Multivariate data

The traditional acute, subchronic, and chronic toxicity studies performed in rodents and other species also can be considered to constitute multiple end point screens. Although the numerically measured continuous variables (body weight, food consumption, hematology values) generally can be statistically evaluated individually by traditional means, the same concerns of loss of information present in the interrelationship of such variables apply. Generally, traditional multivariate methods are not available, efficient, sensitive, or practical [25]. 

Consideration of the results from a simple multi-element analysis will serve to illustrate terms and parameters associated with the techniques used. This example will also introduce some features of matrix operators basic to handling multivariate data. In the scientific literature, matrix representation of multivariate statistics is common. For those readers unfamiliar with the basic matrix operations, or those who wish to refresh their memory, the Appendix provides a summary and overview of elementary and common iriatrix operations. 

The data shown in Table 1.7 are a portion of a multi-element analysis of mineral water samples. The data from such an analysis can conveniently be arranged in an by w array, where n is the number of objects, or samples, and m is the number of variables measured. This array is referred to as the data matrix and the purpose of using matrix notation is to allow us to handle arrays of data as single entities rather than having to specify each element in the array every time we perform an operation on the data set. Our data matrix can be denoted by the single symbol X and each element by Xy, with the subscripts i and j indicating the number of the row and column respectively. A matrix with only one row is termed a row vector, e,g., r, and with only one column, a column vector, e.g., c. 

Each measure of an analysed variable, or variate, may be considered independent. By summing elements of each column vector the mean and 

Aflfi and V. Qark, Computer Aided Multivariate Analysis , Lifetime Learning, California, USA, 1984. 

Flury and H. Riedwyl, Multivariate Statistics, A Practical Approach , Chapman and Hall, London, UK, 1988. 

Matrices for Statistics , Oxford University Press, Oxford, UK, 1986. 

Variables (mg kg ) Samples Sodium Potassium Calcium Magnesium 

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For example, the objects may be chemical compounds. The individual components of a data vector are called features and may, for example, be molecular descriptors (see Chapter 8) specifying the chemical structure of an object. For statistical data analysis, these objects and features are represented by a matrix X which has a row for each object and a column for each feature. In addition, each object win have one or more properties that are to be investigated, e.g., a biological activity of the structure or a class membership. This property or properties are merged into a matrix Y Thus, the data matrix X contains the independent variables whereas the matrix Ycontains the dependent ones. Figure 9-3 shows a typical multivariate data matrix. 

Figure 9-3. Multivariate data matriK X, containing n objects each represented by m features. The matrix Y contains the properties of the objects that are to be investigated.

Sections 9A.2-9A.6 introduce different multivariate data analysis methods, including Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares regression (PLS). 

A detailed description of multivariate data analysis in chemistry is given in Chapter IX, Section 1.2 of the Handbook. 

Because of the usually multidimensional character of chemical information, we will only describe information visualization techniques here that have been developed to handle multivariate data. These techniques may be classified into ... 

K. Varmuza, Multivariate data analysis in chemistry, in Handbook of Chemo-informatics - From Data To Knowledge, J. Gasteiger (Ed.), Weinheim, WOey-VCH, 2003. 

Other methods consist of algorithms based on multivariate classification techniques or neural networks they are constructed for automatic recognition of structural properties from spectral data, or for simulation of spectra from structural properties [83]. Multivariate data analysis for spectrum interpretation is based on the characterization of spectra by a set of spectral features. A spectrum can be considered as a point in a multidimensional space with the coordinates defined by spectral features. Exploratory data analysis and cluster analysis are used to investigate the multidimensional space and to evaluate rules to distinguish structure classes. 

Multivariate data analysis usually starts with generating a set of spectra and the corresponding chemical structures as a result of a spectrum similarity search in a spectrum database. The peak data are transformed into a set of spectral features and the chemical structures are encoded into molecular descriptors [80]. A spectral feature is a property that can be automatically computed from a mass spectrum. Typical spectral features are the peak intensity at a particular mass/charge value, or logarithmic intensity ratios. The goal of transformation of peak data into spectral features is to obtain descriptors of spectral properties that are more suitable than the original peak list data. 

Spectral features and their corresponding molecular descriptors are then applied to mathematical techniques of multivariate data analysis, such as principal component analysis (PCA) for exploratory data analysis or multivariate classification for the development of spectral classifiers [84-87]. Principal component analysis results in a scatter plot that exhibits spectra-structure relationships by clustering similarities in spectral and/or structural features [88, 89]. 

Evidence of the appHcation of computers and expert systems to instmmental data interpretation is found in the new discipline of chemometrics (qv) where the relationship between data and information sought is explored as a problem of mathematics and statistics (7—10). One of the most useful insights provided by chemometrics is the realization that a cluster of measurements of quantities only remotely related to the actual information sought can be used in combination to determine the information desired by inference. Thus, for example, a combination of viscosity, boiling point, and specific gravity data can be used to a characterize the chemical composition of a mixture of solvents (11). The complexity of such a procedure is accommodated by performing a multivariate data analysis. 

P.. Lewi, Multivariate Data Analysis in Industrial Practice, Research Studies Press,John Wiley Sons, Inc., Chichester, UK, 1982. 

However, it is not obvious that when we work with multivariate data, our training set must span the concentration ranges of interest in a multivariate (as opposed to univariate) way. It is not sufficient to create a series of samples where each component is varied individually while all other components are held constant. Our training set must contain data on samples where all of the various components (remember to understand "components" in the broadest sense) vary simultaneously and independently. More about this shortly. 

Figure 2 is a multivariate plot of some multivariate data. We have plotted the component concentrations of several samples. Each sample contains a different combination of concentrations of 3 components. For each sample, the concentration of the first component is plotted along the x-axis, the concentration of the second component is plotted along the y-axis, and the concentration of the third component is plotted along the z-axis. The concentration of each component will vary from some minimum value to some maximum value. In this example, we have arbitrarily used zero as the minimum value for each component concentration and unity for the maximum value. In the real world, each component could have a different minimum value and a different maximum value than all of the other components. Also, the minimum value need not be zero and the maximum value need not be unity. 

Figure 3. The wrong way to span a multivariate data space.

We have already touched on the need to visualize our multivariate data in a multivariate way. We will now consider this topic in more detail. Earlier, we mentioned that Figure 1 contains the most important concepts in this entire book. The next series of figures are the second most important. Once you understand the concepts in the next few figures, you will be well on your way to mastering the factor-based techniques. 

BI- AND MULTIVARIATE DATA Table 2.1. Linear Regression Equations... 

Winiwarter, S., Bonham, N. M., Ax, F., Hallberg, A., Lennemas, H., Karlen, A. Correlation of human jejunal permeability (in vivo) of drugs with experimentally and theoretically derived parameters. A multivariate data analysis approach. J. Med. Chem. 1998, 41, 4939-4949. 

Analytical results are often represented in a data table, e.g., a table of the fatty acid compositions of a set of olive oils. Such a table is called a two-way multivariate data table. Because some olive oils may originate from the same region and others from a different one, the complete table has to be studied as a whole instead as a collection of individual samples, i.e., the results of each sample are interpreted in the context of the results obtained for the other samples. For example, one may ask for natural groupings of the samples in clusters with a common property, namely a similar fatty acid composition. This is the objective of cluster analysis (Chapter 30), which is one of the techniques of unsupervised pattern recognition. The results of the clustering do not depend on the way the results have been arranged in the table, i.e., the order of the objects (rows) or the order of the fatty acids (columns). In fact, the order of the variables or objects has no particular meaning. 

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