32 1 Vol. 32, No ACTA GEODAETICA et CARTO GRAPHICA SIN ICA Feb.,2003. The Model of Error Entropy of Area Unit in GIS

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1 32 Vol. 32, No ACTA GEODAETICA et CARTO GRAPHICA SIN ICA Feb.,2003 : (2003) : P208 :A GIS,2,, 2, (., ; 2., ) The Model of Error Entropy of Area Unit in GIS L I Da2jun,2, GON G Jian2ya, XIE Gang2sheng 2, DU Dao2sheng (. N ational L aboratory f or Inf orm ation Engineering in S urveying, M apping and Remote Sensing, W uhan U ni2 versity, W uhan , China ; 2. Depart ment of S urveying, East China Institute of Technology, Fuz hou , China) Abstract : In order to measure the area unit s positional uncertainty, the average entropy band of line segments is firstly presented based on average entropy of the whole line in this paper, and then is extended to error entropy donut of area unit. The band width of error entropy donut can be determined by weighted average error entropy of all line segments. A comparison between error entropy donut and existing error donut is given and their results are visualized and some conclusion is drawn. Key words :area unit ;positional uncertainty ;information entropy ; error entropy band ;error entropy donut : 2,,,, : ; ;; ; GIS,( ), GIS,, [ ],Blakemore 2, 2, 5 ; [ 2 ] Monte Carlo ; [ 3 ],, 2,, 2,, : ; : : ( ) ; 973 ( G ) :(9652),,,,,GIS

2 ( R. Clau2 [3 ], sius) 854,,,(C. E. Shanon),,,,,, [4 ], GIS,,,,, [5 ],, [ 6 ]; [ 7 ],H2,, Perkal 956 2, Chrisman (982) Blackemore (984), 2 2, 2, 4 4. Z = [ x, y ] T, Z 2 = [ x 2, y 2 ] T, 2,, Z 2 = [ x, y, x 2, y 2 ] T 4 [8 ],2 D Z Z 2 = x y x y x x 2 x y 2 x x 2 y x 2 y x 2 y y 2 x y 2 y y 2 x2 y 2 x2 y 2 2 y 2 Z t 2 D Zt = 2 x ( t) xy ( t) 2 yx ( t) 2 x ( t) = ( - t) ( - t) t x x 2 + t 2 = ( - t) ( - t) t y y 2 + t 2 2 y 2 xy ( t ) = yx ( t ) = ( - () (2) (3) t ) 2 x y + ( - t ) t ( x + x y 2 ) + t 2 x2 y 2 (4) xoy x oy,, y, x Z t x oy Z t = [ x t, y t ] T, 2 2 x 2 y D Zt = 2 x ( t) ( t) x y ( t) ( t) y x 2 y (5) ( t) = sin 2 2 x ( t) - sin 2 xy ( t) + cos 2 ( t) = cos 2 2 x ( t) + sin 2 xy ( t) + sin 2 ( t) = sin cos 2 x ( t) + cos 2 xy ( t) - x y

3 : GIS 33 sin acos a (6) [9 ], Z t x, f ( x ) = (2( t) ) / 2 exp [ - ( x - u x ) / 2 2 x ( t) ] 2 x (7) x = sin [ ( - t) x + tx 2 ] - cos [ ( - t) y + ty 2 ] u x = sin [ ( - t) u + ru 2 ] - cos [ ( - t) v + tv 2 ], Z t x,. 67H2 H ( x t ) = f ( x ) ln f ( x ) d x= - ln (2e 2 x ( t) ) (8) H ( x t ), ( H ave ) H ave = H ( x t ) d t = 0 2 ln (2e) + 0 ln ( t) d t (9) x, = 2 eh ave (0) :, 2,, 2, :,, = = = = 2, x y = x2 y 2 = 0 (2) (3) (4) (6) (8), H ( x t ) = 2 ln[2e 2 ( ( - t) 2 + t 2 ) ] () (9), H ave = ln ( 2e ) + 2 ln[ (2 ) / 2 e (- 2) / 4 ] 0 ln[ ( - t) 2 + t 2 ]d t = (0), ave ave = 2 (2 ) / 2 e (- 2) / 4 =. 67 (2) H ( x t ) [ 0, ], H ( x t ), t = 0,, H max = ln ( 2 e ), max = 2 2 e t = / 2, H min = ln ( e ), min = 2 e = max / 2. 46,H2, H2, 2. 07;, 4. 2,, 2, i i, p i = l i, l i i = p i i / n p i i (3) i =,, ;,,,,,,,, :, 2 5, 2 ([ 3 ]) 4 : ; ; ; 3,

4 34 32 Tab. The coordinates of polygon m Z Z 2 Z 3 Z 4 Z 5 Z 6 X Y X Y X Y X Y X Y X Y Tab. 2 The variance and co2variance of polygon cm 2 2 x 2 y xy 2 x 2 y xy 2 x 2 y xy 2 x 2 y xy Z Z Z Z Z Z Tab. 3 Computation results cm i ( i =,,6) 6, 4, cm, cm, cm, cm,,, cm, 2,,,, 6, : Fig. Error donut and entropy error donut of polygon. 2, 2, 2. 2,, 3.

5 : GIS 35, 4. GIS,, : [ ] BLA KEMORE M. Generalization and Error in Spatial Databases[J ]. Cartographica,984,2 :3239. [ 2 ] DU TTON G. Handling Positional Uncertainty in Spa2 tial Databases [ A ]. Proceeding of 5 th International Symposium on Spatial Data Handling[ C]. Charleston : IGU Commission of GIS,992,2 : [ 3 ] L IU Wen2bao, DAI Hong2lei, et al. The Model of Ploygon s Positional Error Dount in GIS [ J ]. Acta Geodatica et Cartographica Sinica,998,27 (4) : (in Chinese) [4 ] ZHAN G Xue2wen. What is Exactly the Entropy [A ]. Entropy and Cross2 Science [ C]. Beijing : China Mete2 orological Press,988. [5 ] L IU Da2jie,L IU Chun. The Status of Researching on Uncertainty of Spatial Data and Quality Control in GIS [J ]. Engineering of Surveying and Mapping,200,0 () :620. (in Chinese) [6 ] SUN Hai2yan. Entropy and Interval of Uncertainty [J ]. Journal of Wuhan Technical University of Sur2 veying and Mapping, 994, 9 () : (in Chi2 nese) [7 ] FAN Ai2min, GUO Da2zhi. The Uncertainty Band Model of Error Entropy[J ]. Acta Geodatica et Carto2 graphica Sinica,200,30 () : (in Chinese) [8 ] SHI Wen2zhong,L IU Wen2bao. A Stochastic Process2 based Model for the Positional Error of Line Segments in GIS[J ]. IN T J Geographical Informatiom Science, 2000,4 () :5266. [9 ] SHI Wen2zhong. Theory and Methods for Handling Errors in Spatial Data [ M ]. Beijing : Science Press, 989. [ 0 ] KAN G Guang2rong, et al. Error Estimation of Mea2 surement Result [ M ]. Beijing : Publishing House of China Computation and Measurement,990. :2292,,,,,,,, 5, 8. 00, , 2-92 : : ,

Uncertainty in Spatial Data Mining and Its Propagation

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