Foundations of Chemometrics

All technologies are bnilt npon a fonndation of previously-existing technologies, principles and protocols. There are several of these that were critical to the development of chemometrics, and the ones that are most relevant to PAT applications will be discussed in this section. 

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With the foundations of chemometrics as a background, this section provides some detailed discussion of several specific chemometric tools. A truly complete discussion of all chemometric methods would require at least a book s worth of material. As a result, this discussion will focus on the tools that have been found... 

It was mentioned earlier that PCA is a useful method for compressing the information contained in a large number of x variables into a smaller number of orthogonal principal components that explain most of the variance in the x data. This particular compression method was considered to be one of the foundations of chemometrics, because many commonly used chemometric tools are also focused on explaining variance and dealing with colinearity. However, there are other compression methods that operate quite differently than PCA, and these can be useful as both compression methods and preprocessing methods. 

The main aim of the book is to introduce the reader to the foundations of relevant chemometric methods and likewise to show for selected and typical case studies what can be achieved by using these methods in environmental analysis. It cannot be the scope of such a book, written for nonchemometricians, to give more than an introduction to chemometrics. So, highly sophisticated chemometric methods are not treated. 

Natural scientists of all disciplines, including biologists, geologists and chemists, have caught on to these approaches over the past few decades. Within the chemical community, the first major applications of PCA were reported in die 1970s, and form die foundation of many modem chemometric methods described in this chapter. 

The notation chemometrics was introduced in 1972 by the Swede Svante Wold and the American Bruce R. Kowalski. The foundation of the International Chemometrics Society in 1974 led to the first description of this discipline. In the following years, several conference series were organized, for example. Computer Application in Analytics (COMPANA), Computer-Based Analytical Chemistry (COBAC), and Chemometrics in Analytical Chemistry (CAC). Some journals devoted special sections to papers on chemometrics. Later, novel chemometric journals were started, such as the Journal of Chemometrics (WUey) and Chemometrics and Intelligent Laboratory Systems (Elsevier). 

This chapter and Chapter 5 will attempt to demystify some of the more basic tenets of chemometric spectral analysis methods. The intent is not to completely explain all the complex mathematics behind the techniques (although the equations are provided in almost all cases). Instead, they are designed to give a basic understanding of how and why the mathematics work in order to properly apply them to solving analytical problems. These chapters may also serve as a foundation to learning more about chemometrics and, hopefully, to lead the reader to other sources of information. 

PCA is a method based on the Karhunen-Loeve transformation (KL transformation) of the data points in the feature space. In KL transformation, the data points in the feature space are rotated such that the new coordinates of the sample points become the linear combination of the original coordinates. And the first principal component is chosen to be the direction with largest variation of the distribution of sample points. After the KL transformation and the neglect of the components with minor variation of coordinates of sample points, we can make dimension reduction without significant loss of the information about the distribution of sample points in the feature space. Up to now PCA is probably the most widespread multivariate statistical technique used in chemometrics. Within the chemical community the first major application of PCA was reported in 1970s, and form the foundation of many modem chemometric methods. Conventional approaches are univariate in which only one independent variable is used per sample, but this misses much information for the multivariate problem of SAR, in which many descriptors are available on a number of candidate compounds. PCA is one of several multivariate methods that allow us to explore patterns in multivariate data, answering questions about similarity and classification of samples on the basis of projection based on principal components. 

This excitement about second-order sensor calibration has led to a search by chemometric researchers to find equivalent instrumentation that gives rise to second-order data. The definition of second-order instruments is slowly solidifying and currently a foundation has been established to classify which techniques are true second-order devices. This definition of second-order instruments is simply two sensor arrays which are independent of each other. However, in order for the arrays to be independent, one of the arrays must modulate the sample s analyte concentrations. The best known instrument... 

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