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Optimal Classification/Rypka/Equations/Separatory

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Theoretical and Empirical Separatory Equations Edit

Element-related equations Edit

Maximum number of pairs of elements to separate Edit

Maximum number of pairs of elements to separate refers to matrix triangularization of the matrix to permit comparison of each element with every other element to determine the number of pairs that are separable or disjoint. Pairs are separable or disjoint whenever the logic values of the elements that make up a pair are different. In theory, therefore the maximum possible number of pairs that can be separated is determined by the following equation:[1]

 p_{max} = \frac{\left[{G (G-1)}\right]}{2} , where:[2]
  • pmax is the maximum number of pairs to separate, and
  • G is the number of elements in the bounded class.


Order of elements Edit

The elements are arranged in descending order according to their truth table value, i.e., the value calculated as the sum of each characteristic's logic state value times the highest value of logic raised to the power of the order of the characteristic.[3] The element notation truth table value allows elements to be sorted and identified as unique or equivalent.

 e_i = \sum_{j=0}^C \left[v_{i,j} V^{(C-j)}\right], where:
  • ei is the element truth table value in the group,
  • V is the highest value of logic in the group,
  • v is the value of logic of each characteristic in the group,
  • j is the jth characteristic index, where:
j = 0..C and where:
  • C is the number of characteristics in the group,
  • i is the ith element index, where:
i = 0..G and where:
  • G is the number of elements in the bounded class.

Characteristic-related equations Edit

Theoretical separation Edit
The general identification equation Edit
 S_j = \frac{1-{V^{-j}}}{1-V^{-C}}, where:[4]
  • Sj is the theoretical separatory value per jth characteristic,
  • C is the highest number of characteristics in the group,
  • V is the highest value of logic in the group and
  • j is the jth characteristic index in the target set, where:
j = 0..K and where:
  • K is the number of characteristics in the target set.
Minimal number of characteristics to result in theoretical separation Edit
 t_{min} =  \frac{\log G}{\log V}, where:[5]
  • tmin is the minimal number of characteristics to result in theoretical separation,
  • G is the number of elements in the in the bounded class and
  • V is the highest value of logic in the group.


Empirical separation Edit
Target set truth table values Edit

 t_i = \sum_{j=0}^K v_{i,j} V^{(K-j)}, where:[6]

  • vi,j is the element's attribute value,
  • i is the ith element's index value, where,
i = 0...G' where G is the number of elements in the bounded class, and,
  • j is the j'th characteristic's index value, where,
j = 0...K and where,
  • K is the number of characteristics in the target set,
  • V is highest value of logic in the group,
  • V(K-j) is the positional value of the jth characteristic.

 n_{t_i} = n_{t_i} + 1 , where,

nti contains the multiset count for each notation truth table value[7].
Separatory equation Edit
Initial separation Edit

 S_j = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}, where:[8]

  • Sj is the initial empirical separatory value for each characteristic, where,
j = 0...C and is the index of the jth characteristic in the group and C is the number of characteristics in the group, and,
l = 0...R and is the truth table value of the jth characteristic, where R is the truth table size, where:
R = V0, and,
  • V is the highest value of logic in the group and,
  • 0 is the target set exponent for a single characteristic, and,
  • G is the number of elements in the bounded class.
Subsequent separation Edit

 S = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}, where:

  • Sj is the initial empirical separatory value for each characteristic, where,
l = 0...R and is the target set truth table index value, where R is the target set truth table size value, where:
R = VK, and,
  • V is the highest value of logic in the group and,
  • K is the number of characteristics in the target set, and,
  • G is the number of elements in the bounded class.

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