Concept Similarity

CSPR — Methodology

Concept Similarity

The following examples are based on the vehicle ontology depicted infFigure vch. The expression ⊗^C denotes the set of features contained as Power Set Concepts in the class ^C. How meaningful is the Knowledge Subjects Equivalence Axiom (KSE)? The knowledge graph, KG = {(^PassengerVehicle, ◊is, ^Vehicle), (^Truck, ◊is, ^Vehicle)} consists of only two assertions. The KSE axiom states that Σ^PassengerVehicle and Σ^Truck are equivalent because both have exactly only the pair (◊is, ^Vehicle) in common and there are no other properties and thus a certain degree of agreement is recognized. But that corresponds exactly to reality or what was intended with the modeling. If further properties are added, the equivalence is lost at the latest when one of the properties no longer occurs in the other knowledge subject. Until then, however, the equivalence check can be used to determine whether the modeling does need to be further differentiated, or whether two knowledge subjects may have accidentally been given different names, although they should actually be identical. The conceptual feature similiarities between all classes of the vehicle ontology is explained in the following. At first we have to detect the feature sets of all classes:

Fig. VCHS: Vehicle Ontology
  • FS(∵^Vehicle) = FS({^Vehicle}) = {(.∆ModelName, :String), (.LicensePlateNbr, :String), (◊hasMotor, ^Motor)}
  • FS(∵^PassengerVehicle) = FS({^Vehicle, ^PassengerVehicle}) =  {(.∆ModelName, :String), (.LicensePlateNbr, :String), (◊hasMotor, ^Motor), (.^MaxNbrOfPassengers, :Integer)}
  • FS(∵^Truck) = FS({^Vehicle, ^Truck}) =  {(.∆ModelName, :String), (.LicensePlateNbr, :String), (◊hasMotor, ^Motor), (.LoadingBed_sqm, ^Float)}
  • FS(∵^Pickup) = FS({^Vehicle, ^PassengerVehicle, ^Truck, ^Pickup}) = {(.∆ModelName, :String), (.LicensePlateNbr, :String), (◊hasMotor, ^Motor), (.^MaxNbrOfPassengers, :Integer), (.LoadingBed_sqm, :Float), (.isOffroadCapable_TF, :Boolean)}
In this case, the feature sets are identical to the Data Property Definitions (DPDs) of a class and its super classes. The following similarity analysis is performed based on the similarity measure defined in section Feature Similarity.
Then we have:
  • X = FS(SPC(^Vehicle)), Y = FS(SPC(^PassengerVehicle)), Z = FS(SPC(^Truck)), V = FS(SPC(^Pickup)) 
  • X∩Y = {(.Model_Name, :String), (.LicensePlateNbr, :String), (◊hasMotor, ^Motor)}, CX∩Y = 3
  • X∪Y = {(.Model_Name, :String), (.LicensePlateNbr, :String), (.LoadingBed_sqm, :Float)}, CX∪Y = 3
  • CZ∩V = 5, CZ∪V = 6
  • CY∩Z = 3, CY∪Z = 5
  • CX∩V = 3, CX∪V = 6
The Feature Similarity Ratios (FSR) between pairs of classes are computed as follows:
  • FSR(X,Y) = FSR(∴^Vehicle, ∴^PassengerVehicle) = CX∩Y / CX∪Y = 3 / 4 ≘ 75%
  • FSR(Y,Z) = FSR(∴^PassengerVehicle, ∴^Truck) = CY∩Z / CY∪Z = 3 / 5 ≘ 60%
  • FSR(Z,V) = FSR(∴^Truck, ∴^Pickup) = CZ∩V / CZ∪V = 5 / 6 ≘ 83,3%
  • FSR(Y,V) = FSR(∴^PassengerVehicle,∴^Pickup) = CY∩V / CY∪V = 5 / 6 ≘ 83,3%
  • FSR(X,V) = FSR(∴^Vehicle, ∴^Pickup) = CX∩V / CX∪V = 3 / 6 ≘ 50%

The results already show for a relatively small number of features that Feature Similarity Measure (FSM) delivers a very good assessment of class similarities. In this case, the minimum similarity of classes in the vehicle class hierarchy is already 50%. For the pairs (^PassengerVehicle, ^Pickup) and (^Truck, ^Pickup) we already get FSR = 83.3%. With the addition of further features, the evaluation becomes more and more precise. While in the vehicle ontology we can only use a total of six features for the similarity determination, in the following example of the species ontology there are already twenty features and we naturally expect a higher significance.

Be Props(c) the set of names of the properties contained in the class c. Then the following values result for the concepts of ^Species:
  • Props(^Species) = {.Age_Years, .Height_cm, .^^NbrOfLivingInstances, .^^AverageHeight_cm, .^LifeExpectancy_Years, .^^MigrationPeriod, ^Habitat, ^Region, ^Eater}
  • Props(^Species_by_Threat-Phase) = {.ThreatPhase}
  • Props(^Carnivore) = {^Eater, ^Meat}
  • Props(^Piscivore) = {^Eater, ^Fish}
  • Props(^Herbivore) = {^Eater, ^Plant}
  • Props(^Fish) = {^HBT-Maritim}
  • Props(^Bird) = {.^^BeakPattern, ^HBT-Aerial, ^HBT-Terrestial}
  • Props(^SeaBird) = {^HBT-Maritim, ^Sea}
  • Props(^EmperorPinguin) = {^Bird, threatenend, ^Eater, ^Fish, ^Pole, ^Region, ^South}
  • Props(^GoldenEagle) = {^Bird, ^Eater, ^Meat, ^Robbery, ^Region, ^North}
  • Props(^Bear) = {^HBT-Terrestial}
  • Props(^Polar_Bear) = {^Bear, ^HBT-Maritim, ^Eater, ^Meat, ^Fish, ^Pole, ^Region, ^North}
  • Props(^Panda_Bear) = {^Bear, threatenend, ^Eater, ^Plant, ^Region, ^East}
Properties that are already contained in superclasses have each been removed from the sets, such as ^Bear, ^Bird and ^Eater. This results in the following cardinalities for concept feature sets:
  • SP = FS(^Species), CSP = 9
  • SPT = FS(^Species_by_Threat-Phase), CSPT = 10
  • BD = FS(^Bird), CBD = 13
  • SBD = FS(^SeaBird), CSBD = 15
  • EP = FS(^EmperorPinguin), CEP = 20
  • GE = FS(^GoldenEagle), CGE = 19
  • BR = FS(^Bear), CBR = 11
  • PLB = FS(^Polar_Bear), CPLB = 17
  • PDB = FS(^Panda_Bear), CPDB = 15
Feature Similarity Space
All pairs of concepts from the vehicle and the species ontology can now be arranged in a Feature Similarity Space (FSS) as shown in FIgure fsr: The X-axis contains the total number CX∪Y of features of both concepts. The Y-axis corresponds to the Feature Similarity Ratio FSR(X,Y) and has the value range [0,1] or 0 to 100%. So, at the top right, the pairs of concepts are mapped that have a large feature set with a high degree of agreement. This allows quick conclusions to be drawn about the quality of the modelling. If the FSM and FSR values are significantly higher or lower than expected, an extension of the concepts of one of the involved classes is likely required. It should be noted that the further apart the concepts are in the class hierarchy, the arrangement with regard to the feature similarity ratio on the Y-axis must inevitably be lower.
pdf:fsr:1.0
Figure fsr: Feature Similarity Ratios
For selected pairs of concepts we get the Feature Similarty Ratios (FSR) represented in Figure fsr:
  • CSP∩SPT = 9, CSP∪SPT = 10 → FSR(SP,SPT) = 90.0%
  • CSPT∩BD = 10, CSPT∪B = 13 → FSR(SPT, BD) = 76.9%
  • CBD∩EP = 13, CBD∪EP = 18 → FSR(SPT, EP) = 72.2%
  • CBD∩GE = 13, CBD∪GE = 17 → FSR(BD, GE) = 76.4%
  • CEP∩GE = 13, CEP∪GE = 20 → FSR(EP, GE) = 65.0%
  • CSPT∩BR = 10, CSPT∪BR = 11 → FSR(SPT, BR) = 90.1%
  • CBR∩PLB = 11, CBR∪PLB = 17 → FSR(BR, PLB) = 64.7%
  • CBR∩PDB = 11, CBR∪PDB = 15 → FSR(BR, PDB) = 73.3%
  • CPLB∩PDB = 13, CPLB∪PDB = 19 → FSR(PLB, PDB) = 68.4%
  • CEP∩PLB = 13, CEP∪PLB = 17 → FSR(EP, PLB) = 58.8%

Extension: deriver.app

Back to Concept Similarity; canonical overview on taoke.de — CSPR. Deriver documentation.

Source: taoke.de — CSPR Concept Similarity.