On the design of metric relations

Transcription

1 On the design of metric relations Lluís A Belanche, Jorge Orozco On the design of metric relations Lluís A Belanche* Corresponding author, Jorge Orozco Languages and Systems Information Department Technical University of Catalonia Barcelona, Spain {belanche, jorozco}@lsiupcedu Abstract Metric distances and the more general concept of dissimilarities are widely used tools in instance-based learning methods and very especially in the nearestneighbor classification technique This paper contributes to the design of general dissimilarity measures to increase their utility The ability to understand the main properties of a hand-crafted dissimilarity measure and to alter them if necessary greatly widens its applicability Grounded upon a formal definition for a dissimilarity measure, together with a set of fundamental properties, a main body of results is presented concerning equivalence between dissimilarities Moreover, the very important concepts of transitivity and aggregation are studied Results in preserving transitivity under transformations and/or in aggregations are presented, with emphasis in the relationship between aggregation and transitivity Further, the issue of dealing with special values (eg missing values) is studied Although the focus is in dissimilarities, the particular case of metric distances (ie dissimilarities fulfilling the triangular inequality) is specifically covered, given its importance Several examples of the use and potential utility of the results are worked out throughout the text Keywords Dissimilarity Measures, Metrics, Aggregation, Transitivity, Missing Values 1 Introduction The importance of a good choice of the metric distance in instance-based learning methods is beyond doubt Together with the more general concept of dissimilarity, these are widely used tools in machine learning and artificial intelligence, and very especially in the nearest-neighbor classification rule Nevertheless, dissimilarity and its companion similarity are hard concepts to establish as illustrated by the several different definitions present in the literature [1-4] Many of their properties are still under discussion, very especially transitivity [11, 12] Moreover, one of the fundamental aspects of these relations is aggregation and how it keeps desired properties Many authors have studied the effects of aggregation on transitivity [5-10], sometimes from different points of view We collect in this document some of the most relevant results, trying to write them in a common format that can be useful to design or choose aggregation operators that keep or achieve a desired transitivity These results are a generalization of those achieved with metrics, since metrics are a particular case of dissimilarity relations The paper is structured as follows In the next section, we introduce a working definition for dissimilarity relations, a notion of equivalence and how it affects the defining properties In section 3 we introduce aggregation, its definition and properties and explain the relationship between aggregation and transitivity Finally, in section 4 the issue of special values (notably missing values) is studied in the particular context of metrics We present several practical examples of the use and utility of the results throughout the text 2 Definition of dissimilarity In a general sense, a dissimilarity expresses the degree of disagreement or unlikeness between two elements of a common reference set Since the objective is to measure or calculate this value between any two elements of the set, it is reasonable to treat them as functions of two arguments A dissimilarity function is defined in this work as follows: Let be a non-empty set (the reference set) where there is defined a equality relation Let be a function (1) 70

2 Journal of Convergence Informarion Technology Vol 3 No 3, September 2008 Assume that is lower bounded, exhaustive and total This implies that is lower bounded and also that exists 1 Define 2 The function can fulfill the following axioms (definitional properties), for any : Property 1 (Reflexivity) property implies This Property 2 (Strong reflexivity) Property 3 (Symmetry) Property 4 (Upper boundedness) A dissimilarity is upper bounded when, we have that exists Given that Property 5 (Upper closedness) Given an upper bounded dissimilarity function, let denote The property asks for the existence of such that This is equivalent to ask that Property 6 (Complement) An upper closed dissimilarity defined in has complement function, where if,, If is reflexive, necessarily On the other hand, has unitary complement function if, In this case, : 3 (symmetry) 4 (associativity) Property 7 (Transitivity) A dissimilarity defined on is -transitive if there is a transitivity operator such that the following inequality holds: This property applies to the particular case of Definition 1 when is the null element of In this case it can be shown that and, for all Along this document we make use of the following terminology in order to identify the different dissimilarities: A dissimilarity is a function satisfying properties 2 and 3 above We denote the set of all the dissimilarities having as reference set A pseudo-dissimilarity is a function satisfying 1 and 2 but not 3 A (pseudo)dissimilarity is bounded (resp closed) if it satisfies 4 (resp 5) A dissimilarity or pseudo-dissimilarity is - transitive if it satisfies property 7 for some dissimilarity transitivity operator We illustrate the previous properties with a simple example of a dissimilarity: Example 1 Let be the set of polygons with or less vertexes, where Consider the relation as has the same number of vertexes than Call the number of vertexes of Let us define the following dissimilarity 4 : Let us define a transitivity operator in order to introduce transitivity in dissimilarity functions Definition 1 (Transitivity operator) Let be non-empty subsets of, such that and Let also be a fixed element of A transitivity operator is a function, satisfying the following properties 3, for all : 1 (null element) 2 (monotonicity) 1 The set can be either finite, infinite countable or infinite uncountable 2 We take the assumption, though it is not strictly necessary A non-negative can be obtained with the transformation 3 This definition is coincident with the uninorms [13] when Where The reader can verify that this dissimilarity is strongly reflexive ( ) and symmetric Given that, for a fixed the dissimilarity between a triangle and a polygon of vertexes is, this dissimilarity is also closed and upper bounded However, the set is empty for those polygons with a number of vertexes greater than 3 and lower than For the rest, for all such that : 4 This dissimilarity is not related to the usual concept of similarity in geometry It is only for illustrative purposes 71

3 On the design of metric relations Lluís A Belanche, Jorge Orozco Therefore, this dissimilarity does not have a complement function Finally, this dissimilarity is transitive for the operator, meaning that for all : Consider now the set of all ordered pairs of elements of and denote it For a fixed, consider the preorder relation in defined as to belong to a class of equivalence with greater or equal dissimilarity value This preorder depends on because it is induced by In symbols, define in as,, It is not difficult to see that only the monotonically increasing and invertible functions keep an induced preorder Proposition 6 Let be a -transitive dissimilarity Let be an equivalence function The equivalent dissimilarity is transitive, where Proof We necessarily have Using that is transitive as indicated we know that, for all, Applying to this inequality we get We recall that, in a preorder relation, does not imply and Definition 2 (Equivalent dissimilarities) Two dissimilarities (in the same reference set ) are equivalent if they induce the same preorder in The previous definition can be expressed in various ways Proposition 3 The three next definitions are analogous For all and : (i) are equivalent (ii) (iii) Note that this notion of equivalence between dissimilarities is itself an equivalence relation Are the properties of dissimilarities kept under equivalence? Yes, with the exception of boundedness Proposition 4 Given two equivalent dissimilarities, is reflexive only if is reflexive is strong reflexive only if is strong reflexive is symmetric only if is symmetric is upper closed only if is upper closed has (unitary) complement only if has (unitary) complement, with the same complement function Definition 5 (Equivalence function) Let be a dissimilarity An equivalence function is a monotonically increasing and invertible function such that and is a dissimilarity on equivalent to Using that, Simply taking as is defined in the Proposition we get the desired transitivity expression for : 3 On transitivity and aggregation We collect in this section several results in preserving transitivity when aggregating a number of dissimilarities, intending to clarify the relationship between aggregation and transitivity The aim is to either keep or achieve a given transitivity for a given dissimilarity 31 Aggregation of dissimilarities Let us define an aggregation operator as a function, where, and is a set of dissimilarity values 5 From here on, we shall use the abbreviated notation to stand for The properties that an aggregation operator should fulfill,, follow: Property 8 (Idempotency) is idempotent if implies Property 9 (Symmetry) is symmetric if, for all permutations of 5 Using the fact that dissimilarities are functions from to, we could also consider an aggregation operator as another dissimilarity defined in the new reference set This vision is not pursued further in this work 72

4 Journal of Convergence Informarion Technology Vol 3 No 3, September 2008 Property 10 (Monotonicity) is monotonic if for some entails Property 11 (Boundedness) is bounded if Actually boundedness is a consequence of the other properties Property 12 (Neutral element) has neutral element if implies deal with this, the concept of domination has been introduced [14] We restrict here the definition to the specific case of dominating dissimilarity transitivity operators Definition 8 (Domination) Consider an aggregation operator and a transitive operator for dissimilarities We say that dominates if for all with : for any In an aggregation process, it is of great interest that the original properties of the involved dissimilarities are kept Reflexivity or symmetry are easy to deal with, but transitivity is delicate: given a set of -transitive dissimilarities, not any aggregation operator keeps the -transitivity Actually, the transitivity of the aggregated dissimilarity cannot be stronger than the transitivity of the original dissimilarities Definition 7 Given two transitivity operators, we say that is stronger than if for all, This relation is expressed as Alternatively, it can be said that operator is weaker than In consequence, a weak transitivity is less informative than a strong one The objective is then to keep the strongest transitivity possible when aggregating It is immediate to realize that any - transitive dissimilarity is also -transitive, where 32 Transitivity in Aggregation Consider a set of -transitive dissimilarities, where for all and And let be an operator that aggregates these dissimilarities Trivially, given that is monotonic However it cannot be assured that, for all in : Theorem 9 ([14, Theorem 9) An aggregation operator preserves -transitivity if and only if dominates Two questions arise: given a set of -transitive dissimilarities, which aggregation operators do keep the transitivity? Given a set of different -transitivity dissimilarities, what will the final aggregated transitivity be? Let us answer the second question first Corollary 10 Given a set of dissimilarity functions, each one -transitive, denote the weakest transitivity operator among the For any aggregation operator, we have that dominates if and only if is -transitive Further, is not - transitive (for all ) if and only if does not dominate This result is very pessimistic because we cannot guarantee a stronger transitivity than the weakest of any dissimilarity being aggregated However, for the first question there are interesting results for certain choices of and (see [5, 9]), although not a general formula Despite this, rather general results for continuous transitivity operators are collected in the following two theorems These results are based on the concept of an Archimedean norm (from Fuzzy Logic) Recall that, for all, holds in transitive operators for dissimilarities Then is called idempotent if, for all A continuous uninorm that is not idempotent is called Archimedean in fuzzy theory [15] It is first refreshed that any Archimedean transitive operator can be defined using a one-place monotonic function (called its generator), which is increasing in the case of transitive operators for dissimilarities Proposition 11 (Archimedean transitive operator) Let be a transitive operator on an interval ; then is Archimedean only if there exists a function such that: which is precisely the equivalent of affirming that the aggregated dissimilarity is -transitive In order to 73

5 On the design of metric relations Lluís A Belanche, Jorge Orozco If is a dissimilarity transitive operator, then must be an strictly increasing mapping such that The following result can be applied to any Archimedean transitive operator Substituting and : Theorem 12 ([14, Theorem 14]) Given an Archimedean transitive operator with a generator An aggregation operator dominates if and only if for all : Applying in some strategic places: This theorem is a generalization of that in [10, Theorem 33] With it we can check whether our favorite aggregation operator preserves a certain transitivity Example 2 Let, then Using Theorem 12 any subadditive aggregation operator preserves metric transitivity The only continuous and not Archimedean transitivity operators are and They form a particular case that is covered next Theorem 13 The only aggregation operator that dominates (and is also dominated by) the transitivity operator is The only aggregation operator that dominates (and is also dominated by) the transitivity operator is For certain families of aggregation operators, further useful results can be inferred Consider the following family: (3) where is a monotonic function and Next we enunciate a pair of useful results in combination of this family The first one is a generalization of a theorem in [9] for dissimilarity transitivity operators Theorem 14 Consider the aggregation operator family in (3) and a dissimilarity transitivity operator Given a continuous strictly increasing function, define Applying to both sides, and defining (6) We get that dominates If we initially suppose that does not dominate, making the same substitutions the result is that does not dominate Corollary 15 Let be an aggregation operator and a transitivity operator If dominates for any transitivity operator such that, then dominates Proof Using Theorem 9, if dominates then the aggregation is -transitive If, then any - transitive dissimilarity is also -transitive Using again Theorem 9 we get that necessarily also dominates Using Theorem 14 and Expression (3), several domination relations can be extracted In Table 1 some results are shown obtained with a few choices of function for the aggregation operators family described in Expression (3) Table 1 Aggregation operators that dominate transitive operators (4) (5) Then if and only if Proof Given the aggregation operator and a dissimilarity transitivity operator, suppose that dominates This means that for all : The foregoing theorems let us choose some functions in Expression 4, such that a certain transitivity is kept 74

6 Journal of Convergence Informarion Technology Vol 3 No 3, September 2008 Example 3 Let on given by be three dissimilarities defined must be true Therefore, the resulting aggregation is only -transitive, but it is so for both aggregation operators considered Where are given by Example 4 Let be an aggregation operator What is the transitivity of the aggregation of these three dissimilarities? Consider the aggregation operator Since dominates and, using Corollary 10 we know that the aggregated dissimilarity is not a metric because Take three elements x, y, z with components in The aggregated dissimilarity will be metric only if: where we have defined and To prove that the dissimilarity is a not metric, recall Theorem 9 and the concept of domination The operator dominates the transitivity operator only if: and a dissimilarity transitivity operator It is easy to see that dominates because (8) Note that it is an strict equality relation, not a lower or equal relation This implies that also dominates The previous example uses addition as a dissimilarity transitivity operator Dissimilarities that fulfill this transitivity (also called triangle inequality) are commonly known as metrics Example 5 Consider the following dissimilarities: which happens to be false Hence, the aggregated dissimilarity cannot be a metric Considering the same aggregation operation, does the aggregated dissimilarity comply with -transitivity? This will be so if holds Recalling again Theorem 9 and the concept of domination, the operator dominates the transitivity operator only if: which is true as expected, for any value in If we choose another aggregation operator (which also dominates all three transitivity operators), using Corollary10, Where, and the sets to are subsets of Consider their aggregation by the special average: The first six dissimilarities are sum-transitive (ie they fulfill triangle inequality) However, tho other two are -transitive Since the sum operator is weaker than the operator, the aggregation is sumtransitive In fact, it is verified that, : Therefore, this aggregated dissimilarity is a metric defined in, is closed and strongly reflexive 75

7 On the design of metric relations Lluís A Belanche, Jorge Orozco 33 Obtaining and keeping metric properties The following are basic results for the particular case of metrics [22]: Proposition 1 Let be a pseudo-metric in and a monotone and subadditive function with Then: 1 is a pseudo-metric in 2 if for and is a metric in, then is a metric in Proposition 2 Let an injective function Then the function is a metric in These are useful results for designing datadependent metrics in a general set Some wellknown families of subadditive monotone functions are The following proposition introduces a criterion to know if some aggregation operator keeps the triangle inequality Proposition 16 Let be an aggregation operator of the family defined in Expression (3) If is subadditive then preserves the triangle equality Proof Given, and, we know that dominates Using Theorem 14 we know that any modified aggregation operator given by Expression (4) dominates any modified transitivity operator given by Expression (5) Using Theorem 15 we know that if the modified transitivity operator is stronger that, then the modified aggregation operator dominates Thus, This aggregation keeps metric transitivity It is easy to see that this is an aggregation operator that belongs to the family represented in Expression (3) with In the following example we compare the structure of two trees with a non-metric dissimilarity and apply an equivalence function to get an equivalent metric Figure 1 A simple coding of binary trees The reason of going bottom-up is to have the less significant digits close to the root of the tree The choice of making the left nodes more significant than the right ones is arbitrary The symbol represents the empty tree Example 7 Consider a dissimilarity function between binary trees It does not measure differences between nodes but the structure of the tree First, let us describe a tree coding function, that assigns a unique value for each tree This value is first coded as a binary number and later interpreted as a natural number, and contains bits, where is the height of the tree The value is calculated such that the most significant bit corresponds to the leftmost and bottommost tree node (see figure 1) Note that is not a bijection, since there are numbers that do not code any valid binary tree Consider now the following dissimilarity function, where and are binary trees; represents the empty tree and its value is 0 Example 6 Given a set of metric dissimilarities, (with and ), where, and an aggregation operator given by 6 In all these functions, only the segment is used, since their argument is a distance This is a strongly reflexive, symmetric, unbounded dissimilarity defined on with If we impose a limit to the height of the trees (call it ), then is upper bounded and closed with It is also transitive with the product operator (which is a dissimilarity transitivity operator on, with 1 as 76

8 Journal of Convergence Informarion Technology Vol 3 No 3, September 2008 null element) This means that for any three trees : To see this, suppose first that neither of are the empty tree; substituting in the previous expression and operating with and the product one gets: which is true If, then the inequality is which is trivially true It is also true for or If we now apply the equivalence function to we get an equivalent dissimilarity function By Proposition 4 the properties of are kept in The transitivity operator is changed, using Expression (6), as: Therefore, we have a metric dissimilarity function over these trees and fully equivalent to the initial choice of 4 Special values 41 Methods for treating missing data In this last section we concentrate on metric dissimilarities and the problem of dealing with special values, notably missing values We identify up to four kinds of special values: missing, xor, don t care and illegal There could be others, but these are enough to cover most of the situations encountered in practice Note we somewhat abuse terminology by calling them values These special values potentially apply to any kind of variable (continuous, nominal, structured, etc) Missing : the value is not known It could be any value in the domain (but only one of them) Example: a value not recorded because of technical difficulties Xor : the value is known to be one (and only one) of a reduced list Example: the color of a car in a police invoice (uncertain witness declaration) It could be one of dark blue, dark grey or black Don t care : the value could be any, but we simply don t care what the value is Example: the age at which a specific patient starts to develop an illness Illegal : no value makes sense here Example: the number of pregnancies of a male A value of zero does not tell that this is the only possible value When comparing it to a female with no pregnancies, the distance would be zero In this work we concentrate on missing values, although what follows could be applied to other special values The way they are treated is a source of semantics For example, defining the distance between two don t care" values to be the maximum distance is dangerous, because it gives an overall lower dissimilarity value We believe that a better way of dealing with these values is by ignoring them From a statistical point of view, the type of missing data randomness can succinctly be categorized following [16]: 1 Missing completely at random (MCAR) It occurs when the probability of an example having a missing value does not depend on the known values of the example 2 Missing at random (MAR) It occurs when the former probability does depend on the known values of the example, but not on the missing datum itself 3 Not missing at random (NMAR) It occurs when the former probability depends on the known values of the example, as well as on the missing datum itself However, from the machine learning (ML) point of view it may be more useful to categorize the situations under a different computational perspective In an inductive learning scenario, one is usually interested in computing a certain function that may be affected by missing data in one or more of its arguments Let be the function that one needs to compute in the context of a ML method (eg a metric distance) We identify three policies: Type 1 : Estimate the value itself (ie, fill in the hole"), prior and independently of Type 2 : Extend the function to cope with missing values This is done by extending its domain The function accepts missing values and in case any of its arguments is missing, it yields a specific and constant value in its co-domain Type 3 : Estimate what the value of could be, without estimating the missing value itself This method can be seen as a generalization of Type 2 methods The difference is that the value that yields in case any of its arguments is missing depends on and on other arguments (thus is non-constant, in the likely case that not all the information needed by is missing) 42 Metric treatment of missing values There has been considerable work on Type 1 methods In this work we concentrate on the other two In Type 2 methods, we require the extension of the domain of the function being computed To settle the 77

9 On the design of metric relations Lluís A Belanche, Jorge Orozco idea, let us concentrate on a metric distance, and consider the extension with, where behaves as an incomparable element wrt any ordering relation in The Type 2 method requires to define the quantities and for all, which can be done as follows: We assume that is upper-bounded (though not necessarily closed) If it is not, the equivalence function with does the job to and keeps the metric properties In any case, define It is worth mentioning here previous related work A typically encountered choice is, as in [17] Some authors advocate for different choices For example, [21] advocate for in their HVDM distance for continuous values For nominal values, and,, where VDM is the value-difference metric [20] That is, the special value is treated like any other value A similar proposal is in [19], but using instead the SVDM (simplified version of the VDM), as follows: where are two values of some feature, is the number of classes and is the fraction of examples having and of class among those having For continuous values, [19] advocates for, to ignore a feature when its value is unknown The question is: for what values of is a metric distance? We give a precise answer in the following result Proposition 3 Provided is a distance in, the function is a distance in if and only if and Proof The proof is straightforward by developing the eight possible cases arising ( equal or not to ) and solving for the inequalities Again, interesting different semantics can be obtained for the valid choices of to deal in a userdependent meaningful way with special values In Type 3 methods, the problem of missing values is tackled by giving an estimation of the value of a function of interest when either or both of are unknown In the present case, is a dissimilarity (eg a distance) This approach is of great interest since the value of is what is needed in many practical situations, and not the value of themselves The difference with Type 2 methods is that the estimated values are not constant but dependent on and either of An assessment of the estimation error (or at least a bound of it) is especially welcome Given a distance in, define: In analogy with previous settings, a simple choice can be: (9) (10) Proposition 4 Provided is a distance in, the function with the choices in (9), (10) is a distance in Proof The proof is again immediate by developing the eight possible cases arising ( equal or not to ) and solving for the inequalities In doing so, it turns out that these settings are the tightest that still make a distance In this particular case the estimation error can be (very crudely) bounded as follows: Proposition 5 In case (resp ) is missing, and with the choices in (9), (10), the function has a maximum deviation from the true value" equal of (resp ) If both are missing, the maximum deviation is Proof We are interested in 78

10 Journal of Convergence Informarion Technology Vol 3 No 3, September 2008 In addition, Example 8 Let be the standard metric in and take ; then we obtain and 43 The case of Gower-like aggregation If additional information is used, then better estimates can in principle be given In practice one has a collection of variables and is only interested in the aggregated dissimilarity Then the values for other variables can be used to derive estimations of a missing computation This idea is used in Gower s similarity index, well-known in the literature on multivariate data analysis [1] In this proposal, the similarity between two values, one or both of which being unknown, is given by the average similarities computed on the known values (both values known) Specifically, for any two objects given by tuples of cardinality, Gower s heterogeneous similarity score is given by the expression: (11) where is the similarity according to variable, and is a binary function expressing whether the objects are comparable or not according to variable, equal to 1 if and to 0 otherwise The ignorance of the absent elements normalized by the number of present ones has been found superior to other treatments in standard data analysis experiments [18] This treatment corresponds to taking the average similarities computed on the known values To see this, let the number of variables for which both values are available, the sum of known similarity values and set It suffices to see that: This is a simple, intuitive and easy-to-implement idea (a Type 3 method using arithmetic mean as aggregation) However, it has a serious drawback: it does not keep metric properties We illustrate this with a counter-example, using a distance Example 9 Let and ; note that Consider the three observations, and Then,, but clearly The reason is that the variables for which their respective distances are computed are different for different pairs of examples We are interested in giving more general results concerning Type 3 missing value treatments, that are valid for any dissimilarity (regardless of its transitivity) and any aggregation operator 44 Extension to general transitivities Proposition 6 Let dissimilarity and define dissimilarity: is still -transitive if and only if a -transitive upper-bounded Then the and Proof The proof develops again the eight possible cases arising ( equal or not to ) and makes use of the properties of (Def 1) for solving the inequalities 45 Extension to general aggregations If is an aggregator that maintains transitivity, then it is immediate to realize that the aggregation of extended measures of Type 2 is again -transitive, since the show the same transitivity as the corresponding from which they come 46 Experimental evaluation In this last section we carry out a little experimental comparison between several missing value treatments, for illustrative purposes The problem used is the wellknown IRIS data set [23], a three-class classification task in which there are continuous features and no missing data To assess performance, we use the one-nearest-neighbor rule and the leave-one-out crossvalidation method (LOOCV) Scaling the features to the interval and using Euclidean distance: the obtained LOOCV baseline performance is 940% Then increasing amounts of missing values are introduced at random locations uniformly distributed among rows and columns, from 5% to 50%, in 5% steps In order to setup a realistic scenario, first the missing values are introduced and then the data is 79

11 On the design of metric relations Lluís A Belanche, Jorge Orozco scaled (using only the known values) A total of seven missing-value treatments (MVT) are tested, as follows: MVT1 Gower method, no imputation MVT2 Filled with a 0 MVT3 Filled with the mean of the feature MVT4 constant MVT5 constant MVT6 constant MVT7 as in (9), constant For each missing value percentage, 1000 random data sets were created, LOOCV performance was computed and the average result was taken The obtained results are presented in Table 2 Table 2 Results of the missing value treatment experiment MVT1 2 MVT 3 MVT MV MV MV MV T4 T5 T6 T Several interpretations can be made in light of these results First, it is noted how different treatments can yield a different performance, sometimes very acute Second, there are methods, like MVT4, that seem to work well for small amounts of missing data, but quickly degrade when the percentage is increased Third, the best methods seem to be MVT5 and MVT6, which are better than either of MVT1, MVT2 or MVT3 And fourth, the introduced method MVT7 does not seem to be better than the rest 5 Conclusions We have presented a body of work concerning the design and properties of dissimilarity measures The studied properties include transitivity and the aggregation of a number of different dissimilarities A special emphasis has been put in the relationship between aggregation and transitivity Moreover, several ways of dealing with special values (eg missing values) have been studied Although the focus is in general dissimilarities, the particular case of metric distances (ie dissimilarities fulfilling the triangular inequality) has been specifically covered, given its importance in machine learning and pattern recognition, among other disciplines The results presented have the purpose of helping to understand the main properties of hand-crafted dissimilarity measures (and to alter them if necessary) in order to widen their applicability We would like to bear to mind the subtitle of this paper: things one should know about a dissimilarity measure, meaning that some properties (like the triangular inequality, a case of transitivity) are very delicate and can be easily lost with apparently innocent transformations 6 References [1] J Gower, A general coefficient of similarity and some of its properties, Biometrics, no 27, 1971 [2] F Klawonn and J L Castro, Similarity in fuzzy reasoning, Mathware & Soft Computing, vol 2, pp , 1995 [3] Tversky, Features Of Similarity Psychological Review, July 1977, vol 84(4), pp [4] S Santini and R Jain, Similarity measures, in IEEE Transactions On Pattern Analysis and Machine Intelligence 21(9), pp , 1999 [5] P Fonck, J Fodor, and M Roubens, An application of aggregation procedures to the definition of measures of similarities between fuzzy sets, Fuzzy Sets & Systems, vol 97, pp 67 74, 1998 [6] D Bridge, Defining and combining symmetric and asymmetric similarity measures, in Advances in Case-Based Reasoning (Procs of the Fourth European Workshop on Case-Based Reasoning), BSmyth and PCunningham, Eds Springer, 1998, pp [7] Ferguson and D Bridge, Generalised weighting:a generic combining form for similarity metrics, in Procs of the 11th Irish Conference on Artificial Intelligence & Cognitive Science (AICS 2000), J Griffith and C O Riordan, Eds, 2000, pp [8] R Mesiar and M Komorníková, Aggregation operators, in Proceedings of the XI Conference on applied Mathematics PRIM 96, D Herceg and K Surla, Eds Novi Sad: Institute of Mathematics, 1997, pp [9] S Ovchinnikov, Aggregating transitive fuzzy binary relations, International Journal of Uncertainty and Fuzziness in Knowledge-based Systems, vol 3, no 1, pp 47 55, 1992 [10] J Fernández Salido and S Murakami, On -precision aggregation, Fuzzy Sets and Systems, vol 139, 2003 [11] F Klawonn, Should fuzzy equality and similarity satisfy transitivity? Fuzzy Sets & Systems, vol 133, pp , 2003 [12] A Tversky and I Gati, Similarity, Separability and the Triangle Inequality Psychological Review, July 1982, vol 86, pp

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