Gb3 spatial analysis

Prof. Dr.-Ing. Ralf Bill GI Basics

Spatial Analysis 1

Rostock University, Chair for Geodesy and Geoinformatics X 2007

Introduction

Geom. Meth.

Topol. Meth.

Set Methods

Statistic Meth.

Models

Summary

Spatial Analysis

Prof. Dr.-Ing. Ralf Bill and Dr. Edward NashRostock University

Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 2 -

Content

- Basic terms- Geometrical methods- Topological methods- Set methods- Statistical methods- Models

- Basic terms- Geometrical methods- Topological methods- Set methods- Statistical methods- Models

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Introduction


Spatial Analysis 2


Some definitions of the term 'Analysis'

• Analysis = scientific study of problems or correlations• Analysis = division, decomposition of compounds into their

components (opposite of synthesis!)• Analysis = systematic study of an object• Analysis = scientifically dissolving and studying

=> qualitative analysis = according to properties etc. => quantitative analysis = according to amount, number, order etc.

Introduction


Basic problem of spatial analysis

• Given:• User-defined task and an information system with observations A, B, C, ...

• Search:• establish function(s) through which the available data may be involved

and manipulated to provide the required output (e.g. presentations such as maps, graphs, reports, …) related to the problem

Link: U = f (A, B, C ...)

• Functions f• Selection• Boolean operations• Algebraic terms• Reclassification• Polygon overlay with functions between data

Introduction


Spatial Analysis 3


5 questions a GIS can answer!

• 1. Location: What is at a given location?• The first of these questions seeks to find out what exists at a particular location.

A location can be described as a place name, zip code or address.• 2. Condition: Where does something occur?

• Using spatial analysis the second question seeks to find a location where certain conditions are satisfied (e.g., an unforested section of land at least 2,000 square meters in size, within 100 meters of a road, and with soils suitable for supporting buildings).

• 3. Trends: What has changed since ...?• The third question might involve a combination of the first two and seeks to find

the differences within an area over time.• 4. Patterns: What spatial patterns exist?

• You might ask this question to determine whether cancer is a major cause of death among residents near a nuclear power station. Just as important, you might want to know how many anomalies there are that don't fit the pattern and where they are located.

• 5. Modeling: What if ...?• "What if ..." questions are posed to determine what happens, for example, if a

new road is added to a network. Answering this type of question requires geographic as well as other information.

Source: http://volusia.org/gis/whatsgis.htm

Summ

ary


Analysis-Synthesis-Simulation-Prognosis

• Analysis = dissecting, decomposing compounds into its components

• Synthesis = merging single components to a higher order

• Simulation = realistic imitation of technical processes

• Prognosis = assessment in advance (forecast)

AnalysisSynthesis

SimulationPrognosis

Introduction


Spatial Analysis 4


Spatial analysis

• has mathematical foundations• Coordinate geometry• Numerical methods• Topology and graph theory• Set theory• Relational algebra • Statistics• …

• offers in contrast to CAD/DB/IS ..• Polygon overlay• Geo-statistical analysis• Spatial aggregation• Selective spatial search• Topological analysis• …

Introduction


Spatial analysis methods

• Geometrical Methods• Computed geometry• Polygon overlay• Generation of zones• Triangulation/neighbour-

hood graphs

• Topological Methods• Network analysis• Neighbourhood analysis• Site planning

• Statistical Methods• Descriptive statistics• Analytical statistics• Geostatistics

• Set Methods• Boolean, relational and

fuzzy-algebra• Sort and search• Aggregation

• Models and Simulations• Cartographic modeling• System analytical

approaches• Dispersion- and

simulation models

Introduction


Spatial Analysis 5


Geometric methods

• Mathematical background• Metric and coordinate systems • Computed geometry

• Cross-sections in 2 and 3 dimensions• Spatial search and clipping-algorithms

• Specific analysis functions• Point-in-polygon problem• Polygon overlay• Triangulation and Thiessen-polygons• …

Geom

etricm

ethods


Geometry and coordinate systems

• Vector data (points, lines and polygons in x,y,[z])

• Raster data (pixels)

x |H|Northing

y |R|Easting

Geodeticalcoordinate system

100 gon

y

x

90

Mathematicalcoordinate system

Coordinate systems

[0,0]1,1

[0,n]1,n

n,1[n,0]

Column

Row

Geom

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ethods


Spatial Analysis 6


Metrics and distance definitions

• Different possibilities to describe the distance d between points P,Q• Commonly used distance functions:

• Vector data: Euclidean Distance: dE = sqrt((xi-xj)*(xi-xj)+ (yi-yj)*(yi-yj))

• Raster data: with d1=|i-k|, d2=|j-l| for P(i,,j), Q(k,l) in pixel coordinates• City-Block-Distance: d4 = d1+d2

• Chessboard distance: d8 = max(d1,d2)

• Euclidean Distance: dE = sqrt (d1*d1+d2*d2)

A metric on a set X is a projection d : X*X on R0 with the following properties for any P, Q, T from X:

d(P,Q) = 0 if P=Q Idempotenced(P,Q) = d(Q,P) Symmetryd(P,Q) <= d(P,T) + d(T,Q) Triangle inequality

A pair (X,d) is called metric space.

Geom

etricm

ethods

N.4

N.8


Problem of definition? Which distance to take?

Centroid distanceAny distance

Minimum distance

Geom

etricm

ethods


Spatial Analysis 7


Approximation of spatial objects

• Origin

• Simplification ofgeometry:

• to search and to index storage for the computer

• to approximate geometric algorithms

x

2D-approximation

3D-approximation

4D-approximation

y

x

y

x

y

x

yHigher approximation

x

Geometryy

x

Point Line Polygon

Geom

etricm

ethods


Rough tests with MER

• MER = minimum enclosing axis-parallel rectangle(also called BoundingBox)

• Concept for quick data access and preprocessing• e.g. used for point in polygon test and intersection

Point in Polygon IntersectionMER

Point1

2

3

4

g1

g2MER

MER

Polygon

Geom

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ethods


Spatial Analysis 8


Computational geometry

• Point in rectangle

• Geometric intersection

• Minimum enclosing rectangle

• Spatial operators• Area size and perimeter

• Intersection and shortest distance

• Union and difference

• Inclusion

XMIN,YMIN

RP(x,y)

XMAX,YMAX

2-dimensional

3 2

1 4

h

g

s

x

y

3 2

1 4h

g

s

y

H

Gz

x

3-dimensional

Geom

etricm

ethods


Point-in-polygon (vector)

• Jordan Theorem:• Every polygon R separates a plane into 2 disjunct regions (inner and

outer). If the number of real intersections of a ray through X with the edges of the polygon odd-numbered, then X is inside R, else outside.

• Algorithm • Choose test ray through X parallel to coordinate axis.• Choose point t on test ray outside R.• Check if TX touches vertex.

Yes: shift T in y-direction (only real intersections are to be taken).• Count number of real intersections of TX with (n-1) polygon edges.

• If mod(number,2)=0, X is outside.If mod(number,2)=1, X is inside

• with [mod (N,2) = N-(int)(N/2)*2

y

x

MER

PT

i X

R

Geom

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ethods


Spatial Analysis 9


Point-in-polygon (raster)

• Assumptions:All cells inside polygon have the attribute 1All cells outside polygon have the attribute 0

• Algorithm:Check for row l (l=1,m) if row index l = point

row index i.Yes: check for col index k (k=1,n) if col index

k = point col index j.

Yes: check if cell attribute = 1.Yes: Point is inside, else outside.

If point (i,j) is not in the same row / col of polygon increase row index and restart.

123456789

101112

= 0= 1

Row

sl =

1,m

1 72 3 4 5 6 8 9 10 11 12Columns k = 1,n

= point to be checked

R

Geom

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ethods


Polygon overlay: Two polygons

• 1. Edge intersections:• Divide all intersecting edges of the

starting polygons at their intersections.

=> List of all nodes and edges: no further intersection of polygons.

• 2. Polygon formation:• Link individual edges to form new,

closed polygons.=> List of all polygons.

• 3. Overlay identification:• Check polygons: which were the

original polygons?=> transfer attributes (copy or link).

12

3 4

1 34

56

81011

11 Objects

POLYGON OVERLAY

RESULT-Object class 3

Object class 1 Object class 2AREA- AREA-

4 Objects 4 Objects

7

2

4

9

32

1

Geom

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ethods


Spatial Analysis 10


Variations for polygon overlay

Source: Alan Murta (http://www.cs.man.ac.uk/aig/staff/alan/software/gpc.htm)l

Difference Intersection

Exclusive-or Union

Geom

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ethods


Polygon overlay: Line with polygon

Class 1: - All power lines of an energy provider

Class 2: - All parcels that are

owned by the municipality

Class 3: - All lines on parcels owned by the municipality

LINE-Object class 1

AREA-Object class 2

3 Objects 4 Objects

6 Objects

23

4

1

2

3

1 2 34

56

RESULT-Object class 3

INTERSECTION

1

Geom

etricm

ethods


Spatial Analysis 11


Polygon overlay: Raster with raster

y

x

z Feature class 1

+

=

Feature class 2

Feature class 3

Geom

etricm

ethods


Buffer- or zone generation (vector)

• Example:• Travel time problem

(geometric)

• All in the yellow area need 20 min to the next station

• All in the read area only 10 min

Point

Line

Polygon Inner buffer

Circular buffer

Narrow buffer

Outer buffer

Square buffer

Wide buffer

20

20

20

10

10

10

Geom

etricm

ethods


Spatial Analysis 12


Buffer- or zone generation (Raster)

a) Original matrix b) Buffer zone 20mraster size 10m city block metric

- - - - - - - - - - - - - - - - - - - - - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - 0 - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -- - - - - - - - - - - - - - - 0 0 0 0 0 - - - -

Geom

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ethods


Interpolation, Neighbourhood graphs, Centroids

70

80

90

90

100

110

70

8090

100

110

Point values Points Polygon values

Centroid-generation

Neighbourhood-graph

Isoline-Inter-polation

Geom

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ethods


Spatial Analysis 13


Delaunay-Triangulation/Thiessen-Diagrams

3

2

1

4

5

0

Concept: Circumference of 3 pointsdoes not contain any further points

(or: Voronoi-Diagram and Dirichlet-Tesselation)

Enlargement

2

34

5 0

Input data

Thiessen-Polygon

Geom

etricm

ethods


Neighbourhood graphs (Raster)

a) Original matrix b) Distance transformation c) Main axiswith special given metric transformation

- - - - - - - - - - - - 18 15 12 11 10 9 10 11 12 15 18 21 - - - - - - - - - - - -- - - - - - - - - - - - 17 14 11 8 7 6 7 8 11 14 17 20 - - - - - - - - - - - -- - - - - - - - - - - - 16 13 10 7 4 3 4 7 10 13 16 19 - - - - - - - - - - - -- - - - - 0 - - - - - - 14 12 9 6 3 0 3 6 9 12 15 18 - * - - - 0 - - - - - -- - - - - - - - - - - - 11 10 9 7 4 3 4 7 10 13 16 17 - - * * - - - - - * * -- - - - - - - - - - - - 8 7 6 7 7 6 7 8 11 12 13 14 - - - * * - - - * * * -- - - - - - - - - - - - 7 4 3 4 7 9 10 11 10 9 10 11 - - - - * * * * * - - -- - 0 - - - - - - - - - 6 3 0 3 6 9 11 8 7 6 7 8 - - 0 - - - * - - - - -- - - - - - - - - - - - 7 4 3 4 7 10 10 7 4 3 4 7 - - - - - * * - - - - -- - - - - - - - - 0 - - 8 7 6 7 8 11 9 6 3 0 3 6 - - - - - * - - - 0 - -- - - - - - - - - - - - 11 10 9 10 11 12 10 7 4 3 4 7 - - - - - * - - - - - -- - - - - - - - - - - - 14 13 12 13 14 14 11 8 7 6 7 8 - - - - * * - - - - - -

Geom

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ethods


Spatial Analysis 14


Neighbourhood graphs (Raster)

6

45

2

3

1

* * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * ** * * * 6 6 6 6 6 6 6 6 4 4 4 4 45 5 5 5 5 6 6 6 6 6 6 4 4 4 4 4 45 5 5 5 5 5 6 6 6 6 4 4 4 4 4 4 45 5 5 5 5 5 5 6 6 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 2 4 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 2 4 4 4 4 45 5 5 5 5 2 2 2 2 2 2 2 2 4 4 4 45 5 5 5 2 2 2 2 2 2 2 3 3 3 3 3 *1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 *1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 * * * * * * * * *1 1 1 1 1 1 1 1 * * * * * * * * *

Point distribution Neighbourhood graph* * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * * * * * * 6 6 6 6 6 6 6 6 6 * * * ** * * * 6 6 6 6 6 6 6 6 4 4 4 4 45 5 5 5 5 6 6 6 6 6 6 4 4 4 4 4 45 5 5 5 5 5 6 6 6 6 4 4 4 4 4 4 45 5 5 5 5 5 5 6 6 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 4 4 4 4 4 4 4 45 5 5 5 5 5 5 2 2 2 4 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 4 4 4 4 4 45 5 5 5 5 5 2 2 2 2 2 2 4 4 4 4 45 5 5 5 5 2 2 2 2 2 2 2 2 4 4 4 45 5 5 5 2 2 2 2 2 2 2 3 3 3 3 3 *1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 3 *1 1 1 1 1 1 2 2 2 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 3 3 3 3 3 3 3 3 *1 1 1 1 1 1 1 1 * * * * * * * * *1 1 1 1 1 1 1 1 * * * * * * * * *

Starting pointDistance 1 unitDistance 2 unitsDistance 3 unitsDistance 4 units

Quasi-Thiessen-Diagram resp. Delaunay-triangulation

Geom

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2.5D- and 3D-analysis functions

• 3D-Interpolation (poss. with time 4D)• Isolines• Slope and gradient• Exposition, aspect, hill-shading• Analysis inside objects• Buffer in 3D• Surface, network flows and path analysis• Volumes and deposit calculation• Cross-sections in2D and 3D, profiles• Line-of-sight• ..

Profile from View1

Elev

atio

n

Distance

Vertical exaggeration 7.3 X

770.20

800.

171

3.8

150.0 300.0 450.0 600.0

738.

876

3.8

#

Geom

etricm

ethods


Spatial Analysis 15


Topological methods

• Mathematical background• Adjacency and incidence

• Algorithms and applications• best path, best site, Travelling Salesman Problem• Shortest path• Floyd-Warshall-Algorithm• Dijkstra-Algorithm

Topologicalmethods


Basics of topology (Adjaceny/Incidency)

Edge1234567

fromAABEEED

toBDEACDC

weight7613125

AB

E

CD

1(p=7)

3(p=

1)

4(p=3)

5(p=1)7(p=5)

6(p=2)

2(p=

6)

Assessment matrix :

A

00300

B

70000

E

01000

C

00105

D

60200

ABECD

Adjacence matrix BTB :

A

3....

B

-12...

E

-1-14..

C

00-12.

D

-10-1-13

ABECD

Incidence matrix B :

A

110-1000

1234567

B

-1010000

E

00-11110

C

0000-10-1

D

0-1000-11

A

114300

ABECD

Shortest path sums according to Floyd-Warshall:

B E C D

7111000

811100

92105

63200

Shortest path (Floyd-Warshall-Algorithm) Topologicalmethods


Spatial Analysis 16


Network Analysis: 3 Categories of tasks

Best Path

Start-pointt

End point

Best Location

Location

Start-point

Traveller-Problem

Travelling Salesman Problem:

- Operations research- Linear Optimisation- Graph Theory

Best site in terms of reachability and commuter-belt:- topologic algorithms- polygon overlay- 2D- Median

Best path- geometrically shortest path- topologically best path- cheapest path

Topologicm

ethods


Goals for shortest distance

• Minimise distance (ABC=57)• Minimise travel time (ABC=57)• Minimise junctions (AC=61)• Minimise turns (especially left turns) (AC=61)• Minimise cost including fixed points (via D => ADC=59)

=> Optimal path is dependent on the factors considered=> Application: Routing, car navigation systems

D C

A B

14

2121

45

3661 32

Example: Path from A to C

Topologicalmethods


Spatial Analysis 17


Path problems in networks and graphsTopologic

methods

Round-trips

Steiner network

Min.spanning tree

Distance tree

Shortest path Centre problems

Path problems


Shortest path between 2 nodes

Example: Hike starting in A to mountain cabin H. Estimation of walking time [h]. From D to E you can take the chair-lift. Which is the shortest walking time from A to H?

Example: Hike starting in A to mountain cabin H. Estimation of walking time [h]. From D to E you can take the chair-lift. Which is the shortest walking time from A to H?

Solution: Stepwise approachAccumulation of sequential shortest ways

C

F

H

E

G

A

BD

2,0

3,0

1,5

2,5

1,0 3,5

0,5

2,0

2,5

1,53,0

1,5

1,0

0,5

0,5

Topologicm

ethods


Spatial Analysis 18


Shortest path between two nodes

The shortest path from A to H is ABDEGH = 5,5 hrs

Topologicm

ethods

--H (ABDEGH, 5.5)8

L(ABDEFH)=7.0HF (ABDEF, 4.5)7

L(ABDEGH)=5.5HG (ABDEG, 4.0)6

L(ABDEF)=4.5; L(ABDEG)=4.0F, GE (ABDE, 3.0)5

L(ACE)=3.5; L(ACF)=5.0; L(ACG)=5.0E, F, GC (AC, 3.0)4

L(ABDE)=3.0; L(ABDG)=5.5; L(ABDH)=6.0E, G, HD (ABD, 2.5)3

L(ABC)=4.0; L(ABD)=2.5; L(ACD)=3.5C, DB (AB, 1.5)2

L(AB)=1.5; L(AC)=3.0B, CA (A, 0)1

Outgoing Paths and Total LengthUnvisitedNeighbours

Source Node(Path, Length)

Step


Minimum spanning tree

• Example: 6 cities should be linked via glass-fibre network.

• Goal: All cities have to be connected to the network with minimum cost.

• Solution: Order the edges according to their evaluation

• PU(22), RT(24), PT(30), TU(36), QR(38), PQ(39), RS(42), QU(43), ST(57), RU(60), QT(61), PR(62), SU(65), PS(78), QS(84)

U T

P

Q R

S

36

38

2230

39

43 6178

60

60

42

5765

24

U

P

Q R

S

T

62

Topologicalmethods


Spatial Analysis 19


Minimum spanning tree

• If a circle is created, eliminate the relevant edges

• PU, RT, PT, (TU eliminated), QR, (PQ eliminated), RS

• Remaining sequence is the minimum spanning tree

Algorithm definition :

Given a connected network (E,K) resp. a planar graphwhere each edge ki K is evaluated with di >= 0.

A graph with the edges k1, k2, … kr is calledminimum spanning tree, if

Σ di = minimum.r

i=1

Topologicalmethods


Round trip problem (Traveling salesman problem)

• Example: Business man from Frankfurt (F) has to go by car to Kassel (K), Nürnberg (N), Stuttgart (S) and Würzburg (W) and return to F.

K

F

S

NW

190

190

310

220

120

230

105

165

R = min ?3

R : F - K - N - W - S - F

190+310+105+165+220=990

R : F - K - N - S - W - F

190+310+190+165+120=975

R : F - K - W - N - S - F

190+230+105+190+220=935

R : F - W - K - N - S - F

120+230+310+190+220=1070

R : F - K - W - S - N - F

190+230+165+190+225=1000

Topologicalmethods


Spatial Analysis 20


Traveling salesman problem TSP

• In general: complete net (graph) with n nodes

• n --> 1/2*(n-1)*(n-2)*...*2*1 = 1/2(n-1)! , ne N

• Procedure becomes very time-consuming with an increasingnumber of nodes

Example: Calculation time for round trip problem, dependent on n

nodes n 6 10 11 12 13 14

time t 0,001 [s] 4 [s] 40 [s] 8 [min] 2 [h] 1 [day]

+

Topologicm

ethods


Search for optimal site

• Along a line (polygon)• Inside polygon• In space

=> Application in infrastructure planning

• Sites for companies, schools, hospitals• Sites for airports, sewage plants,...

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Topologicalmethods


Spatial Analysis 21


Centre problems

Solution: Mean:X = 1/7Σ pos(i) = 7Minimum of quadratic distance = 34Median: M = 50%-Quantil = 5Minimum of absolute values = 32

A B C D E F G0 1 2 3 5 7 10 15 16

• given: road in residential area with residents A,B,C,D,E,F,G• find site with minimal distance to all residents

Topologicalmethods


Set methods

• Set theory• Boolean logic• Kleenean logic• Fuzzy-Set theory• Relational algebra• Sort and search• Mathematical functions• Aggregation• others

Set m

ethods


Spatial Analysis 22


Basics of set theory

Complementary law:A 0 = AA S = AA Ā = SA Ā = 0

U

U

U

U

A U B = B U A

Absorption law:

Distributive law:

A (B C) = (A B) C

A (B C ) = (A B) C

U U U U

U U U U

A (A B) = AU U

A ( A B ) = AU

U

A ( B C ) = (A B) (A C)

A ( B C ) = (A B) (A C)

U

U U

U U UU U

UU

A B = B A

U U

Commutative law:

Associative law:

A and B are sets.

A B is the intersection of the two sets. The intersection contains all elements that are in both sets, A and B.

If A B have no common elements then this is the empty set 0.

A U B is the union of the two sets, containing all elements that appear at least in one of A and B.

The complimentary set Ā contains all elements from the universal set R that are not included in A.

U

U

Set m

ethods


Boolean logic – binary decisions

Truth tables for Boolean operators in programmingA B NOT A A AND B A OR B A XOR B1 1 0 1 1 01 0 0 0 1 10 1 1 0 1 10 0 1 0 0 0

1 = ”true"; 0 = "false".

Venn Diagrams

A AND B

A OR B

(A AND B) OR C

A AND (B OR C)

A NOT B

A XOR BBinary logic

(true=1/false=0)

False

True

Set m

ethods


Spatial Analysis 23


Example: Set operations in raster data

Identify areas where the following three criteria are fulfilledt:- Feasible soil conditions- Water depth smaller 3 m- more than 200 m away from mangroves

Mathematical assumptions related to metrics- Raster cell size is 200 m- Chess board distance resp. N.8-Neighbourhood

Water depthin m

0 1 2 3 4 40 1 2 3 4 40 1 2 2 3 30 1 1 2 2 20 0 1 1 1 10 0 0 0 0 00 0 0 0 0 0

Mangroves(1=Mangroves, 0=no,2= 200m-Zone)

1 1 0 0 0 01 1 0 0 0 00 0 0 0 0 0 0 0 0 0 0 00 0 0 0 0 00 1 1 0 0 01 1 1 0 0 0

AND =

Result(1=true, 0=false)

0 0 0 0 0 00 0 0 0 0 00 0 0 0 0 00 0 1 0 0 10 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0

Soil conditions(1=good, 0=bad or

no data)

0 0 1 1 1 10 1 1 0 1 10 0 1 0 1 10 0 1 0 0 1 0 0 0 0 0 10 0 0 0 0 00 0 0 0 0 0

AND

Set m

ethods


Kleenean logic

• Sometimes true/false is not enough:• Is the point inside the polygon? (statement “the point is inside the

polygon”)

true

false

maybe

Set m

ethods


Spatial Analysis 24


Fuzzy logic

• The world is not always clear cut• This uncertainty is modelled using multi-valued logic (“Fuzzy Sets”)

• Even true/false/maybe is not always enough

False

True

Probability-value s

0

1

True False

If ( A with s=x) AND ( B with s=y) THEN ( C with s=z)

Set m

ethods


Selective queries in GIS-data base

DESCRIPTIVE

- all parcels with value > 100.000.-

TOPOLOGIC- all parcels bordering x-road on left-hand side

ExamplesGEOMETRIC- all landmarks in

COMBINATIONall parcels along x-road (left side) with value > 100.000 €inside

125 126 127100.000.-

80.000.-

150.000.-

x - Straße

70.000.- 100.000.- 90.000.-

145 146 147

125 126 127100.000.-

80.000.-

150.000.-

x - Straße

70.000.-100.000.-

90.000.-

145 146 147

125 126 127100.000.-

80.000.-

150.000.-

x - Straße

70.000.-100.000.- 90.000.-

145 146 147

DATA

125 126 127100.000.-

80.000.-150.000.-

x - Straße

70.000.- 100.000.- 90.000.-

145 146 147

Set m

ethods


Spatial Analysis 25


Relational operators

Selection Projection Product Union

abc

xy

aabbcc

xyxyxy

Intersection Difference (natural) join Division

a1a2a3

b1b1b2

a1a2a3

c1c2c3

a1a2a3

b1b1b2

c1c1c2

aaabc

xyzxy

xz

a

Set m

ethods


Searches in large data base

• Common approaches • e.g. name=Bill in 10,000 data sets

• linear search (O|n|, mean of 5,000 operations, worst case 10,000)

• logarithmic search after build-up of ordered lists (O|log(n)|, 14 operations)

• Tree search methods• used when search space can be represented as a tree• start at the root and run along edges to the next node

representing a hit• can be divided into order in which nodes of the tree are

visited

Set m

ethods


Spatial Analysis 26


Sorting large data bases

• Sorting procedures are some of the most common procedures in IT-applications• by name• by size• by frequency

• Sorting process such as quicksort, heapsort, shellsort• Order O|n²| to O|n|

• Sorting with Divide/Sort and Merge

Set m

ethods


Classification (Generation of spatial clusters)

- Given dataProperty AProperty BProperty CProperty D

- Extreme values- Parallel epiped class- Minimum-Box-

Classification

- Distance Classification(Euclidean Distance)

- Shortest Distance-classification

- Maximimum-Likelihood-Classification (Lines ofequal probability)

Set m

ethods


Spatial Analysis 27


Reclassification

ac

ab bcad

ae

A

A BA

A

AB

1 2 3

Legend : a,c,d,e – Deciduous Forestb – Coniferous Forest

Reclassification :

Set m

ethods


Aggregation

Set m

ethods

Region

Governmentdistrict

Federal state

County

Parish

District X

State Y

County V

Commune U

Region W

X

W

V

U

Data collection and management

Commune codesin Germany (8 digits)

0 8 1 1 8 0 0 1= Affalterbach

Fede

ral s

tate

Gov

ernm

.dis

trict

Reg

ion

Cou

nty

Par

ish

Agg

rega

tion


Spatial Analysis 28


Statistic methods

• Mathematical foundations• Describing Statistics• Analytical Statistics• Univariate -, bivariate and multivariate Statistics

• Advanced geostatistical methods• Interpolation methods• Variogram and Kriging

Statisticalm

ethods


Describing statistics/diagrams

Project Part Nodes Lines Curves Arcs Circles Areas

ATKIS 1 3380 4016 795 12 0 1383Hochheim 1 3090 3671 685 0 0 1208

45004000350030002500200015001000500

0

Number of geometric primitives:

Nodes Lines Curves Arcs Circles Areas

Statisticalm

ethods


Spatial Analysis 29


Analytical statistics and testing

• Univariate statistics• Average, standard deviation …

• Bivariate statistics• Correlation, covariance• Regression• Cross-correlation

• Multivariate statistics• Cluster analysis• Factor analysis• Multivariate regression• …

Statisticalm

ethods


Interpolation

• in raster

• in triangle

• in lines

525

3 (x,y,z)

1 (x,y,z)2 (x,y,z)

P (x,y,?)

1 2

3 4

P (x,y,?)

1

2

(x,y,z)(x,y,z)

(x,y,z)

(x,y,z)

(x,y,z)

(x,y,z)

P (x,y,?)

Dig

ital T

erra

in M

odel

s

Statisticalm

ethods


Spatial Analysis 30


Interpolation approaches

01 5 10 15

10

5

1

Linear interpolation

01 5 10 15

10

5

1

Polynomial interpolation

01 5 10 15

5

1

10

Compound cubic poly.

01 5 10 15

10

5

1

Akima interpolation

Statisticalm

ethods


Interpolation/Approximation of surfaces

• TIN-Interpolation• Interpolation with area summation• Interpolation with minimum least squares methods• Piece wise linear polynomes• Polynom interpolation• Kriging

Nearest neighbour

Minimal curvature

Inverse Distance

Spline

Polynomregression

Area-Summation

Kriging

TIN-Interpolation

Statisticalm

ethods


Spatial Analysis 31


Example: statistical methods in DTM

Dots: digitised elevation samples (topographic map) Statisticalm

ethods


Interpolation – an example

X Y Z7 6 85 1 21 1 24 3 20 4 24 5 17 3 42 6 106 3 43 3 11 3 3

Statisticalm

ethods


Spatial Analysis 32


Triangle interpolation

• natural coordinate system • Interpolation approach

y

x

v

u

12

33 (0,1)

Arbitrarycoordinate system triangular coordinates

Natural

1 (0,0) 2 (1,0)

1 2

3

L = 0 3

L = 1/33

L = 2/33

L = 13

L=

2/3

L=1/3

L=0

L=1

11 1 1

L= 2/

32

L= 1/

32

11

L= 0

2

L= 12

x y

z

1

2

3

x y

z

1

2

3

P P

a. linear b. cubic


Triangle interpolation

• Interpolation approach

• Problem: triangulationz=0 z=10

z=5

z=0

2.5

5.07.5

5.02.5

7.5

CubicInterpolation

z=0 z=10

z=5

z=0

2.5

5.0

7.5

LinearInterpolation

2.5

7.5

5.0

z=0 z=10

z=5

z=0

2.5

LinearInterpolation

z=0 z=10

z=5

z=0

2.5

5.0

7.5

LinearInterpolation

2.5

7.5

5.0

0.02.55.0

7.5

Statisticalm

ethods


Spatial Analysis 33


Triangle interpolation - linearS

tatisticalmethods


Interpolation/approximation in grids/raster

• Interpolation by area summation• Interpolation by least squares method• Piecewise linear polynomes• Polynom interpolation• Kriging

Nearest neighbour

Minimal curvature

Inverse Distance

Spline

Polynom regression

Area-Summation

Kriging

Other methods

Statisticalm

ethods


Spatial Analysis 34


Interpolation - nearest neighbour

• Transferring the z-component from nearest neighbour• Assumes a sufficiently dense distribution of points

Statisticalm

ethods


Interpolation - minimum curvature

• Application especially in geo-sciences• Thin deformable plate through all points• Smooth surface• Iterative solution of an equation system

Statisticalm

ethods


Spatial Analysis 35


Interpolation - inverse distance weightingS

tatisticalmethods

Weight: w(di) = 1/dip

Height: h = Σi=1,n hi*wi / Σ widi = distance to point ip = power (1, 2, .., inf)hi = height of point in = # of neighboring points


Approximation - polynomial regression

Statisticalm

ethods


Spatial Analysis 36


Multilog

0,8

1

1,2

1,4

1,6

1,8

2

0 0,2 0,4 0,6 0,8 1 1,2

Area summation - multilogarithmic kernelS

tatisticalmethods


Thin Plate Spline

0,6

0,8

1

1,2

1,4

1,6

1,8

2

0 0,2 0,4 0,6 0,8 1 1,2

Area summation - thin plate spline as kernel

Statisticalm

ethods


Spatial Analysis 37


Area summation - cubic splines as kernel

Natural Cubic Spline

0,8

11,2

1,4

1,6

1,82

2,2

2,42,6

2,8

33,2

3,4

0 0,2 0,4 0,6 0,8 1 1,2

Statisticalm

ethods


Area summation - multiquadratic kernel

MultiquadricInvers Multiquadric

0,6

0,8

1

1,2

1,4

1,6

0 0,2 0,4 0,6 0,8 1 1,2

Statisticalm

ethods


Spatial Analysis 38


Splines - cubic polynomesS

tatisticalmethods


Geostatistics: Variogram I

Assumption: The spatial variability of a random sample Z can be explained as a sum of 3 components:

Z(x) = m(x) + e'(x) + e''(x)

with: m(x) = trend, e‘(x) = random component, e‘‘(x) = random noise/error

Variogram determines the influenceof each point on a random variable

g(h)=1/(2n) Σ(z(xi)-z(xi+h))²

Analogous: Correlation functions

C

C a

h

g(h)

0

1

a = rangec0= nugget (noise)c1= sill (maximum variance)

Statisticalm

ethods


Spatial Analysis 39


Geostatistics: Variogram II

• Typical estimation function for Variogram

Linear Regression: γ(h) = c0 + b hSpheric Model:

γ(h) = c0 + c1{3h/2a - 0.5 (h/a)³) für 0 < h < aγ(h) = c0 + c1 für h >= a

Gauss Model: γ(h) = c0 + c1 (1 - exp(-h/a)²)

200

100

g(h)

00 10 20 30 40 50 h

g(h)=13,16+4,15h

Example:Variogram estimationwith linear model

Statisticalm

ethods


Geostatistics: Kriging I

• Kriging is an exact interpolator. Individual samples are used with a range-dependent weight derived from the variogram.

• Example: 5 Samples with measured values (3,4,2,4,6) and distance between each others and to target point 0.

1 2 3 4 5 01 0.0 5.0 9.8 5.0 3.2 4.32 5.0 0.0 6.3 3.6 4.4 2.93 9.8 6.3 0.0 5.0 7.2 5.54 5.0 3.6 5.0 0.0 2.3 1.05 3.2 4.4 7.2 2.3 0.0 2.0

• Function shall be a spheric model with c0=2.5, c1=7.5 and a=10.0.

Statisticalm

ethods


Spatial Analysis 40


• To be solved: A-1b =

1 2 3 4 5 1 2.500 7.656 9.996 7.656 5.977 1.000 7.039 0.0189 2 ... 2.500 8.650 6.375 7.131 1.000 5.671 0.17623 ... ... 2.500 7.656 9.200 1.000 b = 8.064 = -0.01094 ... ... ... 2.500 5.401 1.000 3.621 0.62125 ... A ... ... 2.500 1.000 4.720 0.1945

... ... ... ... ... 0.000 1.000 -0.1676

• Interpolation of the target point 0 and variance according to

z(x0) = Σ λi z(xi) = 0.0189*3+0.1762*4-0.0109*2+0.6212*4+0.1945*6 = 4.392

σ² = Σ λi bi + h = 0. 0189*7.039+0.1762*5.671-0.0109*8.064+ 0.6212*3.621+0.1945*4.720-0.1676 = 4.044

Geostatistics: Kriging II

λh

λh

Statisticalm

ethods


Kriging with linear variogram asumption

Statisticalm

ethods


Spatial Analysis 41


Comparing quality of interpolation methods

Area: ca. 63haHeight difference: 60mCaptured with: DGPS – 850 pointsMeasuring time: ca. 14 hours

Statisticalm

ethods


Quality comparison: Computing time

1

:

5

:

20

Statisticalm

ethods


Spatial Analysis 42


Quality comparison: Contour line digitising vs. DGPS

Mean terrainslope: 7.2°

Standard deviationmeasured:sG = 1.88mallowed ZIR10: sG = 2.10m

Statisticalm

ethods


Quality comparison: Standard deviation (m)based on 80% of points, 20% true error points

0.33 1.49 3.17

0.22 0.69 1.81

0.29 0.77 1.87

1.49 3.17

0.69 1.81

0.77 1.87

Statisticalm

ethods


Spatial Analysis 43


Models

• Point models (interpolation..)• Line models (net flow calculations..)• Area models (dispersion..)• Simulation• others

Modellling


Classification of models

Models

Physicalmodels

Mathematicalmodels

Electricalmodels

Analyticalmodels (flows)

Numericalmodels(finite elements)

Deterministicmodels

Stochasticmodels

Source: G. Teutsch, 1992

Modellling


Spatial Analysis 44


Geographic models I: Stochastic approaches

• Behaviour of geographic systems is determined to a considerable extent by random processes. For such systems initial hypothesis are defined by probability theory.

Spatial probability models

Geographic decision support

models

- spatial pattern of factory sites- correlation of sales and number of employees,

literacy and social status- autocorrelation between voters of special parties

- Decision about areas for cultivating grain

Modellling


Geographic models II: Deterministic approaches

• Behaviour of geographic systems is determined by (pseudo-)physical laws and thus can be predicted exactly.

Cascading models

Space-time-models

- Population migration- Ecosystem stability

- Temperature distribution in soil profiles- Water flow in soil- Heating-up of cities

Models for spatial interaction

Models forspatial assignment

(also called gravitational models)- Movement of consumer capital between regions- Migration of employees (residence to work)(also called transport models)

- Consumers to vendors- Pupils to schools

Modellling


Spatial Analysis 45


Cartographic modeling

• C.D. Tomlin (1983), (1990), MAP (Map Analysis Package)• Goal: Division of workflow into parts that can be combined and a

definition of a map algebra to process that workflow.• Terms:

Carto-graph.Model

Map-sheet

Map-sheet

Map-sheet

Title

Orientation

Zone

Zone

Zone

Mark

Value

Position

Position

Position

Column-coor-dinate

Row-coor-dinate

Scale

Modellling


Workflow in cartographic modeling terms

- Data inter-pretation :

Sheet

Operation

Input Processing Output

SheetSheet

Sheet

- Procedure : I

H

F

C

G

E

A

AlgebraicExpression :

(F+C+(E/A)) 2

- Data interpretationoperations :

1 = f (Position)2 = f (Neighbourhood)3 = f (Zone)

1 2 3

Modellling


Spatial Analysis 46


Example: Best location for a sports field

• Conditions for a site suitable for a sports field:

• A: slope < 7 %.• B: area > 40.000 sqm.• C: outside residential area (> 50m).• D: traffic infrastructure (< 50m away from existing road

network/public transport).

Modellling


Solution: Sports field

A B

C D

E = A AND B AND C AND D

E

Set theory andcartographic model

Modellling

SettlementareasLand use Selection

occupiedBuffer50m outer

Bufferedsettlementareas

DTM Slope Slopemap

Selection< 7 % A

NOT

C

Road net Buffer bothsides 50 m D

Polygon overlay

Recommareas

Area size> 40000 qm

E=Potent.candidates

B


Spatial Analysis 47


Soil erosion model

R = rain factor = f(precipitation)

K = Erodability of soil = f(soilparticle distribution); soil map

L = slope length factor = f(plot size)

S = slope factor = f(slope) from DTM

C = land use factor = f(crop rotation)

P = Erosion protection

A = mean annual erosion [t/ha]

Universal Soil Loss Equation A = R*K*L*S*C*Pfrom K. Kraus (1991) after Wischmeier/Smith (1978)

Modellling

R

K

A

*

=

L*

S*

C*

P*


Erosion model: cascading

Gradient = Δz/Δd mit Δd = Δx+Δy (City-Block-Distanz)

78 72 6974 67 5669 63 44

5.5 5 17 X -111 4 -11.5

78 72 69 71 58 49

74 67 56 49 46 50

69 63 44 37 38 48

64 58 56 29 31 34

68 61 47 21 18 19

74 60 34 12 10 12

z.B.

Modellling


Spatial Analysis 48


Cost functions for utility planning

Approach:Raster data

Problem: Cost function

Cost efficient path Cost surface

Modellling

New fabrique

Existing cables

Agriculture 1Street 1Uncultivated land 1Coniferous forest 4Decidous forest 5Water 1000Settlement 1000

Weight


Status of GIS data analysis

• Geometric operations usually realised• Polygon overlay is a basic function• Topologic operations rather restricted• Set methods such as sort, search, query etc. realised• Simple descriptive statistics realised, interpolations for DTM,

geostatistics rare• Models usually externally realised for special applications

Summ

ary

Gb3 spatial analysis

Technology

Transcript of Gb3 spatial analysis