K-means clustering

L. Kiernan, J.D. Mason and K. Warwick, Robust initialisation of Gaussian radial basis function networks using partitioned k-means clustering. Electron. Lett., 32 (1996) 671-672. 

The objective functions for both k-means clustering and the F-nearest neighbor heuristic given by Eqs. (20) and (21) use information only from the inputs. Because of this capacity to cluster data, local methods are particularly useful for data interpretation when the clusters can be assigned labels. 

Beauchaine, T. P., Beauchaine, R. J. (2002). A comparison of maximum covariance and k-means cluster analysis in classifying cases into known taxon groups. Psychological Methods, 7, 245-261. 

Figure 6.8 shows clustering results for a synthetic data example in two dimensions with three groups. The plot symbols indicate the result of k-means clustering for k 2 (left), k 3 (middle), and k 4 (right). Here it is obvious that the choice k = 3 gave the best result since it directly corresponds to the visually evident data groups. 

Cluster Algorithm. The Forgy variety of k-means cluster analysis ( ) is chosen because of its speed for large data sets. Forgy k-means cluster analysis is an iterative process. In the first iteration observations are assigned to the nearest centroid. This defines the initial clusters. The composition of the observations in each cluster are then averaged to find approximate centroids. Let xk be the centroid vector for cluster k. with components xkj. for... 

K-means cluster analysis is an excellent method for the reduction of individual-partide datai if extra clusters are used to allow for the non-spherical shape and natural variability of atmospheric particles. The "merge" method for choosing seedpoints is useful for detecting the types of lew abundance particles that are interesting for urban atmospheric studies. Application to the Phoenix aerosol suggests that the ability to discriminate between various types of crustal particles may yield valuable information in addition to that derived from particle types more commonly associated with anthropogenic activity. 

Prediction of BOD value. In the ten clusters Identified by the K-means clustering procedure, two clusters were represented by chemicals with only low BOD values and one cluster with nearly all (18 of 19 or 95 %) high BOD values (Table III). Therefore, no discrimination was attempted within these clusters. In the remaining clusters there were 202 high BOD chemicals and 97 low BOD chemicals. Of these, approximately 75 % (152 of 202) were correctly classified Into the high BOD class, while 73 Z (71 of 97) were correctly classified Into the low BOD class. Using both the clustering and discrimination analyses, 77 % (170 of 220) and 78 % (93 of 120) of the chemicals In the data base were correctly classified. Within each of the clusters, between 2 and 4 molecular connectivity Indices were used In the final discriminant functions to separate the two classes of BOD. Within each cluster a different combination of variables were used as discriminators. Because of Che exploratory nature of this analysis, we lowered the F-ratlo Inclusion level Co 1.0. In several of the clusters, the F-ratlos for variables Included In Che discriminant functions were subsequently small(e.g., < 4.0). 

Principal component analysis (PCA) was employed to identify the underlying motive dimensions. K-means cluster analysis was used to classify the respondents according to their travel experience levels. A number of checks on the importance of the items contributing to the travel experience levels were also conducted (Lee Pearce, 2002). The factor scores from the PCA analysis results were computed and independent t-tests were... 

In this research the concept of a travel career is viewed as the combined level and stage of travel experience, life-stage, and age (see Table 3.2). K-means cluster analysis was used to identify the different travel career levels. Travel experience level, life-stage and age variables were included in the analysis. As age was measured in an open format in the questionnaire, the responses were classified into five groups with each group containing similar number of respondents. The five age groups were under 26,26-30, 31-40,41-50, and over 50. 

The algorithm originally proposed by Moody and Darken (1989) uses k-means clustering to determine the centers of the clusters. The hypersphere around each cluster center is then determined to ensure sufficient overlap between the clusters for a smooth fit by criteria such as the P-nearest neighbor heuristic,... 

The selection of cluster number, which is generally not known beforehand, represents the primary performance criterion. Optimization of performance therefore requires trial-and-error adjustment of the number of clusters. Once the cluster number is established, the neural network structure is used as a way to determine the linear discriminant for interpretation. In effect, the RBFN makes use of known transformed features space defined in terms of prototypes of similar patterns as a result of applying k-means clustering. 

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