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 Nash Rostock University Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis -2- Content - Basic terms - Geometrical methods - Topological methods - Set methods - Statistical methods - Models Ha Be Aa Fr St Ba Dü Nü Mü Introduction
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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 -
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 3 -
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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 4 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 3
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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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 6 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 4
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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
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Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 31 -
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
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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
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Spatial Analysis 17
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Path problems in networks and graphsTopologic
methods
Round-trips
Steiner network
Min.spanning tree
Distance tree
Shortest path Centre problems
Path problems
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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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 18
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Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 41 -
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
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Set methods
• Set theory• Boolean logic• Kleenean logic• Fuzzy-Set theory• Relational algebra• Sort and search• Mathematical functions• Aggregation• others
Set m
ethods
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Spatial Analysis 22
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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
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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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 23
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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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 58 -
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
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Spatial Analysis 30
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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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 60 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
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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
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Kriging with linear variogram asumption
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 41
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 81 -
Comparing quality of interpolation methods
Area: ca. 63haHeight difference: 60mCaptured with: DGPS – 850 pointsMeasuring time: ca. 14 hours
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 82 -
Quality comparison: Computing time
1
:
5
:
20
Statisticalm
ethods
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 42
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 83 -
Quality comparison: Contour line digitising vs. DGPS
Mean terrainslope: 7.2°
Standard deviationmeasured:sG = 1.88mallowed ZIR10: sG = 2.10m
Statisticalm
ethods
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 84 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 43
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 85 -
Models
• Point models (interpolation..)• Line models (net flow calculations..)• Area models (dispersion..)• Simulation• others
Modellling
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Classification of models
Models
Physicalmodels
Mathematicalmodels
Electricalmodels
Analyticalmodels (flows)
Numericalmodels(finite elements)
Deterministicmodels
Stochasticmodels
Source: G. Teutsch, 1992
Modellling
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 44
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 87 -
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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 88 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 45
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 89 -
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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 90 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 46
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 91 -
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
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 92 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 47
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 93 -
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*
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 94 -
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
Prof. Dr.-Ing. Ralf Bill GI Basics
Spatial Analysis 48
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 95 -
Rostock University, Chair for Geodesy and Geoinformatics SpatialAnalysis - 96 -
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