Factor analysis interpretation

The above can be illustrated by data taken from several multi-element air pollution biomonitor surveys carried out at IRI The air pollution surveys commonly include a number of soil-associated elements (e.g. Al, Fe, Sc, Cr, Th) and several rare earth elements (Kuik et al., 1993a,b). In the Factor Analysis interpretation of the data on all selected 20 elements, the soil indicator elements serve to extract a soil-factor (De Bruin and Wolterbeek, 1984), based on which, for all individual elements, site-specific soil-associated fractions of the total concentrations can be calculated (Kuik et al., 1993a,b). 

Because of peak overlappings in the first- and second-derivative spectra, conventional spectrophotometry cannot be applied satisfactorily for quantitative analysis, and the interpretation cannot be resolved by the zero-crossing technique. A chemometric approach improves precision and predictability, e.g., by the application of classical least sqnares (CLS), principal component regression (PCR), partial least squares (PLS), and iterative target transformation factor analysis (ITTFA), appropriate interpretations were found from the direct and first- and second-derivative absorption spectra. When five colorant combinations of sixteen mixtures of colorants from commercial food products were evaluated, the results were compared by the application of different chemometric approaches. The ITTFA analysis offered better precision than CLS, PCR, and PLS, and calibrations based on first-derivative data provided some advantages for all four methods. ... 

Factor analysis is a statistical technique that has been used to interpret numerous types of data. Hamer (1989), Rastogi et al. (1990, 1991, 1992), Fotopoulos et al. (1994), and Bonvin and Rippin (1990) have used it successfully for the identification of stoichiometries of complex reactions. The technique is applied to Eqn. (A-1) which are rewritten in matrix form ... 

The aim of factor analysis is to calculate a rotation matrix R which rotates the abstract factors (V) (principal components) into interpretable factors. The various algorithms for factor analysis differ in the criterion to calculate the rotation matrix R. Two classes of rotation methods can be distinguished (i) rotation procedures based on general criteria which are not specific for the domain of the data and (ii) rotation procedures which use specific properties of the factors (e.g. non-negativity). 

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. 

The interpretation of a multivariate image is sometimes problematic because the cause for pictorial structures may be complex and cannot be interpreted on the basis of images of single species even if they are processed by filtering etc. In such cases, principal component analysis (PCA) may advantageously be applied. The principle of the PCA is like that of factor analysis which has been mathematically described in Sect. 8.3.4. It is represented schematically in Fig. 8.33. 

Nowadays, generating huge amounts of data is relatively simple. That means Data Reduction and Interpretation using multivariate statistical tools (chemometrics), such as pattern recognition, factor analysis, and principal components analysis, can be critically important to extracting useful information from the data. These subjects have been introduced in Chapters 5 and 6. 

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. 

So far in this chapter, all our elaborations were completely abstract there has been no attempt at an interpretation or understanding of the results of Factor Analysis in chemical terms. Abstract Factor Analysis is the core of most applications of Factor Analysis within chemistry, but, nevertheless, much more insight can be gained than the results of the rank analysis we have seen so far. How can we relate the factors U and V to something chemically meaningful Very sensibly these factors are called abstract factors, in contrast to real factors such as the matrices C and A containing the concentration profiles and pure component spectra. Is there a useful relationship between U, V, C and A ... 

In contrast to PCA which can be considered as a method for basis rotation, factor analysis is based on a statistical model with certain model assumptions. Like PCA, factor analysis also results in dimension reduction, but while the PCs are just derived by optimizing a statistical criterion (spread, variance), the factors are aimed at having a real meaning and an interpretation. Only a very brief introduction is given here a classical book about factor analysis in chemistry is from Malinowski (2002) many other books on factor analysis are available (Basilevsky 1994 Harman 1976 Johnson and Wichem 2002). 

On the other hand, factor analysis involves other manipulations of the eigen vectors and aims to gain insight into the structure of a multidimensional data set. The use of this technique was first proposed in biological structure-activity relationship (i. e., SAR) and illustrated with an analysis of the activities of 21 di-phenylaminopropanol derivatives in 11 biological tests [116-119, 289]. This method has been more commonly used to determine the intrinsic dimensionality of certain experimentally determined chemical properties which are the number of fundamental factors required to account for the variance. One of the best FA techniques is the Q-mode, which is based on grouping a multivariate data set based on the data structure defined by the similarity between samples [1, 313-316]. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or covariance) relationships explored with R-mode factor analysis. The measure of similarity used is the cosine theta matrix, i. e., the matrix whose elements are the cosine of the angles between all sample pairs [1,313-316]. 

W. Windig, P. G. Kistenmaker, and J. Haverkamp, Chemical interpretation of differences in pyrolysis—Mass spectra of simulated mixtures of biopolymers by factor analysis with graphical rotation, J. Anal. Appl. Pyrolysis 3(3), 199-212 (1981/1982)... 

The second step in factor analysis is interpretation of the principal components or factors. This is accomplished by examining the contribution that each of the original measured variables makes to the linear combination describing the factor axis. These contributions are called the factor loadings. When several variables have large loadings on a factor they may be identified as being associated. From this association one may infer chemical or physical interactions that may then be interpreted in a mechanistic sense. 

The advent of analytical techniques capable of providing data on a large number of analytes in a given specimen had necessitated that better techniques be employed in the assessment of data quality and for data interpretation. In 1983 and 1984, several volumes were published on the application of pattern recognition, cluster analysis, and factor analysis to analytical chemistry. These treatises provided the theoretical basis by which to analyze these environmentally related data. The coupling of multivariate approaches to environmental problems was yet to be accomplished. 

Although the software used was not a full-featured factor analysis program, portions of the printed output are useful in studying the spectral data set. Table VI shows some information obtainable from PCR models (large data set) with 5, 10 and 13 factors. In this case, the "factors" are principal components derived entirely from the sample data set. PLS factors are not interpretable in the same manner. 

Among these techniques are various forms of a statistical method called factor analysis. Several forms of factor analysis have been applied to the problem of aerosol source resolution. These different forms provide several different frameworks in which to examine aerosol composition data and Interpret it in terms of source contributions. 

Prior Applications. The first application of this traditional factor analysis method was an attempt by Blifford and Meeker (6) to interpret the elemental composition data obtained by the National Air Sampling Network(NASN) during 1957-61 in 30 U.S. cities. They employed a principal components analysis and Varimax rotation as well as a non-orthogonal rotation. In both cases, they were not able to extract much interpretable information from the data. Since there is a very wide variety of sources of particles in 30 cities and only 13 elements measured, it is not surprising that they were unable to provide much specificity to their factors. One interesting factor that they did identify was a copper factor. They were unable to provide a convincing interpretation. It is likely that this factor represents the copper contamination from the brushes of the high volume air samples that was subsequently found to be a common problem ( 2). 

Hopke, et al. (4) and Gaarenstroom, Perone, and Moyers (7) used the common factor analysis approach in their analyses of the Boston and Tucson area aerosol composition, respectively. In the Boston data, for 90 samples at a variety of sites, six common factors were identified that were interpreted as soil, sea salt, oil-fired power plants, motor vehicles, refuse incineration and an unknown manganese-selenium source. The six factors accounted for about 78 of the system variance. There was also a high unique factor for bromine that was interpreted to be fresh automobile exhaust. Large unique factors for antimony and selenium were found. These factors may possibly represent emission of volatile species whose concentrations do not oovary with other elements emitted by the same source. 

Sievering and coworkers ( ) have made extensive use of factor analysis in their interpretation of midlake aerosol composition and deposition data for Lake Michigan. 

The second difference is that the correlations between samples are calculated rather than the correlations between elements. In the terminology of Rozett and Peterson ( ), the correlation between elements would be an R analysis while the correlation between samples would be a Q analysis. Thus, the applications of factor analysis discussed above are R analyses. Imbrle and Van Andel ( 6) and Miesch (J 7) have found Q-mode analysis more useful for interpreting geological data. Rozett and Peterson (J ) compared the two methods for mass spectrometric data and concluded that the Q-mode analysis provided more significant informtlon. Thus, a Q-mode analysis on the correlation about the origin matrix for correlations between samples has been made (18,19) for aerosol composition data from Boston and St. Louis. 

It is clear that several forms of factor analysis can be very useful in the interpretation of aerosol composition data. The traditional forms of factor analysis that are widely available permit the identification of sources, the screening of data for noisy results, and the identification of interferences or analytical procedure problems. 

There are two general types of aerosol source apportionment methods dispersion models and receptor models. Receptor models are divided into microscopic methods and chemical methods. Chemical mass balance, principal component factor analysis, target transformation factor analysis, etc. are all based on the same mathematical model and simply represent different approaches to solution of the fundamental receptor model equation. All require conservation of mass, as well as source composition information for qualitative analysis and a mass balance for a quantitative analysis. Each interpretive approach to the receptor model yields unique information useful in establishing the credibility of a study s final results. Source apportionment sutdies using the receptor model should include interpretation of the chemical data set by both multivariate methods. 

What factor analysis allows initially is a determination of the number of components required to reproduce the adsorbance or data matrix A. Factor analysis allows us to find the rank of the matrix A and the rank of A can be interpreted as being equal to the number of absorbing components. To find the rank of A, the matrix ATA is... 

The complexity and volume of. available diffraction data requires that other than manual tediniques be used to match unknown to known spectra. Available computer programs have indeed simplified the problem of identifying an unknown substance (Refs 9,15,16,21 22). The work of Abel and Kemmey (Ref 16) in this area is worthy of note. Data taken from this report is presented as Table 4. The authors use values of 26 (<90°) to identify phase location instead of values of d in A. Major computer programs of this type endeavor to identify the crystal structure of an. unknown and cite a general factor of certainty to support the credibility of the analysis interpretation... 

Factor analysis was applied, and it integrated data for both DCM and MeOH extracts at the same time. By using the same interpretation... 

By using the factor analysis method, the mutagenicity data were applied only on water after disinfection. Then, comparisons were made between treatment lines (Figure 4). By using the same interpretation (i.e., assigning weights of 3, 2, or 1) as that used for the ozone and GAC treatment, the conclusions based on ordered classifications of treatment lines for each month and for the year are the following Treatment line 1 is the best line of the pilot plant and yields a 92 relative ideal complete treatment line, treatment line 2 is 66 , and treatment line 4 is 41 (Table VII). 

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