Data multivariate methods

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). 

Matrix (3-72) is essentially the same as mahix (3-70), but it is not exactly the same because it was obtained by the multivariate method from a different data set. 

Canonical Correlation Analysis (CCA) is perhaps the oldest truly multivariate method for studying the relation between two measurement tables X and Y [5]. It generalizes the concept of squared multiple correlation or coefficient of determination, R. In Chapter 10 on multiple linear regression we found that is a measure for the linear association between a univeiriate y and a multivariate X. This R tells how much of the variance of y is explained by X = y y/yV = IlylP/llylP. Now, we extend this notion to a set of response variables collected in the multivariate data set Y. 

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... 

Among nonlocal methods, those based on linear projection are the most widely used for data interpretation. Owing to their limited modeling ability, linear univariate and multivariate methods are used mainly to extract the most relevant features and reduce data dimensionality. Nonlinear methods often are used to directly map the numerical inputs to the symbolic outputs, but require careful attention to avoid arbitrary extrapolation because of their global nature. 

Neural networks are helpful tools for chemists, with a high classification and interpretation capacity. ANNs can improve and supplement data arrangements obtained by common multivariate methods of data analysis as shown by an example of classification of wine (Li-Xian Sun et al. [1997]). 

A definition of Chemometrics is a little trickier of come by. The term was originally coined by Kowalski, but nowadays many Chemometricians use the definition by Massart [4], On the other hand, one compilation presents nine different definitions for Chemometrics [5, 6] (including What Chemometricians do , a definition that apparently was suggested only HALF humorously ). But our goal here is not to get into the argument over the definition of the term, so for our current purposes, it is convenient to consider a perhaps somewhat simplified definition of Chemometrics as meaning multivariate methods of data analysis applied to data of chemical interest . 

The multivariate method MCR-ALS has been used to analyse data in order to identify the main sources of organic pollution affecting the Ebro River delta. Subsequently, an interpolation procedure has been also applied to obtain distribution maps from the punctual resolved data (corresponding to the score values obtained from MCR-ALS). 

Pollard, A.M. (1982). A critical study of multivariate methods as applied to provenance data. In Proceedings of the 22nd Symposium on Archaeometry, University of Bradford, 30th March-3rd April 1982, ed. Aspinall, A. and Warren, S.E., University of Bradford Press, Bradford, pp. 56-66. 

Pollard, A.M. (1986). Multivariate methods of data analysis. In Greek and Cypriot Pottery A Review of Scientific Studies, ed. Jones, R.E., British School at Athens Fitch Laboratory Occasional Paper 1, Athens, pp. 56-83. 

Multivariate IQC. Multivariate methods in IQC are still the subject of research and cannot be regarded as sufficiently established for inclusion in the guidelines. The current document regards multianalyte data as requiring a series of univariate IQC tests. Caution is necessary in the interpretation of this type of data to avoid inappropriately frequent rejection of data. 

In this example, we apply D-PLS (PLS discriminant analysis, see Section 5.2.2) for the recognition of a chemical substructure from low-resolution mass spectral data. This type of classification problems stood at the beginning of the use of multivariate data analysis methods in chemistry (see Section 1.3). 

Basic understanding and efficient use of multivariate data analysis methods require some familiarity with matrix notation. The user of such methods, however, needs only elementary experience it is for instance not necessary to know computational details about matrix inversion or eigenvector calculation but the prerequisites and the meaning of such procedures should be evident. Important is a good understanding of matrix multiplication. A very short summary of basic matrix operations is presented in this section. Introductions to matrix algebra have been published elsewhere (Healy 2000 Manly 2000 Searle 2006). 

This paper presents a method to decide the handling of seemingly Inconsistent data (outliers). The univariate and multivariate methods recommended are strongly based on statistics and the experience of the author In using them. 

Shapiro Wilks W-test for normal data Shapiro Wilks W-test for exponential data Maximum studentlzed residual Median of deviations from sample median Andrew s rho for robust regression Classical methods of multiple comparisons Multivariate methods... 

The potential of modern chemical instrumentation to detect and measure the conposition of coirplex mixtures has made it necessary to consider the use of methods of multivariable data analysis in the overall evaluation of environmental measurements. In a number of instances, the category (chemical class) of the compound that has given rise to a series of signals may be known but the specific entity responsible for a given signal may not be. This is true, for example, for the polychlorinated biphenyls (PCB s) in which the clean-up procedure and use of specific detectors eliminates most possibilities except PCB s. Such hierarchical procedures simplify the problem somewhat but it is still advantageous to apply data reduction methods during the course of the interpretation process. 

There are two general types of aerosol source apportionment methods dispersion models and receptor models. Receptor models are divided into microscopic methods and chemical methods. Chemical mass balance, principal component factor analysis, target transformation factor analysis, etc. are all based on the same mathematical model and simply represent different approaches to solution of the fundamental receptor model equation. All require conservation of mass, as well as source composition information for qualitative analysis and a mass balance for a quantitative analysis. Each interpretive approach to the receptor model yields unique information useful in establishing the credibility of a study s final results. Source apportionment sutdies using the receptor model should include interpretation of the chemical data set by both multivariate methods. 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

Above mentioned examples clearly show that if multivariate data processing methods are applicable, analytical information can be derived with a minimal amount of pre-information and a foreseeing of a maximum of problems. When the sampled object is homogenous, multivariate methods are only applicable when the analytical method itself produces multivariate signals. This is the case when several signals (e.g. spectra) are obtained for the sample as a function of another variable (e.g. time, excitation wavelength). For e mple in GC-MS, a mass spectrum is m sured of the eluents every. 1 a 1 second. In excitation-emission spectroscopy, spectra are measured at several excitation-wavelengths. The potentials of the application of multivariate... 

In the paragraphs below, some of the above mentioned multivariate methods will be discussed in somewhat more detail, with respect to the data processing of signals obtained for hyphenated methods of the type chromatography-spectrometry and spectrometry-spectrometry. 

The term factor is a catch-all for the concept of an identifiable property of a system whose quantity value might have some effect on the response. Factor tends to be used synonymously with the terms variable and parameter, although each of these terms has a special meaning in some branches of science. In factor analysis, a multivariate method that decomposes a data matrix to identify independent variables that can reconstitute the observed data, the term latent variable or latent factor is used to identify factors of the model that are composites of input variables. A latent factor may not exist outside the mathematical model, and it might not therefore influence... 

In Germany, as well as generally in Europe, multivariate methods (PCA and PMF) are the predominantly applied tools (see [1]). Their main advantage is that no information about emission sources is needed, and sources or processes so far not registered in an emission inventory may be detected. On the other hand, the analytical effort to be invested is considerably higher than for the previous methods since these models need enough data to disentangle the sources and source... 

---