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== Equations == |
== Equations == |
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| − | === Truth table size-related equations === |
+ | === [[/TruthTableSize/|Truth table size-related equations]] === |
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| − | These equations determine the truth table size ''from'' the highest value of logic and the number of characteristics in the set and in the group for the '''bounded''' class<ref>See page 158, Table II of the '''primary reference'''</ref> of elements and denote the range of the truth table values for the set and the group.<ref>See page 172 of the '''primary reference'''.</ref> |
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| − | ==== Group size ==== |
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| − | : <math> Q = V^C </math>, where: |
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| − | ::* '''''Q''''' is the group size<ref>group - Columns C1 to C5 (0 to C, C=5)</ref> |
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| − | ::* '''''V''''' is the highest value of logic in the group, |
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| − | ::* '''''C''''' is the highest number of characteristics in the group. |
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| − | ==== Set size ==== |
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| − | : <math> R = V^K</math>, where: |
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| − | ::* '''''R''''' is the set size<ref>[http://en.wikipedia.org/wiki/Set Target set] - Columns C1 to C3 (0 to K, K=3)</ref> |
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| − | ::* '''''V''''' is the highest value of logic in the group, |
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| − | ::* '''''K''''' is the highest number of characteristics in the set. |
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| − | ==== Element-related equations ==== |
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| − | ===== Maximum number of pairs of elements to separate ===== |
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| − | Maximum number of pairs of elements to separate refers to [http://en.wikipedia.org/wiki/Triangular 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:<ref>See page 176 Table XI of the '''primary reference'''.</ref> |
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| − | : <math> p_{max} = \frac{\left[{G (G-1)}\right]}{2} </math>, where:<ref>[http://en.wikipedia.org/wiki/Triangular number Triangular Number]</ref> |
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| − | ::*'''''p<sub>max</sub>''''' is the maximum number of pairs to separate, and |
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| − | ::*'''''G''''' is the number of elements in the '''bounded''' class. |
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| − | ===== Order of elements ===== |
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| − | 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.<ref>See [http://www.cstc.org/data/resources/60/convexp.html Number Systems] - essentially using the characteristic values to compute a network or memory address, followed by sorting.</ref> The element [http://en.wikipedia.org/wiki/Positional notation truth table value] allows elements to be sorted and identified as unique or equivalent. |
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| − | : <math> e_i = \sum_{j=0}^C \left[v_{i,j} V^{(C-j)}\right]</math>, where: |
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| − | ::*'''''e<sub>i</sub>''''' is the element truth table value in the group, |
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| − | ::*'''''V''''' is the highest value of logic in the group, |
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| − | ::*'''''v''''' is the value of logic of each characteristic in the group, |
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| − | ::*'''''j''''' is the '''''jth''''' characteristic index, where: |
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| − | :::'''''j = 0..C''''' and where: |
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| − | ::::*'''''C''''' is the number of characteristics in the group, |
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| − | ::*'''''i''''' is the '''''ith''''' element index, where: |
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| − | :::'''''i = 0..G''''' and where: |
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| − | ::::*'''''G''''' is the number of elements in the '''bounded''' class. |
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| − | ==== Characteristic-related equations ==== |
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| − | ===== Theoretical separation ===== |
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| − | ====== The general identification equation ====== |
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| − | :<math> S_j = \frac{1-{V^{-j}}}{1-V^{-C}}</math>, where:<ref>See page 153, page 167, Fig. 2. & page 175 of the '''primary reference'''</ref> |
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| − | ::*'''''S<sub>j</sub>''''' is the theoretical separatory value per '''''jth''''' characteristic, |
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| − | ::*'''''C''''' is the highest number of characteristics in the group, |
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| − | ::*'''''V''''' is the highest value of logic in the group and |
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| − | ::*'''''j''''' is the '''''jth''''' characteristic index in the target set, where: |
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| − | :::'''''j = 0..K''''' and where: |
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| − | ::::*'''''K''''' is the number of characteristics in the target set. |
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| − | ====== Minimal number of characteristics to result in theoretical separation ====== |
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| − | : <math> t_{min} = \frac{\log G}{\log V}</math>, where:<ref>See page 157, ''Primary Schemes'' footnote of the '''primary reference'''</ref> |
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| − | ::*'''''t<sub>min</sub>''''' is the minimal number of characteristics to result in theoretical separation, |
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| − | ::*'''''G''''' is the number of elements in the in the '''bounded''' class and |
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| − | ::*'''''V''''' is the highest value of logic in the group. |
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| − | ===== Empirical separation ===== |
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| − | ====== Target set truth table values ====== |
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| − | <math> t_i = \sum_{j=0}^K v_{i,j} V^{(K-j)}</math>, where:<ref>As the characteristic with the greatest separatory value is moved to the next most significant position, K is incremented to expand the target set size from two characteristics to the number of characteristics in the group or the the number of characteristics which result in 100% separation. For the initial target set with one characteristic the separatory value is computed individually for each characteristic in the group to find the initial characteristic with the highest separatory value.</ref> |
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| − | :*'''''<big>t<sub>i</sub></big>''''' is the [http://en.wikipedia.org/wiki/Positional notation truth table value] of the target set, |
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| − | ::*'''''<big>v<sub>i,j</sub></big>''''' is the element's attribute value, |
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| − | :::*'''''i''''' is the '''''i'''''th element's index value, where, |
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| − | ::::'''''i = 0...G'''''' where '''''G''''' is the number of elements in the '''bounded''' class, and, |
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| − | :::*'''''j''''' is the '''''j''''th characteristic's index value, where, |
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| − | ::::'''''j = 0...K''''' and where, |
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| − | :::*'''''K''''' is the number of characteristics in the target set, |
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| − | ::::*'''''V''''' is highest value of logic in the group, |
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| − | ::::*'''''V<sup>(K-j)</sup>''''' is the positional value of the '''''j'''''th characteristic. |
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| − | <math> n_{t_i} = n_{t_i} + 1 </math>, where, |
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| − | :<big>'''''n'''''<sub>'''''t'''''<sub>'''''i'''''</sub></sub></big> contains the [http://en.wikipedia.org/wiki/multiset multiset] count for each [http://en.wikipedia.org/wiki/Positional notation truth table value]<ref>coefficient of association, <math> coa = \frac{n_{t_i}}{n_R} </math>, see page 172 of the '''primary reference'''</ref>. |
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| − | ====== Separatory equation ====== |
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| − | ====== Initial separation ====== |
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| − | <math> S_j = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}</math>, where:<ref>Must be applied initially to each characteristic. (The equations have been implemented in [http://en.wikipedia.org/wiki/Mathcad] and using the Visual Basic programming language. See [[Optimal_Classification#Application_Example|Application Example]</ref> |
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| − | :*'''''S<sub>j</sub>''''' is the initial empirical separatory value for each characteristic, where, |
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| − | ::'''''j = 0...C''''' and is the index of the '''''j'''''th characteristic in the group and '''''C''''' is the number of characteristics in the group, and, |
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| − | ::'''''l = 0...R''''' and is the truth table value of the '''''j'''''th characteristic, where '''''R''''' is the truth table size, where: |
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| − | ::::'''''R''''' = '''V<sup>0</sup>''', and, |
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| − | <!-- :::'''<math> R = V^{0} </math>''', where: --> |
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| − | ::::*'''''V''''' is the highest value of logic in the group and, |
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| − | :::;*'''''0''''' is the target set exponent for a single characteristic, and, |
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| − | :*'''''G''''' is the number of elements in the '''bounded''' class. |
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| − | ====== Subsequent separation ====== |
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| − | <math> S = \frac{\left[(G^{2})-\sum_{l=0}^{R} n_l^{2}\right]}{2}</math>, where: |
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| − | :*'''''S<sub>j</sub>''''' is the initial empirical separatory value for each characteristic, where, |
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| − | ::'''''l = 0...R''''' and is the target set truth table index value, where '''''R''''' is the target set truth table size value, where: |
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| − | ::::'''''R''''' = '''V<sup>K</sup>''', and, |
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| − | <!-- :::'''<math> R = V^{K} </math>''', where: --> |
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| − | ::::*'''''V''''' is the highest value of logic in the group and, |
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| − | :::;*'''''K''''' is the number of characteristics in the target set, and, |
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| − | :*'''''G''''' is the number of elements in the '''bounded''' class. |
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