1. Faculty of Civil Engineering Institute of Construction Informatics, Prof. Dr.-Ing. Scherer Institute of Construction Informatics, Prof. Dr.-Ing. Scherer Technische Universität Dresden GIS 1 Geo Information Systems Part 5 Prof. Dr.-Ing. Raimar J. Scherer Institute of Construction Informatics Dresden, 05.07.2006

2. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 2 Metrical adjacency sizes x x optional minimal centre of gravity (Distance of centres) Example of misleading distances: Epicentral Distance e.g. earthquake M=7 with approx. 70x20 km rupture plane x Building x Epicentre strong earthquake damages occure in the radius of ca. 70 km 70 x 20 = 1.400  70² -  = 15.000 i.e.: about 10 % of all places would have distance d = 0 but their epicentral distance given is d ≤ 35 km all other distances are overestimated by up to 35 km or even up to 70 km if epicentre is at the border of the rupture plain. This lead to big errors in the PGA Peak Ground Acceleration attenuation laws: PGA = f(distance) Distance between 2 areas

3. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 3 Shortest Way 2-step solution 1)compute the shortest distance between each 2 neighbouring vertices using the triple algorithm 2)search the shortest distance in number of edges and add all smallest distances of these edges The triple algorithmn computes all shortes ways in a weighted graph. It defines a matrix with all possible connections and assigns to each cell the sum of its ways. For a given problem this matrix can be computed in advance. It is a matrix with the dimension n² (n=number of nodes). After this, a way between two nodes can be read directly from the matrix. But this shows also the problem of this approach: Even if only one shortes way shall be determined, the algorithm needs to handle all node tripels. Furthermore the way table has to be stored, which especially in case of minor occupied adjacent matrices leads to enormous memory. From the combination of the way table with the incidence matrix results the shortest connections with the net.

4. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 4 Shortest Way, Triple-Algorithm Compute for each neighbouring vertices the shortest distance by evaluating all possible ways between the 2 vertices (i,j) with 1 additional vertix (k) and keep the shortes distance of the two possibilities, namely d(i,j) or [d(i,k)+d(k,j)]. Doing this for all possible i,j we end up in an iterative way with the shortest distance between each (i,j) For i = 1,n For j = 1,n For k = 1,n d(i,j) = min {d(i,j), d(i,k)+d(k,j)} Triple Algorithm (s. D. Jungnickel, 1987): The algorithm solves this problem in order O(|n|³), where n is the number of vertices i j k

5. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 5 Shortest Way, Example For the given graph the shortest distance between each 2 vertices is sought. A B D E C 1(p=7) 3(p=1)2(p=6) 7(p=5) 6(p=2) 5(p=1) p = speed  t = d/v, Sum time in [min]

6. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 6 Shortest Way, Example edgeVertixweight Fromto 1AB7 2AD6 3BE1 4EA3 5EC1 6ED2 7DC5 Vertix ABECD A 07006 B 00100 E 30012 C 00000 D 00050 Given graphWeighting matrix A B D E C 1(p=7) 3(p=1)2(p=6) 7(p=5) 6(p=2) 5(p=1)

7. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 7 Shortest Way, Example Vertix ABECD A 3 0 B 2 00 E ´-14 C 00 2 D 0 3 Adjacent matrix Incidence matrix Vertix EdgeABECD 11000 21000 301 00 4 0100 5001 0 60010 7000 1 A B D E C 1(p=7) 3(p=1)2(p=6) 7(p=5) 6(p=2) 5(p=1)

8. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 8 Shortest Ways, Example Vertix ABECD A117896 B4 123 E3101112 C00000 D00050 Totals of Shortest Ways Vertix FromToviaTotal of way AAB,E11 ABDirekt7 AEB8 ACB,E9 ADDirekt6 BAE4 BBE,A11 BEDirekt1 BCE2 BDE3 CBE,A11 EADirekt3 EBA10 EEA,B11 ECDirekt1 ED 2 CA-- CB-- CE-- CC-- CD-- DA-- DDDDDDDD BECDBECD - Direkt - --5---5-

9. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 9 Example for direct computing 1) Develope tree with support of adjacent matrix 2) Pick all pathes with leaves A 3) Compute distance ABCEFG A2 B3 C1 Adjacent Matrix A B E F G C.. AMPASTUAXY...AMPASTUAXY...

10. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 10 Presentation Data-/ Information-/ Knowledge-Presentation = Visualisation through  geometrical graphic (mathematical exactly)  photorealistic graphic (human readable)  Symbolism (representation of non-visual facts)  diagrams  alphanumeric tables  reports

11. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 11 Object oriented data structure Object class Object Object ID Graph. Representation Thematic Topologie Geometrie T1T1 TnTn Geometric Topologie (prim) Thematic Topologie (prim) 0-C Point Line Polyline=face 3-C Volume 0-C 3-C Vector Raster 2D 2D + 1D unlinked model 2,5 D point with height attr. 3D 4 D 0D Metric Primary Metric Secondary Metric 2 D Point exact IDs Names Addresses

12. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 12 Information acquisition Amount of data/information is very huge exactness / resolution efficiency photogrammetry – surface model of site visual interpretation of pictures is currently most often used – objects, meta-information remote sensing – object identification, meta-information, attributes information of vegetation only semi-automatic supervised classification (lerned with testbeds) other acquisition methods - data acquisition on site: measuring or counting - interviews - continous measurement acquisition (gaging station) - positioning - seismic methods - boring

13. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 13 Information retrieval Classification:1 correlating value Intersection:2 correlating values Topography erection phase a max - distribution intensity distribution intensity -> vector coast line -> grid city ??-> grid calculated max a-> vector (circles) (=damage) Map of Los Angeles 1994 earthquake

14. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 14 Data Analysis Goal identification of spatial interdependencies in one theme (statistic analysis) identification of location based interdependencies between different themes identification of spatial interdependencies between different themes identification of interactions and resulting changes (=simulation)

15. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 15 Metric metric = distance function d(P,Q) Definition: d(P,Q) = 0 -> P  Q > 0-> P  Q = d(Q,P) =< d(P,R) + d(R,Q) possible metrics are: Euklid d e (  (p i – q i )²) 1/2 (to long)City Block d 4  (p i – q i ) (to short)Chess board d 8 max { |(p i – q i )| } the following relations can be given:

16. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 16 Correlation of Topics m Statistical methods  describing statistics  analytical statistics m Measurement scales  nominal scale: description (word, letter, number)  ordinal scale: hierarchy  (metric scales)  interval scale (without zero-point  multiplikation not possible)  rational scale (with zero-point)

17. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 17 Correlation of Topics m Analytical statistics  univariate statistics: 1 variable  bivariate statistics: 2 variables  multivariate statistics: n variables

18. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 18 Correlation of themes x y z object class 1 object class 2 object class 3 It is easy, if there are location-related objects in each theme. It is the case for raster data. Remind, raster data have very little information content.

19. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 19 Overlay of Points and Areas 17 4 6 8 53 2 + + + + + ++ + 1 2 + + + + +++ + + ++ + + Points – Object class 1 Area – Object class 1 8 objects2 objects Overlay Result – Object class 3 5 objects

20. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 20 Overlay of Curve and Area 1 2 Curve – Object class 1 Area – Object class 1 3 objects4 objects Overlay Result – Object class 3 1 2 3 3 4 1 2 3 4 5 6 6 objects

21. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 21 Overlay of Area and Area 1 2 Area – Object class 1 4 objects Overlay Result – Object class 3 4 3 3 4 1 6 objects 2 1 23 4 5 7 6 8 10 9 11

22. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 22 Vertex in Area a)rectangle inside if:y,x min  y,x  y,x max b)polygon:test-ray: vertex – test-vertex, test-vertex must be outside if the number of cuts with polygon: odd-numbered = inside even = outside fast pre-test on outside with envelope rectangle (a) vertex Test 1 2 3 Vertex Test 1 2

23. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 23 70 80 90 100 110 70 80 90 100 110 point cluster isoline interpolation polygon cluster neighbourship graph zentroid determination

24. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 24 Statistical methods fundamental terms and coherences see lecture Prof. Herz measured values point estimator: mean value: modally value: median value: x = x 50% I F(x) = 0,50 quantile value: x q = x I F(x) = q%

25. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 25 Statistical methods interval estimator: span with: D = x max – x min mean deviation: variance: deviation: variation coefficient:

26. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 26 Probability P(a<x  b) = f x (x)dx = F(a) – F(b) P(a=x) = 0

27. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 27 Normal distribution standardised (i.e. σ U =1,  U =0) due to symmetry it applies, that

28. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 28 Classification For the assignment (classification) of an object (area etc.) to a set it is required e.g. that min. X% of the objects' elements hold the characteristic properties. P(-  ≤u≤  P(-2  ≤u≤2  P(-3  ≤u≤3   ≙ P(u  1.65   ≙ P(u  1. 96   ≙ P(u  2.58   ≙ P(u  3.29 

29. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 29 Dependency between 2 variables (bivariate statistical methods) Mean value center Standard distance Correlation coefficient Covariance

30. Institute of Construction Informatics, Prof. Dr.-Ing. Scherer GIS Technische Universität Dresden 30 Regression line (Fitting of a straight line) Condition:  y x ?