### Table 1: Types for crisp spatial data

2002

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### Table 5 Cluster validity values for the AD-3D data set. Optimal cluster number is 4. c

"... In PAGE 6: ... Three of the clusters were placed adjacent to each other, and one cluster was separated from them. Table5 shows the results obtained using the various validity indexes with re- spect to c = 2, 3,.... ..."

### Table 3 Example dataset: crisp decisions

"... In PAGE 20: ...Table 3 Example dataset: crisp decisions To illustrate the operation of the techniques involved, an example dataset is given in Table3 . The table contains three real-valued conditional attributes and a crisp-valued decision attribute.... In PAGE 24: ... Again, this tends not to be applied to larger datasets as the cost of evaluating these larger feature subsets is too great. Application to the Example Dataset Figure 13 Fuzzifications for conditional features Using Table3 and the fuzzy sets defined in Figure 13 (for all conditional attributes), and setting A={a}, B={b}, C={c} and Q={q}, the following equivalence classes are obtained: U/A = {Na, Za} U/B = {Nb, Zb} U/C = {Nc, Zc} U/Q = {{1,3,6}, {2,4,5}} The first step is to calculate the lower approximations of the sets A, B and C, using Equation 20. To clarify the calculations involved, Table 4 contains the membership degrees of objects to fuzzy equivalence classes.... In PAGE 32: ... Application to the Example Dataset To illustrate the operation of the tolerance QUICKREDUCT algorithm, it is applied to the example data given in Table3 . For this example, the similarity measure is the same as that given in Equation 27 and Equation 28 for all conditional attributes, with = 0.... ..."

### Table 1: Crisp, Soft and Hardened 2-partitions of n objects

"... In PAGE 2: ... Each column of U sums to 1, and each row of U must have at least one non- zero entry. Table1 contains an example of crisp (U1) and soft (U2) 2- partitions of n=6 objects. U1 identifies 3 objects in each of the two crisp clusters.... In PAGE 2: ... We denote the hardening of U by H(U). The last column in Table1 has an example for the hardening of U2. If we compare H(U2) to U1, we see that there is one mismatch (object 3).... ..."

### Table 4: Experimental results for 3D and METIS data distributions

"... In PAGE 23: ...5 was implemented, which was based on graph distribution algorithms im- plemented in METIS library [23]. In Table4 , the results of computations are presented and the two strategies of data distribution are compared. The computations were performed on Virgo cluster of computers, but in this case a Gigabit Ethernet network is used.... ..."

### Table 2: Elliptical body phantom data 2D and 3D Ellipse 2D Ellipse 3D

"... In PAGE 53: ...Chapter H. Fast Curve Estimation Using Pre-Conditioned Generalized Radon Transform The estimated parameter vectors are given in Table2 , where the clustering process has used a value of = 1:5. As seen from Table 2, eight groups of curve parameters are found, and the estimated curve parameters are rather close to the true parameters.... In PAGE 53: ... Fast Curve Estimation Using Pre-Conditioned Generalized Radon Transform The estimated parameter vectors are given in Table 2, where the clustering process has used a value of = 1:5. As seen from Table2 , eight groups of curve parameters are found, and the estimated curve parameters are rather close to the true parameters. Curve Number Parameter 1 True (0:202 s;2:7027 10?4 s=m) 1 Estimated (0:200 s;2:705 10?4 s=m) 2 True (0:202 s;6:2500 10?4 s=m) 2 Estimated (0:200 s;6:261 10?4 s=m) 2 Estimated (0:204 s;6:245 10?4 s=m) 3 True (0:232 s;2:5000 10?4 s=m) 3 Estimated (0:232 s;2:503 10?4 s=m) 4 True (0:232 s;6:2500 10?4 s=m) 4 Estimated (0:232 s;6:245 10?4 s=m) 4 Estimated (0:236 s;6:245 10?4 s=m) 5 True (0:902 s;5:0000 10?4 s=m) 5 Estimated (0:904 s;4:982 10?4 s=m) 6 True (0:932 s;4:5455 10?4 s=m) 6 Estimated (0:932 s;4:530 10?4 s=m) 6 Estimated (0:932 s;4:546 10?4 s=m) 7 True (1:302 s;2:7027 10?4 s=m) 7 Estimated (1:300 s;2:721 10?4 s=m) 7 Estimated (1:300 s;2:737 10?4 s=m) 8 True (1:302 s;2:5000 10?4 s=m) 8 Estimated (1:304 s;2:456 10?4 s=m) 8 Estimated (1:304 s;2:472 10?4 s=m) Table 2: The results of the FCE-algorithm along with the true curve parameters.... In PAGE 53: ... As seen from Table 2, eight groups of curve parameters are found, and the estimated curve parameters are rather close to the true parameters. Curve Number Parameter 1 True (0:202 s;2:7027 10?4 s=m) 1 Estimated (0:200 s;2:705 10?4 s=m) 2 True (0:202 s;6:2500 10?4 s=m) 2 Estimated (0:200 s;6:261 10?4 s=m) 2 Estimated (0:204 s;6:245 10?4 s=m) 3 True (0:232 s;2:5000 10?4 s=m) 3 Estimated (0:232 s;2:503 10?4 s=m) 4 True (0:232 s;6:2500 10?4 s=m) 4 Estimated (0:232 s;6:245 10?4 s=m) 4 Estimated (0:236 s;6:245 10?4 s=m) 5 True (0:902 s;5:0000 10?4 s=m) 5 Estimated (0:904 s;4:982 10?4 s=m) 6 True (0:932 s;4:5455 10?4 s=m) 6 Estimated (0:932 s;4:530 10?4 s=m) 6 Estimated (0:932 s;4:546 10?4 s=m) 7 True (1:302 s;2:7027 10?4 s=m) 7 Estimated (1:300 s;2:721 10?4 s=m) 7 Estimated (1:300 s;2:737 10?4 s=m) 8 True (1:302 s;2:5000 10?4 s=m) 8 Estimated (1:304 s;2:456 10?4 s=m) 8 Estimated (1:304 s;2:472 10?4 s=m) Table2 : The results of the FCE-algorithm along with the true curve parameters. The width of the cluster band is set to 3, i.... In PAGE 73: ... Slice Normalized Variance Figure 5 Normalized variance of Case KL (2D/3D), case MN (2D/3D), and Brain phantom (2D/3D) as a function of NEC per slice. Table2 shows noise estimates obtained from di erent ROIs. The emission parameter a and the transmission parameter c have been estimated for the Brain Phantom (3D) in four di erent ROIs.... In PAGE 73: ... Note the approximately constant ra- tio between a and c, indicating the major in uence of the average on the coe cients. Table2 Brain phantom data (3D) with di erent ROIs. ROI a [k counts] c [sec] Average Head 17 1.... ..."

### Table 1. Tabular method for finding the shortest path in the crisp case.

"... In PAGE 11: ... First, we put the edge weights from Fig. 3 into the correct entries in Table1 , i.... In PAGE 12: ... We look for inequalities that satisfy (19). Clearly, any row with two or more entries in Table1 entails an inequality. Hence, we get rows i =1,2, and 5 and obtain the following inequalities: when i =1,c12 + f(2) lt; c13 + f(3), i.... In PAGE 13: ... The tabular method for finding the shortest path in a fuzzy network is given in Table 2. Using the same approach as in Table1 , the shortest path in the fuzzy sense obtained from ) 1 ( * f = c12 +f(2) = * 12 c+* 25 c+ ) 5 ( * f = * 12 c+* 25 c+* 58 c+ ) 8 ( * f = * 12 c+* 25 c+* 58 c ,is 1, 2, 5, 8 with length 13.575.... In PAGE 17: ...ries of the Table. Next, we calculate f(j) using the same tabular method as in Example 2. The tabular method for finding the shortest path in the fuzzy sense is shown in Ta- ble 3. Using the same approach as in Table1 , the shortest path based on statistical data obtained from ) 1 ( f = 12 c+ ) 2 ( f = 12 c+25 c+ ) 5 ( f = 12 c+25 c+58 c+ ) 8 ( f = 12 c+25 c+ 58 c , is 1, 2, 5, 8 with length 13.27.... In PAGE 18: ...s constructed, as shown in Fig. 7. The tabular method for finding the shortest path is shown in Table 4. Using the same approach as in Table1 , the shortest path in the fuzzy sense obtained from ) 1 ( o f = o 12 c + ) 2 ( o f = o 12 c+o 25 c+ ) 5 ( o f = o 12 c+o 25 c + o 58 c+ ) 8 ( o f = o 12 c+o 25 c+o 58 c , is 1, 2, 5, 8 with length 13.3855.... ..."

### Table 2. Some observed patterns obtained from 2D and 3D cluster plots

"... In PAGE 4: ... This can be achieved by 2D and 3D cluster plots. We have found that there are certain definite patterns for some categories as summarized in Table2 . Fig.... In PAGE 5: ...quantization scales and finally choose a 10-interval scale which is the coarsest scale that can still distinguishes each data point. The plots for each triplet of features (with a* and b* being always kept together) re-confirm the observations summarized in Table2 . Fig.... ..."

### Table 2. Test results of classi cation of 8 vowel data clusters projected from 12 dimensions to 2D and 3D.

### Table 2. Crisp and fuzzy information in systems. system input data resulting mathematical framework

"... In PAGE 8: ... A fuzzy system can simultaneously have several of the above attributes. Table2 gives an overview of the relationships between fuzzy and crisp system descriptions and variables. In this text we will focus on the last type of systems, i.... ..."