducing the dimensionality to a smaller number of dimensions and hence the ability to display multivariate data. The most important technique in this context is called principal component analysis (PCA). 

Displaying multivariate data in low-dimensional space can be useful for visual clustering of items. For example, plotting the scores of the first few pairs of principal components as biplots of the first versus the second or the third principal components can cluster normal process operation and operation under various faults. Examples of biplots and their interpretation for fault diagnosis are presented in Chapter 7. 

In this chapter the fundamentals of chemometrics will be presented by means of a quick overview of the most relevant techniques for data display, classification, modeling, and calibration. The goal of the chapter is to make people aware of the great superiority of multivariate analysis over the commonly used univariate approach. Mathematical and algorithmical details will not be presented, since the chapter is mainly focused on the general problems to which chemometrics can be successfully applied in the field of environmental chemistry. 

Statistical analyses were performed with ADE-4 [25], a powerful statistical software program designed specifically for the analysis of environmental data. ADE-4 includes the main linear multivariate analyses and numerous graphical tools for optimal data display. 

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