Principal component analysis versus fuzzy principal component analysis A case study: the quality of danube water (1985-1996).

Principal component analysis (PCA) is a favorite tool in environmetrics for data compression and information extraction. PCA finds linear combinations of the original measurement variables that describe the significant variations in the data. However, it is well-known that PCA, as with any other multivariate statistical method, is sensitive to outliers, missing data, and poor linear correlation between variables due to poorly distributed variables. As a result data transformations have a large impact upon PCA. In this regard one of the most powerful approach to improve PCA appears to be the fuzzification of the matrix data, thus diminishing the influence of the outliers. In this paper we discuss and apply a robust fuzzy PCA algorithm (FPCA). The efficiency of the new algorithm is illustrated on a data set concerning the water quality of the Danube River for a period of 11 consecutive years. Considering, for example, a two component model, FPCA accounts for 91.7% of the total variance and PCA accounts only for 39.8%. Much more, PCA showed only a partial separation of the variables and no separation of scores (samples) onto the plane described by the first two principal components, whereas a much sharper differentiation of the variables and scores is observed when FPCA is applied.

Assessment of heart disease using fuzzy classification techniques.

In this paper we discuss the classification results of cardiac patients of ischemical cardiopathy, valvular heart disease, and arterial hypertension, based on 19 characteristics (descriptors) including ECHO data, effort testings, and age and weight. In this order we have used different fuzzy clustering algorithms, namely hierarchical fuzzy clustering, hierarchical and horizontal fuzzy characteristics clustering, and a new clustering technique, fuzzy hierarchical cross-classification. The characteristics clustering techniques produce fuzzy partitions of the characteristics involved and, thus, are useful tools for studying the similarities between different characteristics and for essential characteristics selection. The cross-classification algorithm produces not only a fuzzy partition of the cardiac patients analyzed, but also a fuzzy partition of their considered characteristics. In this way it is possible to identify which characteristics are responsible for the similarities or dissimilarities observed between different groups of patients.

Fuzzy robust estimation of central location.

The problem of a new robust algorithm for estimation of central location has been described in a mathematically simpler way using the fuzzy sets theory. It was compared with ordinary mean estimator and other robust estimators - median, 5% trimmed mean and Huber-, Tukey-, Hampel-, and Andrews-type M-estimators. The performance of Fuzzy 1-means algorithm (FM) proposed is demonstrated by applying it to different data sets from published literature and has been shown to exceed that of conventional ordinary mean estimator and equals or often exceeds that of the most robust estimators.

Fuzzy clustering analysis of the first 10 MEIC chemicals.

In this paper, we discuss the classification results of the toxicological responses of 32 in vivo and in vitro test systems to the first 10 MEIC chemicals. In this order we have used different fuzzy clustering algorithms, namely hierarchical fuzzy clustering, hierarchical and horizontal fuzzy characteristics clustering and a new clustering technique, namely fuzzy hierarchical cross-classification. The characteristics clustering technique produces fuzzy partitions of the characteristics (chemicals) involved and thus it is a useful tool for studying the (dis)similarities between different chemicals and for essential chemicals selection. The cross-classification algorithm produces not only a fuzzy partition of the test systems analyzed, but also a fuzzy partition of the considered 10 MEIC (multicentre evaluation of in vitro cytotoxicity) chemicals. In this way it is possible to identify which chemicals are responsible for the similarities or differences observed between different groups of test systems. In another way, there is a specific sensitivity of a chemical for one or more toxicological tests.

A new fuzzy regression algorithm.

A new fuzzy regression algorithm is described and compared with conventional ordinary and weighted least-squares and robust regression methods. The application of these different methods to relevant data sets proves that the performance of the procedure described in this paper exceeds that of the ordinary least-squares method and equals and often exceeds that of weighted or robust methods, including the two fuzzy methods proposed previously (Otto, M.; Bandemer, H., Chemom. Intell. Lab. Syst. 1986, 1, 71. Hu, Y.; Smeyers-Verbeke, J.; Massart, D. L. Chemom. Intell. Lab. Syst. 1990, 8, 143). Moreover, we emphasize the effectiveness and the generality of the two new criteria proposed in this paper for diagnosing the linearity of calibration lines in analytical chemistry.