Abstract and true factors

In Chapter 31 we stated that any data matrix can be decomposed into a product of two other matrices, the score and loading matrix. In some instances another decomposition is possible, e.g. into a product of a concentration matrix and a spectrum matrix. These two matrices have a physical meaning. In this chapter we explain how a loading or a score matrix can be transformed into matrices to which a physical meaning can be attributed. We introduce the subject with an example from environmental chemistry and one from liquid chromatography. 

Let us suppose that dust particles have been collected in the air above a city and that the amounts of p constituents, e.g. Si, Al, Ca. Pb have been determined in these samples.The elemental compositions obtained for n (e.g. 100) samples, taken over a grid of sampling points, can be arranged in a data matrix X (Fig. 34.1). Each row of the table represents the elemental composition of one of the samples. A column represents the amount of one of the elements found in the sample set. Let us further suppose that there are two main sources of dust in the neighbourhood of the sampled area, and that the particles originating from each source have a specific concentration pattern for the elements Si to Pb. These concentration patterns are described by the vectors s, and Sj. For instance the dust in the air may originate from a power station and from an incinerator, having each a specific concentration pattern, sj = [Si, Al, Ca , ... PbJ with k = 1,2. 

Obviously, each sample in the sampled area contains particles from each source, but in a varying proportion. Some of the samples mainly contain particles from the power station and less from the incinerator. Other samples may contain an equal amount of particles of each source. In general, one can say that the composition x, of any sample i of dust is a linear combination of the two source patterns Sj and S2 given by x, = c, s, + c,2 2. In this expression c, gives the contribution of the first source and the contribution of the second dust source in sample i. For all n samples these contributions can be arranged in a nx2 matrix C giving X = CS where S is the px2 matrix of the source patterns. If the concentration patterns of the 

Matters become more complex when the concentration patterns of the sources are not known. It becomes even more complicated when the number and the origin of the potential sources of the dust are unknown. In this case the number of sources and the concentration patterns of each source have to be estimated from the measured data table X. This operation is caWed factor analysis. In the terminology of factor analysis, the two sources of dust in our example are called factors, and the concentration patterns of the compounds in each source are calledfactor loadings. 

In Chapters 17 and 31 we explained that a matrix X can be decomposed by S VD in a product of three matrices the two matrices of singular vectors U and V and a diagonal matrix of singular values A such that  

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