Multivariate statistical techniques principal components analysis

Data generated from process analysis techniques are commonly displayed on control charts and the term statistical process control (SPC) is often used to describe the use of such data visualisation [5], As the amount of data available increases, due both to different measurements and greater frequency of measurements, then combinations of different data provide improved methods of monitoring the processes concerned. The procedures and concepts of multivariate SPC incorporating principal component analysis (PCA) and partial least squares (PLS) analysis then become important [6]. The different SPC approaches are all aimed at providing better process control and improved process understanding. 

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]. 

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. 

Among the multivariate statistical techniques that have been used as source-receptor models, factor analysis is the most widely employed. The basic objective of factor analysis is to allow the variation within a set of data to determine the number of independent causalities, i.e. sources of particles. It also permits the combination of the measured variables into new axes for the system that can be related to specific particle sources. The principles of factor analysis are reviewed and the principal components method is illustrated by the reanalysis of aerosol composition results from Charleston, West Virginia. An alternative approach to factor analysis. Target Transformation Factor Analysis, is introduced and its application to a subset of particle composition data from the Regional Air Pollution Study (RAPS) of St. Louis, Missouri is presented. 

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). 

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. 

There are several books on pattern recognition and multivariate analysis. An introduction to several of the main techniques is provided in an edited book [19]. For more statistical in-depth descriptions of principal components analysis, books by Joliffe [20] and Mardia and co-authors [21] should be read. An early but still valuable book by Massart and Kaufmann covers more than just its title theme cluster analysis [22] and provides clear introductory material. 

Differentiation between the stereoisomeric dexamethasone and betamethasone (Figure 13.2) has been investigated in detail [47-49]. While De Wasch et al. [47] differentiated from differences in relative abundances, Antignac et al. [48] proposed the use of multivariate statistical techniques, based on principal component analysis. Baseline separation between betamethasone, dexamethasone, and various related esters was reported by Arthur et al. [49] using a Cg column and a step gradient. 

Multivariate statistical techniques are commonly employed in near-IR quantitative and qualitative analysis because these approaches have been proven useful for extracting desired information from near-IR spectra, which often contain up to 1200 wavelengths of observation per spectrum. Principal component analysis/principal component regression (PCA/ PCR) is one such multivariate approach. Descriptions of this... 

In both studies, nonmetric clustering outperformed the metric tests, although both principal components analysis and correspondence analysis yielded some additional insight into large-scaled patterns, which was not provided by the nonmetric clustering results. However, nonmetric clustering provided information without the use of inappropriate assumptions, data transformations, or other dataset manipulations that usually accompany the use of multivariate metric statistics. The success of these studies and techniques led to the examination of community dynamics in a series of two multispecies toxicity tests. 

Principal Components Analysis (PCA) is a multivariable statistical technique that can extract the strong correlations of a data set through a set of empirical orthogonal functions. Its historic origins may be traced back to the works of Beltrami in Italy (1873) and Jordan in Prance (1874) who independently formulated the singular value decomposition (SVD) of a square matrix. However, the first practical application of PCA may be attributed to Pearson s work in biology [226] following which it became a standard multivariate statistical technique [3, 121, 126, 128]. 

Principal components analysis (PGA), a multivariate statistical technique, was used for data reduction and pattern recognition [8]. PGA represents the variation present in many variables using a small number of principal components (PCs). PCA functions by finding a new set of axes, which more efficiently describe the variance in the data. Samples are no longer described by their intensities in... 

Throughout this chapter, reference will be made to techniques and approaches described elsewhere in this book, and a certain familiarity with these topics will be assumed Methods of representing molecular conformation, and different coordinate systems (Chapter 1), ways of dealing with symmetry aspects (Chapter 2), data retrieval from the Cambridge Structural Database (CSD Chapter 3) [3], and multivariate statistical techniques such as principal component analysis (PCA) and cluster analysis (CA Chapter 4). 

Derivative spectrophotometry is also proving quite useful for the simultaneous determination of two or more components in mixtures. With mixtures, several methods have been proposed for quantitative analysis. The peak-lo-peak height has been used ns has the peak height at the zero crossing wavelengths for the individual components. More recently, multivariate statistical techniques, such as partial least squares and principal components analysis, have been used to determine concentrations. Derivative methods have been used to determine trace metals in mixtures. For example, trace amounts of Mn and Z.fi can be determined in mixtures by forming complexes with. S,S-dihydroxy-l.4-naphthoquinone. > Derivative methods have also been widely applied to pharntaceulical preparations and to vitamin mixtures. ... 

Water is the most important chemical constituent of fruits and vegetables and water highly absorbs NIR radiation, so the NIR spectrum of such materials is dominated by water. Further, the NIR spectrum is essentially composed of a large set of overtones and combination bands. This, in combination with the complex chemical composition of a typical fruit or vegetable causes the NIR spectrum to be highly convoluted. Multivariate statistical techniques are required to extract the information about quality attributes which is buried in the NIR spectrum Developments in multivariate statistical techniques such as partial least squares (PLS) regression and principal component analysis (PCA) are then applied to extract the required information from such convoluted spectra (Cozzolino et al., 2006b McClure, 2003 Naes et al., 2004 Nicolai et al., 2007 ). 

The chemo(bio)diversity analysis of maize landraces and propolis produced in southern regions of Brazil was successfully assessed by using a typical metabolomic platform involving spectroscopic techniques (FTIR, iR- and 13C-NMR, and UV-visible) and chemometrics. The huge amount of data afforded by those spectroscopic techniques was analyzed using multivariate statistical methods such as principal component analysis and cluster analysis allowing obtaining extra information on the metabolic profile of the complex matrices in study. 

Principal components analysis is a well-established multivariate statistical technique that can be used to identify correlations within large data sets and to reduce the number of dimensions required to display the variation within the data. A new set of axes, principal components (PCs), are constructed, each of which accounts for the maximum variation not accounted for by previous principal components. Thus, a plot of the first two PCs displays the best two-dimensional representation of the total variance within the data. With pyrolysis mass spectra, principal components analysis is used essentially as a data reduction technique prior to performing canonical variates analysis, although information obtained from principal components plots can be used to identify atypical samples or outliers within the data and as a test for reproducibihty. 

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