Assessment of Various Fuzzy c-Mean Clustering Validation Indices for Mapping Mineral Prospectivity: Combination of Multifractal Geochemical Model and Mineralization Processes

  • Mehrdad Daviran
  • Abbas MaghsoudiEmail author
  • David R. Cohen
  • Reza Ghezelbash
  • Huseyin Yilmaz
Original Paper


This paper describes the application of an unsupervised clustering method, fuzzy c-means (FCM), to generate mineral prospectivity models for Cu ± Au ± Fe mineralization in the Feizabad District of NE Iran. Various evidence layers relevant to indicators or potential controls on mineralization, including geochemical data, geological–structural maps and remote sensing data, were used. The FCM clustering approach was employed to reduce the dimensions of nine key attribute vectors derived from different exploration criteria. Multifractal inverse distance weighting interpolation coupled with factor analysis was used to generate enhanced multi-element geochemical signatures of areas with Cu ± Au ± Fe mineralization. The GIS-based fuzzy membership function MSLarge was used to transform values of the different evidence layers, including geological–structural controls as well as alteration, into a [0–1] range. Four FCM-based validation indices, including Bezdek’s partition coefficient (VPc) and partition entropy (VPe) indices, the Fukuyama and Sugeno (VFS) index and the Xie and Beni (VXB) index, were employed to derive the optimum number of clusters and subsequently generate prospectivity maps. Normalized density indices were applied for quantitative evaluation of the classes of the FCM prospectivity maps. The quantitative evaluation of the results demonstrates that the higher favorability classes derived from VFS and VXB (Nd = 9.19) appear more reliable than those derived from VPc and VPe (Nd = 6.12) in detecting existing mineral deposits and defining new zones of potential Cu ± Au ± Fe mineralization in the study area.


Mineral prospectivity mapping Multifractal inverse distance weighting Fuzzy c-means clustering Clustering validation indices Normalized density index 


  1. Abedi, M., Norouzi, G. H., & Bahroudi, A. (2012). Support vector machine for multiclassification of mineral prospectivity areas. Computers & Geosciences, 46, 272–283.CrossRefGoogle Scholar
  2. Afzal, P., Alghalandis, Y. F., Khakzad, A., Moarefvand, P., & Omran, N. R. (2011). Delineation of mineralization zones in porphyry Cu deposits by fractal concentration–volume modeling. Journal of Geochemical Exploration, 108(3), 220–232.CrossRefGoogle Scholar
  3. Agterberg, F. P., & Bonham-Carter, G. F. (2005). Measuring the performance of mineral-potential maps. Natural Resources Research, 14(1), 1–17.CrossRefGoogle Scholar
  4. Aitchison, J. (1986). The statistical analysis of compositional data. London: Chapman & Hall.CrossRefGoogle Scholar
  5. Behroozi, A. (1987). Geological map of Iran 1:100,000 series, Feizabad (7760). Tehran: Geological Survey of Iran.Google Scholar
  6. Bezdek, J. C. (1973a). Cluster validity with fuzzy sets. Journal of Cybernetics, 3(3), 58–73.CrossRefGoogle Scholar
  7. Bezdek, J.C. (1973b.). Fuzzy mathematics in pattern classification. Ph.D. Thesis, Cornell University.Google Scholar
  8. Bezdek, J. C. (1981). Objective function clustering. In Pattern recognition with fuzzy objective function algorithms (pp. 43–93). Springer.Google Scholar
  9. Bezdek, J. C., Coray, C., Gunderson, R., & Watson, J. (1981). Detection and characterization of cluster substructure. I. Linear structure: Fuzzy c-lines. SIAM Journal of Applied Mathematics, 40(2), 339–357.CrossRefGoogle Scholar
  10. Bezdek, J. C., Ehrlich, R., & Full, W. (1984). FCM: The fuzzy c-means clustering algorithm. Computers & Geosciences, 10(2–3), 191–203.CrossRefGoogle Scholar
  11. Bonham-Carter, G. F. (1994). Geographic information systems for geoscientists-modeling with GIS. Computer Methods in the Geosciences (Vol. 13). Elsevier.Google Scholar
  12. Bonham-Carter, G. F., Agterberg, F. P., & Wright, D. F. (1989). Weights of evidence modeling: A new approach to mapping mineral potential. In F.P. Agterberg, G.F. Bonham-Carter (Eds.), Statistical applications in the earth sciences. Geological Survey of Canada Paper, 89(9), 171–183.Google Scholar
  13. Carranza, E. J. M. (2008). Geochemical anomaly and mineral prospectivity mapping in GIS. In Handbook of exploration and environmental geochemistry (Vol. 11). Elsevier.Google Scholar
  14. Carranza, E. J. M. (2009). Objective selection of suitable unit cell size in data-driven modeling of mineral prospectivity. Computers & Geosciences, 35(10), 2032–2046.CrossRefGoogle Scholar
  15. Carranza, E. J. M., Hale, M., & Faassen, C. (2002). Where are porphyry copper deposits spatially localized? A case study in Benguet province, Philippines. Natural Resources Research, 11(1), 45–59.CrossRefGoogle Scholar
  16. Cheng, Q. (1999). Multifractal interpolation. In Proceedings of the 5th annual conference of the international association of mathematical geology, Aug 1999, Trondheim, Norway (pp. 245–250).Google Scholar
  17. Cheng, Q. (2000). Interpolation by means of multiftractal, kriging and moving average techniques. GAC/MAC meeting of GeoCanada, May 2000, Calgary.Google Scholar
  18. Cheng, Q. (2007). Mapping singularities with stream sediment geochemical data for prediction of undiscovered mineral deposits in Gejiu, Yunnan Province, China. Ore Geology Reviews, 32(1–2), 314–324.CrossRefGoogle Scholar
  19. Clare, A. P., & Cohen, D. R. (2001). A comparison of unsupervised neural networks and k-means clustering in the analysis of multi-element stream sediment data. Geochemistry: Exploration, Environment, Analysis, 1(2), 119–134.Google Scholar
  20. Demir, N., Kaynarca, M., & Oy, S. (2016). Extraction of coastlines with fuzzy approach using SENTINEL-1 SAR image. International Archives in Photogrammetry, Remote Sensing and Spatial Information Sciences, 41, 747–751.CrossRefGoogle Scholar
  21. Egozcue, J. J., Pawlowsky-Glahn, V., Mateu-Figueras, G., & BarceloVidal, C. (2003). Isometric logratio transformations for compositional data analysis. Mathematical Geosciences, 35, 279–300.Google Scholar
  22. Fukuyama, Y., & Sugeno, M. (1989). A new method of choosing the number of clusters for the fuzzy c-mean method. In Proceedings of the 5th fuzzy systems symposium, Japanese Fuzzy System Association (pp. 247–250).Google Scholar
  23. Geva, A. B., Steinberg, Y., Bruckmair, S., & Nahum, G. (2000). A comparison of cluster validity criteria for a mixture of normal distributed data. Pattern Recognition Letters, 21(6–7), 511–529.CrossRefGoogle Scholar
  24. Ghezelbash, R., & Maghsoudi, A. (2018a). A hybrid AHP-VIKOR approach for prospectivity modeling of porphyry Cu deposits in the Varzaghan District, NW Iran. Arabian Journal of Geosciences, 11(11), 275.CrossRefGoogle Scholar
  25. Ghezelbash, R., & Maghsoudi, A. (2018b). Comparison of U-spatial statistics and C-A fractal models for delineating anomaly patterns of porphyry-type Cu geochemical signatures in the Varzaghan district, NW Iran. Comptes Rendus Geoscience, 350(4), 180–191.CrossRefGoogle Scholar
  26. Ghezelbash, R., Maghsoudi, A., & Carranza, E. J. M. (2019a). An improved data-driven multiple criteria decision-making procedure for spatial modeling of mineral prospectivity: adaption of prediction-area plot and logistic functions. Natural Resources Research, 28, 1299–1316.CrossRefGoogle Scholar
  27. Ghezelbash, R., Maghsoudi, A., & Carranza, E. J. M. (2019b). Performance evaluation of RBF-and SVM-based machine learning algorithms for predictive mineral prospectivity modeling: Integration of SA multifractal model and mineralization controls. Earth Science Informatics (pp. 1–17).Google Scholar
  28. Ghezelbash, R., Maghsoudi, A., & Carranza, E. J. M. (2019b). Mapping of single-and multi-element geochemical indicators based on catchment basin analysis: Application of fractal method and unsupervised clustering models. Journal of Geochemical Exploration, 199, 90–104.CrossRefGoogle Scholar
  29. Ghezelbash, R., Maghsoudi, A., & Daviran, M. (2018). Prospectivity modeling of porphyry copper deposits: Recognition of efficient mono-and multielement geochemical signatures in the Varzaghan district, NW Iran. Acta Geochimica, 38, 131–144.CrossRefGoogle Scholar
  30. Ghezelbash, R., Maghsoudi, A., & Daviran, M. (2019d). Combination of multifractal geostatistical interpolation and spectrum–area (S–A) fractal model for Cu–Au geochemical prospects in Feizabad district, NE Iran. Arabian Journal of Geosciences, 12(5), 152.CrossRefGoogle Scholar
  31. Ghezelbash, R., Maghsoudi, A., Daviran, M., & Yilmaz, H. (2019e). Incorporation of principal component analysis, geostatistical interpolation approaches and frequency-space-based models for portraying the Cu-Au geochemical prospects in the Feizabad district. Geochemistry: NW Iran.CrossRefGoogle Scholar
  32. Harris, D., & Pan, G. (1999). Mineral favorability mapping: A comparison of artificial neural networks, logistic regression, and discriminant analysis. Natural Resources Research, 8(2), 93–109.CrossRefGoogle Scholar
  33. Hu, D., Liu, D., & Xue, Sh. (1995). Explanatory text of geochemical map of Feizabad (7760). Tehran: Geological Survey of Iran.Google Scholar
  34. Jain, A. K., Murty, M. N., & Flynn, P. J. (1999). Data clustering: A review. ACM Computing Surveys (CSUR), 31(3), 264–323.CrossRefGoogle Scholar
  35. Karimpour, M. H., Saadat, S., & Malekzadeh-shafaroudi, A. (2003). Discovery of several Fe-oxides Cu-Au deposits along Khaf, Kashmar, and Bardaskan volcanic plutonic belt (Northestern Iran). In Abstract of the 21st symposium on geosciences, Geological Survey of Iran (pp. 144–145).Google Scholar
  36. Kohonen, T. (1984). Phonotopics maps insightful representation of phonological features of speech recognition. In Proceedings of the 7th international conference on pattern recognition, montreal (pp. 182–185).Google Scholar
  37. Kohonen, T. (1997). Exploration of very large databases by self-organizing maps. In Proceedings of International Conference on Neural Networks (ICNN’97) (Vol. 1, pp. PL1–PL6). IEEE.Google Scholar
  38. Liu, Y., Cheng, Q., & Zhou, K. (2019). New insights into element distribution patterns in geochemistry: A perspective from fractal density. Natural Resources Research, 28, 5–29.CrossRefGoogle Scholar
  39. Liu, Y., Zhou, K., & Cheng, Q. (2017). A new method for geochemical anomaly separation based on the distribution patterns of singularity indices. Computers & Geosciences, 105, 139–147.CrossRefGoogle Scholar
  40. Maghsoudi A., Rahmani, M., & Rashidi, B. (2005). Gold deposits and indications of Iran. Pars (Arian Zamin). Geology research center.Google Scholar
  41. Maulik, U., & Bandyopadhyay, S. (2002). Performance evaluation of some clustering algorithms and validity indices. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(12), 1650–1654.CrossRefGoogle Scholar
  42. Mazloumi, A. R., Karimpour, M. H., Rassa, I., Rahimi, B., & Vosoughi Abedini, M. (2008). Kuh-E-Zar gold deposit in Torbat-e-Heydaryeh new model of gold mineralization. Iranian Journal of Crystallography and Mineralogy, 16(30), 363–367.Google Scholar
  43. McLachlan, G. (2004). Discriminant analysis and statistical pattern recognition (Vol. 544). London: Wiley.Google Scholar
  44. Mihalasky, M. J., & Bonham-Carter, G. F. (2001). Lithodiversity and its spatial association with metallic mineral sites, Great Basin of Nevada. Natural Resources Research, 10, 209–226.CrossRefGoogle Scholar
  45. Moon, W. M. (1990). Integration of geophysical and geological data using evidential belief function. IEEE Transactions on Geosciences and Remote Sensing, 28(4), 711–720.CrossRefGoogle Scholar
  46. Pal, N. R., & Bezdek, J. C. (1995). On cluster validity for the fuzzy c-means model. IEEE Transactions on Fuzzy Systems, 3(3), 370–379.CrossRefGoogle Scholar
  47. Pal, N. R., & Bezdek, J. C. (1997). Correction to “on cluster validity for the fuzzy c-means model” [Correspondence]. IEEE Transactions on Fuzzy Systems, 5(1), 152–153.CrossRefGoogle Scholar
  48. Pan, G., & Harris, D. P. (2000). Information synthesis for mineral exploration (spatial information systems). Oxford: Oxford University Press.Google Scholar
  49. Parsa, M., Maghsoudi, A., & Ghezelbash, R. (2016). Decomposition of anomaly patterns of multi-element geochemical signatures in Ahar area, NW Iran: A comparison of U-spatial statistics and fractal models. Arabian Journal of Geosciences, 9(4), 260.CrossRefGoogle Scholar
  50. Parsa, M., Maghsoudi, A., & Yousefi, M. (2017a). An improved data-driven fuzzy mineral prospectivity mapping procedure; cosine amplitude-based similarity approach to delineate exploration targets. International Journal of Applied Earth Observations and Geoinformation, 58, 157–167.CrossRefGoogle Scholar
  51. Parsa, M., Maghsoudi, A., Yousefi, M., & Sadeghi, M. (2017b). Multifractal analysis of stream sediment geochemical data: Implications for hydrothermal nickel prospection in an arid terrain, eastern Iran. Journal of Geochemical Exploration, 181, 305–317.CrossRefGoogle Scholar
  52. Porwal, A., Carranza, E. J. M., & Hale, M. (2003). Knowledge-driven and data-driven fuzzy models for predictive mineral potential mapping. Natural Resources Research, 12(1), 1–25.CrossRefGoogle Scholar
  53. Ranjbar, H., Masoumi, F., & Carranza, E. J. M. (2011). Evaluation of geophysics and spaceborne multispectral data for alteration mapping in the Sar Cheshmeh mining area, Iran. International Journal of Remote Sensing, 32(12), 3309–3327.CrossRefGoogle Scholar
  54. Rantitsch, G. (2000). Application of fuzzy clusters to quantify lithological background concentrations in stream-sediment geochemistry. Journal of Geochemical Exploration, 71(1), 73–82.CrossRefGoogle Scholar
  55. Rezaee, M. R., Lelieveldt, B. P., & Reiber, J. H. (1998). A new cluster validity index for the fuzzy c-mean. Pattern Recognition Letters, 19(3–4), 237–246.CrossRefGoogle Scholar
  56. Sillitoe, R. H. (2010). Porphyry copper systems. Economic Geology, 105(1), 3–41.CrossRefGoogle Scholar
  57. Tangestani, M. H., & Moore, F. (2001). Comparison of three principal component analysis techniques to porphyry copper alteration mapping: A case study, Meiduk area, Kerman, Iran. Canadian Journal of Remote Sensing, 27(2), 176–182.CrossRefGoogle Scholar
  58. Taqadosi, H., & Malekzadeh Shafaroudi, A. (2018). Evidence for probable porphyry Cu-Au mineralization in the Namegh area, Northeast of Kashmar: Geology, alteration, mineralization, geochemistry, and fluids inclusion studies. Geosciences, Geological Survey of Iran, 108, 105–114.Google Scholar
  59. Templ, M., Filzmoser, P., & Reimann, C. (2008). Cluster analysis applied to regional geochemical data: Problems and possibilities. Applied Geochemistry, 23(8), 2198–2213.CrossRefGoogle Scholar
  60. Vriend, S. P., Van Gaans, P. F. M., Middelburg, J., & De Nijs, A. (1988). The application of fuzzy c-means cluster analysis and non-linear mapping to geochemical datasets: Examples from Portugal. Applied Geochemistry, 3(2), 213–224.CrossRefGoogle Scholar
  61. Xie, X. L., & Beni, G. (1991). A validity measure for fuzzy clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(8), 841–847.CrossRefGoogle Scholar
  62. Yousefi, M., & Carranza, E. J. M. (2015). Prediction–area (P–A) plot and C-A fractal analysis to classify and evaluate evidential maps for mineral prospectivity modeling. Computers & Geosciences, 79, 69–81.CrossRefGoogle Scholar
  63. Zekri, H., Cohen, D. R., Mokhtari, A. R., & Esmaeili, A. (2018). Geochemical prospectivity mapping through a feature extraction–selection classification scheme. Natural Resources Research, 27, 1–17.CrossRefGoogle Scholar
  64. Zekri, H., Mokhtari, A. R., & Cohen, D. R. (2019). Geochemical pattern recognition through matrix decomposition. Ore Geology Reviews, 104, 670–685.CrossRefGoogle Scholar
  65. Zuo, R., Carranza, E. J. M., & Wang, J. (2011). Support vector machine: A tool for mapping mineral prospectivity. Computers & Geosciences, 37(12), 1967–1975.CrossRefGoogle Scholar
  66. Zuo, R., Cheng, Q., Agterberg, F. P., & Xia, Q. (2009). Application of singularity mapping technique to identify local anomalies using stream sediment geochemical data, a case study from Gangdese, Tibet, western China. Journal of Geochemical Exploration, 101(3), 225–235.CrossRefGoogle Scholar
  67. Zuo, R., & Wang, J. (2016). Fractal/multifractal modeling of geochemical data: A review. Journal of Geochemical Exploration, 164, 33–41.CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2019

Authors and Affiliations

  • Mehrdad Daviran
    • 1
  • Abbas Maghsoudi
    • 2
    Email author
  • David R. Cohen
    • 3
  • Reza Ghezelbash
    • 2
  • Huseyin Yilmaz
    • 4
  1. 1.School of Mining, Petroleum and Geophysics EngineeringShahrood University of TechnologyShahroodIran
  2. 2.Faculty of Mining and Metallurgical EngineeringAmirkabir University of TechnologyTehranIran
  3. 3.School of Biological, Earth and Environmental SciencesUniversity of New South WalesSydneyAustralia
  4. 4.Department of Geological Engineering, Faculty of EngineeringDokuz Eylul UniversitesiBornova, IzmirTurkey

Personalised recommendations