A Distance and Angle Similarity Measure Method

Transcription

1 A Distace ad Agle Similarity Measure Method Ji Zhag* ad Robert R. Korfhage School of Iformatio Scieces, Uiversity of Pittsburgh, 135 N. Bellefield Aveue, Pittsburgh, PA This article presets a distace ad agle similarity measure. The itegrated similarity measure takes the stregths of both the distace ad directio of measured documets ito accout. This article aalyzes the features of the similarity measure by comparig it with the traditioal distace-based similarity measure ad the cosie measure, providig the iso-similarity cotour, ivestigatig the impacts of the parameters ad variables o the ew similarity measure. It also gives the further research issues o the topic. * To whom all correspodece should be addressed. Curret address: Ji Zhag, School of Library ad Iformatio Sciece, Uiversity of Wiscosi Milwaukee, 2400 E. Hartford Ave., Milwaukee, WI Deceased. Received July 17, 1998; revised March 8, 1999; accepted March 8, Joh Wiley & Sos, Ic. Itroductio The similarity measure is a essetial cocept i iformatio retrieval. It is widely used to judge whether a documet matches a query, or to measure the similarity of two documets. I other words, the similarity measure allows a user to arrage or exhibit retrieved documets i decreasig order of similarity with respect to the query; to dowsize a retrieved set by removig the documets with lesser similarity to the query; to measure discrimiative value of idexig terms; ad to dyamically adjust the retrieval strategy by addig more terms with high similarity ad discardig the terms with low similarity. Furthermore, similarity measures ca be applied to costruct visualizatio iterfaces to facilitate iformatio retrieval. A good similarity measure is a importat factor that cotributes to satisfactory precisio ad recall ratios i iformatio retrieval. Differet iformatio retrieval systems usually take differet similarity measures. The distace ad agle itegrated similarity measure itroduced here is a vector-based similarity measure. Accordig to McGill et al. (1979), there are more tha 60 differet similarity measures. These iclude the ier product, Dice coefficiet, cosie coefficiet, Jaccard coefficiet, overlap coefficiet (Frakes & Baeza-Yates, 1992; Korfhage, 1997; Meadow, 1992; Salto, 1968, 1989), the spreadig activatio similarity measure (Joes & Fures, 1987), ad some probability-based similarity measures (Croft & Harper, 1979; Kwok, 1985; Robertso & Sparck, 1976; Va Rijsberge, 1979; as well as Robertso & Walker, 1997). Amog them, the most popular are the distace-based similarity measure ad the agle-based cosie measure. Each similarity measure has its stregths ad weakesses i practice. Although much research has bee doe o similarity measures, the combiatio of differet similarity measures is rarely cosidered. Research o the combiatio of differet similarity measures has the potetial to provide a ew ad uique approach to similarity research. The distace ad agle itegrated similarity measure attempts to orgaically combie a distace-based similarity measure with the agle-based cosie measure, to take advatage of the stregths of both ad to make similarity measuremet more scietific ad accurate. Fudametal, Features, ad Aalysis To better uderstad the ratioale of the proposed distace ad agle itegrated similarity measure, we should aalyze the stregths ad weakesses of both the aglebased cosie measure ad distace-based measure, from which the ew idea shall be elicited. The agle-based cosie measure is a directio-based similarity measure. It measures the similarity betwee a referece poit ad a documet based oly o the directio of the documet i the documet space vis-à-vis the referece poit ad the origi of the coordiate, igorig the impact of the distace betwee the referece poit ad the documet. The cosie measure ca effectively idetify documets i a vector documet space that have the same idexig term distributio withi the each documet; that is, they have the same idexig terms, the same proportio of weights of ay pair of idexig terms betwee two documets. This characteristic ca be employed to idetify documets with the same subject but at differet levels i a documet vector space. JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE. 50(9): , 1999 CCC /99/

2 Suppose R is a referece poit, Di, Dj are two documets: Rx k1, x k2,...,x k ; Dix i1, x i2,...,x i ; Djx j1, x j2,...,x j ; x jr c*x ir, r 1,...,, c is a costat. FIG. 1. Three documets with the same directio. cosier, Di cosier, Dj cosier, Dj x kr 2 1/2 x kr 2 1/2 x kr 2 1/2 x kr x ir x kr x jr (1) 21/2 x ir 21/2 x jr x kr c x ir 21/2 c 2 x ir cosier, Dj cosier, Di (2) Therefore, if Di ad Dj are similar to R i terms of the cosie measure, ad if the weights of the idexig terms are associated with the frequecies of the terms i the documets, the differece betwee the two documets will oly be affected by measures of the legths of the documets. The essece of the cosie measure is that it ca idetify the documets i terms of the idexig term distributio. From the aalysis we kow that the directio of a documet i a documet vector space does affect its similarity to a certai object. However, it is ot the oly factor that ca ifluece its similarity. O the other had, i the distace-based similarity measure, the similarity ca be trasformed from the distace betwee the documet ad the referece poit as follows: s a d (3) where d is the distace, ad a is a costat whose value is greater tha 1. The distace-based similarity measure follows the philosophy that documets close together are likely to be highly similar. I this case, all directios are cosidered equal. The distace-based similarity measure takes oly the impact of the distace ito accout, regardless of the directio of the documet. I other words, documets with the same distace to the referece poit shall have the same similarity. This approach ca resolve the iheret weakess of the cosie measure that is that it caot distiguish documets that have the same directio, but are far from each other i terms of distace i a documet space. Although two documets share the same directio, it is argued that the validity of the high similarity of two documets is reduced to some extet whe they are far apart i terms of distace. For example, there are three documets D1, D2, ad D3 with the same directio i a documet vector space (see Fig. 1). Because documets i the same directio vis-à-vis the origi of a documet vector space have the same keyword distributio with proportioal weights, the differeces amog these documets are reflected the extet to which they address the same topic. The similarity betwee D1 ad D3 is the same as that betwee D2 ad D3 i terms of the agle-based similarity measure. Similarity betwee two objects should be measured by both the topics they address ad the extet to which they address. It is clear that documet D2 should be more similar to D3 tha D1 because documet D2 addresses the same topic i more detail tha documet D1. Obviously, the farther apart documets D1 ad D2 are, the bigger the differece should be. However, due to the iheret weakess of the agle-based similarity measure, igorig the impact of the distace o the similarity, it caot discer the differece i measurig the similarities of a group of documets with the same directio i a documet vector space. It is possible that two documets are quite similar i terms of the distace-based similarity measure but they are absolutely ot similar i terms of the agle-based similarity measure. Oce a query, a distace-based similarity measure, ad a documet are selected, a cotour is defied ad documets withi the cotour are more relevat tha that documet. We ca get a ice, symmetric moutai of relevat documets with the most relevat earest the query. What modificatio by a agle-based similarity measure dose is to cotour the surface of this moutai, depressig it more i some places tha others, because it is quite likely that a documet withi the cotour is less relevat tha that documet i terms of agle-based similarity measure. JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July

3 If Dp is ay poit i the circle, the correspodig is: arccos x 1i 2 1/2 x 1i x 2i 1/2 (5) x 2i 2 FIG. 2. space. Chage of distace-based similarity measure i a documet where Dp(x 11, x 12,...,x 1 ); R(x 21, x 22,...,x 2 ); ad is the dimesioality of the documet space. The ew distace ad agle similarity measure is defied as follows: The above aalysis demostrates that the two traditioal similarity measures partially reflect the similarity of the compared objects, ad they are complemetary with respect to the distace ad directio. The complemetary feature of two measures suggests that it would be useful to develop a ew similarity measure, takig advatages of both ad discardig the disadvatages. It is this aim that the ew distace ad agle itegrated similarity measure attempts to achieve. Oce the distace betwee a referece poit ad a documet is fixed, the similarity vis-à-vis the referece poit is also fixed with respect to the distace-based similarity measure. I fact, the distace ad the referece poit will determie a hypersphere i the documet space: the ceter of the hypersphere is the referece poit, the radius is the distace betwee the referece poit ad the documet. I this istace, each documet i the circle has aother similarity measure, vis-à-vis the axis, formed by the referece poit ad the origi i terms of the cosie measure. The similarity of the documets o the circle varies with differet positios; i most the cases, they are ot equal. Whe a documet is located at the itersectios betwee the axis ad the circle (there are two such poits D0 ad D1), the cosie measure of the documet reaches the maximum value 1. The miimum value of the similarity depeds o the legth betwee the referece poit ad the documet (see Fig. 2). The pheomeo suggests that we could use the chage of the directio-based similarity measure whe a documet moves alog the circle to modify the distace-based similarity measure so that the ew similarity could reflect ot oly the cotributio of the distace, but also the cotributio from the directio of the measured documet. This is the ratioale for the combied distace ad agle similarity measure method. I Figure 2, r is the radius of the circle, R is the referece poit, D0, D1, Dp, ad Dm are the documets i the circle, h is the distace betwee R ad O, d is the distace betwee Dp ad O. ad are the agles formed by RO ad DpO, DpR, respectively. Oce Dp is selected ad the values of r ad h are fixed, the maximum value of is: max arcsir/h (4) s a r c k (6) The effect of the parameters a ad c will be discussed i detail later. k arccos x 1i 2 1/2 k max x 1i x 2i 1/2 x 2i 2 x 1i x 2i k arccos 1 d h arcsir/h 1 arcsir/h I fact, a r is the distace-based similarity measure, c k is a modifier, where k is defied i Equatio (7). To maitai s i the 0 to 1 rage, we require 0 c 1. The effect of the parameters a ad c will be discussed i detail i the sectio o The Effects of Parameters a ad c o the Similarity Measures. Note that i Equatio (7) the agles rather tha cosie values are used to express the impact of the directio rather tha their cosie values, which reduces the complexity of the computatio, ad simultaeously keeps its basic characteristics. Because the documet space vector elemets are oegative, R is always i the first quadrat i the vector documet space. We assume that r is sufficietly small that the circle lies etirely i the first quadrat. Equatios (6) ad (7) show that the value of the ew similarity measure shall be betwee 0 ad 1. Because the ew similarity measure takes the agle ad the distace ito cosideratio, the problem discussed i Figure 1 ca be avoided. If documets D1, D2, ad D3 have the same directio i a documet vector space, their correspodig agle should be equal to zero i the ew similarity measure. It suggests that the similarity measure is s a r whe measured documets have the same agle with a referece poit. It meas that the similarity betwee the documets D1 ad D3 is differet from that betwee the (7) 774 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July 1999

4 Notice that whe r chages, the maximum ad miimum values of the similarity measure, ad the positio of the miimum value i the X-axis also chage. The smaller the value of r, i.e., the earer the documet to the referece poit, the higher the similarity value, ad vice versa. The Impact of the Parameter c o the Similarity Measure From Figure 2, we have: r si d si (8) FIG. 3. Relatioships amog agle-based, distace-based, ad distaceagle itegrated similarity measures. documets D2 ad D3 due to the differet r values. The differece betwee the two similarities depeds o the distace betwee D1 ad D3 as well as the distace betwee D2 ad D3. I other words, the extet to which they address the same topic is reflected i the ew similarity measure. Now cosider the features of the ew similarity measure i detail. To illustrate the impact, the etire circle should be displayed. However, for simplicity of the display ad due to the symmetry of the circle vis-à-vis RO, oly oe-half of the circle is displayed, that is, rages from zero to. The Relatioship of the New Similarity Measure to the Distace-Based Similarity Measure ad the Cosie Measure I Figure 2, whe ay documet Dp moves from D0 to D1 alog the circle the similarity varies with differet similarity measures. For the distace-based similarity measure, it is a costat, depedig o the distace betwee Dp ad R, (See Fig. 3); for the cosie measure, as Dp moves from D0 tod1, it decreases from D0 to Dm, the icreases from Dm to D1. The miimum value is cos[arcsi(r/h)], the maximum value is 1 (see Fig. 3); the ew similarity measure has characteristics of both similarity measures: first, it is chageable; secod, it has a miimum value at the same positio as the cosie measure does. This value is smaller tha that of the cosie measure; fially, its maximum value is equal to that of the distace-based similarity measure (see Fig. 3). I Figure 3, x-axis is the agle ad y-axis is the similarity. From Equatios (8) ad (9): r cos d cos h (9) r cos r si cos h si arcta si h/r cos (10) From Equatios (6), (7), ad (10): s a r c arctasi/h/rcos/max (11) The impact of the parameter c o the similarity measure is illustrated i Figure 5, where h 5, a 1.11, r 3, max /6. The four curves are associated with c 0.2, 0.4, 0.6, ad 0.8, respectively. Notice that whe c 1, Equatio (6) becomes the distace-based similarity measure. Figure 5 idicates that the smaller the value of c, the greater the ifluece of c k as a modifier o the similarity measure, ad vice versa. Each curve reaches its miimum value at the same positio. They have the same maximum value a r. The Impact of the Parameter a o the Similarity Measure Equatio (11) yields Figure 6, where values of h, r, ad max are the same as above, c 0.5. The Impact of the Parameter r o the Similarity Measure Suppose that r chages, but the ceter of the circle is statioary; i other words, h is fixed. Equatio (13) is used to geerate Figure 4. I Figure 4, h 9, a 1.11, ad c 0.5. The four curves are associated with r 1, 3, 5, ad 7, respectively, i Figure 4. FIG. 4. Impact of r o the similarity measure. JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July

5 FIG. 5. Impact of c o the similarity measure. FIG. 7. Impact of max o the similarity measure. The four curves are associated with a 1.1, 1.3, 1.5, ad 1.7 respectively. The smaller the value of a, the greater the similarity, ad the greater the differece betwee the miimum ad the maximum. Each a determies differet miimum ad maximum similarity values, but the positio at which the curves reach their miimum poits is same. The Impact of the Parameter max o the Similarity Measure Suppose as max chages, the ceter of the circle does ot move, i.e., h stays the same. The chage of max will the affect the radius r; r is a fuctio of the max ; thus: s a hsimax c {arcta[si/((1/simax)cos)]/max} (12) The four curves ( max 3, /4, /5, ad /6) are preseted i Figure 7. The parameter values are h 5, a 1.11, ad c 0.5. Whe max chages, the positios at which the curves achieve miimum values vary with the differet max. The smaller the max, the smaller the miimum value, ad vice versa. The Impact of the Parameter h o the Similarity Measure Whe the value of h chages, the radius of the circle does ot chage. The chage would affect max [see Equatio (4)]; therefore: s a r c arctasi/h/rcos/arcsir/h (13) The four curves with h 3, 5, 7, ad 9, respectively, are show i Figure 8, where a 1.11, r 2, c 0.5. As value of h chages, the positios at which the curves gai their miimum values also chage, the correspodig values chage, but the maximum value does ot. The smaller the value of h, the larger the miimum value; ad vice versa. Iso-similarity Cotour Aalysis Iso-similarity cotour with respect to the parameter c From Equatio (11): log s loga r arcta si h/r cos log c max c 10 max logs ar /arctasi/h/rcos (14) The four cotours (s 0.1, 0.3, 0.5, ad 0.7) are give i Figure 9, where h 5, a 1.11, max /6. Figure 9 shows that the smaller the value of s, the lower the iso-similarity cotour; ad vice versa. FIG. 6. Impact of a o the similarity measure. FIG. 8. Impact of h o the similarity measure. 776 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July 1999

6 FIG. 9. Iso-similarity aalysis of c. FIG. 11. Effect of c o the similarity measure. Iso-similarity cotour with respect to the parameter a From Equatio (11): a r 1 s c{arctasi/h/rcos/amax} a 10 1/rlogs1 c arctasi/h/rcos (15) The four cotours with s 0.1, 0.3, 0.5, ad 0.7 are exhibited i Figure 10, where h 5, max /6, c 0.8. Figure 10 idicates that the smaller the value of s, the higher the cotour. The Effects of Parameters a ad c o the Similarity Measures The parameters a ad c are artificial, affectig the display of similarity. We ote that a is related to the distace r, ad c is related to the agle. The two display parameters are restricted i rage: a 1, ad 0 c 1. Whe either a or c is set to 1, the similarity measure is idepedet of the associated documet parameter. To show the iteractio betwee a or c, we assume a hypothetical query ad documet, thus fixig the parameters h, r, ad. Let us discuss the effect of parameter a o the similarity measure whe c chages. Accordig to Equatio (11), the four cotours with a 1.1, 1.3, 1.5, ad 1.7 are geerated i Figure 11, where h 5, max /6, r 3, /3, ad c from0to1. Figure 11 shows that for a fixed a, whe c icreases, the correspodig similarity value icreases. The lower the value of a is, the more the similarity value icreases. The effect of parameter c o the similarity measure whe a chages is as follows. Accordig to Equatio (11), the four cotours with c 0.2, 0.4, 0.6, ad 0.8 are give i Figure 12,where h 5, max /6, r 3, /3, ad a from1to3. Figure 12 demostrates that whe a icreases, the similarity value of c decreases quickly to zero. Figures 11 ad 12 show that there is a rage of a ad c values that will yield this similarity value. The user s choice for these display parameters will reflect his emphasis o distace (low a value) or agle (high c value) as the domiat similarity factor. Coclusio The distace ad agle similarity measure presets a ew approach to itegratig both a distace-based similarity measure ad a directio-based similarity measure. It takes the effects of the distace ad directio of documets o the similarity measure ito accout. The cotributios of both the distace ad agle to the similarity value are adjustable by cotrollig the correspodig parameters. FIG. 10. Iso-similarity aaysis of a. FIG. 12. Effect of a o the similarity measure. JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July

7 The aalysis of the parameters such as a, c, max, r, ad h i the similarity measure tells the users how to apply the similarity measure appropriately: the parameter c could be used to cotrol the stregth of the directio of the measured documets. The effect of the directio o this similarity measure is based o the stregth of the distace of the documet. The parameter a is applied to adjust the stregth of the distace. The parameter h, which is the distace from the origi to the referece poit, is idirectly associated with the impact of the directio. The parameter r, which is the distace from a documet to the referece poit, iflueces the stregth of both distace ad directio; it is oe of the key variables i the similarity measure, as it also determies the maximum ad miimum values of the similarity measure values. The iso-similarity aalysis shows that the value of c, ad to a lesser extet the value of a, impact the perceived similarity value of a documet. This could help users to select the parameters to best advatage. The aalysis of effects of iteractio betwee two parameters a ad c o the similarity measures presets more useful iformatio for the selectio of a ad c. The way of measurig the agle of a documet iflueces the determiatio of the maximum agle max.itis importat whe this ew similarity measure is applied to the distace agle-based visual eviromet. Basically, i this similarity measure the four parameters ca be classified ito two groups. Group 1 cotais two parameters relatig a documet ad a query (referece poit) positios ( h ad r ) i a documet vector space, they are determied oly by the documet ad the query, ot by ay similarity measure, ad they impact a similarity measure. Group 2 cotais parameters relatig this ew similarity measure ( a ad c ); users are allowed to maipulate them to cotrol the impact of distace ad agle o the similarity measure, ad they are determied by users rather tha the documet ad the query. I the ew distace ad agle itegrated similarity measure the distace-based measure is take as the primary oe, ad it is reduced by the agle-based similarity measure whe the maximum similarity value is used as the compared object (startig poit is D0orD1 i Fig. 2). However, i the same situatio whe the miium similarity value is used as the compared object (differet startig poit Dm i a vector space i Fig. 2), it is icreased rather tha decreased by the agle-based similarity measure. Directios for further research iclude itegratig other distace-based similarity measures with the directio-based similarity measure, for istace, substitutig a rr for a r i the distace ad agle similarity measure; coordiatig the use of the differet parameters, etc. This article oly focuses o discussig the properties of the ew similarity measure. It is ecessary to coduct a experimetal study to ivestigate the performace amog the distace-based similarity measure, the agle-based similarity measure, ad this ew similarity measure i future research, allowig people to uderstad the ew similarity measure from a differet perspective. Refereces Croft, W., & Harper, D. (1979). Usig probabilistic models of iformatio retrieval without relevace iformatio. Joural of Documetatio, 35, Frakes, W.B., & Baeza-Yates, R., Eds. (1992). Iformatio retrieval: Data structure ad algorithms, Eglewood Cliffs, NJ: Pretice Hall. Joes, W.P., & Fures, G.W. (1987). Pictures of relevace: A geometric aalysis of similarity measures. Joural of the America Society for Iformatio Sciece, 38(6), Korfhage, R. (1997). Iformatio storage ad retrieval, New York: Wiley Computer Pub. Kwok, K.L. (1985). A probabilistic theory of idexig ad ximilarity measure based o cited ad citig documets. Joural of the America Society for Iformatio Sciece, 36(5), McGill, M., Koll, M., & Noreault, T. (1979). A evaluatio of factors affectig documet rakig by iformatio retrieval systems, Syracuse, NY: School of Iformatio Studies, Syracuse Uiversity. Meadow, C.T. (1992). Text iformatio retrieval systems, Sa Diego, CA: Academic Press. Robertso, S.E., & Sparck, J.K. (1976). Relevace weightig of searchig terms. Joural of the America Society for Iformatio Sciece, 27, Robertso, S.E., & Walker, S. (1997). O relevace weights with little relevace iformatio. I Proceedigs of the Twetieth Aual Iteratioal ACM SIGIR Coferece o Research ad Developmet i Iformatio Retrieval (pp ), Philadelphia, PA: ACM. Salto, G. (1968). Automatic Iformatio Orgaizatio ad Retrieval, New York: McGraw-Hill. Salto, G. (1989). Automatic text processig: The trasformatio, aalysis, ad retrieval of iformatio by computer, New York: Addiso-Wesley. Va Rijsberge, C.J. (1979). Iformatio retrieval (2d ed.), Lodo: Butterworths. 778 JOURNAL OF THE AMERICAN SOCIETY FOR INFORMATION SCIENCE July 1999