Classical and Fuzzy Principal Component Analysis of Some Environmental Samples Concerning Pollution with Heavy Metals

1. Classical and Fuzzy Principal Component Analysis of Some Environmental Samples Concerning Pollution with Heavy Metals COSTEL SÂRBU Department of Chemsitry, Babeş-Bolyai University Cluj-Napoca ROMANIA costelsrb@yahoo.co.uk

2. Principal Component Analysis

3. Soft Computing Methods Fuzzy Logic Fuzzy Sets Soft Computing Approximate Reasoning PCA, PCR, PLS, ANN Genetic Algorithms Chaos Theory Rough Sets

4. Aim : To exploit the tolerance for imprecision uncertainty, approximate reasoning and partial truth to achieve tractability, robustness, low solution cost, and close resemblance with humanlike decision making To find an approximate solution to an imprecisely/precisely formulated problem. What is Soft Computing ? • Soft Computing is a collection of methodologies (working synergistically, not competitively) which, in one form or another, reflect its guiding principle: • Exploitthe tolerance for imprecision, uncertainty, approximate reasoning and partial truth to achieve tractability, robustness, and close resemblancewith human like decision making. • Provides flexible information processing capability for representation and evaluation of various real life ambiguous and uncertain situations.Real World Computing • It may be argued that it issoft computing rather than hard computingthat should be viewed as the foundation for Artificial Intelligence(AI).

5. Soft Computingvs Hard Computing • Hard computing requires programs to be written; soft computing can evolve its own programs • Hard computing uses two-valued logic; soft computing can use multivalued or fuzzy logic • Hard computing is deterministic; soft computing incorporates stochasticity • Hard computing requires exact input data; soft computing can deal with ambiguous and noisy data • Hard computing is strictly sequential; soft computing allows parallel computations • Hard computing produces precise answers; soft computing can yield approximate answers

6. Fuzzy Sets and Fuzzy Logic • In 1965*Zadeh published his seminal work "Fuzzy Sets" which described the mathematics of Fuzzy Set Theory, and by extension Fuzzy Logic. • It deals with the uncertainty and fuzziness arising from interrelated humanistic types of phenomena such subjectivity, thinking, reasoning, cognition, and perception. This type of uncertainty is characterized by structure that lack sharp boundaries. This approach provides a way to translate a linguistic model of the human thinking process into a mathematical framework for developing the computer algorithms for computerized decision-making processes. *L. A. ZADEH, Fuzzy Sets, Information Control, 1965, 8, 338-353.

7. Fuzzy Sets Theory • A Fuzzy Set is a generalized set to which objects can belongs with various degrees (grades) of memberships over the interval [0,1]. • Fuzzy systems are processes that are too complex to be modeled by using conventional mathematical methods. • In general, fuzziness describes objects or processes that are not amenable to precise definition or precise measurement. Thus, fuzzy processes can be defined as processes that are vaguely defined and have some uncertainty in their description. The data arising from fuzzy systems are in general, soft, with no precise boundaries.

8. Lotfi A. Zadeh betwen Orient and Occident

9. The Impact of Application of Fuzzy Sets Theory in Science and Technical Fields “In 1999, Japan exported products at a total of $35 billion that use Fuzzy Logic or NeuroFuzzy. The remarkable fact that an emerging key technology in Asia and Europe went unnoticed by the U.S. public until recently, combined with its unusual name and revolutionary concept has led to a controversial discussion among engineers.” Constantine von Altrock Inform Software Corp., Germany

10. Reasoning Styles in China and West

11. School of Athens

12. Fuzziness in Everyday World • John is tall; • Temperature is hot; • Mr. B. G. is young (the paradox of Mr. B.G.); • The girl next door is prettty; • The Romanian Leu is getting relatively strong; • The people living close to Bucharest; • My car is slow, your car is fast;

13. Fuzziness in Chemistry • Water is an acid; • Germanium is a metal; • Those drugs are very effective; • Varying peaks in chromatograms; • Varying signal heights in spectra from the same substance; • Varying patterns in QSAR pattern recognition studies;

14. Fuzzinessin Everyday World(OrientversusOccident)

15. Fuzziness in Everyday World(Fuzzy girl-students in chemsitry)

16. P: X  {0,1} P(x) = 1 if x  X P(x) = 0 if x  X A : X  [0,1] A= {X, A(x)} if x  X Characteristic Function in the Case of Crisp Sets andFuzzy SetsRespectively

17. Girl-Student Membership Function for “Young”

18. Mr. B. G. Membership Function for “Young”

19. Generalized Fuzzy c-Means Algorithm

20. Fuzzy 1-Line Regression Algorithm

21. Fuzzy Principal Component Analysis Algorithm

22. Fuzzy Approaches • Fuzzy divisive hierarchical clustering; • Fuzzy horizontal clustering; • Fuzzy cross-clustering; • Fuzzy robust regression; • Fuzzy robust estimation of mean and spread

23. Data Set 1 The data collection was performed in the northern part of Romanian Carpathians Mountains : the western part of Bistriţa Mountains (b), the south-western part of Maramureş Mountains (m) and the north-western part of Igniş-Oaş Mountains (i), according to standardized methods for sampling, sample preparation and analysis. Thirteen different soil ion concentration were checked:lead, copper, manganese, zinc, nickel, cobalt, chromium, cadmium, calcium, magnesium, potassium, ironandaluminum

24. Eigenvalue and Proportion Considering the First Five Principal Components for PCA andFPCA

25. Eigenvectors Corresponding to the First Four Principal Components for PCA andFPCA

26. Loading Plot PC1-PC2-PC3(PCA andFPCA-1)

27. Loading Plot PC1-PC2-PC3(PCA andFPCA-o)

28. Score Plot PC1-PC2(PCA andFPCA-1)

29. Score Plot PC1-PC3(PCA andFPCA-1)

30. Score Plot PC1-PC4(PCA and FPCA-1)

31. Score Plot PC2-PC3(PCA andFPCA-1)

32. Score Plot PC2-PC4(PCA andFPCA-1)

33. Score Plot PC3-PC4(PCA andFPCA-1)

34. Score Plot PC1-PC2(FPCA-1andFPCA-o)

35. Score Plot PC1-PC3(FPCA-1andFPCA-o)

36. Score Plot PC1-PC4(FPCA-1andFPCA-o)

37. Score Plot PC2-PC3(FPCA-1andFPCA-o)

38. Score Plot PC2-PC4(FPCA-1andFPCA-o)

39. Score Plot PC3-PC4(FPCA-1andFPCA-o)

40. Data Set 2 The data set consists of 234 differently polluted sampling locations (East Germany) characterized by four variables: soil lead content (sPb), plant lead content (pPb), traffic density (tD), and distance from the road (dR). As an additional feature a classification number resulting from the a-priori knowledge of the loading situation at the particular sampling location according to the following list is given: Loading situation Class number Samples number Unpolluted 1 175 Moderately polluted2 40 Polluted3 10 Extremely polluted4 9

41. Eigenvalue and Proportion Considering the First Five Principal Components for PCA andFPCA

42. Eigenvectors Corresponding to the First Three Principal Components for PCA andFPCA

43. Loading Plot PC1-PC2-PC3(PCA andFPCA-1)

44. Loading Plot PC1-PC2-PC3(FPCA-1 andFPCA-o)

45. Score Plot PC1-PC2(PCA andFPCA-1)

46. Score Plot PC1-PC3(PCA andFPCA-1)

47. Score Plot PC1-PC4(PCA andFPCA-1)

48. Score Plot PC2-PC3(PCA andFPCA-1)

49. Score Plot PC2-PC4(PCA andFPCA-1)

50. Score Plot PC3-PC4(PCA andFPCA-1)