2006-2007 > Academic research: refereed > Publication in refereed journalVersion of RecordPublishe

Jin, FX

Shi, WZ

Shi, YF

PolyU Institutional Repository

? 31? ? 1?2006 ? 1 ?? ? ? ? ? ?  ? ? ? ? ?Geomatics and Informa tion Science of Wuhan Univer sityVol. 31 No. 1Jan. 2006????:2005-10-16。????:????????(1. 34. 37.A222) ;??????????????????????????????(SD2003-10)。????:1671-8860(2006)01-0082-04 ?????:AGIS中空间数据不确定性的混合熵模型研究史玉峰1 , 2 , 3 　史文中3 　靳奉祥4(1　???????????? ,????????? 12?, 255049)(2　??????????????????????? ,?????? 129? , 430079)(3　??????????????????????????? ,??????)(4　??????????? ,?????????????? 579? , 266510)?　?:基于信息理论和模糊集合理论 , 针对 GIS 中部分空间数据既具有随机性又具有模糊性的特点 ,建立了空间数据不确定性的混合熵模型。以 GIS 中线元不确定性为例 , 讨论了线元不确定性的统计熵 、模糊熵和混合熵估计方法 ,并针对特例给出了线元不确定性的熵带分布。???:不确定性;空间数据;混合熵;统计熵;模糊熵;线元??????:P208　　????????????????????? 、???? 、???????[ 1] ,??????????? 、?????? 、?????? 、????????????? ,??????????????? G IS ????????????。?? ,????????????????????????? 、??????? 。???????? ,????????? GIS ??? 、????????????[ 1-10] 。???? ,?????????????????????? ,??????????????????? ,????????????????????????。?? GIS ???????????????? ,?????? ,?? ,??????????????????????? ,?? ,???????????? ,???????????????????????????????????? ,??????????????? ,????????????????? 。???????????? ,????????????????????????? ,?? ,????????????。1　?????????????????? ,????????????[ 11] 。???????????(??), [ X  P] :{X :a1 , a2 , … , an ;P(X):p1 ,p2 , … , pn}, p i ??? a i ????? ,? p i ≥0 ,∑ni=1 p i =1 , ?H s(X) =E[ - lgp i] =- ∑ni=1p i lgp i (1)???????。???????????????????????????? 。?????????????????? , ??????????[ 12-14] 。?????????????? , Deluca ?Termini????????[ 12]?:H f(A) =- k∑ni=1{μA(xi)lgμA(xi)+(1 - μA(xi))lg(1 - μA(xi))}(2)?? , k ??? 0??? ,?? k=1。?????? ,???????????????? ,?????????。????????????? ,Deluca? Termini?????DOI牶牨牥牣牨牫牪牥牫牤j牣whugis牪牥牥牰牣牥牨牣牥牪牪　? 31 ?? 1 ? ????:GIS ??????????????????????????????? , ?????(total entropy)[ 1 1] ,????:H total =H r(A)+H f(A) (3)?? , Hr(A)???????;H f(A)???? 。?(3)?????????????????????????? ,?????????????。??????????????????????? ,? GIS ????????[ 3 , 5] ,???????? ,??????? ,?? ,???????????????????????? ,??????????(R)?????(F)????????? R ×F ,?? ,???????????????????????????????:f:R ×F →[ 0 , 1] (4)　　???????????????????????? Hh(R ,F)??? ,??????????:??????? , H h(R , F)????????;???????? , H h(R ,F)????????。???????? ,?? Shannon ?? De-luca-Termini????? , ????????Hh(R ,F)?:H h(R , F) =- ∑ni=1{p iμi lg(p iμi)+p i(1 - μi)lg(p i(1 - μi))}(5)?? ,???????。??(5)???? ,??:H h(R , F) =- ∑ni=1p i lgp i - ∑ni=1{μi lgμi +(1 - μi)lg(1 - μi)}+∑ni=1(1 - p i){μi lgμi +(1 - μi)lg(1 - μi)}=H s +H f - H sf (6)?? ,?(6)??? H s 、H f?????(1)??(2)??????????????。?? H sf???????????? ,????????????????? ,?H sf =- ∑ni=1(1 - p i){μi lgμi +　　(1 - μi)lg(1 - μi)} (7)　　??(6)???? ,????????? ,?μ(x)={0 ,1},?(6)?????????;????????? ,? p(x)=1? ,?(6)?????????。?????????????????? ,??????:Hh(R ,F) =-∫+∞- ∞{p(x)μ(x)lg{p(x)μ(x)}+p(x)(1 - μ(x))lg{p(x)(1 - μ(x))}}d x(8)?? , μ(x)? p(x)??????????????????????。??(8)????:Hh(R ,F) =-∫+∞- ∞p(x)lgp(x)dx -∫+∞-∞{μ(x) lgμ(x)+(1 - μ(x))lg(1 - μ(x))}d x +∫+∞- ∞(1 - p(x)){μ(x)lgμ(x)+(1 - μ(x)) lg(1 - μ(x))}dx =H s +H f - H s f (9)?? ,H s =-∫+∞-∞p(x)lgp(x)dx (10)H f =-∫+∞-∞{μ(x)lgμ(x)+(1 - μ(x))lg(1 - μ(x))}d x (11)Hs f =-∫+∞-∞(1 - p(x)){μ(x)lgμ(x)+(1 - μ(x))lg(1 - μ(x))}d x (12)??(6)??(9)???? ,??????????????????? ,????????(???)????????????????????。2　?????????????????? ,???????????????? 、?????? ,???????????????? 。??????????????? 1?? , A 、B ???????? , l???????。??????????????? ,??????? ,??????????? p(x)=12πσexp -x22σ2 ,???(10)?????????:H s(x) =-∫Rp(x)lnp(x)dx =lnσ 2eπ　(13)? 1　?????????Fig. 1　Poly gon Data Classing Model　　??(13)???? ,???????????????????。????[ 15] ,????83??????  ????? 2006 ? 1 ???????????:Rentropy = 12eHs(x) = eπ2σ≈2. 066 4σ(14)?? ,??(- 2. 066 4σ,2. 066 4σ)????????????????? ,????? ,???????????????? ,?????????????。????????? l ???? , A 、B ???????????? ,????????:μA(x) =1 , x <a1(x +a1) /2a1 , a1 ≤ x ≤0(b1 - x) /2b1 , 0 ≤ x ≤b10 , x >b1μB(x) =0 , x <a2(a2 - x) /2a2 , a2 ≤ x ≤0(b2 +x) /2b2 , 0 ≤ x ≤b21 , x >b2(15)　　???????? 2?? ,????? l? ,?????? ,??????0. 5;??????l????? ,???????? ,???????? 1??????? 0 。???(11)?????????:H f(A) =(b1 - a1)/2 (16)H f(B) =(b2 - a2) /2 (17)　　??(16)、?(17)???? ,???????????????????? ,?????????????? 。????????? ,????????????????:H f ={(b1 - a1)+(b2 - a2)}/2??????:H sf =-∫b1a1(1 - p(x)){μA(x)lgμA(x)+　(1 - μA(x))lg(1 - μA(x))}dx -∫b2a2 (1 - p(x)) ? 2　?? A、B ?????????Fig . 2　Distributions of Fuzzy Membe rshipFunction o f Ca teg or y A and B{μB(x)lgμB(x)+(1 - μB(x)lg(1 - μB(x))}d x(18)?(18)?????????? ,?????????????????????????。??????????????????????? ,?σ=±1;?? A 、B ????????? ,?????? ,? a1 =a2 =- 1 ,b1 =b2 =1 ,???? H s=ln 2eπ≈1. 418 9 ,??? H f =2。?? Matlab ?????????(18),?Hs f ≈0. 720 5 , ? Hh (R , F)=1. 418 9 +2 -0. 720 5=2. 698 4。??(14)?????????? 、??????????????????? 2. 006 3 、3. 694 5?5. 616 7。? 3 ??????? 、?????????????????????? 。? 3　??????? 、?????????Fig . 3　Statistic Entropy Band , Fuzzy Entr opy Bandand H ybrid Entropy Band o f Line Segment3　?　??????? GIS ???????????????????????????????? ,??????????????????? ,??????????? ,?????????????????? ,??????? 。?　?　?　?[ 1] 　Goodchild M F . Geog raphical Information Science[ J] . International Journal of Geog raphical Info rma-tio n Sy stem , 1992(6):31-46[ 2] 　Cheung C K , Shi Wenzhong , Zhou X. A P robabili-ty-based Uncertainty Model for Point-in-Poly gon A-naly sis in G IS[ J] . Geoinforma tica , 2004 , 8(1):71-98[ 3] 　Foody G M . Approaches for the P roduction and E-valua tion of Fuzzy Land Cover Classifications fromRemo te ly-sensed Data[ J] . Int. J. Remo te Sensing ,1996 , 17:1 317-1 340[ 4] 　Goodchild M F , H unter G F . A Simple Po sitio nalAccuracy Measure for Linear Features[ J] . Interna-tio nal Journal o f Geog raphica l Info rmation Sy stem ,1997 , 11:299-306[ 5] 　Zhang J X , Goodchild M F. Unce rtainty in Geo-84　? 31 ?? 1 ? ????:GIS ?????????????????g raphical I nfo rmation[ M] . London:Taylor&F ran-cis , 2002[ 6] 　??? ,??? . GIS ???????????????[ J] . ???? , 1998 , 27(1):37-44[ 7] 　??? , ??? , ??? , ?. G IS ????????[ J] . ?????? ????? , 2002 , 27(5):462-465[ 8] 　??? , ???. ?????????????????[ J] . ???? , 1998 , 28(3):244-249[ 9] 　??? , ???. ?????????[ J] . ???? , 2001 , 30(1):48-53[ 10] ??? ,??? ,???. GIS ??????????????????????[ J] . ???????? , 2002 , 4(3):239-249[ 11] Shannon C E. A Ma thematical Theo ry o f Communi-cation[ J] . Bell System Technical Journal , 1948 ,21:379-423 , 623-656[ 12] Deluca A , Termini S. A Definition o f Nonprobabi-listic Entropy in Setting of Fuzzy Se ts Theor y[ J] .Information Contro l , 1972 , 20:301-312[ 13] Yager R R. Measure of Entropy and Fuzziness Re-la ted to Agg regation [ J] . Info rmation Science ,1995 , 82:147-166[ 14] Zadeh L A . P robability Measures o f Fuzzy Events[ J] . Journal of Mathema tics Analy sis and Applica-tio n , 1968 , 23:421-427[ 15] ???? , ????. ????????[ M] . ??:??????? , 1990??????:??? ,??? ,??。??????????????? 、??????????。E-m ail:yufeng788@163. comHybrid Entropy Model of Spatial Data Uncertainty in GISS HI Yu f eng1 , 2 , 3 　S HI Wenz hong3 　J IN Feng x iang 4(1　School of Archi tecture Engineering , Shandong University of Technology , 12 Zhangzhou Road ,Zhangdian Dist rict , Zib o 255049 , C hina)(2　Key Lab oratory of Geospace En vi ronmen t and Geodesy , Minis t ry of Educat ion , Wuhan Universi ty ,129 Luoyu Road , Wuhan 430079 , China)(3　Advanced Research C enter for S patial Informat ion Techn ology , Departmen t of Land Surveying and Geo-informatics ,T he H ong Kong Polytech nic University , Hung hom , Kow loon , H ong Kong)(4　Presiden tial Secretariate , Shandong Universi ty of Science &T echnology , 579 Qianw angang Road ,Qingdao 266510 , China)Abstract:Based on the info rmation theory and fuzzy set theory , in the light of the character-istics of randomicity and fuzziness of part spatial data in GIS , this pape r proposes a hybridentropy model of spatial data uncertainty , and hybrid entropy can be used to measure the to-tal uncertainty of spat ial data caused by stochastic uncertainty and fuzziness unce rtainty .Taking the uncertainty o f line segment as an example , this paper discusses the evaluat ingmethods of stat ist ic ent ropy , fuzziness entropy , and hybrid entropy of line segment uncer-tainty. And the entropy-band dist ribut ion of line segment uncertainty is show n.Key words:uncertainty ;spat ial data;hybrid entropy;stat ist ic entropy;fuzzy entropy ;linesegmentAbout the first author:SHI Yufeng , associate professor , Ph. D ,majors in the theory and applicat ion of information pattern recognit ion andspat ial data uncertainty.E-mail:yufeng788@163. com85

Hybrid entropy model of spatial data uncertainty in GIS

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