Spa$al Analysis and Modeling (GIST 4302/5302) Guofeng Cao Department of Geosciences Texas Tech University

Transcription

1 Spa$al Analysis and Modeling (GIST 4302/5302) Guofeng Cao Department of Geosciences Texas Tech University

2 Outline Last week, we learned: Review of map projec$on Characteris$cs of spa$al data Types of spa$al data This week, we will learn: Concepts of database Representa$on of different types of spa$al data in GIS Commonly used spa$al operators in GIS

3 Class Outlines Spatial Point Pattern Regional Data (Areal Data) Continuous Spatial Data (Geostatistical Data) Representation and Operations of Spatial Data What is Special about Spatial? Characteristics of Spatial Data and Map Projection

4 ArcGIS Desktop Installa$ons Version 10.2 Two steps: Download the installa$on packages from: \\unicomplex\shared\arcgis Send me an for the ac$va$on code

5 Database Fundaments

6 Review: Bits and Bytes Data stored in a computer system is measured in bits each bit records one of two possible states 0 (off, false) 1 (on, true) Bits are amalgamated into bytes (8 bits) Each byte represents a single character A character may be encoded using 7 bits with an extra bit used as a sign of posi$ve or nega$ve Ques%on: Can I use byte to represent the eleva%on of the Everest in meters? Megabytes (2^20 bytes)

7 Database A database is a collec$on of data organized in such a way that a computer can efficiently store and retrieve data A repository of data that is logically related A database is created and maintained using a general- purpose piece of so^ware called a database management system (DBMS)

8 Common Database Applica$ons Home/office database Simple applica$ons (e.g., restaurant menus) Commercial database Store the informa$on for businesses (e.g. customers, employees) Engineering database Used to store engineering designs (e.g. CAD) Image and mul$media database Store image, audio, video data Geodatabase/spa$al database Store a combina$on of spa$al and non- spa$al data

9 Query language Query compiler Run$me database processor Constraint enforcer Stored data manager System catalog/data dic$onary Elements of a DBMS

10 An illustra$on of a database Database

11 The Rela$onal Model The rela$onal model is one of the most commonly used architecture in database A rela%onal database is a collec$on of rela%ons, o^en just called tables Each rela$on has a set of a@ributes The data in the rela$on is structured as a set of rows, o^en called tuples Each tuple consists of data items for each aaribute Each cell in a tuple contains a single value A rela%onal database management system (RDBMS) is the so^ware that manages a rela$onal database

12 An Example Rela$on Rela$on Aaribute Tuple Data item

13 Rela$ons A rela%on is basically a table A rela%on scheme is the set of aaribute names and the domain (data type) for each aaribute name A database scheme is a set of rela$on schemes In a rela$on: Each tuple contains as many values as there are aaributes in the rela$on scheme Each data item is drawn from the domain for its aaribute The order of tuples is not significant Tuples in a rela$on are all dis$nct from each other The degree of a rela$on is its number of columns The cardinality of a rela$on is the number of tuples

14 Rela$on Scheme A candidate key is an aaribute or minimal set of aaributes that will uniquely iden$fy each tuple in a rela$on One candidate key is usually chose as a primary key

15 Opera$ons on Rela$ons Fundamental rela%onal operators: Union, intersec$on, difference, product and restrict: usual set opera$ons, but require both operands have the same schema Selec$on: picking certain rows Projec$on: picking certain columns Join: composi$ons of rela$ons Together, these opera$ons and the way they are combined is called rela%onal algebra combined: An algebra whose operands are rela$ons or variables that represent rela$ons

16 Project Operator The project operator is unary It outputs a new rela$on that has a subset of aaributes Iden$cal tuples in the output rela$on are coalesced Relation Sells: bar beer price Chimy s Bud 2.50 Chimy s Miller 2.75 Cricket s Bud 2.50 Cricket s Miller 3.00 Prices := PROJ beer,price (Sells): beer price Bud 2.50 Miller 2.75 Miller 3.00

17 Select Operator The select operator is unary It outputs a new rela$on that has a subset of tuples A condi$on specifies those tuples that are required Relation Sells: bar beer price Chimy s Bud 2.50 Chimy s Miller 2.75 Cricket s Bud 2.50 Cricket s Miller 3.00 ChimyMenu := SELECT bar= Chimy s (Sells): bar beer price Chimy s Bud 2.50 Chimy s Miller 2.75

18 Join Operator The join operator is binary It outputs the combined rela$on where tuples agree on a specified aaribute (natural join) Sells(bar, beer, price ) Bars(bar, address) Chimy s Bud 2.50 Chimy s 2417 Broadway St. Chimy s Miller 2.75 Cricekt s 2412 Broadway St. Cricket s Bud 2.50 Cricket s Coors 3.00 BarInfo := Sells JOIN Bars Note Bars.name has become Bars.bar to make the natural join work. BarInfo(bar, beer, price, address ) Chimy s Bud Broadway St. Chimy s Milller Broadway St. Cricket s Bud Broadway St. Cricket s Coors Broadway St.

19 Join Operator Join is the most $me- consuming of all rela$onal operators to compute In general, rela$onal operators may not be arbitrarily reordered (le^ join, right join) Query op$miza$on aims to find an efficient way of processing queries, for example reordering to produce equivalent but more efficient queries

20 Complex Rela$onal Operator Example How to find the loca$ons that sell Bud? Sells(bar, beer, price ) Bars(bar, address) Chimy s Bud 2.50 Chimy s 2417 Broadway St. Chimy s Miller 2.75 Cricekt s 2412 Broadway St. Cricket s Bud 2.50 Cricket s Coors 3.00 Step 1 BarInfo(bar, beer, price, address ) Chimy s Bud Broadway St. Chimy s Milller Broadway St. Cricket s Bud Broadway St. Cricket s Coors Broadway St. Step 2 BarInfo(beer, address ) Bud 2417 Broadway St. Milller 2417 Broadway St. Bud 2412 Broadway St. Coors 2412 Broadway St.

21 SQL in One Slide Structured Query Language The standard for rela$onal database management systems (RDBMS) Example: Relation Sells: bar beer price Chimy s Bud 2.50 Chimy s Miller 2.75 Cricket s Bud 2.50 Cricket s Miller 3.00 ChimyMenu := SELECT bar= Chimy s (Sells): bar beer price Chimy s Bud 2.50 Chimy s Miller 2.75 Select * from Sells where bar= Chimy s Select * from Sells where bar= Chimy s orderby price asc

22 An Example of Spa$al Data How would you select the coun$es with popula$on larger than 100?

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24 Representa$on of Spa$al Data

25 Representa$on of Spa$al Data Models Object- based model: treats the space as populated by discrete, iden$fiable en$$es each with a geospa$al reference Buildings or roads fit into this view GIS So^wares: ArcGIS Field- based model: treats geographic informa$on as collec$ons of spa$al distribu$ons Distribu$on may be formalized as a mathema$cal func$on from a spa$al framework to an aaribute domain Paaerns of topographic al$tudes, rainfall, and temperature fit neatly into this view. GIS So^ware: Grass

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27 Rela$onal Models Tuples recording annual weather condi$ons at different loca$ons The field-based and object-based approaches are attempts to impose structure and pattern on such data.

28 Object- based Approach Clumps a rela$on as single or groups of tuples Certain groups of measurements of clima$c variables can be grouped together into a finite set of types

29 Object- based Example

30 Field- based approach Treats informa$on as a collec$on of fields Each field defines the spa$al varia$on of an aaribute as a func$on from the set of loca$ons to an aaribute domain

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32 Object- based Approach

33 En$ty Object- based models decompose an informa$on space into objects or en99es An en$ty must be: Iden$fiable Relevant (be of interest) Describable (have characteris$cs) The frame of spa$al reference is provided by the en$$es themselves

34 Example: House object Has several aaributes, such as registra$on date, address, owner and boundary, which are themselves objects

35 Spa$al objects Spa$al objects are called spa$al because they exist inside space, called the embedding space A set of primi$ve objects can be specified, out of which all others in the applica$on domain can be constructed, using an agreed set of opera$ons Point- line- polygon primi$ves are common in exis$ng systems

36 Example: GIS analysis For Italy s capital city, Rome, calculate the total length of the River Tiber which lies within 2.5 km of the Colosseum First we need to model the relevant parts of Rome as objects Opera$on length will act on arc, and intersect will apply to form the piece of the arc in common with the disc

37 Example: GIS analysis A process of discre$za$on must convert the objects to types that are computa$onally tractable A circle may be represented as a discrete polygonal area, arcs by chains of line segments, and points may be embedded in some discrete space

38 Primi$ve Objects Euclidean Space: coordina$zed model of space Transforms spa$al proper$es into proper$es of tuples of real numbers Coordinate frame consists of a fixed, dis$nguished point (origin) and a pair of orthogonal lines (axes), intersec$ng in the origin Point objects Line objects Polygonal objects

39 Points A point in the Cartesian plane R 2 is associated with a unique pair of real number a = (x,y) measuring distance from the origin in the x and y direc$ons. It is some$mes convenient to think of the point a as a vector. Scalar: Addition, subtraction, and multiplication, e.g., (x 1, y 1 ) (x 2, y 2 ) = (x 1 x 2, y 1 y 2 ) Norm: Distance: ab = a-b Angle between vectors:

40 Lines The line incident with a and b is defined as the point set {λa + (1 λ )b λ R} The line segment between a and b is defined as the point set {λa + (1 λ )b λ [0, 1]} The half line radia$ng from b and passing through a is defined as the point set {λa + (1 λ)b λ 0}

41 Polygonal objects A polyline in R 2 is a finite set of line segments (called edges) such that each edge end- point is shared by exactly two edges, except possibly for two points, called the extremes of the polyline. If no two edges intersect at any place other than possibly at their end- points, the polyline is simple. A polyline is closed if it has no extreme points. A (simple) polygon in R 2 is the area enclosed by a simple closed polyline. This polyline forms the boundary of the polygon. Each end- point of an edge of the polyline is called a vertex of the polygon. A convex polygon has every point intervisible A star- shaped or semi- convex polygon has at least one point that is intervisible

42 Polygonal objects

43 Convexity Visibility between points x, y, and z

44 Example: Triangula$on Every simple polygon has a triangula$on. Any triangula$on of a simple polygon with n ver$ces consists of exactly n 2 triangles Art Gallery Problem How many cameras are needed to guard a gallery and how should they be placed? Upper bound N/3

45 Related: Convex Hull

46 Related: Voronoi Diagram

47 Voronoi Diagram on Road Network

48 John Snow, Pumps and Cholera Outbreak

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51 Primi$ve GIS Opera$ons in Euclidean spaces Length, bearing, area Distance between objects (points, lines, polygons) Centroid Point in polygon Buffer Intersec$on/overlay In topological spaces Spa$al rela$ons (within, touch, cover, )

52 Distance and angle between points Length of a line segment can be computed as the distance between successive pairs of points The bearing, θ, of q from p is given by the unique solu$on in the interval [0,360] of the simultaneous equa$ons:

53 Distance from point to line from a point to a line implies minimum distance For a straight line segment, distance computa$on depends on whether p is in middle(l) or end(l) For a polyline, distance to each line segment must be calculated A polygon calcula$on is as for polyline (distance to boundary of polygon)

54 Area Let P be a simple polygon (no boundary self- intersec$ons) with vertex vectors: (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) where (x 1, y 1 ) = (x n, y n ).Then the area is: In the case of a triangle pqr

55 Area of a simple polygon Note that the area may be posi$ve or nega$ve In fact, area(pqr) = - area(qpr) If p is to the le^ of qr then the area is posi$ve, if p is to the right of qr then the area is nega$ve

56 Centroid The centroid of a polygon (or center of gravity) of a (simple) polygonal object (P = (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) where (x 1, y 1 ) = (x n, y n )) is the point at which it would balance if it were cut out of a sheet of material of uniform density:

57 Point in polygon Determining whether a point is inside a polygon is one of the most fundamental opera$ons in a spa$al database Semi- line method (ray cas9ng) : checks for odd or even numbers of intersec$ons of a semi- line with polygon Winding method: sums bearings from point to polygon ver$ces

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59 Collinearity and point on segment Boolean opera$on colinear(a,b,c) determine whether points a, b and c lie on the same straight line Colinear(a,b,c) = true if and only if side (a,b,c) =0 Opera$on point_on_segment(p,l) returns the Boolean value true if p l (line segment l having end- points q and r) 1 Determine whether p, q, r are collinear 2 If yes, thenp l if and only if p (minimum bounding box) MMB (l)

60 Segment intersec$on Two line segments ab and cd can only intersect if a and b are on opposite sides of cd and c and d are on opposite sides of ab Therefore two line segments intersect if the following inequali$es hold

61 Point of intersec$on Intersec$ng line segments l and l in parametric form: Means that there exists an α and β such that: Which solving for α and β give:

62 Primi$ve GIS opera$ons: Overlay Union Intersect Erase Iden$ty Update Spa$al Join Symmetrical Difference

63 Overlay Union Computes a geometric union of the input features. All features and their aaributes will be wriaen to the output feature class.

64 Overlay Intersect Computes a geometric intersec$on of the input features. Features or por$ons of features which overlap in all layers and/or feature classes will be wriaen to the output feature class.

65 Overlay Erase Creates a feature class by overlaying the Input Features with the polygons of the Erase Features. Only those por$ons of the input features falling outside the erase features outside boundaries are copied to the output feature class

66 Overlay Iden$ty Computes a geometric intersec$on of the input features and iden$ty features. The input features or por$ons thereof that overlap iden$ty features will get the aaributes of those iden$ty features

67 Overlay Update Computes a geometric intersec$on of the Input Features and Update Features. The aaributes and geometry of the input features are updated by the update features in the output feature class.

68 Overlay Spa$al join Joins aaributes from one feature to another based on the spa$al rela$onship. The target features and the joined aaributes from the join features are wriaen to the output feature class.

69 Overlay Symmetrical difference Features or por$ons of features in the input and update features that do not overlap will be wriaen to the output feature class.

70 Spa$al join Joins aaributes from one feature to another based on the spa$al rela$onship. The target features and the joined aaributes from the join features are wriaen to the output feature class. Overlay

71 Quiz Input Feature Overlay Feature Which of the following is the result of iden$ty, intersect, symmetrical difference, union and update respec$vely? A B C D E

72 Buffer

73 Primi$ve operators you might already realized that these primi$ve operators are o^en used collabora$vely with each other, and other analy$cal methods (e.g., dissolve, surface analysis, interpola$on) that we will introduce in the coming lectures.

74 Topological spa$al opera$ons: spa$al rela$onship Object types with an assumed underlying topology are point, arc, loop and area Opera$ons: boundary, interior, closure and connected are defined in the usual manner components returns the set of maximal connected components of an area extremes acts on each object of type arc and returns the pair of points of the arc that cons$tute its end points is within provides a rela$onship between a point and a simple loop, returning true if the point is enclosed by the loop

75 Topological spa$al opera$ons for X meets Y if X and Y touch externally in a common por$on of their boundaries areas X overlaps Y if X and Y impinge into each other s interiors

76 Topological spa$al opera$ons for X is inside Y if X is a subset of Y and X, Y do not share a common por$on of boundary areas X covers Y if Y is a subset of X and X, Y touch externally in a common por$on of their boundaries

77 Topological spa$al opera$ons There are an infinite number of possible topological rela$onships that are available between objects of type cell

78 hap://resources.esri.com/help/9.3/arcgisdesktop/com/gp_toolref/data_management_toolbox/select_by_loca$on_colon_graphical_examples.htm

79 Contain vs CONTAINS_CLEMENTINI: the results of CONTAINS_CLEMENTINI will be iden$cal to CONTAINS with the excep$on that if the feature in the Selec$ng Features layer is en$rely on the boundary of the Input Feature Layer, with no part of the contained feature properly inside the feature in the Input Feature Layer, the input feature will not be selected.

80 Spaghey SpagheG data structure represents a planar configura$on of points, arcs, and areas Geometry is represented as a set of lists of straight- line segments

81 Spaghey- example Each polygonal area is represented by its boundary loop Each loop is discre$zed as a closed polyline Each polyline is represented as a list of points A:[1,2,3,4,21,22,23,26,27,28,20,19,18,17] B:[4,5,6,7,8,25,24,23,22,21] C:[8,9,10,11,12,13,29,28,27,26,23,24,25] D:[17,18,19,20,28,29,13,14,15,16]

82 Issues There is NO explicit representa$on of the topological interrela$onships of the configura$on, such as adjacency Data consistence issues Silver polygons Data redundancy

83 NAA: node arc area Each directed arc has exactly one start and one end node. Each node must be the start node or end node (maybe both) of at least one directed arc. Each area is bounded by one or more directed arcs. Directed arcs may intersect only at their end nodes. Each directed arc has exactly one area on its right and one area on its le^. Each area must be the le^ area or right area (maybe both) of at least one directed arc.

84 NAA: planar decomposi$on Arc Begin End Left Right a 1 2 A X b 4 1 B X c 3 4 C X d 2 3 D X e 5 1 A B f 4 5 C B g 6 2 D A h 5 6 C A i 3 6 D C

85 Finding what s Inside and what s nearby Involves rela$onship between two or more layers Toxic gas Popula$on Ancient Forest Watersheds

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87 Overlay: Finding What s Inside What coun$es are within 50 miles of Interstate 10?

88 Spa$al Index Efficient spa$al query for massive datasets, e.g., spa$al grid index, quadtree, R- tree,.. An example of quadtree

89 Field- based Approach

90 Spa$al fields If the spa$al framework is a Euclidean plane and the aaribute domain is a subset of the set of real numbers; The Euclidean plane plays the role of the horizontal xy- plane The spa9al field values give the z- coordinates, or heights above the plane Regional Climate Varia%ons Imagine placing a square grid over a region and measuring aspects of the climate at each node of the grid. Different fields would then associate locations with values from each of the measured attribute domains.

91 Proper$es of the aaribute domain The aaribute domain may contain values which are commonly classified into four levels of measurement Nominal airibute: simple labels; qualita$ve; cannot be ordered; and arithme$c operators are not permissible Ordinal airibute: ordered labels; qualita$ve; and cannot be subjected to arithme$c operators, apart from ordering Interval aiributes: quan$$es on a scale without any fixed point; can be compared for size, with the magnitude of the difference being meaningful; the ra$o of two interval aaributes values is not meaningful Ra9o aiributes: quan$$es on a scale with respect to a fixed point; can support a wide range of arithme$cal opera$ons, including addi$on, subtrac$on, mul$plica$on, and division

92 Con$nuous and differen$able fields Con9nuous field: small changes in loca$on leads to small changes in the corresponding aaribute value Differen9able field: rate of change (slope) is defined everywhere Spa$al framework and aaribute domain must be con$nuous for both these types of fields Every differen$able field must also be con$nuous, but not every con$nuous field is differen$able

93 One dimensional examples Fields may be ploaed as a graph of aaribute value against spa$al framework Continuous and differentiable; the slope of the curve can be defined at every point

94 One dimensional examples The field is continuous (the graph is connected) but not everywhere differentiable. There is an ambiguity in the slope, with two choices at the articulation point between the two straight line segments. Continuous and not differentiable; the slope of the curve cannot be defined at one or more points

95 One dimensional examples The graph is not connected and so the field in not continuous and not differentiable. Not continuous and not differentiable

96 Two dimensional examples The slope is dependent on the par$cular loca$on and on the bearing at that loca$on

97 Isotropic fields A field whose proper$es are independent of direc$on is called an isotropic field Consider travel $me in a spa$al framework The $me from X to any point Y is dependent only upon the distance between X and Y and independent of the bearing of Y from X

98 Anisotropic fields A field whose proper$es are dependent on direc$on is called an anisotropic field. Suppose there is a high speed link AB For points near B it would be beaer, if traveling from X, to travel to A, take the link, and con$nue on from B to the des$na$on The direc9on to the des9na9on is important

99 Spa$al autocorrela$on Spa$al autocorrela$on is a quan$ta$ve expression of Tobler s first law of geography (1970) Everything is related to everything else, but near things are more related than distant thing Spa$al autocorrela$on measures the degree of clustering of values in a spa$al field Also termed as spa$al dependency, spa$al paaern, spa$al context, spa$al similarity, spa$al dissimilarity

100 Autocorrela$on If like values tend to cluster together, then the field exhibits high positive spatial autocorrelation If there is no apparent rela$onship between aaribute value and loca$on then there is zero spa%al autocorrela%on If like values tend to be located away from each other, then there is nega%ve spa%al autocorrela%on

101 Representa$ons of Spa$al Fields Points Contours Raster/Layce Triangula$on (Delaunay Trangula$on)

102 Example Contour lines and raster

103 Trangula$ons Example

104 Side Note: Delaunay Triangula$on and Dual Graph Voronoi Diagram

105 Opera$ons on fields A field opera$on takes as input one or more fields and returns a resultant field The system of possible opera$ons on fields in a field- based model is referred to as map algebra Three main classes of opera$ons Local Focal Zonal

106 Neighborhood func$on Given a spa$al framework F, a neighborhood func9on n is a func$on that associates with each loca$on x a set of loca$ons that are near to x

107 Local opera$ons Local opera9on: acts upon one or more spa$al fields to produce a new field The value of the new field at any loca$on is dependent on the values of the input field func$on at that loca$on is any binary opera$on

108 Focal opera$ons Focal opera9on: the aaribute value derived at a loca$on x may depend on the aaributes of the input spa$al field func$ons at x and the aaributes of these func$ons in the neighborhood n(x) of x

109 Zonal opera$ons Zonal opera9on: aggregates values of a field over a set of zones (arising in general from another field func$on) in the spa$al framework For each loca$on x: 1 Find the Zone Z i in which x is contained 2 Compute the values of the field func$on f applied to each point in Z i 3 Derive a single value ζ(x) of the new field from the values computed in step 2

110 Summary: Object- based vs Field- based models Object- based models: Greater precision Less redundant informa$on (smaller storage footprints) Complex data structures Field- based models: Simpler data structures More redundant informa$on (larger storage footprints) Less precision Raster is faster, but vector is corrector

111 Raster <- > Vector Vector- > Raster Interpola$on Inverse distance weighted, Kriging, Spline Density surface Kernel density Rasteriza$on Raster- >Vector Watershed Vectoriza$on (raster to polygon)

112 End of this topic