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Identifying Training Tuples in n-Dimensional Ptrees Using Vote Histograms

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This document outlines a method for identifying training tuples within a 0-ring around a specific point using vote histograms derived from Ptrees. With the focus on exact matches and the usage of 1-bit representations, the analysis seeks to determine the presence of training points in various rings, up to the 2-ring level. The proposed algorithms leverage complement notation and structured evaluations to ascertain the vote counts for specific configurations, ultimately providing insights into training data distributions.

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Identifying Training Tuples in n-Dimensional Ptrees Using Vote Histograms

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  1. 0 1 Vote histogram (so far) UF NNC Ptree Ex. 1 using 0-D Ptrees(sequences) a=a5 a6 a1’a2’a3’a4’=(000000) Identifying all training tuples in the distance=0 ring or 0ring, centered at a (exact matches ) as1-bitsof the Ptree, P=a5^a6^a1’^a2’^a3’^a4’ (we use _ for complement) There are no training points in a’s0ring! We must look further out, i.e., a’s1ring a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  2. (a5 a6a1’a2’a3’a4’) 0 1 OR UF NNC Ptree ex-1 (cont.) a’s 1ring? a=a5 a6 a1’a2’a3’a4’ = (000000) (001000) (100000) (010000) (000001) (000100) (000010) Training pts in the 1ring centered at a are given by 1-bits in the Ptree, P, constructed as follows: a5^a6^a1’^a2’^a3’^a4’ a5^a6^a1’^a2’^a3’^a4’ a5^a6^a1’^a2’^a3’^a4’ a5^a6^a1’^a2’^a3’^a4’ a5^a6^a1’^a2’^a3’^a4’ a5^a6^a1’^a2’^a3’^a4’ a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 The C=1 vote count = root count of P^C. a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 The C=0 vote count = root count of P^C. (never need to know which tuples voted) a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 C 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 P 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  3. a’s 2-ring? a=a5 a6 a1’a2’a3’a4’ = (000000) 0 1 (101000) (100010) (100100) (110000) (100001) For each of the following Ptrees, a 1-bit corresponds to a training point in a’s 2-ring: Pa5a6a1’a2‘a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6a1‘a2’a3‘a4‘Pa5a6a1‘a2‘a3’a4‘Pa5a6a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3‘a4’ 1st line first: a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 Stop here? But the other 10 Ptrees should also be considered. The fact that the 2-ring includes so many new training points is “The curse of demensionality”. a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5

  4. Enfranchising the rest of a’s 2-ring? a=a5 a6 a1’a2’a3’a4’ = (000000) 0 1 For each of the following Ptrees, a 1-bit corresponds to a training point in a’s 2-ring: Pa5a6a1’a2‘a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6a1‘a2’a3‘a4‘Pa5a6a1‘a2‘a3’a4‘Pa5a6a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3‘a4’ 2nd line: a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5

  5. Enfranchising the rest of a’s 2-ring (cont.) a=a5 a6 a1’a2’a3’a4’ = (000000) 0 1 For each of the following Ptrees, a 1-bit corresponds to a training point in a’s 2-ring: Pa5a6a1’a2‘a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘ Pa5a6a1‘a2’a3‘a4‘Pa5a6a1‘a2‘a3’a4‘Pa5a6a1‘a2‘a3‘a4’ Pa5a6 a1‘a2’a3‘a4‘Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3’a4‘Pa5a6 a1‘a2‘a3‘a4’ Pa5a6 a1‘a2‘a3‘a4’ 3rd line: a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5

  6. Pts in the 47-disk, each Gets at least a vote of 1/24 Pts in the 47-disk, each Gets at least a vote of 1/12 Pts in the 47-disk, each Gets at least a vote of 1/48 PNNCvote = 1/(1/d) d(p,q) = {wi : p & q differ at i; i in the relevant_attribute_set} One way to address the curse of dimensionality is to require that all relevant attribute weights be different (except for small groups (3?) of equally weighted attributes, and that the next weight always be larger than the previous weight-sum. a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 Weights: 2 12 2 6 24 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 Vote Weight .02 .02 .02 .02 .02 .02 .02 .02 .02 .02 Vote Weight .02 .06 .06 .06 .06 .02 .02 .02 .06 .02 Vote Weight .02 .12 .12 .12 .06 .02 .02 .02 .12 .02 0-ring at 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1 0 1 1 0 1 1 0 0 0 0 0 1 1 0 0 1 1 0 1

  7. PNNCusing weights (about the only way to address the curse of dimensionality) “Gaussian” type of vote weighting = 1/e-dis2 d(p,q) = {weighti : p & q differ at i} a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 1 5 2 2 9 4 4 9 9 2 1 6 6 9 9 9 9 9 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  8. a’s 1ring? a=a5 a6 a1’a2’a3’a4’ = (010010) (110010) (000010) attribute weights (1, 1, 3, 3, 3, 3) vote weight = 1/(1+distance) d(p,q) = {weighti : p & q differ at i} a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  9. a’s 2ring? a=a5 a6 a1’a2’a3’a4’ = (010010) (100010) Distance fctn: d(p,q) = {weighti : p & q differ at i} vote function: vote = 1/(1+distance) a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 P 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  10. Appendix: scratch slides a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a4 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a7 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a8 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a9 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a5‘ 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 a6‘ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 a7‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a8‘ 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 a9‘ 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 d1 d2 t1 t2 t1 t3 t1 t5 t1 t6 t2 t1 t2 t7 t3 t1 t3 t2 t3 t3 t3 t5 t5 t1 t5 t3 t5 t5 t5 t7 t6 t1 t7 t2 t7 t5 C' 1 1 0 0 1 0 1 1 1 0 1 1 0 0 1 1 0 a2 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a3 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1

  11. Appendix: scratch slides a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a4‘ 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a3‘ 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a2‘ 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a1‘ 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a6 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0

  12. Appendix: scratch slides a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a4‘ 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a3‘ 0 1 1 0 0 0 0 0 1 1 0 1 1 0 0 0 1 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a2‘ 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a1‘ 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a6 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1

  13. Appendix: Comprehensive Attribute types example R( a0, a1, a2, a3, a4, a5, B6,B7, C8,C9,C) Class Label Bit-maps for categorical values from, possibly several, flat categorical attributes. Numeric attributes: domains {0..7},{0..3} Hierarchical categorical (e.g., leaf_weights; inode_weights=sums) C8 C9 dairy sundries / | \ / | \ milk egg butter crafts knits toys / \ needles pins 3 5 1 2 1 1 1 2 1 1 Assume bfr=3, so RID = (3quotient,3remainder) a0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a3 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a4 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 a4 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 a2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 B61 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B62 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B63 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 B71 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 B72 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 C81 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 C82 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 C91 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 B61 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B62 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B63 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 B71 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 B72 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 C81 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 C82 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 C91 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 C93 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 C83 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C921 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C922 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C93 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 C83 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C921 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C922 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 RRN 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

  14. RRN a0 a1 a2 a3a4 a5 B6B7C8C9C 00 1 0 1 0 0 0 6 1 {m, b}{c }1 01 1 0 1 0 0 0 6 1 { b}{ t}1 02 1 0 1 0 0 0 6 1 { e }{ n,p }1 03 1 0 1 0 0 0 6 2 {m, b}{ t}1 04 0 1 1 0 1 1 0 2 {m }{ n,p }1 05 0 1 1 0 1 1 0 0 {m,e }{ n,p }1 06 0 1 0 0 1 0 1 2 {m }{ n,p }1 07 0 1 0 0 1 0 1 1 {m, b}{c }1 08 0 1 0 0 1 0 1 1 { b}{ t}1 09 0 1 0 0 1 0 1 1 { e }{ n,p }1 10 0 1 0 1 0 0 6 2 {m }{ n,p }0 11 0 1 0 1 0 0 6 1 { b}{ t}0 12 0 1 0 1 0 0 6 1 { e }{ n,p }0 13 0 1 0 1 0 0 6 0 {m,e }{ n,p }0 14 1 0 1 0 1 0 1 2 {m }{ n,p }0 15 0 0 1 1 0 0 6 1 {m, b}{c }0 16 0 0 1 1 0 0 6 1 { e }{ n,p }0 Distance fctn: d(p,q)={wi : p & q differ at i} must have: w61=2*w62=4*w63w71=2*w72 w93=2*w921=2*w922=2*w91w83=w82=w81 Vote function: vote= 1/(1+distance) wi : 1 0 0 2 9 0 8 4 26 33 3 3 2 2 2 4 a0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a3 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a4 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 a4 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 a2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 B61 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B62 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B63 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 B71 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 B72 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 C81 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 C82 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 C91 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 B61 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B62 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B63 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 B71 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 B72 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 C81 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 C82 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 C91 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 C93 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 C83 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C921 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C922 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C93 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 C83 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C921 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C922 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 RRN 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 16

  15. C93 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 C922 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C921 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 0 1 0 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 unclassified sample 0-ring: C91 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 C83 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C82 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 C81 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 B72 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 B71 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 B63 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 Distance fctn: d(p,q)={wi : p & q differ at i} must have: w61=2*w62=4*w63w71=2*w72 w93=2*w921=2*w922=2*w91w83=w82=w81 Vote function: vote= 1/(1+distance) B62 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B61 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a4 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 1 0 0 3 9 0 8 4 2 6 3 3 3 3 2 2 2 4 wi a3 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a0 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 0 0 a3 0 0 0 0 0 0 0 0 0 0 1 1 1 1 0 1 1 a4 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 a5 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 a3 1 1 1 1 1 1 1 1 1 1 0 0 0 0 1 0 0 a4 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 a5 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 a0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 a1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 a2 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 C 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 a0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 1 1 a1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 a2 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 B61 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B62 1 1 1 1 0 0 0 0 0 0 1 1 1 1 0 1 1 B63 0 0 0 0 0 0 1 1 1 1 0 0 0 0 1 0 0 B71 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 B72 1 1 1 0 0 0 0 1 1 1 0 1 1 0 0 1 1 C81 1 0 0 1 1 1 1 1 0 0 1 0 0 1 1 1 0 C82 0 0 1 0 0 1 0 0 0 1 0 0 1 1 0 0 1 C91 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 B61 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B62 0 0 0 0 1 1 1 1 1 1 0 0 0 0 1 0 0 B63 1 1 1 1 1 1 0 0 0 0 1 1 1 1 0 1 1 B71 1 1 1 0 0 1 0 1 1 1 0 1 1 1 0 1 1 B72 0 0 0 1 1 1 1 0 0 0 1 0 0 1 1 0 0 C81 0 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 1 C82 1 1 0 1 1 0 1 1 1 0 1 1 0 0 1 1 0 C91 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 0 1 C93 1 0 1 0 1 1 1 1 0 1 1 0 1 1 1 1 1 C83 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C921 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C922 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C93 0 1 0 1 0 0 0 0 1 0 0 1 0 0 0 0 0 C83 0 0 1 0 1 1 1 0 0 1 1 0 1 1 1 0 1 C921 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0 C922 1 1 0 1 0 0 0 1 1 0 0 1 0 0 0 1 0

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