Knowledge Subject Similarity

In the following we analyze, how the similarity of knowledge subjects can be determined by measuring the distances within the Conceptual Spaces (CS) spanned by several quality and quantity dimensions. As an example we want to compare four people in terms of the features sets height and age. This could also be the same person at different stages of life. Each feature set FS_i represents the height which the person has at the respective age and the position of the knowledge subject Σ_i in the conceptual space CSPC(.Age, .Height_cm):

• FS₁ = {(.Age, 20),  (.Height_cm, 170)}
• FS₂ = {(.Age, 17),  (.Height_cm, 165)}
• FS₃ = {(.Age, 12),  (.Height_cm, 130)}
• FS₄ = {(.Age, 7),  (.Height_cm, 95)}

Based on the Knowledge Subject Distance definition KSD the abolute value of the similarity of the four persons is computed as:

• ⋈₁₂² = (20-17)²+(170-165)²= 3²+5² = 9+25 = 34; ⋈_1,2 = 5.83
• ⋈₂₃² = (17-12)²+(165-130)²= 5²+35² = 25+1225 = 1250; ⋈_2,3 = 35.36
• ⋈₃₄² = (12-7)²+(130-95)²=5²+35² = 25+1225 = 1250; ⋈_3,4 = 35.36

This means concretely that, for example, the distance between persons 2 and 3 is identical to the distance between persons 3 and 4.

The measure of relative similarity can be between 0% and 100% and is computed as Knowledge Subject Distance Ratio according to definition KSDR with:

• dr_1,1² = (20/20)²+(170/170)²=1²+1²= 2; ⋈%_1,1 = √(2/2)=1 ≘ 100%
• dr_1,2² = (17/20)²+(165/170)² = 0.85² + 0.97² = 0.7225 + 0.942 = 1.6645; ⋈%_1,2 = √(1.6635 / 2 ) = √(0.83225) = 0.912 ≘ 91.2%
• dr_2,3² = (12/17)²+(130/165)² = 0.85²+0.7878² = 0.7225 + 0.62075 = 1.34325; ⋈%_2,3 = √(1.34325/2) = √(0.671625) = 0.8195 ≘ 81.95%
• dr_3,4² = (7/12)² + (95/130)² = 0.583²+0.731² = 0.33988+0.5340 = 0.86388; ⋈%_3,4 = √(0.86388/2) = √(0.43694) = 0.6610 ≘ 66.10%

Of course, the similarity of the same persons like ⋈%_1,1 must be 100%. Since the relative similarity measure normalizes to the scale from 0% to 100%, the ranking is much easier to understand by the user. Based on the example, it is therefore easy to see that with advancing age, growth slows down or even stagnates, leading to higher similarities as ⋈%_1,2 ≘ 91.2% between individuals 1 and 2.