Factor Analysis FA

The term Factor Analysis, FA, has a very wide range of interpretations there is no general agreement of its exact meaning. From an abstract 

There is still a long list of different interpretations for the expression Factor Analysis. All the meanings of the term can be explained on the basis of the Singular Value Decomposition. 

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The goal of factor analysis (FA) and their essential variant principal component analysis (PCA) is to describe the structure of a data set by means of new uncorrelated variables, so-called common factors or principal components. These factors characterize frequently underlying real effects which can be interpreted in a meaningful way. 

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... 

The mathematical techniques are part of multivariate statistics. They are closely related and often exchangeable. Two main approaches can be distinguished Least Squares Optimization (LSO), and Factor Analysis (FA). 

A new method of interpreting Auger electron spectroscopy (AES) sputter profiles of transition metal carbides and nitrides is proposed. It is shown that the chemical information hidden in the shape of the peaks, and usually neglected in depth profiles, can be successfully extracted by factor analysis (FA). The various carbide and nitride phases of model samples were separated by application of FA to the spectra recorded during AES depth profiles. The different chemical states of carbon, nitrogen and metal were clearly identified. 

Factorial methods - factor analysis (FA) - principal components analysis ( PCA) - partial least squares modeling (PLS) - canonical correlation analysis Finding factors (causal complexes)... 

Principal components analysis (PCA) and factor analysis (FA) are aimed at finding and interpreting hidden complex, and possibly causally determined, relationships between features in a data set. Correlating features are converted to the so-called factors which are themselves noncorrelated. 

The aim of the application of factor analysis (FA) to environmental problems is to characterize the complex changes which occur to all the features observed in partial systems of the natural environment. These common factors explain the complex state of the environment more comprehensively and causally and so enable extraction of the essential part of the information contained in a set of data. 

The purpose of application of factor analysis (FA) is the characterization of complex changes of all observed features in partial systems of the environment by determination of summarized factors which are more comprehensive and causally explicable. The method extracts the essential information from a data set. The exclusive consideration of common factors in the reduced factor analytical solution seems to be particularly promising for the analytical process. The specific variances of the observed features will be separated from the reduced factor analytical results by means of the estimation of the communalities. They do not falsify the influence of the main pollution sources (see also Tab. 7-2). The mathematical fundamentals of FA are explained in detail in Section 5.4.3 (see also [MALINOWSKI, 1991 WEBER, 1986]). 

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

Trace element compositions of airborne particles are important for determining sources and behavior of regional aerosol, as emissions from major sources are characterized by their elemental composition patterns. We have investigated airborne trace elements in a complex regional environment through application of receptor models. A subset (200) of fine fraction samples collected by Shaw and Paur (1,2) in the Ohio River Valley (ORV) and analyzed by x-ray fluorescence (XRF) were re-analyzed by instrumental neutron activation analysis (INAA). The combined data set, XRF plus INAA, was subjected to receptor-model interpretations, including chemical mass balances (CMBs) and factor analysis (FA). Back trajectories of air masses were calculated for each sampling period and used with XRF data to select samples to be analyzed by INAA. 

In this study we have employed the simultaneous collection of atmospheric particles and gases followed by multielement analysis as an approach for the determination of source-receptor relationships. A number of particulate tracer elements have previously been linked to sources (e.g., V to identify oil-fired power plant emissions, Na for marine aerosols, and Pb for motor vehicle contribution). Receptor methods commonly used to assess the interregional impact of such emissions include chemical mass balances (CMBs) and factor analysis (FA), the latter often including wind trajectories. With CMBs, source-strengths are determined (1) from the relative concentrations of marker elements measured at emission sources. When enough sample analyses are available, correlation calculations from FA and knowledge of source-emission compositions may identify groups of species from a common source type and identify potential marker elements. The source composition patterns are not necessary as the elemental concentrations in each sample are normalized to the mean value of the element. Recently a hybrid receptor model was proposed by Lewis and Stevens (2) in which the dispersion, deposition, and conversion characteristics of sulfur species in power-plant emissions... 

Comparison and ranking of sites according to chemical composition or toxicity is done by multivariate nonparametric or parametric statistical methods however, only descriptive methods, such as multidimensional scaling (MDS), principal component analysis (PCA), and factor analysis (FA), show similarities and distances between different sites. Toxicity can be evaluated by testing the environmental sample (as an undefined complex mixture) against a reference sample and analyzing by inference statistics, for example, t-test or analysis of variance (ANOVA). 

Chemometrics stands in this context for analysis of multivariate chemical data by means of statistical methods such as principal component analysis (PCA) or factor analysis (FA) cf. Section 3.5. 

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). 

Principle components analysis (PCA), a form of factor analysis (FA), is one of the most common unsupervised methods used in the analysis of NMR data. Also known as Eigenanalysis or principal factor analysis (PEA), this method involves the transformation of data matrix D into an orthogonal basis set which describes the variance within the data set. The data matrix D can be described as the product of a scores matrix T, and a loading matrix P,... 

Principal component analysis is used to reduce the information in many variables into a set of weighted linear combinations of those variables it does not differentiate between common and unique variance. If latent variables have to be determined, which contribute to the common variance in a set of measured variables, factor analysis (FA) is a valuable statistical method, since it attempts to exclude unique variance from the analysis. 

Large data tables may hide information which is not easily detected by simple inspection of the various columns. Principal component analysis and some closely related techniques such as factor analysis (FA), correspondence factor analysis (CFA) and non-linear mapping (NLM), reduce a data matrix to new supervariables retaining a maximum of information or variance from the original data matrix. These new variables are called latent variables or principal components, and are orthogonal vectors composed of linear combinations of the original variables. This concept is shown schematically in Fig. 22.15. 

Factor analysis (FA) is very similar to PCA (for reviews see, for example, [1,12, 13]). The only, but essential, difference is that in FA only part of the data variance is considered to be common to all variables. The remaining part is attributed to unique properties of one variable at a time. With this in mind equation (8) may be written to give the model of FA as... 

PCA has been often employed to explore the relationships among variables in a data set (19 20). Nevertheless, it is generally accepted that Factor Analysis (FA) is better suited than PCA to study these relationships (1 7). This is because FA algorithms are designed to distinguish between shared and unique variability. The shared variability, the so-called communalities in the FA community, reflect the common factors-common variability-among observable variables. The unique variability is only present in one observable variable. The common factors make up the relationship structure in the data. PCA makes no distinction between shared and unique variability and therefore it is not suited to find the structure in the data. 

Transformation of the original data to a new coordinate system is another possibility of data pretreatment. The methods are based on principal component analysis (PCA) or factor analysis (FA). The first step for these transformations is the formation of a data matrix that is derived from the original data matrix and that reflects the relationships among the data. Such derived data matrices are the variance-covariance matrix and the correlation matrix. 

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