K. Hausdorff K K O X = SDA. symbolic data analysis SDA SDA. Vol. 16 No. 3 Mar JOURNAL OF MANAGEMENT SCIENCES IN CHINA

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1 JOURNAL OF MANAGEMENT SCIENCES IN CHINA Vol 16 No 3 Mar K K Hausdorff K K K O1 4 A X = 5 36 K SDA 1 symbolic data aalysis SDA 3 5 SDA guojp@ tju edu c

2 a 11 b 11 a 1p b 1p X = 1 7 Mahalaobi distaces 8 K 9 a 1 b 1 a p b p 1 X j city-block distaces X Hausdorff j = 1 μ kj k = 1 10 Wasserstei X kj μ kj X kj μ kj X j 11 1 X j S j = 1 σ kj + X j - μ kj 3 k = 1 μ kj X kj σ kj X kj X kj S kj K K K 1 Hausdorff Hausdorff 1 R P A = a b B = c d 9 13 Hausdorff H A B = c A - c B + r A - r B 1 1 X 4 a b X c X X r X X X = A B a b 4 Hausdorff c X X 1 X p p r X k = 1 a kj b kj X j k

3 3 K 3 Hausdorff μσ A B A X = x 1 x x p T B μσ D S A B = A - B + 槡 3 S A - S B 5 A B A B = c 1 d 1 c d c p d p T S A S B A B p 5 槡 3 d S X Y = p d x i y i = A B 槡 i =1 X = x + x p x i - y i + 槡 3 S xi - S yi 槡 i =1 7 S X = x - x 6 = r X μσ K 槡 3 槡 3 K 5 Hausdorff μσ Hausdorff 1 3 A B 1 B A = K B 1 = 18 B = μσ K B 1 B A B 1 X 1 X K P 1 A P K y 1 y k K B A J B 1 A B A 4 Hausdorff D H A B 1 = D H A B = 5 μσ D S A B 1 = D S A B = D S A B 1 < D S A B μσ = a 1 b 1 a b a p b p T Y = y 1 y y p T 7 Hausdorff x i i = 1 y k k = 1 K 13 K J = K k = 1 x i y k i P k x i y k x i y k 1 K y 1 y K test = 0

4 μσ P j y j U V 1 x i P j 1 P h P h * μσ h h * test = 1x i P h * P h * = P h * i P h = P h \ i V CR 4 k x i = a i b i x i P k P k y k k = 1 K 3 3 P k y k y k S yk 5 test = 0 test = CR corrected rad idex 14 x y U = u 1 u i u R μ = μ 1 V = v 1 v j v C [ ] Σ = σ 1 0 [ ] μ 0 σ R C 3 CR = [ 1 R R C i =1 j =1 ( io i =1 ( ( ij + C ( ( oj ] - j =1-1 R io ( ( oj i =1 C j =1 -( -1 R ( io i =1 C j =1 ( oj x i i = U CR CR x i μσ K data 1 μ 1 data μ 1 = μ = 64 μ = = ij u i v j x y io u i o j x y x - r 1 x + r 1 y - r y + r v j CR σ 1 = 30 σ 1 = 100 σ = 9 = 9 σ = 36 data 3 μ 1 = 40 μ = 4 σ 1 = 9 σ = 9 r 1 r

5 3 K r 1 r 60 K CR 1 1 K CR Table 1 The average CR values of K-meas clusterig of ormally distributed iterval symbolic data r 1 r 1 Fig 1 The radom iterval data K K 1 μ - 58σ μ + 58σ 99 73% CR 0 7% r K N x r 1 / 58 N y r / 58 r 1 r CR K K 5 1 Hausdorff 13 CR ad uiformly distributed iterval symbolic data K Fig The iterative process of K-meas clusterig of geerally

6 k = 3 K K Table Iterval symbolic data of the cars sample /mm /mm /mm / /hp /s /km h SL Z TT Cayma Boxster GT C A6L E G S

7 3 K 7 3 Table 3 The compariso of precisio of the two methods μσ K K 3 3 G E G % 77 7% μσ K K K 5 1 J Hu Ya Wag Huiwe A ew data miig method based o huge data ad its applicatio J Joural of Beijig Uiversity of Aeroautics ad Astroautics Social Scieces Editio i Chiese Bock H H Diday E Aalysis of Symbolic Data M New York Spriger-Verlag J Li Wehua Guo Jupeg Methodology ad applicatio of regressio aalysis of iterval-type symbolic data J Joural of Maagemet Scieces i Chia i Chiese 4 Diday E Brito M P Symbolic cluster aalysis C / / Coceptual ad Numerical Aalysis of Data Eds Opitz O Heidelberg Spriger-Verlag De Carvalho F A T Cserel M Lechevallier Y Clusterig costraied symbolic data J Patter Recogitio Letters De Carvalho F A T Brito P Bock H H Dyamic clusterig for iterval data based o L distace J Computatioal Statistics Teorio C P De Carvalho F A T Pimetel J T A partitioig fuzzy clusterig algorithm for symbolic iterval data based o adaptive mahalaobis distaces C / / Proceedigs of 7th Iteratioal Coferece o Hybrid Itelliget Systems De Carvalho F A T Teorio C P Fuzzy K-meas clusterig algorithms for iterval-valued data based o adaptive quadratic distaces J Fuzzy Sets ad Systems De Carvalho F A T Lechevallier Y Partitioal clusterig algorithms for symbolic iterval data based o sigle adaptive distaces J Patter Recogitio Irpio A Verde R Dyamic clusterig of iterval data usig a Wasserstei-based distace J Patter Recogitio

8 J Re Shiji Lü Juhuai Geetic algorithm-based kerel fuctio FCM clusterig algorithm for iterval umbers J Joural of System Egieerig i Chiese 1 J Guo Jupeg Li Wehua Gao Feg Descriptive statistics ad aalysis of iterval symbolic data with geeral distributio J Systems Egieerig - Theory & Practice i Chiese 13 J Fa Bo Spatial clusterig miig method for site selectio problem of emergecy respose ceter J Joural of Maagemet Scieces i Chia i Chiese 14 De Carvalho F A T De Souza R M C R Chavet M et al Adaptive Hausdorff distaces ad dyamic clusterig of symbolic iterval data J Patter Recogitio K-meas clusterig of geerally distributed iterval symbolic data GUO Ju-peg CHEN Yig LI We-hua College of Maagemet ad Ecoomics Tiaji Uiversity Tiaji Chia Abstract The existed clusterig methods of iterval data mostly supposed that the data are uiformly distributed across the iterval However this is ot always practical Takig this ito accout this paper aims to research the k-meas clusterig method of iterval data with a geeral distributio The defiitio of geerally distributed iterval data is proposed ad descriptive statistics was researched based o empirical distributio theory O the basis of Hausdorff distace the paper puts forward a ew distace for iterval data which cosiders the poit data cotaied i the itervals Based o this we preset a algorithm of k-meas clusterig of geerally distributed iterval symbolic data A simulatio experimet is coducted to evaluate the validity of our method The results show that compared with aalysis methods of uiform iterval symbolic data the a- alysis methods of geerally distributed iterval symbolic data are more effective uder all the coditios desiged i our experimet Fially the method is illustrated by a example of real-case data which shows the advatages of our method i the practical applicatio Key words iterval symbolic data geeral distributio symbolic data aalysis clusterig aalysis