Principal Component Linear Discriminant Analysis

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... 

The previously mentioned data set with a total of 115 compounds has already been studied by other statistical methods such as Principal Component Analysis (PCA), Linear Discriminant Analysis, and the Partial Least Squares (PLS) method [39]. Thus, the choice and selection of descriptors has already been accomplished. 

In the method of linear discriminant analysis, one therefore seeks a linear function of the variables, D, which maximizes the ratio between both variances. Geometrically, this means that we look for a line through the cloud of points, such that the projections of the points of the two groups are separated as much as possible. The approach is comparable to principal components, where one seeks a line that explains best the variation in the data (see Chapter 17). The principal component line and the discriminant function often more or less coincide (as is the case in Fig. 33.8a) but this is not necessarily so, as shown in Fig. 33.8b. 

While principal components models are used mostly in an unsupervised or exploratory mode, models based on canonical variates are often applied in a supervisory way for the prediction of biological activities from chemical, physicochemical or other biological parameters. In this section we discuss briefly the methods of linear discriminant analysis (LDA) and canonical correlation analysis (CCA). Although there has been an early awareness of these methods in QSAR [7,50], they have not been widely accepted. More recently they have been superseded by the successful introduction of partial least squares analysis (PLS) in QSAR. Nevertheless, the early pattern recognition techniques have prepared the minds for the introduction of modem chemometric approaches. 

Rezzi, S., Axelson, D. E., Heberger, K., Reniero, F., Mariani, C., and Guillou, C. (2005). Classification of olive oils using high throughput flow 1H NMR fingerprinting with principal component analysis, linear discriminant analysis and probabilistic neural networks. Anal. Chim. Acta 552,13-24. 

They employed principal components analysis (PCA) and linear discriminant analysis (LDA) to distinguish the two types of polyps. The spectra (Fig. 2.9) have bands at similar wave numbers and their features are similar, making it difficult for the untrained eye to distinguish between them. The application illustrates the importance of multivariate analysis in clinical applications of Raman spectroscopy. It is often the case that there are only small differences between normal and diseased tissues. 

Initially an optimised model was constructed using the data collected as outlined above by constructing a principal component (PC)-fed linear discriminant analysis (LDA) model (described elsewhere) [7, 89], The linear discriminant function was calculated for maximal group separation and each individual spectral measurement was projected onto the model (using leave-one-out cross-validation) to obtain a score. The scores for each individual spectrum projected onto the model and colour coded for consensus pathology are shown in Fig. 13.3. The simulation experiments used this optimised model as a baseline to compare performance of models with spectral perturbations applied to them. The optimised model training performance achieved 93% accuracy overall for the three groups. 

The data processing of the multivariate output data generated by the gas sensor array signals represents another essential part of the electronic nose concept. The statistical techniques used are based on commercial or specially designed software using pattern recognition routines like principal component analysis (PCA), cluster analysis (CA), partial least squares (PLSs) and linear discriminant analysis (LDA). 

Figure 7.5 Principal components analysis applied to volatile compounds selected by stepwise linear discriminant analysis (source SEXIA Group-Instituto de la Grasa, Seville, Spain).

Chemometrics is a branch of science and technology dealing with the extraction of useful information from multidimensional measurement data using statistics and mathematics. It is applied in numerous scientific disciplines, including the analysis of food [313-315]. The most common techniques applied to multidimensional analysis include principal components analysis (PCA), factor analysis (FA), linear discriminant analysis (LDA), canonical discriminant function analysis (DA), cluster analysis (CA) and artificial neurone networks (ANN). 

The authors also combined metabolomics results with results obtained from AFP determinations. The model was created using linear discriminant analysis. Principal component analysis was also carried out. Thanks to the created model, it was possible to detect metabolites of potential diagnostic value. Moreover, analysis of metabolomic profiles decreased the number of patients that were incorrectly classified with the use of AFP marker [21]. 

PCA, principal component analysis NN, neural net LOA, linear discriminant analysis LOO, leavc-one-out HLSVD, filtering signals in the time domain ANN, artificial neural networks. 

LDA, linear discriminant analysis NN, neural nets P, principal component (PC-based) CCD, computerized consensus diagnosis N, normalized U, unnormalized R, subregion-based F, fuzzy. 

Linear discriminant analysis (LDA) is aimed at finding a linear combination of descriptors that best separate two or more classes of objects [100]. The resulting transformation (combination) may be used as a classifier to separate the classes. LDA is closely related to principal component analysis and partial least square discriminant analysis (PLS-DA) in that all three methods are aimed at identifying linear combinations of variables that best explain the data under investigation. However, LDA and PLS-DA, on one hand, explicitly attempt to model the difference between the classes of data whereas PCA, on the other hand, tries to extract common information for the problem at hand. The difference between LDA and PLS-DA is that LDA is a linear regression-like method whereas PLS-DA is a projection technique... 

To establish a correlation between the concentrations of different kinds of nucleosides in a complex metabolic system and normal or abnormal states of human bodies, computer-aided pattern recognition methods are required (15, 16). Different kinds of pattern recognition methods based on multivariate data analysis such as principal component analysis (PCA) (8), partial least squares (16), stepwise discriminant analysis, and canonical discriminant analysis (10, 11) have been reported. Linear discriminant analysis (17, 18) and cluster analysis were also investigated (19,20). Artificial neural network (ANN) is a branch of chemometrics that resolves regression or classification problems. The applications of ANN in separation science and chemistry have been reported widely (21-23). For pattern recognition analysis in clinical study, ANN was also proven to be a promising method (8). 

A rapid head-space analysis instrument for the analysis of the volatile fractions of 105 extra virgin olive oils coming from five different Mediterranean areas was put forward by Cerrato-Oliveros and his co-workers. The rough information collected by this system was unraveled and interpreted with well-known multivariate techniques of display (principal component analysis), feature selection (stepwise linear discriminant analysis), and classification (linear discriminant analysis). 93.4% of the samples were correctly classified and 90.5% correctly predicted by the cross-validation procedure, whilst 80.0% of an external test set, aiming at full validation of the classification rule, were correctly assigned. 

We haveemployed a variety of unsupervised and supervised pattern recognition methods such as principal component analysis, cluster analysis, k-nearest neighbour method, linear discriminant analysis, and logistic regression analysis, to study such reactivity spaces. We have published a more detailed description of these investigations. As a result of this, functions could be developed that use the values of the chemical effects calculated by the methods mentioned in this paper. These functions allow the calculation of the reactivity of each individual bond of a molecule. 

Mendonca et al. have used an electrospray ionization mass spectrometry (ESI-MS) method to identify the CGA profile, which allowed the discrimination of green Arabica and Robusta coffee beans [22]. This method also allowed discrimination between defective and nondefective coffee beans (ESI-MS positive mode). For this kind of identification and discrimination, they used principal component analysis and hierarchical cluster analysis [22]. Alonso-Salces et al. also used a linear discriminant analysis and a partial least-squares discriminant analysis based on HPLC and UV spectra of phenolic (CGAs) and methykanthine contents for a number of green Robusta and Arabica coffee beans from different geographical origins [9]. 

Two studies have suggested that the IR spectra of synovial fluid specimens provide the basis to diagnose arthritis and to differentiate among its variants.A NIR study demonstrated that osteoarthritis, rheumatoid arthritis, and spondyloarthropathy could be distinguished on the basis of the synovial fluid absorption patterns in the range 2000-2400 nm.< In that case, the pool of synovial fluid spectra was subject to principal component analysis, and eight principal component scores for each spectrum were employed as the basis for linear discriminant analysis (LDA). On that basis, the optimal LDA classifier matched 105 of the 109 spectra to the correct clinical designation (see Table 7). 

In linear multivariate data analysis (MDA) of a set of analytical data, the most common model is that linear relationships exist within the data that can be revealed, for example, in the principal components analysis (PCA) or linear discriminant analysis (LDA). Analytical data may represent the analyte concentrations of the multicomponent mixture, a spectrum of any kind, the results of clinical laboratory tests, the status of a technological process at a given time, etc. The classification of the investigated objects by the MDA techniques is an important task, which provides interpretation of complex problems, often... 

Principal components analysis followed by linear discriminant analysis of the NMR spectra from 98 instant spray dried coffees, obtained from 3 different producers, correctly attributed 99% of the samples to their manufacturer. Blind testing of the PCA model with a further 36 samples of instant coffee resulted in a 100% success rate in identifying the samples from the 3 manufacturers. Coffees from one manufacturer were also assigned into 2 groups using these techniques... 

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