Fuzzy k-means with extragrades and Noise Clustering
This is the original paper by De Gruijter, J.J., McBratney, A.B., 1988. A modified fuzzy k means for predictive classification. In: Bock,H.H.(ed) Classification and Related Methods of Data Analysis. pp. 97-104. Elsevier Science, Amsterdam.
Paper in pdf (1.3 MB)
The idea of fuzzy k-means with extragrades came from the modification of Ohashi's method, which was presented at SUGI 84:
Ohashi. Y. Fuzzy clustering and
robust estimation. Presentation at the 9th meeting SAS User's Group International. Hollywood
Beach. Florida. 18-21 March 1984.
This paper is never published and cannot be found in the 1984 SUGI proceedings. De Gruijter and McBratney (1988) is the first paper that quoted and listed in detail Ohashi's method. Although Ohashi's paper is being quoted by several papers, I believe not many have read the original paper.
Prof.. Yasuo Ohashi is now at Department of Biostatistics, School of Public Health, University of Tokyo.
Although fuzzy k-means with extragrades is known in Soil Science, it is virtually unknown in the Fuzzy sets & systems literature. However you can find a similar approach called noise clustering or robust clustering, which appeared in:
Dave. RN. Characterization and detection of noise in clustering. Pattern Recognition Letters 12 (1991) 657-664.
Dave RN and Krishnapuram K. Robust Clustering Methods: A Unified View. IEEE TRANSACTIONS ON FUZZY SYSTEMS 5 (1997) 270-293.
The following is an extract from the Dave & Krishnapuram paper:
Here is the cited in [12] reference:
Hey..it mentioned Ohashi (as cited in [12]). How about the De Gruijter & McBratney's method? Obviously they have read the paper.
Eq (10) above, is the same as De Gruijter & McBratney's (1988) solution for fuzzy k-means with extragrades.
We can see that Eq (9) in the
Dave & Krishnapuram is the same as Ohashi's objective function:
And the solution for Ohashi's function is:
which is not the same as Eq. (10) of Dave & Krishnapuram.
The derivation of the objective function minimization to get the membership function for fuzzy k-means with extragrades can be found here.
This is from a CSIRO Division of Soils Report on the program MacFuzzy.
A.W. Ward, W.T. Ward, A.B. McBratney and J.J. De Gruijter, MacFuzzy: A Program for Data Analysis by Fuzzy k-Means. , CSIRO Division of Soils, Glen Osmond, S.A., (1992).
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