Principal component analysis multivariate technique

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]. 

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... 

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. 

However, there is a mathematical method for selecting those variables that best distinguish between formulations—those variables that change most drastically from one formulation to another and that should be the criteria on which one selects constraints. A multivariate statistical technique called principal component analysis (PCA) can effectively be used to answer these questions. PCA utilizes a variance-covariance matrix for the responses involved to determine their interrelationships. It has been applied successfully to this same tablet system by Bohidar et al. [18]. 

Principal Component Analysis (PCA) is the most popular technique of multivariate analysis used in environmental chemistry and toxicology [313-316]. Both PCA and factor analysis (FA) aim to reduce the dimensionality of a set of data but the approaches to do so are different for the two techniques. Each provides a different insight into the data structure, with PCA concentrating on explaining the diagonal elements of the covariance matrix, while FA the off-diagonal elements [313, 316-319]. Theoretically, PCA corresponds to a mathematical decomposition of the descriptor matrix,X, into means (xk), scores (fia), loadings (pak), and residuals (eik), which can be expressed as... 

In the past few years, PLS, a multiblock, multivariate regression model solved by partial least squares found its application in various fields of chemistry (1-7). This method can be viewed as an extension and generalization of other commonly used multivariate statistical techniques, like regression solved by least squares and principal component analysis. PLS has several advantages over the ordinary least squares solution therefore, it becomes more and more popular in solving regression models in chemical problems. 

In some cases, many different spectra (or chromatograms) of the same object are available. For inhomogenous objects, for example, several samples of different constitution can be taken. This allows to apply multivariate data processing techniques. When the signals of the compounds in the sample are specific and linearly additive, the number of compounds which contribute to the signal, can be determined by a Principal Components Analysis (PCA) " (see Sect. 3.2.1). Without knowing the identity of all compounds, which are present and without knowing their spectra, a calibration by partical least squares (PLC) allows to quantify the compounds of interest. 

PLS is related to principal components analysis (PCA) (20), This is a method used to project the matrix of the X-block, with the aim of obtaining a general survey of the distribution of the objects in the molecular space. PCA is recommended as an initial step to other multivariate analyses techniques, to help identify outliers and delineate classes. The data are randomly divided into a training set and a test set. Once the principal components model has been calculated on the training set, the test set may be applied to check the validity of the model. PCA differs most obviously from PLS in that it is optimized with respect to the variance of the descriptors. 

Raman spectroscopy can also directly benefit TE analysis by non-invasively monitoring the growth and development of ECM by different cells on a multitude of scaffold materials exposed to various stimuli (e.g. growth factors, mechanical forces and/or oxygen pressures). Indeed the non-invasive nature of Raman spectroscopy enables the determination of the rate of ECM formation and the biochemical constituents of the ECM formed. Univariate (peak area, peak ratios, etc.) and multivariate analytical techniques (e.g. principal component analysis (PCA)) can be used to determine if there are any significant differences between the ECM formed on various scaffolds and/or cultured with different environmental parameters, and what these biochemical differences are. Least square (LS) modelling, for example, could allow the quantification of the relative components of the ECM formed (Fig. 18.3) [4, 38],... 

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). 

However, since failures may involve a large number of parameters, often not independent from each other, the univariate techniques may be not so efficient therefore, they have been replaced by multivariate techniques, which are powerful tools able to compress data and reduce the problem dimensionality while retaining the essential information. In detail, Principal Component Analysis (PCA) [12, 47] is a standard multivariate technique, whose main goal is to transform a number of... 

Principal component analysis (PCA), factor analysis (FA) and cluster analysis (CA) are some of the most widely used multivariate analysis techniques applied to... 

Principal component analysis is a popular statistical method that tries to explain the covariance structure of data by means of a small number of components. These components are linear combinations of the original variables, and often allow for an interpretation and a better understanding of the different sources of variation. Because PCA is concerned with data reduction, it is widely used for the analysis of high-dimensional data, which are frequently encountered in chemometrics. PCA is then often the first step of the data analysis, followed by classification, cluster analysis, or other multivariate techniques [44], It is thus important to find those principal components that contain most of the information. 

Principal component analysis and Kohonen self-organizing maps allow multivariate data to be displayed as a graph for direct viewing, thereby extending the ability of human pattern recognition to uncover obscure relationships in complex data sets. This enables the scientist or engineer to play an even more interactive role in the data analysis. Clearly, these two techniques can be very useful when an investigator believes that distinct class differences exist in a collection of samples but is not sure about the nature of the classes. 

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