Data analysis multivariate

These techniques reduce a large number of indoor VOCs to a few factors that can account for most of the cumulative variance in the VOC data [54,81 ]. A factor loading matrix, which shows the correlation between the factors and the variables is often obtained. Edwards et al. [81] used this method to reduce 23 indoor VOCs in environmental tobacco smoke (ETS) free microenvironments to six factors and to apportion the most likely sources of the VOCs. A summary of the VOC classes loaded on each factor and the probable sources is presented in Table 5. It is, however, noteworthy that UNMIX and positive matrix factorisation, both of which are based on factor analysis and have been applied frequently to ambient air quality data [82], have not featured prominently in indoor VOC source apportionment reports. 

---

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. 

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. 

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. 

This relationship is of importance in multivariate data analysis as it relates distance between endpoints of two vectors to distances and angular distance from the origin of space. A geometrical interpretation is shown in Fig. 29.2. 

The eigenvectors extracted from the cross-product matrices or the singular vectors derived from the data matrix play an important role in multivariate data analysis. They account for a maximum of the variance in the data and they can be likened to the principal axes (of inertia) through the patterns of points that represent the rows and columns of the data matrix [10]. These have been called latent variables [9], i.e. variables that are hidden in the data and whose linear combinations account for the manifest variables that have been observed in order to construct the data matrix. The meaning of latent variables is explained in detail in Chapters 31 and 32 on the analysis of measurement tables and contingency tables. 

Scaling is a very important operation in multivariate data analysis and we will treat the issues of scaling and normalisation in much more detail in Chapter 31. It should be noted that scaling has no impact (except when the log transform is used) on the correlation coefficient and that the Mahalanobis distance is also scale-invariant because the C matrix contains covariance (related to correlation) and variances (related to standard deviation). 

A question that often arises in multivariate data analysis is how many meaningful eigenvectors should be retained, especially when the objective is to reduce the dimensionality of the data. It is assumed that, initially, eigenvectors contribute only structural information, which is also referred to as systematic information. 

O.M. Kvalheim, K. 0ygard and O. Grahl-Nielsen, SIMCA multivariate data analysis of blue mussel components in environmental pollution studies. Anal. Chim. Acta, 150(1983) 145-152. 

P.J. Lewi, Multivariate data analysis in structure-activity relationships. In Drug Design (E.J. Ariens, Ed.), Vol. X. Academic Press, New York, 1980. 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

Multivariate data analysis and experimental design, 25 (1988) 291 Muscarinic Receptors, 43 (2005) 105... 

Current (3) Intensity MDA (2) Molecular dynamics Multivariate data analysis... 

In multivariate data analysis frequently the covariance matrix S is used... 

In general, the evaluation of interlaboratory studies can be carried out in various ways (Danzer et al. [1991]). Apart from z-scores, multivariate data analysis (nonlinear mapping, principal component analysis) and information theory (see Sect. 9.2) have been applied. 

Evaluation of data and validation multivariate data analysis (MULTI-VAR, Wienke et al. [1991]), evaluation of interlaboratory studies (INTERLAB, Wienke et al. [1991]), ruggedness expert system (RES, van Leeuwen et al. [1991]). 

Alternatively, NIR spectroscopy has been applied to relate NIR data to mechanical properties [4], A multivariate data analysis was performed on a series of commercial ethene copolymers with 1-butene and 1-octene. For the density correlation, a coefficient of determination better than 99% was obtained, whereas this was 97.7% for the flexural modulus, and only 85% for the tensile strength. 

---