Fuzzy PCA

Examining all the pairs of variables for relationships between them, we find, indeed, some rather strong correlations, as may also be the case for their correlation coefficients, r. Nevertheless, these relations should be considered with caution, particularly when generalizing to other elements. As a matter of fact, lanthanides are a rather peculiar group of elements their properties are very similar, that is, the variation of a given property in the series of these elements is rather limited and inferences to all the elements are doubtful. Keeping this in mind, we may, however, discuss the closest correlations (similarities) found here. 

The highest correlation coefficient (0.98) is for enthalpy of vaporization and standard enthalpy of atomization. For metals, the latter one is actually the standard sublimation enthalpy and is given, according to Hess law, by the sum of the standard enthalpies of fusion and vaporization. The enthalpies of vaporization are those at the boiling temperature and, therefore, lower than the standard ones. Other very close correlations are those between the boiling 

A significant correlation, but with a negative correlation coefficient ( — 0.93), appears between the specific heat capacity and the atomic mass. For lanthanides, as for most metals, the rule of Dulong and Petit is applicable (inverse proportionality between the two quantities). 

The correlation between the enthalpy of fusion and the melting point is looser r = 0.87), but better for the ensemble of chemical elements for 95 elements, r = 0.775. 

The negative correlations between the third ionization energy and the enthalpy of vaporization r = —0.87), the enthalpy of atomization r = —0.83), or the boiling point r = —0.84) seem rather fortuitous. The correlation of density and atomic mass r = 0.86) is particular to lanthanides the density increases with the atomic number (with some exceptions Eu, Yb), while the atomic volume is nearly constant (actually it decreases slightly, due the contraction of lanthanides). On the other hand, the correlations of surface tension with the enthalpy of atomization (r = 0.84), the boiling temperature r = 0.78), and the enthalpy of vaporization r = 0.77) are well-known general relations. Surface tension represents the free energy necessary 

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Encouraged by the good results obtained with the Fuzzy (first component) PCA," ° we decided to extend the fuzzy approach one step more. A Fuzzy PCA algorithm was written that would extend the fuzzy clustering scheme with computing each particular principal component, not just the first one. The main idea behind the first algorithm in this series rests with the relation... 

A broken line separates the two classes of light lanthanides, including Yb and the heavier lanthanides, together with Sc and Y. The clusters are framed by solid borders. Going along the series of lanthanides in the direction of the arrows we find all the clusters given by the fuzzy PCA analysis. 

FIGURE 6.23 Fuzzy clustering with six clusters for the original (left) and scaled (right) Hyptis data. The plot symbols correspond to the found clusters with the largest membership coefficient, and their size is proportional to this coefficient. The results are presented in the PCA projection obtained from the original data (Figure 6.19, left). 

Thus, multilinear models were introduced, and then a wide series of tools, such as nonlinear models, including artificial neural networks, fuzzy logic, Bayesian models, and expert systems. A number of reviews deal with the different techniques [4-6]. Mathematical techniques have also been used to keep into account the high number (up to several thousands) of chemical descriptors and fragments that can be used for modeling purposes, with the problem of increase in noise and lack of statistical robustness. Also in this case, linear and nonlinear methods have been used, such as principal component analysis (PCA) and genetic algorithms (GA) [6]. 

Ischemia in the forearm was studied by Mansfield et al. in 1997 [38], In this study, the workers used fuzzy C means clustering and principal component analysis (PCA) of time series from the NIR imaging of volunteers forearms. They attempted predictions of blood depletion and increase without a priori values for calibration. For those with a mathematical bent, this paper does a very nice job describing the theory behind the PCA and fuzzy C means algorithms. 

The major novelty of this algorithm is in the way the other fuzzy principal components are computed. The original data set is projected onto the hyperplane orthogonal to the first fuzzy principal component, that is, determined by all the other principal components, as determined by the Fuzzy First Component PCA algorithm. Practically, this may be done by computing the scores... 

We compared fuzzy principal component analysis (FPCA) to classical PCA methods to characterize lanthanum, the 14 lanthanides, and the other... 

Van Veen and De Loos-Vollebregt reviewed various chemometric procedures which had been developed during the last decade for ICP-OES analysis. In these procedures, data reduction by techniques such as digital filtering, numerical derivatives, Fourier transforms, correlation methods, expert systems, fuzzy logic, neural networks, PCA, PLS, projection methods, Kalman filtering, MLR and generalised standard additions were discussed. 

Supervised and unsupervised classification for example PCA, K-means and fuzzy clustering, linear discriminant analysis (LDA), partial least squares-discriminant analysis (PLS-DA), fisher discriminant analysis (FDA), artificial neural networks (ANN). 

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