Speeding up fuzzy c-means: using a hierarchical data organisation to control the precision of membership calculation

doi:10.1016/S0165-0114(01)00204-4

Fuzzy Sets and Systems

Volume 128, Issue 3, 16 June 2002, Pages 365-376

https://doi.org/10.1016/S0165-0114(01)00204-4 Get rights and content

Abstract

We examine the run-time behaviour of conventional fuzzy c-means implementations. Investigating into FCM termination conditions and membership update equations, we derive an approximative FCM that yields the same results as a conventional implementation within a given precision. We incorporate additional information about the data set by re-organizing the set as a tree. Our modification leads to an FCM algorithm with a significantly different run time behaviour; the gain of using the modified implementation increases with an increasing number of data objects and especially an increasing number of clusters, but is also sensitive to the chosen fuzzifier.

References (10)

R.N. Davé
Characterization and detection of noise in clustering
Pattern Recognition Lett.
(1991)
M.S. Kamel et al.
New algorithms for solving the fuzzy clustering problem
Pattern Recognition
(1994)
J.C. Bezdek
Pattern Recognition with Fuzzy Objective Function Algorithms
(1981)
J.C. Bezdek et al.
Fuzzy Models and Algorithms for Pattern Recognition and Image Processing
(1999)
R.L. Cannon et al.
Efficient implementation of the fuzzy c-means clustering algorithms
IEEE Trans. Pattern Anal. Machine Intelligence
(1986)

There are more references available in the full text version of this article.

Cited by (31)

Delineation of isotopic and hydrochemical evolution of karstic aquifers with different cluster-based (HCA, KM, FCM and GKM) methods
2022, Journal of Hydrology
In this work, a combination of isotopic and hydrogeochemical data of a karstic region was clustered with four distinct clustering analysis (CA) methods to study water evolution in a vulnerable karstic region to improve protection, sustainability, and enhanced water resource management. Four CA methods, including hierarchical cluster analysis (HCA), K-means (KM), and fuzzy logic CA methods, fuzzy C-mean (FCM), and genetic K‐means (GKM), have been utilized to analyze hydrochemical, chemical, and isotopic datasets, including dissolved inorganic carbon (DIC), δ¹³C-DIC, δ¹⁸O, and δ²H datasets of water resources of Paveh-Javanrud (PV-JR) karstic region, located at the western border of Iran and Iraq countries. The utilized dataset contains 34 water samples with varied origination to evaluate the performance of each model and find the best method based on a meaningful categorization of geological, hydrogeochemical, and isotopic characteristics. Finally, the best model results were matched graphically with developed geospatial graphs to visualize the correlation between the region's water resources. Accordingly, the FCM and GKM methods represent the same, yet meaningful results and have the best performance among the four methods. It was also identified that the PV-JR water resources could be generally categorized into five distinct clusters, including FC1 to FC5 and GK1 to GK5, of which two clusters that have mixing, two clusters with solo-origination and no sign of mixing, and finally, a seasonal spring which is categorized as a separate cluster. Potentially, studying water resources via theoretical methods combined with considering isotope hydrology is of particular interest since solving the environmental issues related to karstic regions and their water resource management are shared concerns in most arid and semi-arid countries, especially in the Middle East as this study, thus could lay a basis for the following scientific attempts involving hydrogeochemical studies and advanced statistical analysis.
The combined use of self-organizing map technique and fuzzy c-means clustering to evaluate urban groundwater quality in Seoul metropolitan city, South Korea
2019, Journal of Hydrology
To make an overall assessment of the groundwater quality in Seoul city, we used the self-organizing map (SOM) technique in combination with fuzzy c-means (FCM) clustering. SOM visualizes complicate and multidimensional data structures on a 2D surface while the FCM algorithm creates overlapping cluster boundaries among samples that are continuously distributed over a data space. The combination of SOM and FCM clustering was expected to help characterize highly complicated urban groundwater quality. As a result, the SOM characterized 343 groundwater samples using 91 neurons, which were further classified by FCM clustering into three water groups. Group 1 addressed the least polluted groundwater (17% of the samples (n = 58), average TDS = 194.5 mg/L and NO₃ = 6.9 mg/L) and occurred in the peripheral areas whose land cover is mainly occupied by forests. Increasing pH with increasing sodium and bicarbonate concentrations indicated that the hydrogeochemistry of Group 1 was largely controlled by water-rock interactions. Group 2 included the highly polluted groundwater (24% of the samples (n = 82), average TDS = 326.2 mg/L and NO₃ = 42.6 mg/L), and sporadically occurred in Seoul, with no distinct spatial control. This group seemed to be affected by sewage from broken sewer pipes, which are a primary pollution source of Seoul groundwater and are ubiquitously distributed beneath the city. Group 3 water also represented the highly contaminated groundwater (30% of the samples (n = 103), average TDS = 527.1 mg/L), but contained low nitrate concentrations (average NO₃ = 13.1 mg/L). Based on their spatial locations, intensive groundwater pumping from subway tunnels and other underground spaces at the city center seemed to drive the induced flow of organic contaminants, resulting in local reducing conditions sufficient for denitrification. The remaining 100 samples (29% of the samples) shared the hydrogeochemical properties of two or three groups. This study successfully characterized the spatial pattern of urban groundwater quality that is complicated by various contamination sources and hydrogeochemical processes. The combined use of SOM and FCM clustering was proven as a powerful tool to interpret nonlinear and highly heterogeneous environmental data for which it is difficult to define cluster boundaries. Taken together, our results contribute to a better management of urban groundwater in metropolitan cities under high risks of anthropogenic contamination.
Identification of homogeneous regions for regionalization of watersheds by two-level self-organizing feature maps
2014, Journal of Hydrology
One of the several methods in estimating flood quantiles in ungauged or data-scarce watersheds is regional frequency analysis. Amongst the approaches to regional frequency analysis, different clustering techniques have been proposed to determine hydrologically homogeneous regions in the literature. Recently, Self-Organization feature Map (SOM), a modern hydroinformatic tool, has been applied in several studies for clustering watersheds. However, further studies are still needed with SOM on the interpretation of SOM output map for identifying hydrologically homogeneous regions. In this study, two-level SOM and three clustering methods (fuzzy c-mean, K-mean, and Ward’s Agglomerative hierarchical clustering) are applied in an effort to identify hydrologically homogeneous regions in Mazandaran province watersheds in the north of Iran, and their results are compared with each other. Firstly the SOM is used to form a two-dimensional feature map. Next, the output nodes of the SOM are clustered by using unified distance matrix algorithm and three clustering methods to form regions for flood frequency analysis. The heterogeneity test indicates the four regions achieved by the two-level SOM and Ward approach after adjustments are sufficiently homogeneous. The results suggest that the combination of SOM and Ward is much better than the combination of either SOM and FCM or SOM and K-mean.
Assessment of the impact of anthropogenic activities on the groundwater hydrology and chemistry in Tarsus coastal plain (Mersin, SE Turkey) using fuzzy clustering, multivariate statistics and GIS techniques
2012, Journal of Hydrology
Tarsus coastal plain (TCP) is an economically and ecologically important area situated in between the fertile fluvio-deltaic plains of two rivers, Deliçay and Tarsus (Mersin, SE Turkey), where anthropogenic activities (agricultural, industrial, and domestic) are very intense. Twenty-four water quality parameters were surveyed at 193 groundwater and 10 surface water sites during August 2008. The objective was to characterize the physico-chemical properties of groundwaters in TCP, assess the impact of anthropogenic activities on the groundwater hydrology and chemistry, and identify the major hydrogeochemical processes occurring in the area. Groundwater samples were grouped into hydrochemically distinct and spatially continuous four water classes (i.e., C1, C2, C3, and C4) using the fuzzy c-means (FCM) clustering method, where membership values were interpolated using the ordinary kriging technique. Principal components analysis (PCA) was used to decipher various underlying natural and anthropogenic processes creating these distinct water classes. Four principal components (PCs) were extracted in PCA which explained more than 73% of the total variance in water quality. Major factors responsible for the variations in chemistries of water classes are identified as: (1) water–rock interaction and nitrate contamination; (2) salinization by seawater intrusion and evaporite dissolution; (3) geogenic/anthropogenic Cr, Fe, and Mn; and (4) anthropogenic Zn pollution. Overexploitation of the aquifer is clearly evident, especially at settlements located near the coastal zone, where the water table is lowered 2–5 m below the sea level. Salinization is well known in the area and is attributed not only to seawater intrusion, but also to dissolution of evaporitic series from the Handere formation. Hydrochemical evidence also suggest that in the area subsurface paleo-river channels and the deposits infilling the ancient lagoon area within Quaternary–Recent alluvial deposits act as significant hydrological features where preferential groundwater flow occurs along them.
Single point iterative weighted fuzzy C-means clustering algorithm for remote sensing image segmentation
2009, Pattern Recognition
In this paper, a remote sensing image segmentation procedure that utilizes a single point iterative weighted fuzzy C-means clustering algorithm is proposed based upon the prior information. This method can solve the fuzzy C-means algorithm's problem that the clustering quality is greatly affected by the data distributing and the stochastic initializing the centrals of clustering. After the probability statistics of original data, the weights of data attribute are designed to adjust original samples to the uniform distribution, and added in the process of cyclic iteration, which could be suitable for the character of fuzzy C-means algorithm so as to improve the precision. Furthermore, appropriate initial clustering centers adjacent to the actual final clustering centers can be found by the proposed single point adjustment method, which could promote the convergence speed of the overall iterative process and drastically reduce the calculation time. Otherwise, the modified algorithm is updated from multidimensional data analysis to color images clustering. Moreover, with the comparison experiments of the UCI data sets, public Berkeley segmentation dataset and the actual remote sensing data, the real validity of proposed algorithm is proved.
A hybrid fuzzy-optimization approach to customer grouping-based logistics distribution operations
2007, Applied Mathematical Modelling
This paper presents an integrated fuzzy-optimization customer grouping based logistics distribution methodology for quickly responding to a variety of customer demands. The proposed methodology involves three main mechanisms: (1) pre-route customer classification using fuzzy clustering techniques, (2) determination of customer group-based delivery service priority and (3) en-route goods delivery using multi-objective optimization programming methods. In the process of pre-route customer classification, the proposed method groups customers’ orders primarily based on the multiple attributes of customer demands, rather than by static geographic attributes, which are mainly considered in classical vehicle routing algorithms. Numerical studies including a real-world application are conducted to illustrate the applicability of the proposed method and its potential advantages over existing operational strategies. Using the proposed method, it is shown that the overall performance of a logistics distribution system can be improved by more than 20%, according to the numerical results from the case studied.

View all citing articles on Scopus

^☆: This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant Kl 648/1-1.

View full text

Speeding up fuzzy c-means: using a hierarchical data organisation to control the precision of membership calculation☆

Abstract

Pattern Recognition Lett.

Pattern Recognition

Pattern Recognition with Fuzzy Objective Function Algorithms

Fuzzy Models and Algorithms for Pattern Recognition and Image Processing

Efficient implementation of the fuzzy c-means clustering algorithms

IEEE Trans. Pattern Anal. Machine Intelligence