Abstract factors

Many analytical practitioners encounter a serious mental block when attempting to deal with factor spaces. The basis of the mental block is twofold. First, all this talk about abstract vector spaces, eigenvectors, regressions on projections of data onto abstract factors, etc., is like a completely alien language. Even worse, the techniques are usually presented as a series of mathematical equations from a statistician s or mathematician s point of view. All of this serves to separate the (un )willing student from a solid relationship with his data a relationship that, usually, is based on visualization. Second, it is often not clear why we would go through all of the trouble in the first place. How can all of these "abstract", nonintuitive manipulations of our data provide any worthwhile benefits ... 

So now we understand that when we use eigenvectors to define an "abstract factor space that spans the data," we aren t changing the data at all, we are simply finding a more convenient coordinate system. We can then exploit the properties of eigenvectors both to remove noise from our data without significantly distorting it, and to compress the dimensionality of our data without compromising the information content. 

Malinowski, E.R., "Statistical F-Tests for Abstract Factor Analysis and Target Testing 1,/. Chemo. 1987 (1) 49-60... 

Absorbance matrix, 7,9, 11 creating, 37 Abstract factors, 84 Accuracy... 

E.R. Malinowski, Statistical F-tests for abstract factor analysis and target testing. J. Chemom., 3 (1988)49-60. 

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 columns of V are the abstract factors of X which should be rotated into real factors. The matrix V is rotated by means of an orthogonal rotation matrix R, so that the resulting matrix F = V R fulfils a given criterion. The criterion in Varimax rotation is that the rows of F obtain maximal simplicity, which is usually denoted as the requirement that F has a maximum row simplicity. The idea behind this criterion is that real factors should be easily interpretable which is the case when the loadings of the factor are grouped over only a few variables. For instance the vector f, =[000 0.5 0.8 0.33] may be easier to interpret than the vector = [0.1 0.3 0.1 0.4 0.4 0.75]. It is more likely that the simple vector is a pure factor than the less simple one. Returning to the air pollution example, the simple vector fi may represent the concentration profile of one of the pollution sources which mainly contains the three last constituents. 

Taking into consideration the fact that initial targets have to be formulated and that the projected targets have to be inspected and adapted, we estimate the pure elution profiles instead of the pure spectra. Therefore, the PCA is carried out in the elution time space and the resulting principal components represent abstract elution profiles (see Fig. 34.6). The factors we are looking for are the elution profiles of the different compounds. The first abstract factor closely resembles one... 

The first step in analysing a data table is to determine how many pure factors have to be estimated. Basically, there are two approaches which we recommend. One starts with a PCA or else either with OPA or SIMPLISMA. PCA yields the number of factors and the significant principal components, which are abstract factors. OPA yields the number of factors and the purest rows (or columns) (factors) in the data table. If we suspect a certain order in the spectra, we preferentially apply evolutionary techniques such as FSWEFA or HELP to detect pure zones, or zones with two or more components. 

Depending on the way the analysis was started, either the abstract factors found by a PCA or the purest rows found by OPA, should be transformed into pure factors. If no constraints can be formulated on the pure factors, the purest rows... 

The cure, again, is to embed the formal parts in narrative text. If the formal parts are getting too complex, look for ways to abstract, factoring out various aspects so as to better decouple your documents. 

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 ... 

Factor analysis extracts information from the sample data set (e.g., IR spectra) and does not rely on reference minerals. However, because abstract factors have no physical meaning, reference minerals may be needed in target transformations or other procedures to extract mineralogical information. One valuable piece of information obtainable without the use of extraneous data is the number of components required to represent the data within experimental error. Reported applications of factor analysis to mineralogy by FTIR are few (12). However, one commercial laboratory is offering routine FTIR mineral analyses to the petroleum industry, based on related methods (22). 

Multivariate methods, on the other hand, resolve the major sources by analyzing the entire ambient data matrix. Factor analysis, for example, examines elemental and sample correlations in the ambient data matrix. This analysis yields the minimum number of factors required to reproduce the ambient data matrix, their relative chemical composition and their contribution to the mass variability. A major limitation in common and principal component factor analysis is the abstract nature of the factors and the difficulty these methods have in relating these factors to real world sources. Hopke, et al. (13.14) have improved the methods ability to associate these abstract factors with controllable sources by combining source data from the F matrix, with Malinowski s target transformation factor analysis program. (15) Hopke, et al. (13,14) as well as Klelnman, et al. (10) have used the results of factor analysis along with multiple regression to quantify the source contributions. Their approach is similar to the chemical mass balance approach except they use a least squares fit of the total mass on different filters Instead of a least squares fit of the chemicals on an individual filter. 

A substantial amount of confusion (9,10.13,14) has recently developed as to an approach s dependence on conservation of mass. As Cooper and Watson ( ) have noted, the F j factors refer to the source chemistry as it arrives at the receptor. It is assumed with the conservation of mass that the Fj j as might be measured at a receptor, is the same as have been measured at the source. As noted above, this may not be valid depending on the source and the method used for source sampling. The chemical mass balance method incorporates the F j directly in its calculations and as a result is often perceived as having a greater dependence on this assumption than methods such as factor analysis which do not use Fy values in their calculations. Factor analysis methods, however, identify abstract factors, which explain variability. It is impossible to attribute a common... 

Faber, K. and Kowalski, B.R., Critical Evaluation of Two f-tests for Selecting the Number of Factors in Abstract Factor Analysis Anal. Chim. Acta 1997, 337, 57-71. 

Regression techniques. Principal components are sometimes called abstract factors, and are primarily mathematical entities. In multivariate calibration the aim is to convert these to compound concentrations. PCR uses regression (sometimes called transformation or rotation) to convert PC scores onto concentrations. This process is often loosely called factor analysis, although terminology differs according to author and discipline. 

The eigenvectors in Vt can be used to form a set of orthonormal row basis vectors for A. The eigenvectors are called loadings or sometimes abstract factors or eigenspectra, indicating that while the vectors form a basis set for the row space of A, physical interpretation of the vectors is not always possible (see Figure 4.2). 

The outer vector product, t, v, 1, is the variance explained by the first factor. For the HPLC-UV/Vis data set in Figure 4.1, exactly two terms or abstract factors would be required in Equation 4.4 or Equation 4.5 to explain the data, one for each chemical component. The term abstract factors is stressed here because the factors do not necessarily correspond to the two chemical components. 

This is a plane wave state to the extent as x-axis propagation is concerned. The quantum state includes information about interactions at the slits through amplitudes, phases, and Fresnel integrals ((x,y l) and (x,y 2)) [14]. For us, these quantities are parameters that can be controlled by one way or another. The linear superposition form is what matters. This is the form taken by the abstract factor of the physical quantum state. 

Malinowski,E.R., Abstract factor Analysis- a Theory of Error and its Application to Analytical Chemistry, in Chemometrics, Theory and Application, (ed. B.R. Kowalski), ACS Symposium Series 52, American Chemical Society, Washington, D.C., 1977. 

One aim of chemometrics is to obtain these predictions after first treating the chromatogram as a multivariate data matrix, and then performing PCA. Each compound in the mixture is a chemical factor with its associated spectra and elution profile, which can be related to principal components, or abstract factors, by a mathematical transformation. 

ACD/AutoChrom uses the mutual automated peak matching [33] or UV-MAP approach based on extraction of pure variables from diode array data. The UV-MAP algorithm applies abstract factor analysis (AFA) followed by iterative key set factor analysis to the augmented data matrix in order to extract retention times for each of the selected experiments. 

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