Classical and fuzzy principal component analysis of some environmental samples concerning the pollution with heavy metals

Abstract

In this paper two robust fuzzy principal component algorithms are explained and their robustness and efficiency are demonstrated. The efficiency of the applied procedures was illustrated on two data sets. The first one contains 125 soil samples (Northern Romania) and 13 ion concentrations (lead, copper, manganese, zinc, nicker, cobalt, chromium, cadmium, calcium, magnesium, potassium, iron and aluminum) and the second data set consists of 234 differently polluted sampling locations (East Germany) characterized by four variables: soil lead content, plant lead content, traffic density, and distance from the road. The fuzzy principal component algorithms achieved better results mainly because they are more compressible than classical PCA and very robust to outliers. Considering, for example, a three component model for the first data set, fuzzy principal component analysis-first component (FPCA-1) accounts for 73.82% of the total variance and fuzzy principal component analysis-all components (FPCA-o) 85.44%; PCA accounts only for 68.22%. The first two principal components explain 63.11% of the total variance in the case of FPCA-1 and 77.16% in the case of FPCA-o as compared to only 57.43% for PCA. As a direct consequence, PCA showed only a partial separation of samples onto the plane or in the space described by different combination of two or three principal components, whereas a much sharper differentiation of the samples, from their origin and location point of view, is observed when FPCAs are applied.