Spatial Analysis cont.Density Estimation, Summary Spatial

Statistics, Routing Longley et al., chs. 13,14

Density Estimation

• point interpolation to estimate a continuous surface

vs.• density estimation - surface is estimated

from counts within polygons– (e.g., population density surface derived from total

population counts in each reporting zone)

Objects to Fields

• map of discrete objects and want to calculate their density– density of population

– density of cases of a disease

– density of roads in an area

• density would form a field• one way of creating a field from a set of

discrete objects

Density Estimation and Potential

• Spatial interpolation is used to fill the gaps in a field

• Density estimation creates a field from discrete objects– the field’s value at any point is an estimate of the density

of discrete objects at that point

– e.g., estimating a map of population density (a field) from a map of individual people (discrete objects)

Density Estimation Using Kernels

• Mathematical function• each point replaced by a “pile of sand” of

constant shape• add the piles to create a surface

Width of Kernel

• Determines smoothness of surface – narrow kernels

produce bumpy surfaces

– wide kernels produce smooth surfaces

Example

• Density estimation and spatial interpolation applied to the same data

• density of ozone measuring stations

vs.• Interpolating surface based on locations

of ozone measuring stations

Using Spatial Analyst

Kernal too small?(radius of 16 km)

Kernel radius of 150 km

What’s the Difference?

Summary Spatial Statistics

Longley et al., chs. 5,14

Descriptive Summaries

• Ways of capturing the properties of data sets in simple summaries

• mean of attributes• mean for spatial coordinates, e.g.,

centroid

Spatial Min, Max, Average

An example of the use of centroids to summarize the changes in point patterns through time. The centroids of four land use classes are shown for London, Ontario, Canada

from 1850 to 1960. Circles show the associated dispersions of sites within each class. Note how the industrial class has moved east, remaining concentrated, while the

commercial class has remained concentrated in the core, and the residential class has dispersed but remained centered on the core. In contrast the institutional class moved

to a center in the northern part of the city.

Spatial Autocorrelation

Arrangements of dark and light colored cells exhibiting negative, zero, and positive spatial autocorrelation.

Tobler’s 1st Law of Geography: everything is related to everything else, but near things are more related than distant things

S. autocorrelation: formal property that measures the degree to which near and distant things are related.

Close in space

Dissimilar in attributes

Attributes independent

of location

Close in space

Similar in attributes

Why Spatial Dependence?

• evaluate the amount of clustering or randomness in a pattern– e.g., of disease rates, accident rates, wealth,

ethnicity

• random: causative factors operate at scales finer than “reporting zones”

• clustered: causative factors operate at scales coarser than “reporting zones”

Moran’s Index• positive when attributes of nearby objects

are more similar than expected• 0 when arrangements are random• negative when attributes of nearby objects

are less similar than expected

I = n wijcij / wij (zi - zavg)2

n = number of objects in sample

i,j - any 2 of the objects

Z = value of attribute for I

cij = similarity of i and j attributes

wij= similarity of i and j locations

Moran’s Indexsimilarity of attributes, similarity of

locationExtreme negative SA Dispersed, - SA

Independent, 0 SA

Spatial Clustering, + SA Extreme positive SA

Crime Mapping• Clustering - neighborhood scale

Geary’s c Ratio

• Like Moran’s Index, use a single value to describe spatial distribution – e.g., of elevations in DEM cells

less than 1 (clustered)

1

greater than 1 (random)• e.g., spatial autocorrelation indicator of

information loss during conversions between DEMs and TINs

Moran’s and Geary’s

Lee and Marion, 1994, Analysis of spatial autocorrelation of USGS 1:250,000 DEMs. GIS/LIS Proceedings.

Fragmentation Statistics

• how fragmented is the pattern of areas and attributes?

• are areas small or large? • how contorted are their boundaries? • what impact does this have on habitat,

species, conservation in general?

Note the increasing fragmentation of the natural habitat as a result of

settlement. Such fragmentation can adversely affect the success

of wildlife populations.

19861975

1992

Fragstats pattern analysis for landscape

ecologyhttp://www.innovativegis.com/products/fragstatsarc/

FRAGSTATS Overview

• derives a comprehensive set of useful landscape metrics

• Public domain code developed by Kevin McGarigal and Barbara Marks under U.S.F.S. funding

• Exists as two separate programs – AML version for ARC/INFO vector data

– C version for raster data

FRAGSTATS Fundamentals

• PATCH… individual parcel (Polygon)

A single homogeneous landscape unitwith consistent vegetation characteristics,e.g. dominant species, avg. tree height,horizontal density ,etc.

A single Mixed Wood polygon(stand)

CLASS… sets of similar parcelsLANDSCAPE… all parcels within an area

FRAGSTATS FundamentalsPATCH… individual parcel (Polygon)

• CLASS… sets of similar parcels

All Mixed Wood polygons(stands)

LANDSCAPE… all parcels within an area

FRAGSTATS FundamentalsPATCH… individual parcel (Polygon)CLASS… sets of similar parcels

• LANDSCAPE… all parcels within an area “of interacting ecosystems”

e.g., all polygons within a given geographic area (landscape mosaic)

FRAGSTATS Output Metrics

• Area Metrics (6),

• Patch Density, Size and Variability Metrics (5),

• Edge Metrics (8),

• Shape Metrics (8),

• Core Area Metrics (15),

• Nearest Neighbor Metrics (6),

• Diversity Metrics (9),

• Contagion and Interspersion Metrics (2)

• …59 individual indices(US Forest Service 1995 Report PNW-GTR-351)

More Spatial Statistics Resources

• Spacestat (www.spacestat.com)

• S-Plus• Alaska USGS freeware

(www.absc.usgs.gov/glba/gistools/)

• Central Server for GIS & Spatial Statistics on the Internet

– www.ai-geostats.org• GEO 441/541 - Spatial Variation in

Ecology & Earth Science

Location-allocation Problems

• Design locations for services, and allocate demand to them, to achieve specified goals

• Goals might include:– minimizing total distance traveled

– minimizing the largest distance traveled by any customer

– maximizing profit

– minimizing a combination of travel distance and facility operating cost

Routing Problems

• Search for optimum routes among several destinations

• The traveling salesman problem– find the shortest tour from an origin, through a

set of destinations, and back to the origin

Routing service technicians for Schindler Elevator. Every day this company’s service crews must visit a different set of locations in Los Angeles. GIS is used to

partition the day’s workload among the crews and trucks (color coding) and to optimize the route to minimize time and cost.

Optimum Paths

• Find the best path across a continuous cost surface– between defined origin and destination

– to minimize total cost

– cost may combine construction, environmental impact, land acquisition, and operating cost

– used to locate highways, power lines, pipelines

– requires a raster representation

Solution of a least-cost path problem. The white line represents the optimum

solution, or path of least total cost, across a friction surface

represented as a raster. The area is dominated by a mountain

range, and cost is determined by elevation and slope. The best

route uses a narrow pass through the range. The blue line results from solving the same

problem using a coarser raster.