Q-mode factor analysis

Miesch, A. T. Q-Mode Factor Analysis of Geochemical and Petrologic Data Matrices with Constant Row-Sums,... 

Q-mode factor analysis is based on a major product matrix, XX. Whereas the R-mode analyses focus on interrelationships among variables, Q-mode analyses focus on interrelationships among objects. Accordingly, the major product matrix is usually a distance or similarity matrix. Formally, Q-mode and R-mode factor analyses are closely related because the nonzero eigenvalues of the major product matrix are identical to the eigenvalues of the minor product matrix, and the eigenvectors are easily derived from one another (28). 

Aside from principal components, R-mode factor analysis, and Q-mode factor analysis, other techniques that have been used to reduce dimensionality in ungrouped compositional data include multidimensional scaling (34) and correspondence analysis (35). 

Figure 5. Plot of specimens in the example data set plus northern Valley of Guatemala volcanic ash tempers relative to three components defined by Q-mode factor analysis.

We will proceed, therefore, with an eigenvector analysis of the 5x5 covariance matrix obtained from zero-centred object data. TUs is referred to as Q-mode factor analysis and is complementary to the scheme illustrated pre-... 

M.M. Sena, I.S. Scarminio, K.E. Collins, C.H. Collins, Speciation of aqueous chro-mium(VI) solutions with the aid of Q-mode factor analysis followed by oblique projection, Talanta 53 (2000) 453. 

We will proceed, therefore, with an eigenvector analysis of the 5x5 covariance matrix obtained from zero-centred object data. This is referred to as Q-mode factor analysis and is complementary to the scheme illustrated previously with principal components analysis. In the earlier examples the dispersion matrix was formed between the measured variables, and the technique is sometimes referred to as R-mode analysis. For the current MS data, processing by R-mode analysis would involve the data being scaled along each mjz row (as displayed in Table 3.8) and information about relative peak sizes in any single spectrum would be destroyed. In Q-mode analysis, any scaling is performed within a spectrum and the mass fragmentation pattern for each sample is preserved. 

The various factor analysis methods which became widely available during the 1970s are ideally suited to the examination of conservative mixing (Klovan Imbrie, 1971). They can be used to simultaneously classify sites and identify independently varying compositional components. Dean et al. (1988 1993) apply Q-mode factor analysis to total elemental analyses of lake surface sediments, with a view to regional classification and sediment source characterization. The approach uses a varimax rotation. This is a numerical procedure that rigidly rotates the selected axes to maximize or minimize the variable scores on each axis. This helps in the interpretation of the factors as real end-members. 

The factor analysis methods described above can be used to explore mixing, and to identify components. However, they do not yield estimates of the end-member compositions. Miesch (1981) proposed an alternative method, starting with Q-mode factor analysis, which aims to identify and quantify sensible end-member compositions based on the structure of the data. The concept is developed as follows ... 

In a Q-mode factor analysis of data that have not been standardized, the first axis will have positive values for all elements, but the remaining axes will contain some negative elements. 

Q-mode factor analysis A factor analysis method based on eigen analysis of the cross-product matrix. 

Factor analysis has recently been used in source partitioning modeling of molecular marker investigations [1-4,296-300]. Q-mode factor analysis is based on grouping a multivariate data set based on the data structure defined by the similarity between samples. It is devoted exclusively to the interpretation of the inter-object relationships in a data set, rather than to the inter-variable (or co-variance) relationships e q)lored with R-mode factor analysis. 

Q-mode factor analysis defines the similarity of objects by considering the component proportions. The method searches elements in the A matrix for the most divergent objects, represented by the pure component concentrations or... 

Simple Q-mode factor analysis fails to provide a direct solution to the partitioning problem. This is because (1) the vectors generated by factor analysis are not composition vectors and therefore cannot be used to indicate the absolute composition of the end-members (2) the factor scores only give a relative measure of the importance of each variable in each end-member and also reflect any scaling done on the data set prior to the analysis (such as transforming variable values to percent of range or normahzing variables to equal means) ... 

An extension of Q-mode factor analysis [ 1,303-306] provides a solution to the first, second and fourth problems listed above, those associated with obtaining absolute compositions of the end-members themselves. Normally when... 

Data for MM compositions of different SWMs and their leachates were examined statistically in order to determine any significant compositional variations among samples. Most statistical analyses were performed using the SAS Statistical Package V 6.12 [335]. In this report, the results of Q-mode factor analysis and Unear programming techniques will be presented. The objectives of the statistical analyses were to define the MM characteristics for the different SWMs and their leachates, and to determine their original sources. 

Both Q-mode factor analysis and liner programming techniques (Sect. 3.4) were used in the present study, yielding the following results (Fig. 12) ... 

Stuckless et al. (1981) carried out a Q-mode factor analysis of the chemical compositions of the 49 rock... 

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