Multivariate data, principal-components analysis

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. 

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. 

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... 

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. 

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... 

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,... 

Keywords electronic nose principal component analysis pattern recognition chemical sensors sensor arrays olfaction system multivariate data analysis. 

The scope of Principal Component Analysis (PCA) is a consistent portrayal of a data set in a representation space. Mathematically, PCA is a linear transformation that may be described as S=WX. Here X is the original data set, W is the transformation matrix, and S are the data in the representation space. PCA is the simplest and most widely used method of multivariate analysis. Nonetheless, most users are seldom aware of its assumptions and sometimes results are badly interpreted. 

Nowadays, generating huge amounts of data is relatively simple. That means Data Reduction and Interpretation using multivariate statistical tools (chemometrics), such as pattern recognition, factor analysis, and principal components analysis, can be critically important to extracting useful information from the data. These subjects have been introduced in Chapters 5 and 6. 

Principal component analysis (PCA) is aimed at explaining the covariance structure of multivariate data through a reduction of the whole data set to a smaller number of independent variables. We assume that an m-point sample is represented by the nxm matrix X which collects i=l,...,m observations (measurements) xt of a column-vector x with j=, ...,n elements (e.g., the measurements of n=10 oxide weight percents in m = 50 rocks). Let x be the mean vector and Sx the nxn covariance matrix of this sample... 

Chapter 3 starts with the first and probably most important multivariate statistical method, with principal component analysis (PC A). PC A is mainly used for mapping or summarizing the data information. Many ideas presented in this chapter, like the selection of the number of principal components (PCs), or the robustification of PCA, apply in a similar way to other methods. Section 3.8 discusses briefly related methods for summarizing and mapping multivariate data. The interested reader may consult extended literature for a more detailed description of these methods. 

Principal component analysis (PCA) can be considered as the mother of all methods in multivariate data analysis. The aim of PCA is dimension reduction and PCA is the most frequently applied method for computing linear latent variables (components). PCA can be seen as a method to compute a new coordinate system formed by the latent variables, which is orthogonal, and where only the most informative dimensions are used. Latent variables from PCA optimally represent the distances between the objects in the high-dimensional variable space—remember, the distance of objects is considered as an inverse similarity of the objects. PCA considers all variables and accommodates the total data structure it is a method for exploratory data analysis (unsupervised learning) and can be applied to practical any A-matrix no y-data (properties) are considered and therefore not necessary. 

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