represented by specific non-numeric values that can be or not isfied by the ontology. similarin(criterion.) subset of the above criteria and their respective values This measure will also return values between 0 ande Taking into account the previous definitions, user interests will be [0 epresenting the set of thresholds that should be reached by the returning a similarity value between 0 and 2 is inspire ontologies. Given a set of user interests, the system will size up all collaborative matching measures [18] to not manage the stored evaluations, and will calculate their similarity measures. numbers and facilitate. as we shall show in the next sul negative To explain these similarities we shall use a simple example of six coherent calculation of the final ontology rankings different evaluations(E1, E2, E3, E4, Es and E) of a given The similarity assessment is based on the distance between the ontology. In the explanation we shall distinguish between the value of the criterion n in the evaluation m, and the threshold numeric and the boolean criteria. We start with the boolean ones ndicated in the users interests for that criterion. The more the assuming two different criteria, CI and C2, with three possible value of the criterion n in evaluation m overcomes the threshold values:“A",“B”and“C”: In Table 1 we show the threshold the greater the similarity value shall be values established by a user for these two criteria, "A"for CI and B for C,, and the six evaluations stored in the system. Specifically, following the expression below, if the difference dif revaluation - threshold) is equal or greater than 0, we assign a Table 2. Thresholds and evaluations for Boolean criteria Ci and C2 Evaluations difference maxDif=(maxvalue- threshold) we can achieve wit the given threshold; and else, if the difference dif is lower than 0, nresholds EE, we give a negative similarity in [1, 0), punishing the distance of C1 the value with the threshold “A”林A ∈(0,1]ifdf≥0 use of the threshold of a criterion n is satisfied or similarity(criterion)= 1+ marDi not by evaluation m, their corresponding similarity measur 0 if they have the same value, and 2 otherwise. ∈[-1,0)ifdf<0 hreshold similarity(criterion)= Table 5 summarizes the similarity* values for the three numeric The similarity results for the boolean criteria of the example are Table 5. Similarity* values for numeric criteria C] C and C shown in Table 3 Evaluations Table 3. Similarity values for Boolean criteria C and C3 Criteria Thresholds E, E, E E4 E4 Es E Criteria Thresholds E, Ex E: E Es Es 162/65/6 C 0 0 Comparing the evaluation values of Table 4 with the similarity For the numeric criteria, the evaluations can overcome the values of Table 5, the reader may notice several important facts thresholds to different degrees. Table 4 shows the thresholds 1. Evaluation E4 satisfies criteria Ca and Cs with evaluations of 5 established for criteria C3, Ca and Cs, and their six available Applying the above expression, these criteria receive the same evaluations. Note that E1, E2, E3 and E4 satisfy all the criteria, while similarity of 1. However, criterion Ca has a threshold of 0, and Es and e do not reach some of the corresponding thresholds. Cs has a threshold equal to 5. As it is more difficult to satisfy the restriction imposed to Cs, this one should have a greater Table 4. Thresholds and evaluations for numeric criteria C, Ca and C influence in the final rankins 2. Evaluation E gives an evaluation of o to criteria C3 and Cs,not Criteria Thresholds El E E3 E4 Es Es satisfying either of them and generating the same similarity value of-1. Again, because of their different thresholds, we 0 should distinguish their corresponding relevance degrees in Ca the rankings For these reasons, a threshold penalty is applied, reflecting how difficult it is to overcome the given thresholds The more difficul In this case, the similarity measure has to take into account two to surpass a threshold, the lower the penalty value shall be different issues: the degree of satisfaction of the threshold, and the difficulty of achieving its value. Thus, the similarity between the 1+ threshold value of criterion n in the evaluation m. and the threshold of interest penalty(threshold) ∈(0,1 is divided into two factors: 1)a similarity factor that considers whether the threshold is surpassed or not, and, 2)a penalty factor Table 6 shows the threshold penalty values for the three numeric which penalizes those thresholds that are easier to be satisfied criteria and the six evaluations of the example.represented by specific non-numeric values that can be or not satisfied by the ontology. Taking into account the previous definitions, user interests will be a subset of the above criteria and their respective values representing the set of thresholds that should be reached by the ontologies. Given a set of user interests, the system will size up all the stored evaluations, and will calculate their similarity measures. To explain these similarities we shall use a simple example of six different evaluations (E1, E2, E3, E4, E5 and E6) of a given ontology. In the explanation we shall distinguish between the numeric and the Boolean criteria. We start with the Boolean ones, assuming two different criteria, C1 and C2, with three possible values: “A”, “B” and “C”. In Table 1 we show the threshold values established by a user for these two criteria, “A” for C1 and “B” for C2, and the six evaluations stored in the system. Table 2. Thresholds and evaluations for Boolean criteria C1 and C2 Evaluations Criteria Thresholds E1 E2 E3 E4 E5 E6 C1 “A” “A” “B” “A” “C” “A” “B” C2 “B” “A” “A” “B” “C” “A” “A” In this case, because of the threshold of a criterion n is satisfied or not by a certain evaluation m, their corresponding similarity measure is simply 0 if they have the same value, and 2 otherwise. 0 if ( ) 2 if mn mn bool mn mn mn evaluation threshold similarity criterion evaluation threshold ≠ = = ⎧ ⎨ ⎩ The similarity results for the Boolean criteria of the example are shown in Table 3. Table 3. Similarity values for Boolean criteria C1 and C2 Evaluations Criteria Thresholds E1 E2 E3 E4 E5 E6 C1 “A” 2 0 2 0 2 0 C2 “B” 0 0 2 0 0 0 For the numeric criteria, the evaluations can overcome the thresholds to different degrees. Table 4 shows the thresholds established for criteria C3, C4 and C5, and their six available evaluations. Note that E1, E2, E3 and E4 satisfy all the criteria, while E5 and E6 do not reach some of the corresponding thresholds. Table 4. Thresholds and evaluations for numeric criteria C3,C4 and C5 Evaluations Criteria Thresholds E1 E2 E3 E4 E5 E6 C3 ≥ 3 3 4 5 5 2 0 C4 ≥ 0 0 1 4 5 0 0 C5 ≥ 5 5 5 5 5 4 0 In this case, the similarity measure has to take into account two different issues: the degree of satisfaction of the threshold, and the difficulty of achieving its value. Thus, the similarity between the value of criterion n in the evaluation m, and the threshold of interest is divided into two factors: 1) a similarity factor that considers whether the threshold is surpassed or not, and, 2) a penalty factor which penalizes those thresholds that are easier to be satisfied. * ( ) 1 ( )· ( ) [0, 2] num mn num num mn mn similarity criterion similarity criterion penalty threshold = =+ ∈ This measure will also return values between 0 and 2. The idea of returning a similarity value between 0 and 2 is inspired on other collaborative matching measures [18] to not manage negative numbers, and facilitate, as we shall show in the next subsection, a coherent calculation of the final ontology rankings. The similarity assessment is based on the distance between the value of the criterion n in the evaluation m, and the threshold indicated in the user’s interests for that criterion. The more the value of the criterion n in evaluation m overcomes the threshold, the greater the similarity value shall be. Specifically, following the expression below, if the difference dif = (evaluation – threshold) is equal or greater than 0, we assign a positive similarity in (0,1] that depends on the maximum difference maxDif = (maxValue – threshold) we can achieve with the given threshold; and else, if the difference dif is lower than 0, we give a negative similarity in [-1,0), punishing the distance of the value with the threshold. * 1 (0,1] if 0 1 ( ) [ 1, 0) if 0 num mn dif dif maxDif similarity criterion dif dif threshold + ∈ ≥ + = ∈− < ⎧ ⎪ ⎪ ⎨ ⎪ ⎪⎩ Table 5 summarizes the similarity* values for the three numeric criteria and the six evaluations of the example. Table 5. Similarity* values for numeric criteria C3, C4 and C5 Evaluations Criteria Thresholds E1 E2 E3 E4 E5 E6 C3 ≥ 3 1/4 2/4 3/4 3/4 -1/3 -1 C4 ≥ 0 1/6 2/6 5/6 1 1/6 1/6 C5 ≥ 5 1 1 1 1 -1/5 -1 Comparing the evaluation values of Table 4 with the similarity values of Table 5, the reader may notice several important facts: 1. Evaluation E4 satisfies criteria C4 and C5 with evaluations of 5. Applying the above expression, these criteria receive the same similarity of 1. However, criterion C4 has a threshold of 0, and C5 has a threshold equal to 5. As it is more difficult to satisfy the restriction imposed to C5, this one should have a greater influence in the final ranking. 2. Evaluation E6 gives an evaluation of 0 to criteria C3 and C5, not satisfying either of them and generating the same similarity value of -1. Again, because of their different thresholds, we should distinguish their corresponding relevance degrees in the rankings. For these reasons, a threshold penalty is applied, reflecting how difficult it is to overcome the given thresholds. The more difficult to surpass a threshold, the lower the penalty value shall be. 1 ( ) (0,1] 1 num threshold penalty threshold maxValue + = ∈ + Table 6 shows the threshold penalty values for the three numeric criteria and the six evaluations of the example