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Spatial Analysis What is it?

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Spatial AnalysisWhat is it?

- “…the purpose of geographic inquiry is to examine relationships between geographic features collectively and to use the relationships to describe the real-world phenomena that map features represent.” (Clarke 2001, 182).
- One Definition: the quantitative procedures employed in the study of the spatial arrangement of features (points, lines, polygons and surfaces)

Geographic Information Analysis

- “Geographic information analysis is…concerned with investigating the patterns that arise as a result of processes that may be operating in space” (p. 3).
- “Techniques [that] enable the representation, description, measurement, comparison, and generation of spatial patterns…”

How Do We Represent the World (in Map or Digital Form?)

- Raster – Vector
- A Higher Level of Abstraction? (p. 5)
- Objects and Fields
- The key distinction (according to your authors)
- A slightly different conceptualization

- How do we choose the “best” representation(s)?

Spatial Analysis:What is it?

- What types of relationships exist between geographic features, and how do we express them?
- Properties of spatial features and/or relationships between them: size, distribution, pattern, contiguity, neighborhood, shape, scale, orientation

3 Fundamental Questions Regarding Spatial Relationships

- How can two (or more) spatial distributions be compared with each other?
- How can variations in geographic properties over a single area or data set be described and/or analyzed?
- How can we use what we have learned from an analysis(es) to predict future spatial distributions?
Spatial Analysis can cover the spectrum implied by these questions!

What role does GIS play in Spatial Analysis?

- GIS is a tool with unique capabilities:
- Can handle geographically-referenced data
- Spatial/attribute data entry/update capabilities
- Data conversion functions
- Storage and organization of a variety of spatial and attribute data
- Manipulation of spatial and attribute data (encompasses many different operations)
- Presentation/display capabilities
- Spatial analysis tools (many tools may be used in combination)

Do you remember the 5 functional elements of a GIS?

- Data acquisition
- Preprocessing
- Database Management
- Manipulation/Analysis
- Final product output

These elements are all part of the spatial analysis “equation” (and a GIS professional’s knowledge base).

Our framework this semester for discussing GIS operations/procedures that are useful for spatial analysis…

- The Pitfalls and Potential of Spatial Data
- Maps as Outcomes of Processes
- Point Pattern Analysis
- Describing and Analyzing Fields
- Statistical Analysis of Fields/Spatial Interpolation
- Map Overlay Concepts and Procedures
- Spatial Modeling
- Network Analysis

How can we characterize Spatial Analysis (what skills does it require)?

- Spatial analysis is an artistic and a scientific endeavor (what does this mean?)
- It requires knowledge of the problem and/or question to be answered
- It requires knowledge about the data (how it was collected, organized, coded, etc.)
- It requires knowledge of GIS capabilities
- It may require knowledge of statistical techniques
- It requires envisioning the results of any operation…and the combination of any operations
- It is not completely objective, in fact some argue that it is completely subjective
- Many times there is more than one way to derive information that answers a question

Are Spatial Data “Special,” and if They Are, Why? it require)?

- Spatial Data are Special…
- Why?
- How?
- What are the implications?
- Pitfalls
- Potential

- Why “They” Need “Us”

The “Pitfalls” of Spatial Data it require)?

- Most spatial samples are not random!!
- This situation/problem is known as spatial autocorrelation
- The earth’s surface is not an isotropic plane
- Positive autocorrelation, negative auto correlation, zero autocorrelation

- “…Describing the autocorrelation structure, is of primary importance in spatial analysis.” (p. 29)
- First order, and second order spatial variation

The “Pitfalls” of Spatial Data it require)?

- The Modifiable Areal Unit Problem
- “…aggregation units used are arbitrary with respect to the phenomena under investigation”
- “If spatial units…were specified differently, we might observe very different patterns…” (p. 30)

- The Ecological Fallacy
- Rampant in media reporting

The “Pitfalls” of Spatial Data it require)?

- Scale Issues
- Examples

- Nonuniformity of Space and Edge Effects
- Space is not uniform
- Edge Effects?

The Potential of Spatial Data it require)?

- Quantification of Spatial Relationships
- How? What kind of relationships matter?

- Summarizing spatial relationships
- How?

Spatial data are the building blocks of any spatial analysis it require)?

- Spatial data structures:
- Raster: geographically-referenced matrix of uniform size cells…advantages and disadvantages
- Vector: features on the earth’s surface are represented as geographically-referenced vector objects (points, lines, polygons)…advantages and disadvantages

Representation of it require)?vector spatial objects

- Hierarchical nature of objects (points, lines, polygons)
- Points: different types
- Entity, label, area, node

- Lines:
- Line, arc, link, etc.

- Polygons:
- Area, polygon, complex polygon

- Points: different types

Basic elements of it require)?spatial information required to undertake spatial analysis

- Location
- X,Y coordinate or locational reference

- Attribute data
- Describing the (aspatial) characteristics of locations

- Topology
- Describing the spatial relationships between spatial features

Measurement of Location: it require)?GIS Issues

- A GIS suitable for spatial analysis must have the necessary functions dealing with coordinate systems
- What are these functions?

- What coordinate systems do we normally see or work with in a GIS…and what are their characteristics?

Measurement of Location: it require)?GIS Issues

- Basic measurement of spatial features:
- Points are defined by x,y coordinates
- Lines are represented by an ordered sequence of points…they can be “decomposed” into sections of straight line segments
- The distance between two points on a Cartesian plane is derived through Euclidean distance…the length of a line segment is the sum total of the Euclidean distances of all segments that compose it (p. 105 Chou)
- The area of any feature represented as a polygon an be computed by constructing a trapezoid from every line segment delineating the polygon…then systematically aggregating the trapezoid areas (both positive and negative) (p. 106 Chou)

Attribute Data Measurement it require)?

- Categories: Nominal and Ordinal data
- Numeric: Interval and ratio data
- Measures of Central Tendency (mode, median, mean) and Dispersion (variance, standard deviation)
- Must be cognizant of spatial units and geographic sampling techniques

Topology: What kinds of spatial relationships between spatial features?

- Adjacency: Which polygons are adjacent to which? Often used in the spatial analysis of areal data.
- Containment: Which spatial features are contained within which? Can be used for selection or perhaps geocoding.
- Connectivity: Which line segments are connected? Often used for network analysis.

The Arc-node Data Model: a method of expressing vector topology

- Used for ARC/INFO coverages (we will use this as our example)…a proprietary ESRI vector spatial data structure
- Topological data is stored in “attribute tables”: point attribute tables (PATs), arc attribute tables (AATs), polygon attribute tables (PATs)…what is contained in these tables?

Sample Attribute Tables topology

- Arc Attribute Tables (AATs) - contain the following data fields: arc-ID, Length, F-node, T-node, L-poly, R-poly
- Polygon Attribute Tables (PATs) – contain the following data fields: poly-ID, perimeter, area
- Point Attribute Tables (PATs) – the same fields as above, but zero perimeter and area
** These tables store the topological data needed to quantify the spatial relationships between features

Spatial Data Formats topology

- Spatial data formats are the product of the private sector working to create data files that allow users to:
- Create maps
- Manipulate spatial data
- Perform spatial analysis

- Example ESRI spatial data formats (files): shapefiles, coverages, GRIDs, geodatabases, TINs, Routes

3 Major vector-based topologydatasets used in ArcGIS: Shapefiles, Coverages, Geodatabases

- ESRI Shapefiles:
- Spatial data is stored in binary files
- Attribute data is stored in dBase tables
- Contain one simple feature class
- No topology is developed for spatial features
- Types of shapefiles: point, line, polygon and multi-point

3 Major vector-based topologydatasets used in ArcGIS: Shapefiles, Coverages, Geodatabases

- ESRI ARC/INFO Coverages:
- Spatial data is stored in binary files
- Topological and attribute tables are stored in INFO tables
- Contain topological features classes that define line or polygon topology
- Topology is “built” for lines and polygons - lines: arcs, nodes and routes; polygons: arcs, label points, polygons, regions
- Primary coverage feature classes are: point, arc, polygon, and node; secondary: tic, link, annotation; compound: region, route

ARC/INFO Coverages topology

- ARC coverage files: defined by header files, index files, ARC, PAL, LAB, CNT, PRJ, LOG, TOL
- ARC: arc definitions and vertices; PAL: contains polygon definitions; LAB: contains label point records; CNT: contains polygon centroid information; PRJ: contains projection information; TOL: contains the tolerance values to use when processing a polygon coverage

ESRI GRID file topology

- ESRI’s proprietary raster file structure
- Readable in ArcGIS without any extensions
- The Spatial Analyst extension needed to perform analysis on these files

- Follow conventions we have learned about:
- Uniform raster cell size
- Single value per cell
- Continuous data (including null values)

“Special” Spatial Data Structures: TINs and Routes topology

- Triangulated Irregular Networks (TINs): sample points are connected to form triangles, with the relief inside each represented as a plane or facet
- VIPs (Very Important Points)
- Delaunay Triangulation
- 3-dimensional surface description
- ArcGIS can generate these through the 3-D Analyst extension

“Special” Spatial Data Structures: TINs and Routes topology

- Routes are spatial data structures generated to represent linear features
- Used when the definition of linear features does not meet the needs of a network-based application
- Dynamic segmentation procedure
- New line segments are defined…
- Based on the location of “events”
- Measurements of offsets on segments

- Network Analyst and ARC/INFO

3 Major vector-based topologydatasets used in ArcGIS: Shapefiles, Coverages, Geodatabases

- ESRI Geodatabase
- All spatial, topological, and attribute data is stored in tables in a relational database
- A feature dataset in a geodatabase can contain simple or topological feature classes
- Many feature classes can be associated with a topological role within the geodatabase
- User-defined associations can be created between features in different feature classes
- Types of feature classes: point, line, polygon, annotation, simple junction, complex junction, simple edge, complex edge

The Geodatabase Data Model: “…a better way to associate behavior with [spatial] features was needed”

- An object-oriented data model: data “objects” can have rules, relationships, topology
- Facilitates the creation of “smart features” that are more complex than generic points, lines, or polygons
- All data is stored in a relational database ( as opposed to separate spatial and attribute data)

Centralized management of data behavior with

- Geodatabases organize data into a hierarchy of data objects: object classes, feature classes, feature datasets
- Object class: a table in a geodatabase that stores non-spatial data
- Feature class: a collection of features with the same type of geometry and the same attributes
- Feature dataset: a collection of feature classes that have the same spatial reference system
- “Simple” feature classes can exist either within or outside a feature dataset; topological feature classes must be contained within a feature dataset

Maps as Outcomes of Processes behavior with

- [Spatial] patterns provide clues to a possible causal [spatial] process(es)
- “…Usefulness of maps…remains in their inherent ability to suggest patterns in the phenomena they represent.” p. 52
- Conceptualizing spatial analysis as processes and patterns

Types of Processes: Spatial Processes and their Possible Realizations

- Could the pattern we observe have been generated by this particular process?
- Deterministic processes:
- Processes whose outcome can be predicted exactly from knowledge of initial conditions
- Many times can be mathematically described
- Outcome always the same

- Stochastic processes:
- Processes whose outcome is subject to variation that cannot be given precisely by a mathematical formula
- Introduction of a random (stochastic) element to model the range of potential solutions
- “Chance process with well-defined mechanisms” p. 58

Predicting Patterns: RealizationsExpected Results

- Assumptions
- Example: independent random process (IRP) (or complete spatial randomness (CSR))
- Math used to predict frequency distribution under assumed randomness
- Observed vs. expected
- What is this assumption called in the scientific method?

- Real World – usually not characterized by spatial randomness
- First-order effects: the earth is not an isotropic plane, and therefore some areas will be more attractive of phenomena than others
- Second-order effects: the assumption that events are independent of each other is not realistic…i.e. the location of events will influence the location of other events

Point Pattern Analysis Realizations

- The spatial properties of the entire set of points is analyzed (rather than individual points)
- Requirements/Assumptions according to O’Sullivan and Unwin (pp.78-79)?
- Descriptive statistics for point distributions
- Frequency; density; geometric center; spatial dispersion; spatial arrangement

Point Pattern Analysis Realizations

- Thinking about point patterns…
- How can we describe and analyze them

- The geographical properties of a point pattern are characterized (described) by geometric center and dispersion
- Geometric (mean) center = mean x,y coordinates; dispersion = standard distance of x and y distribution
- Geometric (mean) center is not a reliable measure of central tendency when either the x or y standard distance is large

- What are these measures useful for?

Point Pattern Analysis Realizations

- Density-based and distance-based measures
- i.e. Point Density and Point Separation

- Density: ratio of frequency to area…intensity of a pattern
- depending on distribution within a defined study area may be misleading (pp. 81-82)

- Quadrat Count Methods
- Census or random methods
- Issues?

Density-based measures Realizations

- Quadrat Analysis – based on the frequency of occurrence of points within quadrat units
- Requires overlaying quadrats onto a layer of point features
- Once quadrats are overlayed onto the point layer, frequencies of points per quadrat can be counted
- All quadrats are classified according to observed frequency of points
- Null hypothesis: point features are randomly distributed

Density-based Measures Realizations

- Kernel Density Estimation
- A pattern has a density at any location…
- Continous densities for defined “kernels” to create a continous surface

Distance-based RealizationsPoint Pattern Measures

- The Logic of Distance Measures
- Can be described using types (categories):
- Clustered – points are concentrated in one or more groups/areas
- Uniform – points are regularly spaced with relatively large interpoint distance
- Random – Neither the clustered or uniform pattern is prevalent

- Can be described using types (categories):

Measuring Spatial Arrangement Realizations

- Nearest Neighbor Analysis (Index)
- Measures the degree of spatial dispersion in a point distribution based on minimizing interpoint distances
- Logic: in general the average distance between points in a clustered pattern is less than in a uniform pattern\
- Logic: a random pattern is associated with an avg. interpoint distance larger than a clustered pattern but smaller than a uniform pattern
- The “nearest neighbor” for each point feature must be determined, and the interpoint distance is computed

Measuring Spatial Arrangement Realizations

- Nearest Neighbor Analysis (Index) con’t
- Observed average nearest neighbor distances compared to expected average nearest distances assuming complete spatial randomness [CSR] (1/2 sq.rt. A/n)
- NNI = Ad/Ed p.100
- NNI range: 0 to 2.1491…where 0 indicates perfectly clustered and 2.1491 indicates perfectly uniform (values close to 1 indicate a random pattern)
- To test the statistical significance of an NNI value, a computed z value can be compared to a critical value (1.96)

Measuring Spatial Arrangement Realizations

- Nearest Neighbor Analysis: Pros and Cons
- Pros: relatively simple; easy to compute; straightforward logic
- Cons: is not sensitive to complex patterns unless extended to include more than just nearest neighbors

The Concept of Spatial Autocorrelation Realizations

- Spatial Autocorrelation: measures the extent to which the occurrence of one feature is influenced by the distribution of similar features in the adjacent area Why is this idea important in the context of “classical” statistical analysis?
- Captures some aspects of point spatial distribution not reported by NNI or quadrat analysis
- Spatial auto correlation is characterized as positive (the existence of one feature tends to attract similar features) or negative (the existence of one feature tends to deter the location of similar features)

Types of Area Objects Realizations

- Natural Areas vs. Command Regions
- Who cares?

- Issues with Command Regions?
- Raster
- Pros and Cons?

- Planar-enforced areas…
- GIS-context?

Geometric Properties of Areas Realizations

- Area
- How is it calculated?

- Shape
- Comparison of a polygon to a known shape

- Spatial pattern
- Contact numbers
- Fragmentation (FRAGSTATS)

Spatial Autocorrelation Realizations

- Most common spatial autocorrelation statistic is Moran’s I coefficient
- Similar to a traditional correlation coefficient
- The I coefficient for the most part ranges between –1 and +1; larger negative values indicate a scattered pattern…positive values indicate a clustered pattern

- Also Geary’s C (Geary’s Ratio)
- The C coefficient tends to range between 0 and 2; values approaching 0 imply similar values of a variable tend to cluster (positive spatial autocorrelation)…values approaching 2 indicate that dissimilar values tend to cluster

Spatial Autocorrelation Realizations

- Joins Count approach
- Logic?

The Concept of Fields Realizations

- “…phenomena are continously variable and measureable across space.” (p. 210)
- Scalar fields: All locations are represented by a value…one value per unit
- Vector fields: values are not independent of coordinates (magnitude and direction)

Describing and Analyzing Fields Realizations

- Two steps in the recording and storage process of fields (p. 213):
- Sampling the “real” surface
- The input data

- Interpolation to derive a continuous surface representation
- Types of fields and how they are derived

- Sampling the “real” surface

Sampling the earth’s surface Realizations

- Issues to consider:
- The methodology used to obtain the sample
- How would we find out?

- The spatial resolution of the sample
- In may cases, we may be “stuck” with scalar field sample data…Why?

- The methodology used to obtain the sample

Continuous Surface Description Realizations

- Types of Fields (pp. 214-220):
- Point Systems
- Grid sampling (raster) , surface specific, surface random

- Triangulated Irregular Networks (TINs)
- Contours
- Mathematical Functions

- Point Systems
- Data may need to be processed further to derive “usable” fields…interpolation

Continuous Surface Description: The Raster Data Structure Realizations

- A cell (grid) data structure
- Row, Column coordinates (all positive values)
- Uniform cell size
- Every cell is assigned a value
- Numeric (integer or floating point)
- Categorical (usually in effect integer)

Continuous Surface Description: The Raster Data Structure Realizations

- Cell Value Assignment:
- Centroid Method
- Predominant Type
- Most Important Type
- Hierarchical

- ** In many cases, the data you are working with may already have cell values assigned

Example Continuous Surface RealizationsDescription: DEMs

- Digital Elevation Models (DEMs): a sample of elevation data for a study area represented as evenly-spaced points or raster cells
- Data from a DEMs is often used in land surface analysis, as they are free and data quality can be ascertained
- In most GIS packages, DEMs are converted to a raster format prior to analysis

Derived Measures on Surfaces: Raster Data Processing Realizations

- Local Operations: raster layer is processed on a cell-by-cell basis
- Single layer
- Multiple layer (raster overlay)
- Examples…

Derived Measures on Surfaces: Raster Data Analysis Realizations

- Neighborhood Spatial Operations: cell data is processed based on a focal cell and its neighboring cells
- Neighboring cells become part of an operation based on a distance and/or directional relationship to the focus cell
- Focus cell is usually assigned a value based on the values of neighboring cells
- Common neighborhoods: 3x3 “window”; circle;
- Operations: sum, mean, standard deviation, minimum, maximum
- Examples…

Derived Measures on Surfaces: Raster Data Processing Realizations

- Zonal Operations: apply to groups of cells that belong to the same “zone” or have a common value
- Single layer: geometry of zones (perimeter, area, centroid, etc.)
- Multiple layers (overlay): one layer defines the zones, the other defines variables values…summary statistics are calculated by zone (mean, standard deviation, area, min, max.)
- Examples…

Derived Measures on Surfaces: Raster Data Analysis Realizations

- Global (Distance Measure) Operations: the output value of each cell is calculated based on spatial relationship to a “source” cell
- Distance measurement in a raster layer is based on nodes and links
- Node = centroid
- Link = lateral (1 cell) or diagonal (1.4142 cells) connections to adjacent cells

- Euclidean, Physical (buffer), and Cost Distance Measurement

- Distance measurement in a raster layer is based on nodes and links

Derived Measures on Surfaces: Surface Analysis Realizations

- Involves analyzing a phenomena that is 3-dimensional…the 3rd dimension can be represented as a “z-coordinate” (in addition to x,y coordinates)
- The z-coordinate (or value) can represent almost anything, although it is most often employed to model topography

Derived Measures on Surfaces: Surface Analysis Realizations

- Data Types for Surface Analysis
- Irregularly-spaced point features
- Regularly-spaced cells in a raster layer (for example, DEMs)
- Vector contour lines
- Triangulated Irregular Networks (TINs)

Derived Measures on Surfaces: Surface Analysis Realizations

- Triangulated Irregular Networks (TINs): approximate a 3-dimensional surface using a series of non-overlapping triangles
- Based on an irregular distribution of points that have x,y, and z coordinates
- Sample points are used to generate triangles using either the VIP or max z-tolerance algorithm
- Triangles are generated using rules of Delaunay Triangulation…all nodes are connected to their nearest neighbors, and triangles are as equi-angular as possible
- Triangles have area and angles associated with them

Derived Measures on Surfaces: Surface Analysis Realizations

- Slope and Aspect: Calculated by determining the amount and direction of tilt of a cell’s normal vector
- Surface Curvature: Used to determine if the surface at a cell location is upwardly convex or concave
- Viewshed Analysis: Determining what areas are visible and not visible from a vantage point
- Watershed Analysis: Watershed delineation and drainage characterization based on elevation data

Spatial Interpolation Realizations

- Control points are points with known values…it is best if there is “good coverage” of control points (how often does this happen?)
- Assumptions:
- 1. The surface of the Z variable is continuous
- 2. The Z variable is spatially dependent

Types of Spatial Interpolation Realizations

- Global vs. Local
- The difference is the number of control points used

- Exact vs. Inexact
- How control point values are used and “re-estimated”

- Deterministic vs. Stochastic
- Assessment of prediction errors (with estimated variances)

Simple Spatial RealizationsInterpolation Techniques

- Local Methods: The z value of an unknown point location is estimated from known local point neighbor locations
- Interpolation procedures are used when we have discontinuous datasets and we want (or need) to process them into spatially continuous datasets

“Simple” Deterministic Spatial RealizationsInterpolation Techniques

- Usually used to derive field datasets for further processing:
- Inverse Distance Weighted Spatial Average
- Proximity polygons
- Local Spatial Averaging
- Other Methods

Statistical Spatial Interpolation Realizations

- A process of using locations with known data values to estimate values at other locations.
- Global (Statistical) Methods: Use all available data (control points) to perform estimation

- A “statistical surface” is constructed by interpolating unknown values from known values

Spatial Interpolation Realizations

- Global (Statistical) Methods: The z value of an unknown point location is estimated from all known point data
- Polynomial Trend Surface Analysis (Inexact, Deterministic): approximates points with known values with a polynomial equation
- The equation is used as an “interpolator” to estimate values at other points
- Computed by the least squares method and a “goodness of fit” can be computed for each control point

Spatial Interpolation Realizations

- Local Methods
- Inverse Distance Weighted (Exact, Deterministic): enforces that the estimated value of a point is influenced more by nearby known points than those farther away
- All predicted values are within the range of the maximum and minimum values in the distribution

- Inverse Distance Weighted (Exact, Deterministic): enforces that the estimated value of a point is influenced more by nearby known points than those farther away

Spatial Interpolation Realizations

- Local Methods
- Splines (Exact, Deterministic): create a surface that passes through the control points and has the least possible change in slope at all points (minimum curvature surface)

Spatial Interpolation Realizations

- Local Methods
- Kriging (Exact, Stochastic): a geostatistical method for spatial interpolation where the mean is estimated from the best linear unbiased estimator or best linear weighted moving average
- Assumes that the spatial variation of an attribute is neither totally random nor totally deterministic (a correlated component, a drift, a random error term)

- Kriging (Exact, Stochastic): a geostatistical method for spatial interpolation where the mean is estimated from the best linear unbiased estimator or best linear weighted moving average

How do we Accomplish Spatial Interpolation in ArcGIS? Realizations

- Geostatistical Analyst:
- An ArcGIS extension that provides tools to perform statistically-based spatial interpolation
- Exploratory Data Analysis
- Calculation and Modeling of Surface Properties (Structural Analysis)
- Surface Prediction and Assessment of Results

- An ArcGIS extension that provides tools to perform statistically-based spatial interpolation

Knowing the Unknowable: RealizationsThe Statistics of Fields

- Statistical spatial interpolation techniques…why are they necessary or advantageous? (p. 246-247)
- Control point data has error and varies over time…we are not going to obtain an exact fit from deterministic methods
- If we have sample datasets, we have data pertaining to the spatial distribution of phenomena that can be used in spatial interpolation
- We try to “fit” a mathematical model or function to the semivariogram (Gaussian, linear, spherical, circular, exponential) to be used as an interpolator

“Geostatistical” Spatial Interpolation Realizations

- Kriging: Assumes that the estimation of surface variations is based on the assumption that the surface can be represented by 3 factors:
- The residual of local fluctuation…the level of spatial correlation locally estimated from a polynomial function
- The drift of regional tendency…representing a spatial “trend”
- A random error estimate
- There are different variations of kriging, based on the the presence or absence of a “drift” factor and the interpretation

Spatial Interpolation Realizations

- Types of Kriging:
- Ordinary:
- the drift component is excluded
- Focus on the degree of spatial dependence among sampled known points (semivariance)
- Semivariance =
- Semivariance values are plotted on a semivariogram where the semivariance is recorded on the Y-axis and the distance between known points on the X-axis (nugget, range, sill)
- The semivariogram is fitted to a mathematical model (sherical, circular, exponential, linear, Gaussian)
- Equation for estimating Z:

- Ordinary:

Spatial Interpolation Realizations

- Types of Kriging:
- Universal Kriging: assumes that the spatial variation in z values has a “drift” or “trend” in addition to the spatial correlation between known points
- Co-Kriging: Can be used to improve spatial predictions by incorporating secondary variables, provided they are spatially correlated with the primary variable

Semivariogram Realizations

Covariance

Co-Kriging using multiple variables Realizations

Concept of Cross-correlation

Isotropic vs. Anisotropic Interpolation Techniques Realizations

Single Layer Operations Realizations

- We might consider these operations the “simplest” form of spatial analysis; although this might not always be true
- Single layer (horizontal) operations: procedures that apply to only one data layer at a time
- We are conceptualizing things in this way to simplify our understanding of what analysis operations do…not because this is really how we utilize the operations

- Operations that apply to a single feature type
- Does this change with the geodatabase?

Single Layer Operations Realizations

- Feature Identification and Selection
- Identify, Select Feature, Attribute Query

- Feature Classification
- What type of distribution, how do we determine? Uniform (equal interval, equal frequency); Normal (standard deviation); Multiple Cluster (natural breaks)

Single Layer Operations Realizations

- Feature Manipulation
- Boundary Operations
- That ArcView can perform: Clip, Dissolve, Append?
- That ArcView cannot perform (ARC/INFO required): Erase, Update, Split, Mapjoin, Eliminate

- Proximity Analysis
- ArcView: Buffer
- ArcView cannot: Thiessen polygons

- Boundary Operations

Map Overlay Realizations(Multiple Layer) Operations

- “…arguably, the most important feature of any GIS is its ability to combine spatial datasets…” (p. 285)
- 10 Possible types of Map Overlay

Map Overlay Operations Realizations

- Polygon Overlay operations
- Simplest Form: Boolean Overlay (Sieve mapping)

- 4 Steps (pp. 288-302):
- Determining the Inputs
- Getting the Data
- Getting the Spatial Data into the Same Coordinate System
- Overlaying the Maps

Map Overlay Operations Realizations

- Overlay Operations (in ArcGIS)
- Union
- Intersect
- Identity
- Results?

Erase (Coverage) Realizations

Identity Overlay Realizations

Intersect Overlay Realizations

Symmetrical Difference Realizations

Union Overlay Realizations

Update Overlay Realizations

Spatial Modeling Realizations

- According to Chou (1997), a Spatial Model:
- 1. Analyzes phenomena by identifying explanatory variables that are significant to the distribution of the phenomenon and providing information about the relative weight of each variable
- 2. Is useful for predicting the probable impact of a potential change in “control” factors (independent variables)

Spatial Modeling: RealizationsThinking About Models

- Models can be:
- Descriptive or Prescriptive
- Deterministic or Stochastic
- Static or Dynamic
- Deductive or Inductive

Spatial Modeling Realizations

- General Types of (Spatial) Models
- Descriptive: characterization of the distribution of spatial phenomena
- Explanatory: deal with the variables impacting the distribution of a phenomena
- Predictive: once explanatory variables are identified, predictive models can be constructed
- Normative: models that provide optimal solutions to problems with quantifiable objective functions and constraints

Spatial Modeling Realizations

- More specific types of spatial models:
- Binary models (descriptive): use logical expressions to identify or select map features that do or do not meet certain criteria…How?
- Index models (descriptive): use index values calculated for variables to produce a ranked spatial surface…How?
- Weighted Linear Combination Model

- Regression models (explanatory or predictive): a dependent variable is related or explained by independent variables in an equation…How?
- Linear and logistic regression

- Process (explanatory or predictive): integrate existing knowledge about environmental processes into a set of relationships and equations for quantifying those processes…How?

Spatial Modeling Realizations

- Steps in the Modeling Process
- Define the goals of the model
- Break down the model into elements
- Implementation and calibration of the model
- Model validation
- Sometimes difficult or not feasible

The Role of GIS in Spatial Modeling Realizations

- How can GIS enable spatial modeling?
- GIS is a tool that can integrate a myriad of data sources
- GIS can incorporate raster and/or vector data into modeling schemes
- Modeling may take place within a GIS, or require linking to other computer programs
- Loose coupling
- Tight coupling
- Embedded System

Spatial Modeling Realizations

- Important Issues in Conducting Spatial Analysis:
- Delineation of geographic units of analysis
- How do you choose geographic units of analysis so that spatial analyses are valid?

- Identification of structural and spatial factors that impact spatial analysis
- Structural – impact site
- Spatial – impact situation (absolute and relative location, neighborhood effects)

- Delineation of geographic units of analysis

Stormwater modeling Realizationsproject logic

Based on TR-55

- First issued by the US SCS in 1975, today Natural Resource Conservation Service (NRCS)
- Presents simplified procedures for addressing stormwater during initial overland flow (runoff, peak discharge, hydrographs, and storage volumes for detention ponds)

Stormwater modeling Realizationsproject logic

TR-55

- Stormwater runoff calculation
- based on Runoff Curve Number (CN) method

- CN - empirically derived number
- Product of hydrologic soil group, cover type, treatment, hydrologic condition, and antecedent runoff condition

- Also – Percent impervious surface

Network Analysis Realizations

- Network analysis: the spatial analysis of linear (line) features
- Your text distinguishes between several different types of lines

- Network analysis involves 2 types of problems:
- analyzing structure (connectivity pattern) of networks
- analyzing movement (flow) over the network system

- Network analysis is often a major part of subfields that are related to transportation: transportation geography, transportation planning, civil engineering, etc.

Linear Regression Models: Logic and Assumptions Realizations

- Assumptions (predicted vs. actual values):
- Errors have the expected mean value of zero
- Errors are independent of each other
- Correlations among independent variables should not be high

Network Analysis Realizations

- Concepts:
- Network
- Line segment(s)/Links
- Nodes (and vertices)
- Impedance
- Topology
- Dynamic Segmentation

Network Analysis: RealizationsNetwork Structure

- Evaluation of Network Structure:
- Index: the ratio of the actual number of links to the maximum possible number of links 3(n-2) (n = # of nodes)…range between 0-1
- Index: the ratio of the actual number of circuits to the maximum number of circuits (c/(2n-5))…evaluation in terms of the number of ways to get from one node to another

Network Analysis: RealizationsNetwork Structure

- Network Diameter: the maximum number of steps required to move from any node to any other node using shortest possible routes over as connected network
- Network Connectivity: an evaluation of nodal connectivity over a network based on direct and indirect connections (expressed through the construction of matrices c1, c2, c3)

Network Analysis: RealizationsNetwork Structure

- Network Accessibility: can be evaluated based on nodes or the entire network…the accessibility network is many times called the T matrix
- T matrix is the sum of all connectivity matrices up to the level equal to the network diameter (i.e. c3 or c4)
- Logically this makes sense if you are trying to evaluate total connectivity of a node or the entire network
- How do we read the matrix?

Network Analysis: RealizationsNetwork Structure

- Network Structure in a Valued Graph
- The previously discussed measures of network structure are based on either counting links and/or nodes….what element are we missing with these?
- Q. What is a valued graph? A. A matrix is constructed in which every link (line segment) in a network is coded with an impedance measure (such as what?)
- An often-used type of valued graph is the minimal spanning tree…satisfies 3 criteria:
- Can a GIS construct a minimal spanning tree?

Network Analysis: RealizationsNormative Models of Network Flow

- Normative models are those that are designed to determine a best or optimal solution based on specific criteria
- Simple Shortest Path Algorithm:
- Involves finding the “path” or route with the minimum cumulative impedance between nodes on a network
- Requires an impedance matrix (such as a valued graph) and a set of interative procedures:
- GIS must know which nodes are connected to which…multi-step evaluation of connectivity and least cumulative impedance (distance, time, cost, etc.)

Network Analysis: RealizationsNormative Models of Network Flow

- The Traveling Salesman Problem:
- 2 “constraints” – 1) the salesman must stop at each location once 2) the salesman must return to the origin of travel (there can be variations)
- The objective is to determine the path or route that the “salesman” can take to minimize the total impedance value of the trip
- Often a heuristic method is used…beginning with an initial random tour, a series of locally optimal solutions is run by swapping stops that cause a reduction in cumulative impedance (an iterative procedure is also described in your book on pp. 236-244).

Network Analysis: RealizationsNormative Models of Network Flow

- Various Types of Network Problems:
- Shortest Path Analysis (Best Route)
- Simple shortest path
- Traveling Salesman
- Closest Facility

- Allocation (Define Service Area)
- Location-Allocation: solves problems matching supply and demand by using sets of objectives and constraints
- P-median, max covering, max equity

- Shortest Path Analysis (Best Route)

Network Analysis: RealizationsNormative Models of Network Flow

- Dynamic Segmentation Data Model: The ability to derive the locations of events in relation to linear features dynamically…not reliant upon the existing topology of a network
- Models linear features using routes and events…
- Routes: represent dynamic linear features
- Events: phenomena that occur at locations along line segments

- Dynamic segmentation is used to operationalize network analysis in ArcInfo/ArcGIS

Spatial Interpolation Realizations

With Anisotopy Realizations

Mean= .01694

RMS = 2.862

Avg. Stan Error = 3.441

Mean Stan. = .004232

RMS Stan. = .8324

Without Anisotopy

Mean= .0002331

RMS = 2.857

Avg. Stan Error = 3.424

Mean Stan. = .0006747

RMS Stan. = .8347

Ordinary Kriging ComparisonWith Anisotopy Realizations

Mean= .04253

RMS = 2.595

Avg. Stan Error = 2.354

Mean Stan. = .01806

RMS Stan. = 1.102

Without Anisotopy

Mean= .0001592

RMS = 3.054

Avg. Stan Error = .8181

Mean Stan. = .001031

RMS Stan. = 3.731

Universal Kriging ComparisonRegression Equations Realizations

- TWOYR = -3.538 + 0.06031 * AVGCURV + 0.03331 * PERCIMPV
- TENYR = -4.156 + 0.07806 * AVGCURV + 0.04368 * PERCIMPV

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