Introduction to Visualization Techniques for Spatial Datagrinstei/vis2010/Lecture 6 - Intro to spatial data... · Lecture 6 – Geo-Spatial Data Visualization October 6, 2010

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Institute for Visualization and Perception Research

information visualized

knowledge discovereddecisions made

Lecture 6 – Geo-Spatial Data VisualizationOctober 6, 2010Georges Grinstein, University of Massachusetts Lowell




Introduction to VisualizationTechniques for Spatial Data

Spatial Visualizations

Maps, GIS and Applications

What is 2D Spatial Data?

Data tied to a location in the real world

Sources:

Maps

Aerial Photographs

Field survey notes

GPS data

GIS

Satellite Systems

We Look for Data/Spatial Relationships

But in many applications it is important toseek relationships that involve geographiclocation

Examples

Telephone calls

Environmental records

Crime Data

Census demographics

...

Seeking Spatial RelationshipsExample (1)

Public Health and Safety Analysis

Customer

Analysis

Visualization

of AT&T long

distance call

volume

(AT&T

Swift3D)


2

Basic Definition

Result of accumulating

samples or readings of

some phenomena while

moving along 2d paths

in space

These are discrete samplesof a continuous phenomenon

Bodensee -

original topographic data

resolution 1:500'000

(Visual Spatial Data 2002)

Formalization (1)

A 2-dimensional spatial data item

is an ordered d-tuple of the form

with

(1)

(2) Without loss of generality we set

, the first 2 values, to be the 2d vector

representing the spatial dimensions

),(21

iixx

is a d-dimensional record

DBxi

( )d

iiixxx ,...,

1=

,...1 di

DDx

Formalization (2)

(3)1

1

vxi= :

2

2

vxi=

2

21 ),(;, RvvR

0:,221121=+ vvR

Then 1 = 2 = 0

in other words v1 and v2 are linearly independent

Data Analysis Techniques (1)

Example

Sales Transaction Application contains data about

Customers

Products

Quantities

Time

…

Data Analysis Techniques (2)

Many ways to approach analysis of the data,

including

Building statistical models

Clustering

Finding association rules

...

Earthquake

hazard

assessment

• Turkey

• Icon


3

2D Spatial Data Mining (1)

Goal

Extracting interesting knowledge or general

characteristics from large 2D spatial (location)

databases

Important task in the development of spatial

database systems

2D Spatial Data Mining (2)

Problem:

It is almost impossible for a user to examine the

huge amounts (usually terabytes) of spatial data

obtained from large databases (credit cards

payments, telephone calls, environmental

records,...) in detail and extract interesting

knowledge or general characteristics

WITHOUT SUPPORT

computational

human participation/performance

2D Spatial Data Mining (3) 2D Spatial Data Mining (4)

Key observation

The presentation of data in an interactive,

graphical form often fosters new insights,

encouraging the formation and validation of new

hypotheses to help in better problem-solving and

gaining deeper domain knowledge

That is the purpose of visualization

2D Spatial Data Mining (5) What are Maps? (1)

Definition of a map

A set of points, lines, and areas defined both by

1. position reference in a coordinate system

(spatial attributes) and

2. by their non-spatial attributes

From the U.S. Geological Survey (USGS))

4

What are Maps? (2)

Maps are the world reduced to points, lines,

and areas, using a variety of visual resources

size, shape, value, texture or pattern, color,

orientation, and shape

A thin line may mean something different from

a thick one, and similarly, red lines from blue

ones




Symbolizing 2D Spatial Data

Spatial Dimension (1)

Point Phenomena (zero dimensional)

No spatial extent

Definition: (x, y) – (longitude, latitude)

z – statistical value (data value)

Examples

Locations of religious worship

Oil wells

Locations of nesting sites for eagles

Census Demographics

…


Linear Phenomena (one dimensional)

One dimensional in spatial extent

Have length, but essentially no width

Definition: unclosed series of (x, y) for each

phenomena

Examples:

Boundaries between countries

Path of a stunt plane during an air show

…


Area Phenomena (two dimensional)

Two dimensional in spatial extent

Have both length and width

Definition: series of (x, y)-coordinates that

completely enclose a region

with a statistical value for each phenomena

Examples:

Lakes

Political Units (States,…)

…


2 D Phenomena

Each point is defined by longitude, latitude, and an

associated value above a zero point

Examples

Surfaces

…

5

Smooth Statistical Surface Spatial Dimension (5)

True 3D Phenomena (most common)

Each point is defined by longitude, latitude, and

multiple associated values

Examples:

Longitude, Latitude, Height above sea level, CO2

Concentration, …

3D Model

3D Model of an

open-pit coal

mining site

Source:

Pennsylvania

Department of

Environmental

Protection

How are Maps used? (1)

Three ways

to provide specific information about particular

locations

to provide general information about spatial patterns

to compare patterns in one, two or more maps


(1) to provide specific information about

particular locations

Approximately

2 millions

slaves were

transported

from Africa to

Spanish

America

between 1700

and 1870


(2) to provide general information about

spatial patterns

Low percentage of

inhabitants voted for

Perot in the

southeastern part of the

United States

High percentage voted

for Perot in the central

and northwestern states

6


(3) to compare spatial patterns in one or two

mapsSpatial patterns of the two

maps are quite different

Distribution for Corn:

concentrated in the

traditional corn belt region

of Midwest

Distribution for Wheat:

Concentrated on the Great

Plains

Spatial Visualization and Cartography (1)

How should the term “visualization“ be used in

the field of spatial data?

First Definition MacEachren et. al. (1992)

“Spatial (Geographic) Visualization will be defined

as the use of concrete visual representations to

make spatial contexts and problems visible, so as

to engage the most powerful human-processing

abilities, those associated with vision“

Spatial Visualization and Cartography (2)

Second Definition MacEachren et. al. (1994):

“Cartography-Cube Representation“

Visualization:

Private activity in which

unknowns are revealed in

highly interactive

environments

vs.

Communication:

Public activity in which

knowns are presented in non-

interactive environments

GIS – Relation to other spatial informationsystems

GIS – Structure and Relation toVisualization

Example: Repair Service

7

Organizing Spatial Data

• Reality

• Field based

• Object based

• Digital Landscape Model

• Raster Model

• Vector Model

• Digital Cartographic Model

• Raster Structure

• Vector Structure




Visual Variables for Spatial Data

Visual Variables for Spatial Visualization

Spacing (Texture)

Changes in the distance between the symbols

Size

Change size of the entire symbols

point, linear phenomena

Change size of individual symbols

aerial, 2 1/2D, true 3D phenomena


Perspective Height

Refers to the perspective three-dimensional view ofthe phenomena

Cannot be used for true 3D phenomena becausethree dimensions are needed to locate thephenomena being mapped

Orientation

Refers for linear, aerial, true 3D phenomena tovarious directions of individual symbols

Refers for point phenomena to the direction of theentire symbol


Arrangement

for aerial and true 3D phenomena refers to how the

symbols are distributed (square pattern, randomly

distributed symbols, …)

for linear phenomena refers to how lines are broken

into a series of dots and dashes

for point phenomena refers to changing the position

of the white marker within the black symbol


Visual variables

for

black-and-white

maps matching

Bertin‘s original

seven visual

variables

8

Visual Variables for Black and White MapsVisual Variables for Color Maps

Choices during the visualization process Spatial and Non-spatial Information

Three

dimensional

space-time cube

in which time is

treated as the Z-

axis

Discrete and Continuous

Depictions of discrete and continuous phenomena

Absolute and Relative

9

Discrete vs. Continuous Data ModelSmooth vs. Abrupt Data Model

Discrete:

Presumed to occur

at distinct locations

Continuous:

Region of interest

Smooth:

Change in gradual

fashion

Abrupt:

Change suddenly

Discrete vs. Continuous VisualizationSmooth vs. Abrupt Visualization

Syntax of map

forms

(How Maps Work by

Alan MacEachren)

Subdivision

of map types

based on

measurement

scale,

graphical

variables and

continuity of

the data

(Cartography –

Visualization of

spatial data by

Kraak & Ormeling)




Map Projections

Map Scales Comparison of Globe and Flat Map

Simple Way: Map earth

without distortion on a

globe

Disadvantages:

expensive to make

difficult to reproduce

cumbersome to handle

awkward to store

difficult to measure

difficult to draw

10

Choice of Map Projections (1)

Number of possible projections are unlimited

In practice

several projections combine useful characteristics

Geographer, Historian and Ecologist

more concerned with sizes of areas

Navigator, Meteorologist, Astronaut, Engineer

more concerned with angles and distances

Choice of Map Projections (2)

Problem: no specific method can be given thatwill lead to the right selection of the projection

General generalizations

Spherical Geometry Symmetry and Deformationcharacteristics (atlas maker)

Equivalence, Conformality and Azimuthally(temperature distribution over large areas)

Overall Shape

Projection Classification

Widely used theoretical surfaces on which the earth‘s surface

can be projected

Major Projections

Azimuthal Projections

Cylindrical Projections

Conical Projections

Other Projections

Equivalent Projections

World Projections

Interruption and Condensing

Conformal Projections

Equal Area Projections

Used for presentations

attempts to give a correct visual impression of the

relative sizes

Two important factors

Size of the involved area

Distribution of the angular deformation



11


Theoretical positions of the points for the class of azimuthal

projections: (1) gnomonic, (2) stereographic, (3) equidistant, (4)

equivalent, (5) orthographic


Nearly same areas: left France, right

Madagascar

Using non-equal area projections France

appears larger than it should in comparison

to Madagascar

Azimuthal patterns of distortion:

Contour shows the arrangement of

relative values

Spacing of the contours indicates the

gradients or rates of change

Example: Albers ProjectionU

sed u

nd

er perm

ission

from

F. M

ansm

ann

Cylindrical Projections

Example: Lambert Projection

Used

un

der p

ermissio

n fro

m F

. Man

sman

n

Conical Projections

12

Example: Conical equidistant projection

Source: Hans Havlicek

Other Projection Systems

Equal Area Projection

Azimuthal

equal-area

projection

Source: Hans Havlicek

Aitoff-Hammer Projection

Used

un

der p

ermissio

n fro

m F

. Man

sman

n

Mollweide Projection

Used

un

der p

ermissio

n fro

m F

. Man

sman

n

Cosinusodial Projection

Used

un

der p

ermissio

n fro

m F

. Man

sman

n

13

Some Comparisons Some Comparisons




Visualization Strategies

Visualization Strategy

Map the spatial (location) attributes of data

with two spatial dimensions directly to the

spatial attributes of the screen

The result will be some of the following

visualizations


(1) Dot Map

if the data contains point phenomena(zero dimensional)

Then point objects (road or stream) arerepresented as pixels on the screen

Dot Map

A simple dot map of commercial wireless antennas in the USA

14

Map of the United

States shows the

total dissolved

solid (TDS)

content of water

from oil and gas

wells in the USGS

produced-water

database

(Source:

US Department of

the Interior)

Dot Map Visualization Strategy

(2) Network/Transportation Map

if the data contains linear phenomena (onedimensional), as well as point objects

Then linear phenomena (roads or streams)are represented as a sequence ofconnected coordinates, which are plottedas a series of line segments

WorldCom Networks

Network/Transportation Map Network/Transportation Map

Lower Manhattan Subway Map


(3) Thematic Map

if the data contains aerial phenomena (two

dimensional), as well as point objects

Then area features (lake or political

boundaries) are generally represented as a

closed contour, a set of coordinates where the

first and last points are the same

Thematic Map

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Thematic Map Visualization Strategy

(3) Choropleth Maps

Greeks words: Choro = area

pleth = value

Similar to thematic maps

Uses enumeration units to represent different

magnitudes of a variable

Choropleth Maps Visualization Strategy

(4) Isovalue Map

if the data contains 2 D phenomena

Then boundary information is extracted from

an image and depicts a continuous

phenomena, such as elevation or temperature

Iso-Value Map

Monthly

average of ozone

during the

spring warming

left: Southern

Hemisphere

right: Northern

Hemisphere

(Source:

IBM T.J.

Watson

Research

Center)


(5) Smooth Statistical Surface

if the data contains 2 D phenomena

Then statistical values are mapped to height

above sea level (or some other zero

boundary)

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Smooth Statistical Surface Visualization Strategy

(6) 3D Model

if the data contains true 3D phenomena

Then boundary information is extractedfrom an image and depicts a continuousphenomena, such as elevation ortemperature

3D Model

3D Model of an

open-pit coal

mining site

(Source:

Pennsylvania

Department of

Environmental

Protection)

Transformation

possibilities

among maps

(Cartography –

Visualization of spatial

data, Kraak & Ormeling)

http://www.fao.org/docrep/003/T0446E/T0446E06.htm




Part III: GIS - ComputationalApproach

GIS – Computational Approach

• A computationalapproach tovisualize,manipulate, analyze,and display spatialdata to study theworld

• “Smart Maps”linking a database tothe map, creatingdynamic displays

17

GIS – Relation to other spatialinformation systems

GIS – Structure and Relation toVisualization

Example: Repair Service Example: Environmental DiseaseAnalysis for industrial locations

Organizing Spatial Data

• Reality

Field based

Object based

• Digital Landscape Model

Raster Model

Vector Model

• Digital Cartographic Model

Raster Structure

Vector Structure




Models of Spatial Data

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Reality

Field Objects

Collection of

spatial

distributions

Geo-Objects

Discrete geo-

referenced

entities

Field-Objects

Grid of Rows and Columns: Tessellated

Basic data unit is the cell (spatial)

(x,y) implicit

Entity information must be explicitly encoded

Geo-Objects

Geometric Data Objects (Geo-objects) are

related to a spatial reference point in a

coordinate system

Geo-objects are at least two-dimensional

Geo-objects in general have other spatial and

non-spatial attributes (height information,

name, etc)

X

Y

2-dim. Points 2-dim. Polygons

Wheat

Barley

Sugar beets

Geo-Objects

Geo-Objects

Attribute classes

Spatial Attributes

(x,y)-Coordinates / Spatial Data

Extent of a polygon

Area of the polygon

Topological Attributes

Polygon Pol1 is a neighbor ofpolygon Pol2

Polygon Pol1 and Pol2 have acommon edge

Topic Tables Attributes

In polygon Pol1 barley iscultivated

At the point P2 is the post office78457

K1

Pol1

Pol2

Pol3

X

Y

P1

P2

Digital Landscape Model

Vector Model

Geo-objects aredescribed by theirborders

Border is defined by a setof points

Raster Model

Geo-objects aredescribed by their innerdefinition

Inside is defined as ancollection of pixels fromwithin a grid

P1

P2

P3

P4

P5

(P1, P2, P3, P4, P5)

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Reality Raster Model Raster Resolution Problem

Advantages

Easy to conceptualize

Overlay operations are easy

Represents a two-dimensional array (easy to

implement)

Raster Resolution Problem

Small Grid: Coarse Resolution but limited storage

space

Large Grid: Fine Resolution but large storage space

Raster Resolution Problem Raster vs. Vector

Raster vs. VectorInstitute for Visualization and Perception Research



Visualization of Point DataDot maps

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Dot Map

A simple dot map of commercial wireless antennas in the USA

Map of the United

States shows the

total dissolved

solid (TDS)

content of water

from oil and gas

wells in the USGS

produced-water

database

(Source:

US Department of

the Interior)

Dot Map

Highly non-uniformly distribution

Input: (longitude,lattitude,s) –

geographic location and statistical

value

Observation: spatial data are highly

non-uniformly distributed in real world

data sets

Example: State New York Median

Household Income Year 2000

Database (1% of the data)

8.3 Visualization of Point Data 8.3.1 Dot Maps

Problem Definition

Visualization strategy:

Map the spatial (location) attributes of data with two spatialdimensions directly to the spatial attributes of the screen.

Mapping

to colored

pixels


Visual Exploration Goals

• No Overlap “normal” screen resolutions

Provide effective

visualizations

No lack of information

Provide visualization with

general geo-spatial

relationships

Make interesting geo-

patterns visible

Achieve Pixel-Coherence

Make small clusters visible

• Clustering

• Position Preservation


Simple Mapping – Dot Map

• High degree of overlap in the highly

populated areas

• Low populated areas are virtually empty


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Visualization of Point DataPixel maps

PixelMaps

KDE DetermineKernelDenEst3D(D)

Smoothness of the kernel function needs to vary depending on the

density in the x-y dimension

Kernel function also has to be different for the x-y and third dimension

P DeterminePeaks(KDE)

Determine the peaks at a given point required to make a scan over the

database

C DetermineClusters(P)

Hill-Climbing, ...

For our evaluation we implemented a PixelMap-Algorithm

based on the DenClue-Clustering Algorithm

Only usable for a subset of the data (<50000 ~ 30h)

An Efficient Implementation

Basic Idea: Rescaling of the map

regions to fit the 3D point clouds on the

screen better

Fast-PixelMap: heuristic to approximate

the result of the PixelMap algorithm

Combines the advantages of gridfile and

quadtree

Density-based Distortion

Basic Idea:

Rescaling of the

map regions to fit

better the 3D

points clouds on

the screen

Data Structure: Combination of the gridfile and quadtree

KDE-Display for USA Rescale Operations Series

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Array based Clustering

Constant number of classes NoOfClasses

Partioning the third dimensions in NoOfClasses

intervals beginning with minimal value

Endpoints of each interval are stored in an arrayBinary search to finding the interval for each data point

Pixel Placement

New York Census Year 2000

Traditional

Dot Map

PixelMap

showing local

pattern

New York Census Year 2000

Dot Map of

the USA

US Census Year 2000 Analysis

PixelMap of

the USA

US Census Year 2000 Analysis

23

US Census ExamplesInstitute for Visualization and Perception Research



Visualization of Point DataOther distortion techniques

Other Distortion Techniques

Basic Idea:

Rescaling of the

map regions to fit

better the 3D

points clouds on

the screen

Traditional map of the U.S. The data set is U.S. Year 2000 Median Household Income.

This scatter-plot map shows artifacts, such as spiral effects, especially in dense areas

such as Los Angeles County,Cook County and Manhattan.


HistoScale

HistoScale using the U.S. Year 2000 Median Household Income dataset.

The problem is the non-suffient distortion especially for sparse areas which are in the

same bin with highly dense areas, such as the areas north of Los Angeles.


HistoScale RadialScale AngularScale

US median household income shown by different distortion techniques.

Advantages and disadvantages of the different techniques are observable e.g. Florida,

Atlanta, Chicago, and the coastal areas, when an analysis of income distribution is

conducted or high/low income-areas are compared.


Generalized Scatter Plots

Generalized Scatter Plots of Census Data.The color map indicates the household

income (green: low; blue: medium, and red: high):

(1) Traditional Scatter Plot without distortion and data-induced overlap

(2) 50% distortion and 50% overlap

(3) 90% distortion and 5% overlap

24




Visualization of Line DataNetwork maps

Network Map

if the data contains linear phenomena

(one dimensional), as well as point objects

Linear phenomena (road or stream) are represented as a

sequence of connected coordinates, which are plotted as a

series of line segments

WorldCom Networks

Network Map Network Map

Lower

Manhattan

Subway

Map




Visualization of Line DataFlow maps

Flow Maps

(a) Minard’s 1864 flow map of wine exports from France

(b) Tobler’s computer generated flow map of migration from California from

1995 - 2000.

(c) A flow map produced by our system that shows the same migration data.

Doantam Phan, Ling Xiao, Ron Yeh, Pat Hanrahan, and Terry Winograd, Flow Map Layout,Proceedings of the IEEE Symposium on Information Visualization 2005 (INFOVIS’05)

25

Flow Maps

Input to the Process

Enforce minimum separation distance among the

nodes while preserving their relative positions to

one another.

Agglomerative hierarchical clustering

Modify the produced tree to make a particular node

(the source of the flow) the root of the tree.

Recursively lay out the tree of flows rooted at the

source in a way that edge crossings can be

minimized.

Lay out the edges. Make sure they do not intersect

nodes.

Render the whole map.

Layout Adjustment

minimum separation distance among the nodes in the horizontal and vertical

directions

= maximum width of a flow line

Algorithm: Misue et al.‘s force scan algorithm

Advantage: stable layout (no randomness)

For further details on the algorithm see:Kazuo Misue, Peter Eades, Wei Lai, Kozo Sugiyama. Layout adjustment and the Mental Map.

Journal of Visual Languages and Computing. 1995

Primary Clustering

Spatial representation of the primary hierarchical clustering (PHC) andits equivalent tree.

Flow Maps

Rooted hierarchicalclustering (RHC) modifies

the PHC to produce aflow map for

a particular root.

(b) The RHC for a flowmap from C. The (A,B)

and the ((D,E),F) clustersare kept.

(c) The RHC for a flowmap from D. Only the

(A,B) cluster is preserved.

Spatial Layout

The binary structure of the rooted clustering allows to generate the layout recursively.

Branching points are always placed on the line between the start node and thedestination that has more weight (or flow).

Edge Routing

Spatial layout may

cause anintersection

by placing b3 in away that intersects

c1. The algorithmfinds the

intersection of b1-b3 with c1, and

adds a new nodeand adjusts the

position of b3 to

avoid c1 ifnecessary.

26

Examples

Outgoing migration map from Colorado from 1995-

2000, generated without layout adjustment or edge

routing.

Outgoing migration map from Colorado for 1995-

2000 generated using edge routing but no layout

adjustment.

Examples

Outgoing migration map

from Colorado for 1995-

2000 generated using edge

routing and layout

adjustment.




Visualization of Area DataThematic maps

if the data contains areal phenomena

(two dimensional), as well as point

objects

Area features (lake or political boundary) are generally

represented as a closed contour, a set of coordinates

where the first and last points are the same

Thematic Map

Thematic Map Thematic Map

27




Visualization of Area DataChloropleth maps

Greeks words: Choro = area

pleth = value

thematic maps that use enumeration units to

represent different magnitudes of a variable

Choropleth Maps

Choropleth MapsInstitute for Visualization and Perception Research



Visualization of Area DataCartograms

thematic maps that try to represent a statistical

value by the area of the polygons

Cartograms

28

Cartograms

Cartogram Solvability

Given:

Goal:

,

Checker Board Example Impossible Cartogram Example

29

Continuous Cartogram DrawingProblem

The contiguous Cartogram problem may

be defined as a transformed set of

polygons for which

(Area) (Shape) (Global Shape)

CartoDraw Algorithm

Reduction Algorithm

Cartogram Construction Algorithm

Manual Placement

Automatic Placement


Basic IdeasIncremental Repositioning of Vertices

Scanlines Define Path of Action

Automatic Scanlines versus Manual Scanlines

Algorithm Controlled by Area and Shape Error

Shape Error Defined by Fourier Transform

Area Error

Shape Error

1. Parameterized Representation

of the Polygons

2. Normalized Curvature

3. Compute Fourier-Coefficients

4. Compute Euclidean Distance of Fourier-Coefficients

8.5.3 Cartograms


30

Cartogram Construction Algorithm Cartogram Construction Algorithm



31



Cartogram Construction Algorithm Algorithm

32

Automatic and Manual Scanlines Using Medial Axes as Skeleton

Area vs ShapeInstitute for Visualization and Perception Research



Visualization of Area DataRecMaps

Circular Map Approxiamation

Daniel Dorling. Area Cartograms: Their Use and Creation. Department

of Geography, University of Bristol, England, 1st edition, 1996.

Rectangular Map Approximations

Erwin Raisz. Principles of Cartography. McGraw-Hill, New York,1962.

33

Problem Definition

Objective function: Area Objective function: Shape

Objective function: Topology

The Figures show the pseudo dual of the U.S. map (left) and the pseudo dual

of the map partition (right). The red colored segments demonstrate the

topology error as defined in equation 3.

Objective function: Relative polygonpositions

34

Objective function: Empty space Formulation of the optimization problem

Variant 1: No Empty Space Variant 2: Preserve aspect ratio

User control over the weighting functionof MP2

Cartograms P resulting from different weights for the components

of f for Problem Variant MP2

Results of RecMap (Variant 2) using

US-Census population data on U.S. state

Results of RecMap (Variant 1) using US-

Census population data on U.S. state

Examples for MP1 and MP2

35

Basics of Genetic Algorithm

Population I is a set of individualsThe vth population is also called vthgenerationAn individual is characterized by threeaspects

The genotype

A construction algorithm

The phenotype

Basics of Genetic Algorithm(con‘t)

Three aspects of individual


Genetic Algorithms are based on three

components:

Selection (survival of the fittest)

Replication (combination of father and mother

individuals)

Mutation (alter its genotype and hence its

phenotype)


Creation of the next generation within three steps (with m = 6)

Genetic algorithm for Rectangular MapApproximation

Compute a candidate

cartogram

The RecMap Construction Heuristics TwoSolutions

Quadtree-based

Layout-based

36

Quadtree-Algorithm QT-Algorithm – Important Observation

The splits do not preservethe aspect ratio of thepolygons andneighborhood of thepolygons

Solution:Using different splitsequences which can beevaluated by the objectivefunctions

The QT-algorithm applied to the U.S.

using population data

The Quadtree-based heuristic

Adapted construction algorithm of

Quadtree-based heuristic

Construction algorithm of quadtree-based

heuristic

Example for the U.S. Population Data

Layout-based construction

Basic Idea:Initial Step:

Finding the core polygon pc

Main Step:

Construct a sequence of partial layouts or partialcartograms starting with pc

R-1 polygons are placed around pc one after theother until we found a (complete) cartogram

Extended adjacency graph

Insert an additional

node R+1 for the outer

region

pc equals to a polygon

which has the maximal

BFS number running

from node R+1

37

The Layout-based constructionheuristics Example for the U.S. Population Data

Efficiency

The scatterplots display the errors over time for the U.S.-state partition where both

algorithm variants were used. Note that the time axes were logarithmic scaled. The

whole computation time after 10 iteration took 0.33 seconds for MP1 60 seconds

for MP2.

Conclusion

Fully automatic (+)

Scalable (+)

explicit control of all visualization constrains: (+)

no area error

explicit control of shape

topology

empty space

relative position

Computational Complexity (O(n)*number of iterations)




Spatial Data Types

Spatial Data Types

.

What should be modeled?

Point LineArea

Partition Network

38

.

..

.

...

..

.

..

. ..

... ..

.

..

.

...

..

.

..

. ..

... ..

Euclidean Plane Issues (very general)

Point p=(r1,r2); ri are real

numbers

The Intersection Problem

The coordinates of the

intersection point are

rounded to the nearest

computed precision

Computers cannot represent

arbitrary real numbers

build a grid of possible

points

Intersection point may

not be on one of the lines

Let a finite discrete space with

N-Realm is a set R with

P PN (R-Points), S SN (R-Segments)

" s S : s = (p,q) p P q P

" p P " s S : ¬ ( p in s)

" s, t S, s t : ¬ ( s and t intersect)

NN

{ }1,...,0= nN

A

SPR =

Spatial Data Types - Realms

Realm Structures

R-cycle

R-face

R-faceR-block

• The following are based on an interpretation of a Realm as graph

An R-cycle is a loop in the graph of R

An R-face is an R-cycle, which includes additional disjunctive R-

cycles

An R-block is a connected component in the graph of R

Realm-based Spatial Data Types

. .. . .. . . ... ..

..

.

..

..

.

..

. ..

... ..

.

..

.

.

.. ... ..

... ..

.

..

.

.

.. ... ..

... ....

... ..

. ..

... ..

.

..

.

.....

...

. ..

... ..

.

..

.

...

..

.

..

. ..

... ..

Elements of the point type

Set of R-points

Elements of the line type

Set of different R-blocks

Elements of the area type

Set of different R-faces

.. .

. .

.. ..

Spatial Data Types

Types:

EXT = {lines, regions}

GEO = {points, lines, regions}

OBJ = {cities, highways, . . .}

(depends on the application scenario)

set(OBJ) is a spatial database

Second Order Signature

Spatial Data Types

Topological predicates:

39

Spatial Data Types

Important Queries:

Spatial Data Management Models

Hierarchical Database Structure


Network Database Structure


Relational Database Structure

Spatial Data Types

Queries using relational algebra:

(1) select cities [center inside Bavaria]

(2) select rivers [route intersects Window]

(3) select cities [dist(center,Hagen) < 100 and

population > 500,000]

(4) join cities states [center inside area]

(5) join cities rivers [dist(center,route) < 50]




Signs and Labeling

40

Ucar‘s typology of map signs

Diagrammtitel

written words

dimensions of the plane

symbols

figure icons plan icons

icons

artificial sign with representation function

non-linguistic signs

map sign

(How Maps Work by Alan MacEachren)

Categories of map sign aspects

designate appraise

indicate label

relate

apprise

prescribe emotive

connote aesthetic

stimulate

Sign Aspects

(How Maps Work by Alan MacEachren)

Types of Map Signs

Pictorial,

associative,

geometric

nominal point

symbols

Types of Map Signs

Example of the mimetic to an arbitrary continuum

of map marks for a city

Nyerges‘s meaning triangle Nyerges‘s meaning triangle

41

Generalization Geometric Generalization

Conceptual Generalization Effect of parameter binning and boundariesorShould we believe what we see?

Different sets of bin boundaries applied to the same data yield

different-looking choropleth maps

Should we believe what we see? Should we believe what we see?

42

Should we believe what we see?Institute for Visualization and Perception Research



Spatial Index Structures


Spatial index structure

(MUR, , ...) (MUR, , ...) ......

R-Tree

Quadtree

(EB)

• Normal index structures are not usable for organizing spatial data

• So organize simplified approximations to the geo-objects

- Minimal Bounding Rectangle (MBR)

- Pointer to the detailed description (EB) of the geo-object

P

R

Point Query Window Query


• The following types of queries must be efficiently supported

- Given a query point P or a query rectangle R

- Point Query: Find the Geo-Objects Obj: P (Obj.MBR)

- Window Query: Find the Geo-Objects Obj: R (Obj.MUR)


• Given two sets with minimal bounding rectangles

M1 = {MUR1,1, MUR1,2, …, MUR 1,m} and M2 = {MUR2,1, MUR2,2, …, MUR2,n}

• Spatial Join:

{ ( MUR1, MUR2 ) | MUR1 M1, MUR2 M2 and MUR1 MUR2 }

B1

A2

A3

A4A5

A6

A1

B2

B3

Result Set:

(A5, B1)(A4, B1)

(A1, B2)

(A6, B2)(A2, B3)

Spatial-Join


+ =

Ground load Disease numbers Combination

Support

through

Spatial Join

a1

a3

a4

a2

b3

b2b1

(a1, b1)

(a4, b2)

(a2, b2)

43

Principles of Spatial Data Organization

X

A5A1

A4A3

A6A2

Y

A2

A3

A4A5

A6

A1

3 Classes (Pages):

• Partitioning the minimal bounding rectangles in disjunctive

classes

• Each class is stored exclusively in a page

R-Tree


• Median-Split operation in each dimension (x/y-median splits)

• Points = Pixels in a discrete data space

00 00 00 00

00 01 00 00

00 00 01 0000 00 01 10

00 00 00 00

10 00 00 01

00 00 00 0101 00 01 00

NW NE

SW SE


NW

NE SW

SE

Quadtree

00 00 00 00

00 01 00 00

00 00 01 00

00 00 01 10

00 00 00 00

10 00 00 01

00 00 00 01

01 00 01 00

NW NE

SW SE

Point Query supported by R-Tree

A5A1A4A3A6A2

RST

Point Query

X

Y

A2

A3

A4A5

A6

A1

R S

T

.

Result Set:

Paths to be observed

[]

PointQuery (Page, Point);

FOR ALL Entries Page DO

IF Point IN Entry.Rectangle THEN

IF Page = DataPage THEN

Write (Entry.Rectangle)

ELSE

PointQuery (Entry.Subtree^, Point);

Window Query supported by R-Tree

A5A1A4A3A6A2

RSTWindow Query

X

Y

A2

A3

A4A5

A6

A1

R

S

TResult Set:

[A1, A2]

Paths to be observed

Window Query (Page, Window);

FOR ALL Entries Page DO

IF Window INTERSECTS Entry.Rectangle THEN

IF Page = DataPage THEN

Write (Entry.Rectangle)

ELSE

WindowQuery (Entry.Subtree^, Window);

Literature

Jaccques Bertin, Graphische Darstellungen und die graphische Weiterverarbeitungder Information, 1982, Walter de Gruyter, Chap. B5

Leland Wilkinson, The Grammar of Graphics, Springer, Chap. 10

D. A. Keim, S. C. North, C. Panse: CartoDraw: A Fast Algorithm for GeneratingContiguous Cartograms, IEEE Transactions on Visualization and Computer Graphics(TVCG), Vol. 10, No. 1, pp. 95-110, 2004.

D. A. Keim, S. C. North, C. Panse, M. Sips: Visual Data Mining in Large Geo-SpatialPoint Sets, Special Issue: Visual Analytics, IEEE Computer Graphics and Applications(CG&A), September-October 2004, pp. 36-44, IEEE Press, September, 2004.

D. A. Keim, S. C. North, C. Panse, M. Sips: Pixel based Visual Mining of Geo-SpatialData, Computers & Graphics (CAG), Vol. 28, No. 3, pp. 327-344, Elsevier Science,June, 2004.

Natalia Andrienko, Gennady Andrienko, Exploratory analysis of spatial and temporaldata: A systematic approach, 2006, Springer

J. Dykes, A.M. MacEachren, M.-J. Kraak, Exploring geovisualization, 2005, Elsevier

Documents

Introduction to Visualization Techniques for Spatial Datagrinstei/vis2010/Lecture 6 - Intro to spatial data... · Lecture 6 – Geo-Spatial Data Visualization October 6, 2010