Principal component analysis measurement errors

In a paper that addresses both these topics, Gordon et al.11 explain how they followed a com mixture fermented by Fusarium moniliforme spores. They followed the concentrations of starch, lipids, and protein throughout the reaction. The amounts of Fusarium and even com were also measured. A multiple linear regression (MLR) method was satisfactory, with standard errors of prediction (SEP) for the constituents being 0.37% for starch, 4.57% for lipid, 4.62% for protein, 2.38% for Fusarium, and 0.16% for com. It may be inferred from the data that PLS or PCA (principal components analysis) may have given more accurate results. 

Furthermore, quantitative structural phase analysis, for instance, is important for investigations of solid catalysts, because one frequently has to deal with more than one phase in the active or precursor state of the catalyst. Principal component analysis (PCA) permits a quantitative determination of the number of primary components in a set of experimental XANES or EXAFS spectra. Primary components are those that are sufficient to reconstruct each experimental spectrum by suitable linear combination. Secondary components are those that contain only the noise. The objective of a PCA of a set of experimental spectra is to determine how many "components" (i.e., reference spectra) are required to reconstruct the spectra within the experimental error. Provided that, first, the number of "references" and, second, potential references have been identified, a linear combination fit can be attempted to quantify the amount of each reference in each experimental spectrum. If a PCA is performed prior to XANES data fitting, no assumptions have to be made as to the number of references and the type of reference compounds used, and the fits can be performed with considerably less ambiguity than otherwise. Details of PCA are available in the literature (Malinowski and Flowery, 1980 Ressler et al., 2000). Recently, this approach has been successfully extended to the analysis of EXAFS data measured for mixtures containing various phases (Frenkel et al., 2002). 

Principal components analysis (see also p. 16) involves an examination of set of data as points in n-dimensional space (corresponding to n original tests) and determines (first) the direction that accounts for the biggest variability in the data (first principal component). The process is repeated until n principal components are evaluated, but not all of these are of practical importance because some may be attributable purely to experimental error. The number of significant principal components shows the number of independent properties being measured by the tests considered. 

Wentzell PD, Lohnes MT, Maximum likelihood principal component analysis with correlated measurement errors theoretical and practical considerations, Chemometrics and Intelligent Laboratory Systems, 1999, 45, 65-85. 

When analysing the level of measurement error of LC and LD50 it was realised that the set of data was difficult to use since it is hardly reliable, and therefore of questionable coherence amongst all the figures. In order to find an answer to this a sample of the LC and LD50 values were submitted to an analysis based on principal components (PCA). it would take far too much time to describe this method, besides this goes beyond our subject, its purpose is to look for and classify the different types of information contained in a complex table of quantitative data. 

You will better understand the goals of factor analysis considering first the highly idealized situation with error-free observations and only r < n linearly independent columns in the matrix X. As discussed in Section 1.1, all columns of X are then in an r—dimensional subspace, and you can write them as linear combinations of r basis vectors. Since the matrix X X has now r nonzero eigenvalues, there are exactly r nonvanishing vectors in the matrix Z defined by (1.111), and these vectors form a basis for the subspace. The corresponding principal components z, z2,. .., zr are the coordinates in this basis. In the real life you have measurement errors, the columns of X... 

Seven types of espresso coffee were classified by Pardo and Sberveglieri with a system composed of an electronic nose and an SVM with polynomial and Gaussian RBF kernels.For each coffee type, 36 measurements were performed with an electronic nose equipped with five thin-film semiconductor sensors based on SnOi and Ti-Fe. The output signal from sensors was submitted to a PGA analysis whose principal components (between 2 and 5) represented the input data for the SVM classifier. The error surface corresponding to various kernel parameters and number of input principal components was investigated. 

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