Principal component analysis Reproducibility

Figure 3.13 Principal component analysis of repetitive GC/MS profiles of M. truncatula root (R), stem (S) and leaves (L). The first and second principal component of each GC/MS analysis were calculated and plotted. The relative distance between points is a measure of similarity or difference. The clustering shows good reproducibility within the independent tissues but clear differentiation of tissues. The results also show that roots and stems are more similar to each other than to leaves.

Basic Concepts. The goal of factor and components analysis is to simplify the quantitative description of a system by determining the minimum number of new variables necessary to reproduce various attributes of the data. Principal components analysis attempts to maximally reproduce the variance in the system while factor analysis tries to maximally reproduce the matrix of correlations. These procedures reduce the original data matrix from one having m variables necessary to describe the n samples to a matrix with p components or factors (p[Pg.26]

The NIR spectra of the pure isomers and the extractant are shown in Figure 11.2. Although the spectra of the three isomers and the extractant are quite similar overall, there are distinct, reproducible spectral differences, especially in the 2100-2500 nm region. The authors did extensive studies involving mid-IR spectra, spectral simulations, and principal components analysis (PCA) in order to understand the origins of these differences to ensure that they were related to the isomers themselves and not to coincidental impurities. 

Principal component analysis makes it possible to find a set of representations for mixture spectra in which noise and interactions are taken into account without knowing anything about the spectra of the pure components or their concentrations. The basic idea is to find a set of representations that can be linearly combined to reproduce the original mixture spectra. In PCA, Equation (4.3) is rewritten as... 

Using a different set of standard substances, i.e. substituting 1-butanol, pentan-2-one, and 1-nitropropane for the rather volatile ethanol, butan-2-one, and nitromethane, McReynolds developed an analogous approach [103]. Altogether, he characterized over 200 liquid stationary phases using a total of 10 probes. A statistical analysis of the McReynolds retention index matrix using the principal component analysis method has shown that only three components are necessary to reproduce the experimental data matrix [262]. The first component is related to the polarity of the liquid phase, the second depends almost solely on the solute, and the third is related to specific interactions with solute hydroxy groups [262]. 

Spectrum using linear combinations of spectra from known species. LCF is a fingerprinting technique and is limited by how well the set of standard species represents the actual species in the samples. However, natural samples usually possess a range of species from crystalline to amorphous and doped solids that would be difficult to reproduce exactly with pure standards (Kelly et al. 2008). Thus, LCF benefits from the use of additional information provided by other statistical approaches like principal component analysis and target transformation (Beauchemin et al. 2002) as well as complementary analysis of the sample (e.g. XRD, X-ray fluorescence, ion-coupled plasma mass spectrometry). 

Readers with experience in chemometrics will have noticed that, like principal components analysis (PCA), MDS is a dimensionality reduction method. For each molecule, a large number of attributes (similarity to each other molecule) is reduced to a much smaller number of coordinates in an abstract property space, which reproduce the original data within an established error. The pertinent difference is that PCA uses the matrix of correlations between a set of (redundant) properties, which are usually obtained from a table of those properties for an initial set of molecules. In contrast, MDS uses a matrix of similarities between each pair of molecules (or substituents). 

Chen P, Lu Y, Harrington PB. Biomarker profiling and reproducibility study of MALDI-MS measurements of Escherichia coli by analysis of variance-principal component analysis. Anal Chem. 2008 80(5) 1474 1. 

A sensor array was composed by assembling three different sensors consisting of Zn-, Cd- and non-imprinted materials, and it allowed for discrimination of the title ions, within the range of 10-100 pM concentration, against non-templated ions such as Mg ", Ca " and Al ". The principal component analysis (PCA) showed a clear discriminative and reproducible pattern (Fig. 15). 

It must be concluded that the measurement of fat, protein, and lactose of milk by NIRS can be successful. Our experience with cow and goat milk shows that the main problem is to find a representative sample set, a method of presenting the sample to the instrument that is exactly reproducible, and a method to identify unknown samples with an abnormal spectra. Study of Bertrand et al. [28] illustrates that principal components analysis (PCA) has the potential to select the most relevant calibration samples (independent of the chemical composition) and to identify samples with abnormal spectra as an outlier. 

Figure 5.6 Principal components analysis (PCA) score plot for Raman spectra of (A) A549, (B) BEAS2B and (C) HaCaT cell lines. Percentage labels on each axis denote the variance described by that PC. There is clear separation of cellular spectra fixed with Meth-Ac and Carnoy s fixative relative to live and formalin fixed spectra. A degree of similarity between the spectral content of formalin fixed and live cell spectra is implied by the proximity of their clusters. (Reproduced from reference [52].)...

Figure 17 Cluster analysis of 44 isomers with molecular formula C6H3CI3 by principal component analysis (score plot of the first and second principal component containing 41.2% and 18.8% of the total variance, respectively). The chemical structures have been characterized by 20 binary molecular descriptors. The common structural properties within each cluster are characterized by the maximum common substructure (MCS). [Reproduced from Ref. 133 with kind permission of Gesellschaft Deutscher Chemiker]...

Several studies in the past years have shown how such apparently simple approaches are indeed capable of representing well thermally associated local deformations of protein scaffolds, as well as how they can be used to reproduce principal component analysis coming from more complex all-atom simulations of the same targets. ... 

Figure 6,6 First three principal component analysis scores (correlation) for the analysis of truffles taken from six regions in Italy (Langhe, Marche, Umbria, Lazio, Toscana and Molise). The results for truffles taken from Marche, Umbria and Toscana are particularly well separated. Reproduced with permission from [ 10]. Copyright 2007 John Wiley Sons, Ltd.

Figure 6,7 Principal component analysis biplot of the mean scores and loadings (numbers represent m/z values) for PTR-MS data extracted from sea urchin roe taken from northern (N) and southern (S) locations in New Zealand. Reproduced from [91] with permission from Elsevier.

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. 

Score values of the first principal component following analysis of the 1550 to 600 cm spectral region plotted against the mole fraction water. Source Reproduced from Taylor, L.S., Langkilde, F.W., and Zografi, G. Fourier transform Raman spectroscopic study of the interaction of water vapor with amorphous polymers, /. Pharm. Set., 90,888-901,2001. With permission of the copyright owner.)... 

The data generated from a NIR or Raman spectrum do not immediately provide the concentrations of the species at any time, so there is no predictive capability. Construction of a calibmtion set requires an independent measure of the property, e.g. by HPLC or by NIR of known mixtures of the components. Two such methods are principal-component regression (PCR) and partial least squares (PLS). As soon as quantitative analysis is considered, the question of noise and reproducibility of the data set becomes important. It is therefore necessary to treat the mw data to remove the drift in baseline etc. which will occur over a long period of spectml acquisition. 

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