Discriminant analysis multivariate models

A total of 185 emission lines for both major and trace elements were attributed from each LIBS broadband spectrum. Then background-corrected, summed, and normalized intensities were calculated for 18 selected emission lines and 153 emission line ratios were generated. Finally, the summed intensities and ratios were used as input variables to multivariate statistical chemometric models. A total of 3100 spectra were used to generate Partial Least Squares Discriminant Analysis (PLS-DA) models and test sets. 

Gombar, V.K. and K. Enslein. 1991. A structure-biodegradability relationship model by discriminant analysis. In J. Devillers and W. Karcher, Eds., Applied Multivariate Analysis in SAR and Environmental Studies, pp. 377-414. Kluwer Academic Publ., Dordrecht, Holland. 

Supervised learning methods - multivariate analysis of variance and discriminant analysis (MVDA) - k nearest neighbors (kNN) - linear learning machine (LLM) - BAYES classification - soft independent modeling of class analogy (SIMCA) - UNEQ classification Quantitative demarcation of a priori classes, relationships between class properties and variables... 

Multivariate Analysis of Variance and Discriminant Analysis, and PLS Modeling... 

The principle of multivariate analysis of variance and discriminant analysis (MVDA) consists in testing the differences between a priori classes (MANOVA) and their maximum separation by modeling (MDA). The variance between the classes will be maximized and the variance within the classes will be minimized by simultaneous consideration of all observed features. The classification of new objects into the a priori classes, i.e. the reclassification of the learning data set of the objects, takes place according to the values of discriminant functions. These discriminant functions are linear combinations of the optimum set of the original features for class separation. The mathematical fundamentals of the MVDA are explained in Section 5.6. 

There are many other statistical models which can be used for the evaluation of DICE studies. Inclusion of not only a group factor, but also a time factor in the experiment methods of the analysis of variance (ANOVA) can be applied to find expression changes within the temporal course of the protein expression or to find interactions between the group and time factor. Several multivariate statistical methods are of use, too. Spots with similar expression profiles can be grouped by cluster analysis or, on the other hand, new spots can be assigned to existing groups by the methods of discriminant analysis. 

Gombar VK, Enslein K. Structure-biodegradability relationship model by discriminant analysis. In Devillers J, Karcher W, editors, Applied multivariate analysis in SAR and environmental studies. Dordrecht Kluwer Academic, 1991. 

It is commonly the case that a wide variety of properties can be included in a QSAR analysis and a decision must be made on whether to include all possibilities or limit the number of descriptors. This decision depends on the size of the data set and the correlation matrix between the properties. Farge sets of property data contain a lot of redundancy of information. For example, molecular weight, surface area and molar refraction are always highly correlated, therefore a decision to nse only molecular weight could be made. Some multivariate statistical analysis methods are tolerant of data sets which contain more properties than compounds, for example, PFS, while others are not, for example, linear discriminant analysis (FDA). Ideally, a set of uncorrelated properties is desirable as this is most likely to give a robust, interpretable model. 

PCA is a least square method and therefore its results depend on data scaling. The initial variance of a column variable partly determines its importance in the model. In order to avoid the problem of over- or under-representation of variables, column variables are scaled to unit variance before analysis. The column average is then subtracted from each variable, which, from a statistical point of view, corresponds to moving the multivariate system to the center of the data, which becomes the starting point of the mathematical analysis. The same auto-scaUng and centering procedures are applied in PLS discriminant analysis. 

According to Ennis (1988), the application of the various multivariate analysis techniques (factor, cluster, discriminant analysis, multidimensional scaling) to classification in sensory analysis has been very valuable but is of little help for understanding the modes of perception. Mathematical models are proposed for predicting human sensory responses and the author concludes that they need development before they are able to improve the understanding of the complex perceptions associated with foods and beverages . 

The Mahalanobis distance occurs frequently in chemometric analysis, not only in cluster analysis but also in multivariate calibration, in discriminant analysis, and in modelling. It is appropriate to consider it further here, and the following account is based on the tutorial article by De Maesschalck et al Table 4.4 details ten objects A. .. J) described by two, mean-centred, variables (jcj and JC2). These data are illustrated graphically in Figure 4.6, and it is immediately apparent that the two variables exhibit high, positive correlation. In Figure 4.6(b), contours (circles) of equal Euclidean distance from the centroid are displayed and, for example, objects C and F have similar distance values. This is not the case if Mahalanobis distances are calculated and examined. 

Two fundamentally different statistical approaches to biomarker selection are possible. With the first, experimental data can be used to construct multivariate statistical models of increasing complexity and predictive power - well-known examples are Partial Least Square Discriminant Analysis (PLS-DA) (Barker Rayens, 2003 Kemsley, 1996 Szymanska et al., 2011) or Principal Component Linear Discriminant Analysis (PC-LDA) (Smit et al., 2007 Werf et al., 2006). Inspection of the model coefficients then should point to those variables that are important for class discrimination. As an alternative, univariate statistical tests can be... 

Then the next step consists on application of multivariate statistical methods to find key features involving molecules, descriptors and anticancer activity. The methods include principal component analysis (PCA), hiererchical cluster analysis (HCA), K-nearest neighbor method (KNN), soft independent modeling of class analogy method (SIMCA) and stepwise discriminant analysis (SDA). The analyses were performed on a data matrix with dimension 25 lines (molecules) x 1700 columns (descriptors), not shown for convenience. For a further study of the methodology apphed there are standard books available such as (Varmuza FUzmoser, 2009) and (Manly, 2004). 

Supervised methods for recognizing patterns can also be based on multivariate modeling methods, for example, by use of PLS as discussed in Section 6.2.2. The method is termed discriminant analysis-partial least squares (DA-PLS) analysis where the input feature data from the X matrix and the assignment to a class is described in the Y matrix. To avoid a ranking of classes, the containment of classes is not coded in a single classification vector, for example, classes 1-6, but is described by ones or zeros columnwise in the Y matrix. 

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