Multivariate methods

Multivariate chemometric methods have claimed considerable attention in the last few decades because of their inherent capacity for resolving multicomponent, complex systems. Applications of multivariate methods in different electrochemical techniques have been recently reported by several authors [192-194], 

There have been attempts to deal with the issue of nonlinearity in data sets. Detrended principal components (DPC) use a polynomial expression to remove the nonlinear relationships from the PCA axes. DPC are useful for data sets of moderate nonlinearity. Detrended correspondence analysis uses a more complex algorithm to eliminate the nonlinearity but requires a more complex computation. Nonmetric multidimensional scaling (NMDS) is a robust method that deals with nonlinearities by using ranks. 

A technique derived from a principal components approach is the coupling of PCA with redundancy analysis (RDA) (van der Brink et al. 1996). The utility of the technique is that it provides a depiction of the treatment trajectories in an ecological space, and the statistical significance can be examined using a permutation test. One of the proposed benefits of the technique is that it can determine recovery, a dubious distinction in light of the ground work laid in Chapter 2. In common with other PCA techniques, the technique does assume a linear response. 

Developed for the analysis of ecological data (Matthews and Hearne, 1991), nonmetric clustering and association analysis (NCAA) is a multivariate derivative of artificial intelligence research. NCAA has a fundamentally different approach to discovering patterns in data sets. 

For the purposes of this chapter, it is convenient to consider the complete set of measured quantities (e.g. the different m/z ratios of PyMS ion counts) for a given sample as being a single n-dimensional position vector, = (xi X2. .. with Xi being the measured value of quantity i. A set of m such vectors is represented as a matrix, X, with rows, x]. The value stored at row i, column j of X is denoted Xy. 

A scores matrix is one in which the columns contain the values of transformed variables. An example of a scores matrix is that containing principal component values. A loadings, or weightings, matrix is one by which a data matrix may be multiplied to obtain a scores matrix. 

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To obtain this matrix by the multivariate method, we first generate two absorptivity vectors ap and a2j from a known concentration matrix in parts per million... 

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. 

M.O. Hill, Correspondence analysis a neglected multivariate method. Appl. Statist., 23 (1974) 340-355. 

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. 

Multivariate chemometric techniques have subsequently broadened the arsenal of tools that can be applied in QSAR. These include, among others. Multivariate ANOVA [9], Simplex optimization (Section 26.2.2), cluster analysis (Chapter 30) and various factor analytic methods such as principal components analysis (Chapter 31), discriminant analysis (Section 33.2.2) and canonical correlation analysis (Section 35.3). An advantage of multivariate methods is that they can be applied in... 

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... 

Input mapping methods can be divided into univariate, multivariate, and probabalistic methods. Univariate methods analyze the inputs by extracting the relationship between the measurements. These methods include various types of single-scale and multiscale filtering such as exponential smoothing, wavelet thresholding, and median filtering. Multivariate methods analyze... 

As introduced earlier, inputs can be transformed to reduce their dimensionality and extract more meaningful features by a variety of methods. These methods perform a numeric-numeric transformation of the measured input variables. Interpretation of the transformed inputs requires determination of their mapping to the symbolic outputs. The inputs can be transformed with or without taking the behavior of the outputs into account by univariate and multivariate methods. The transformed features or latent variables extracted by input or input-output analysis methods are given by Eq. (5) and can be used as input to the interpretation step. 

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. 

Wu et al. [46] used the approach of an artificial neural network and applied it to drug release from osmotic pump tablets based on several coating parameters. Gabrielsson et al. [47] applied several different multivariate methods for both screening and optimization applied to the general topic of tablet formulation they included principal component analysis and... 

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]). 

J. J., Prediction of drug absorption using multivariate methods, J. Med. Chem. 2000, 43, 3867-3877. 

Galego and Arroyo [14] described a simultaneous spectrophotometric determination of OTC, hydrocortisone, and nystatin in the pharmaceutical preparations by using ratio spectrum-zero crossing derivate method. The calculation was performed by using multivariate methods such as partial least squares (PLS)-l, PLS-2, and principal component regression (PCR). This method can be used to resolve accurately overlapped absorption spectra of those mixtures. 

Kleinbaum, D.G. and Kupper, L.L., Applied Regression Analysis and Other Multivariable Methods (Duxbury Press, Boston, 1978), pp. 508-520. 

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 . 

Damasio J, Navarro-Ortega A, Tauler R et al (2010) Identifying major pesticides affecting bivalve species exposed to agricultural pollution using multi-biomarker and multivariate methods. Ecotoxicology 19 1084-1094... 

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). 

Grunsky, E.C. Smee, B.W. 1999. The differentiation of soil types and mineralization from multi-element geochemistry using multivariate methods and digital topography. Journal of Geochemical Exploration, 67, 287-299. 

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. 

The traditional acute, subchronic, and chronic toxicity studies performed in rodents and other species also can be considered to constitute multiple endpoint 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 (Young, 1985). 

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. 

A simple strategy for variable selection is based on the information of other multivariate methods like PCA (Chapter 3) or PLS regression (Section 4.7). These methods form new latent variables by using linear combinations of the regressor... 

Eigenvectors of square matrices are frequently used in multivariate methods. Each eigenvector is connected with an eigenvalue, A, and in most applications the eigenvectors are ordered by decreasing A and are normalized to length 1 (see Figure A.2.7). 

Andersson M. Ringberg A. Gustafsson C. Multivariate methods in tablet formulation suitable for early drug development Predictive models from a screening design of several linked responses. Chemometrics and Intelligent Laboratory Systems, 2007, 87 (1), 151-156. 

In a review paper devoted to the chemometrics discussed in previous citations, Shaw et al.28 go into details of the many applications of advanced chemometrics to bioprocess monitoring. Multivariate methods and numerous algorithms are explored in this work. 

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... 

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