barcode generator in vb net free download STEP 4: EXECUTE THE CLUSTER METHOD in Software

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15.4.4 STEP 4: EXECUTE THE CLUSTER METHOD
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The k-means method, a nonhierarchical clustering procedure was applied to group the SIC code groups. The k-means method applies an iterative process to nd the optimal clustering for a speci ed number of groups (k). The k-means clustering method splits a set of objects into a selected number of groups by maximizing between-cluster variation (SSA) relative to within-cluster variation (SSE). The following calculations are used to calculate the sums of squares; the nomenclature of the cluster matrix is shown below as well:
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WASTE GROUPS (CLUSTERS)
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Attribute 1
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k yk1 yk2 ykn
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SST = (yij y ..)2 = total sum of squares
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SSA = n (yi . y ..)2 = within cluster sum of squares
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i =1
SSE = (yij yi .)2 = between cluster sum of squares
i =1 j =1
SST = SSA + SSE
SOLID WASTE CHARACTERIZATION BY BUSINESS ACTIVITIES
where k = SIC code groups number n = number of attributes for the SIC code groups yij = the matrix value for attribute i and SIC code group k y.. = mean of all standardized attributes yi . = mean of standardized attributes for SIC code group k
It is similar to doing a one-way analysis of variance where the groups are unknown and the largest F value is sought by reassigning members to each group (Norusis, 1986). The k-means method starts with one cluster and splits it into two clusters by picking the case farthest from the center as a seed for a second cluster and assigning each case to the nearest center. It continues splitting one of the clusters into two (and reassigning cases) until a speci ed number of clusters are formed. The k-means method reassigns cases until the within-groups sum of squares can no longer be reduced (Norusis, 1986). The k-means method was made possible by the high speed of computer processing available. The k-means method is a rigorous procedure that evaluates all permutations to minimize SSE and maximize SSA. The software program SYSTAT, developed by SPSS, Inc. was used to perform the multivariate cluster analysis. The drawback of this method is determining the number of clusters to use (k). This was handled by applying a variance analysis technique (Everitt, 1980). Thorndike plotted average within cluster distance (SSA/k) against the number of groups (k). With every increase in k, there will be a decrease in this measurement, but Thorndike suggested that a sudden marked attening of the curve at any point indicated a distinctive, correct value for k (Everitt, 1980). Such a point should occur when the number of groups corresponds to the con guration of points and there is relatively little gain from further increase in k. Applying the k-means method to all possible optimal grouping for every k (2 through 65) and graphing the results of the Thorndike method, a k = 22 groups was determined as the optimum. Table 15.3 and graphs in Fig. 15.8 display the results.
TABLE 15.3 CLUSTERS
ANOVA TABLE USED TO DETERMINE OPTIMAL NUMBER OF
NUMBER OF WASTE GROUPS (CLUSTERS)
SSA (BETWEEN)
SSE (WITHIN)
AVERAGE (SSE /K)
2 3 4 5 6 7
280 421 570 695 832 977
1461 1320 1171 1046 909 764
730.50 440.00 292.75 209.20 151.50 109.14
(Continued )
MULTIVARIATE CLUSTER ANALYSIS AND DISCUSSION
TABLE 15.3 ANOVA TABLE USED TO DETERMINE OPTIMAL NUMBER OF CLUSTERS (CONTINUED) NUMBER OF WASTE GROUPS (CLUSTERS)
SSA (BETWEEN)
SSE (WITHIN)
AVERAGE (SSE /K)
8 9 10 11 12 13 14 15 16 17 18 19 20 21 22* 23 24 ... 35
1084 1130 1260 1321 1402 1432 1488 1570 1590 1617 1644 1658 1676 1690 1700 1703 1707 ... 1726
657 611 481 420 339 309 253 171 151 124 97 83 65 51 41 38 34 ... 15
82.13 67.89 48.10 38.18 28.25 23.77 18.07 11.40 9.44 7.29 5.39 4.37 3.25 2.43 1.86 1.65 1.42 ... 0.43
* Using the Thorndike method, 22 waste groups (k = 22) is the optimal cluster number. This may be observed from the relatively little gain in average variance within groups from increases in k. The Thorndike method involves plotting average within cluster variation (noise) against the number of groups. Thorndike suggested that a marked attening of the curve at any point indicates the optimal for k (please see the next page for graphs of the curve, with the optimal point identi ed).
As shown from the statistical analysis in Table 15.4 and multivariate cluster analysis, 22 waste groups is the optimal number of clusters to balance variation between groups and the number of groups. A sum of squares between groups (variation between groups) of 1700.21 and a sum of squares within groups (noise) of 40.96 was achieved using the clustering techniques. The sum of square calculations may be found in Sec. 16.3. The next section discusses the results of the cluster analysis and details of the 22 waste groups.
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