Multivariate data, principal-components

Correlations are inherent in chemical processes even where it can be assumed that there is no correlation among the data. Principal component analysis (PCA) transforms a set of correlated variables into a new set of uncorrelated ones, known as principal components, and is an effective tool in multivariate data analysis. In the last section we describe a method that combines PCA and the steady-state data reconciliation model to provide sharper, and less confounding, statistical tests for gross errors. 

Fig. 5.11 Multivariate analysis (principal components scores plots of the first three principal components) of the response of the two-size CdSe nanocrystals sensor film (a) PC 1 vs. PC 2 and (B) PC 1 vs. PC 3. Regions numbered 1 and 2 are data points of dynamic response from replicate (n = 3) film exposures to methanol and toluene, respectively. Unlabeled data points result from times when the film was exposed to a blank (dry air). Reprinted with permission from Leach and Potyrailo26 copyright 2006 Materials Research Society...

White (1995), not for a sensory analysis but mainly with a view to determining coffee adulterations, used the data of combined headspace GC and high-performance LC for multivariate analysis. Principal component analysis visualized the relationship between samples, and the outlying samples could be identified. The method could be an additional tool for classification and quality control of coffee products. 

Several multivariate techniques are useful in analyzing the chemical sensor data. Principal component analysis (PCA) is a multivariate technique that reduces the dimensionality of the data sets by building a new set of coordinates, principal components or PCs. These PCs are linear combinations of the original variables and they are orthogonal to each other and therefore uncorrelated. They are also built in such a way that each one successively account for the maximum variability of the data set (12). This is useful for visualizing the data to determine if there is inherent structure. 

More sophisticated statistical treatments of sensory data have been more commonly applied in studies of multiple factors of Upid oxidation on quality of foods, including Multivariate and Principal Component analyses. These procedures attempt to simplify complex relationships of several factors and sets of data into more understandable levels. Multivariate analysis is based on the fact that one measured property generally depends on more than one factor and the classical statistical univariate methods dealing with just one variable at a time are inadequate to analyse complex data. 

A NMR-based metabonomic study of transgenic maize sets an example of discrimination possible using multivariate techniques (principal component analysis and partial-least squares-discriminant analysis) to NMR data on unfractionated metabolites. Other metabonomics studies are reviewed under... 

Figure 9. Multivariate statistics principal component plot of cyclohexane conformatioi derived from a synunetry-expanded torsional data-set.

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). 

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]. 

Usually, the raw data in a matrix are preprocessed before being submitted to multivariate analysis. A common operation is reduction by the mean or centering. Centering is a standard transformation of the data which is applied in principal components analysis (Section 31.3). Subtraction of the column-means from the elements in the corresponding columns of an nxp matrix X produces the matrix of... 

J.M. Deane, Data reduction using principal components analysis. In Multivariate Pattern Recognition in Chemometrics, R. Brereton (Ed.). Chapter 5, Elsevier, Amsterdam, 1992, pp. 125-165. 

In order to apply RBL or GRAFA successfully some attention has to be paid to the quality of the data. Like any other multivariate technique, the results obtained by RBL and GRAFA are affected by non-linearity of the data and heteroscedast-icity of the noise. By both phenomena the rank of the data matrix is higher than the number of species present in the sample. This has been demonstrated on the PCA results obtained for an anthracene standard solution eluted and detected by three different brands of diode array detectors [37]. In all three cases significant second eigenvalues were obtained and structure is seen in the second principal component. 

The application of principal components regression (PCR) to multivariate calibration introduces a new element, viz. data compression through the construction of a small set of new orthogonal components or factors. Henceforth, we will mainly use the term factor rather than component in order to avoid confusion with the chemical components of a mixture. The factors play an intermediary role as regressors in the calibration process. In PCR the factors are obtained as the principal components (PCs) from a principal component analysis (PC A) of the predictor data, i.e. the calibration spectra S (nxp). In Chapters 17 and 31 we saw that any data matrix can be decomposed ( factored ) into a product of (object) score vectors T(nxr) and (variable) loadings P(pxr). The number of columns in T and P is equal to the rank r of the matrix S, usually the smaller of n or p. It is customary and advisable to do this factoring on the data after columncentering. This allows one to write the mean-centered spectra Sq as ... 

Principal components analysis can also be used in the case when the compounds are characterized by multiple activities instead of a single one, as required by the Hansch or Free-Wilson models. This leads to the multivariate bioassay analysis which has been developed by Mager [9]. By way of illustration we consider the physicochemical and biological data reported by Schmutz [41] on six oxazepines... 

Beilken et al. [ 12] have applied a number of instrumental measuring methods to assess the mechanical strength of 12 different meat patties. In all, 20 different physical/chemical properties were measured. The products were tasted twice by 12 panellists divided over 4 sessions in which 6 products were evaluated for 9 textural attributes (rubberiness, chewiness, juiciness, etc.). Beilken etal. [12] subjected the two sets of data, viz. the instrumental data and the sensory data, to separate principal component analyses. The relation between the two data sets, mechanical measurements versus sensory attributes, was studied by their intercorrelations. Although useful information can be derived from such bivariate indicators, a truly multivariate regression analysis may give a simpler overall picture of the relation. 

Sets of spectroscopic data (IR, MS, NMR, UV-Vis) or other data are often subjected to one of the multivariate methods discussed in this book. One of the issues in this type of calculations is the reduction of the number variables by selecting a set of variables to be included in the data analysis. The opinion is gaining support that a selection of variables prior to the data analysis improves the results. For instance, variables which are little or not correlated to the property to be modeled are disregarded. Another approach is to compress all variables in a few features, e.g. by a principal components analysis (see Section 31.1). This is called... 

Techniques for multivariate input analysis reduce the data dimensionality by projecting the variables on a linear or nonlinear hypersurface and then describe the input data with a smaller number of attributes of the hypersurface. Among the most popular methods based on linear projection is principal component analysis (PCA). Those based on nonlinear projection are nonlinear PCA (NLPCA) and clustering methods. 

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. 

Multivariate analytical images may be processed additionally by chemo-metrical procedures, e.g., by exploratory data analysis, regression, classifica-tion> and principal component analysis (Geladi et al. [1992b]). 

The results show that DE-MS alone provides evidence of the presence of the most abundant components in samples. On account of the relatively greater difficulty in the interpretation of DE-MS mass spectra, the use of multivariate analysis by principal component analysis (PCA) of DE-MS mass spectral data was used to rapidly differentiate triterpene resinous materials and to compare reference samples with archaeological ones. This method classifies the spectra and indicates the level of similarity of the samples. The output is a two- or three-dimensional scatter plot in which the geometric distances among the various points, representing the samples, reflect the differences in the distribution of ion peaks in the mass spectra, which in turn point to differences in chemical composition of... 

The multivariate statistical data analysis, using principal component analysis (PCA), of this historical data revealed three main contamination profiles. A first contamination profile was identified as mostly loaded with PAHs. A samples group which includes sampling sites R1 (Ebro river in Miranda de Ebro, La Rioja), T3 (Zadorra river in Villodas, Alava) and T9 (Arga river in Puente la Reina, Navarra), all located in the upper Ebro river basin and close to Pamplona and Vitoria cities,... 

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