Class 9. Query, Measurement & Transformation; Spatial Buffers; Descriptive Summary, Design & Inference

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1 Class 9 Query, Measurement & Transformation; Spatial Buffers; Descriptive Summary, Design & Inference

2 Spatial Analysis Turns raw data into useful information by adding greater informative content and value Reveals patterns, trends, and anomalies that might otherwise be missed Provides a check on human intuition by helping in situations where the eye might deceive

3 Types of Spatial Analyses There are literally thousands of techniques Six categories are used in this course, each having a distinct conceptual basis: Queries and reasoning Measurements Transformations Descriptive summaries Optimization Hypothesis testing

4 Queries and Reasoning A GIS can respond to queries by presenting data in appropriate views and allowing the user to interact with each view It is often useful to be able to display two or more views at once and to link them together linking views is one important technique of exploratory spatial data analysis

5 The Catalog View Shows folders, databases, and files on the left, and a preview of the contents of a selected data set on the right. The preview can be used to query the data set s metadata, or to look at a thumbnail map, or at a table of attributes. This example shows ESRI s ArcCatalog.

6 The Map View A user can interact with a map view to identify objects and query their attributes, to search for objects meeting specified criteria, or to find the coordinates of objects. This illustration uses ESRI s ArcMap.

7 The Table View Here attributes are displayed in the form of a table, linked to a map view. When objects are selected in the table, they are automatically highlighted in the map view, and vice versa. The table view can be used to answer simple queries about objects and their attributes.

8 Measurements Many tasks require measurement from maps measurement of distance between two points measurement of area, e.g. the area of a parcel of land Such measurements are tedious and inaccurate if made by hand measurement using GIS tools and digital databases is fast, reliable, and accurate

9 Measurement of Length A metric is a rule for determining distance from coordinates The Pythagorean metric gives the straight-line distance between two points on a flat plane The Great Circle metric gives the shortest distance between two points on a spherical globe given their latitudes and longitudes

10 Issues with Length Measurement The length of a true curve is almost always longer than the length of its polyline or polygon representation

11 Issues with Length Measurement Measurements in GIS are often made on horizontal projections of objects length and area may be substantially lower than on a true three-dimensional surface

12 Measurement of Shape Shape measures capture the degree of contortedness of areas, relative to the most compact circular shape by comparing perimeter to the square root of area normalized so that the shape of a circle is 1 the more contorted the area, the higher the shape measure Pattern metrics landscape, class, patch

13 Shape as an Indicator of Gerrymandering in Elections The 12 th Congressional District of North Carolina was drawn in 1992 using a GIS, and designed to be a majorityminority district: with a majority of African American voters, it could be expected to return an African American to Congress. This objective was achieved at the cost of a very contorted shape. The U.S. Supreme Court eventually rejected the design.

14 Slope Angle & Slope Aspect Calculated from a grid of elevations (a digital elevation model) Slope and aspect are calculated at each point in the grid, by comparing the point s elevation to that of its neighbors usually its eight neighbors but the exact method varies in a scientific study, it is important to know exactly what method is used when calculating slope, and exactly how slope is defined

15 Transformations Create new objects and attributes, based on simple rules involving geometric construction or calculation may also create new fields, from existing fields or from discrete objects

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23 Buffering (Dilation) Create a new object consisting of areas within a user-defined distance of an existing object e.g., to determine areas impacted by a proposed highway e.g., to determine the service area of a proposed hospital Feasible in either raster or vector mode

24 Buffering Polyline Point Polygon

25 Raster Buffering Vary the distance buffered according to values in a friction layer City limits Areas reachable in 5 minutes Areas reachable in 10 minutes Other areas

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30 Point-in-Polygon Transformation Determine whether a point lies inside or outside a polygon generalization: assign many points to containing polygons used to assign crimes to police precincts, voters to voting districts, accidents to reporting counties

31 Polygon Overlay Two complete layers of polygons are input, representing two classifications of the same area e.g., soil type and land ownership The layers are overlaid, and all intersections are computed creating a new layer each polygon in the new layer has both a soil type and a land ownership the attributes are said to be concatenated The task is often performed in raster

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46 Spatial Interpolation Specifying the likely distance decay linear: w ij = -b d ij negative power: w ij = d ij -b negative exponential: w ij = e -bdij Isotropic and regular relevance to all geographic phenomena? Inductive vs. deductive approaches

47 Spatial interpolation: Spatial Interpolation - estimate continuous (or discretized continuous) surfaces from data at known points Why do this? we have sparse data samples that do not completely cover a study area, and we want complete coverage usually done to represent phenomena that are continuous in reality Examples: turn spot/point samples of elevation heights into a DEM turn temperature samples from weather stations into a temperature surface

48 Interpolation Characteristics Exact vs. inexact interpolators exact: at sample points, the interpolation returns the same value as the original sample inexact: do not return same value at sample points (usually optimize some other characteristic, e.g. minimze squared deviations of interpolated values from sampled values) Global vs. local interpolators global: utilize all sample points to create interpolated surface local: utilize subset of sample points in neighborhood of each location being estimated to create interpolated surface All spatial interpolation is based on a simple (but powerful) observation: Everything in the universe is related to everything else, but closer things are more related. (Tobler s Law)

49 Thiessen (Voronoi) Polygons exact, local probably simplest (and crudest) method of interpolation method: assign value of nearest sample point sample point Do this in ArcView: Analysis Assign Proximity (Spatial Analyst extension) many obvious drawbacks including what spatial phenomena does this approximate? discontinuities at polygon edges

50 local, exact Inverse Distance Weighting basic idea: estimates value at location in output surface by weighting the relative influence of input points in the local area by some inverse function of their distance from the location being estimated issues: effect of distance decay exponent choice (number, selection) of input points used for estimation

51 Spatial Interpolation Values of a field have been measured at a number of sample points There is a need to estimate the complete field to estimate values at points where the field was not measured to create a contour map by drawing isolines between the data points Methods of spatial interpolation are designed to solve this problem

52 Inverse Distance Weighting (IDW) The unknown value of a field at a point is estimated by taking an average over the known values weighting each known value by its distance from the point, giving greatest weight to the nearest points an implementation of Tobler s Law

53 point i known value z i location x i weight w i distance d i unknown value (to be interpolated) location x z (x) = i w z i i i w i The estimate is a weighted average w = i 2 1 d i Weights decline with distance

54 Issues with IDW The range of interpolated values cannot exceed the range of observed values it is important to position sample points to include the extremes of the field this can be very difficult

55 Spatial Pattern & Quadrat Counts

56 Kriging: Spatial Interpolation

57 Quadrat Counts & Nearest Neighbor Analysis Spatial pattern can be described as regular or irregular and further subdivided into clustered, random, or anti-clustered. Application is to discern the locational characteristics of the points rather than their attributes.

58 Quadrat Counts Assessing spatial pattern using a consistently sized and shaped sample unit (e.g., rectangle, square, circle); Poisson distribution model is used in which expected (mean number of points/quadrat) vs. observed patterns are considered; Chi-square analysis is used. Size of quadrat affects Chi-square results. Diffusion of innovation or disease, commercial store patterns in urban areas, sightings of birds, dispersal of seeds.

59 Nearest Neighbor Analysis Measures distances between sample points and their nearest neighbors; mean of the NN distance (observed) is compared to the expected mean distance. Distance may be sample location to nearest point, or point to nearest point.

60 Trend Surface Analysis Data exploratory technique. Uses a general linear model or a 2 nd (linear and quadratic) or 3 rd order (powers and cross-products). Mapping residuals from the fitted trend surface may shed light on the underlying spatial structure of the data.

61 Gravity Models A popular spatial interaction model Force of attraction of two bodies (gravity) is proportional to the product of their masses, but inversely proportional to the squared distance separating them. Interactions between cities of two different populations; cities closer to each other have greater interaction than cities further apart; larger cities exert a greater influence than smaller ones. Model may be expanded to include interactions among a set of centers, variables other than population, measures of intervening opportunity and an agglomeration effect such as in the case of shopping centers and clustering of services.

62 Data Mining Analysis of massive data sets in search for patterns, anomalies, and trends spatial analysis applied on a large scale must be semi-automated because of data volumes widely used in practice, e.g. to detect unusual patterns in credit card use

63 Descriptive Summaries Attempt to summarize useful properties of data sets in one or two statistics The mean or average is widely used to summarize data centers are the spatial equivalent there are several ways of defining centers

64 The Centroid Found for a point set by taking the weighted average of coordinates The balance point

65 Dispersion A measure of the spread of points around a center Useful for determining positional error Related to the width of the kernel used in density estimation

66 Fragmentation Statistics Measure the patchiness of data sets e.g., of vegetation cover in an area Useful in landscape ecology, because of the importance of habitat fragmentation in determining the success of animal and bird populations populations are less likely to survive in highly fragmented landscapes

67 Optimization Spatial analysis can be used to solve many problems of design A spatial decision support system (SDSS) is an adaptation of GIS aimed at solving a particular design problem

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74 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

75 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

76 Network Analysis Characterized by a set of geographic locations interconnected in a system by a number of routes; a system of lines topologically structured and connecting points. Network indices -- Alpha (number of circuits vs. maximum number), Beta (relations between the edge and the vertex in the network), Gamma (degree of connectivity), Dispersion (overall distribution of the network), Accessibility (spatial relation between a given element of a structure and the remainder of the network), and Smeed s (efficiency of the road network). Location /Allocation supply, demand, impedance, spatial interactions.

77 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

78 Hypothesis Testing Hypothesis testing is a recognized branch of statistics A sample is analyzed, and inferences are made about the population from which the sample was drawn The sample must normally be drawn randomly and independently from the population

79 Hypothesis Testing with Spatial Data Frequently the data represent all that are available e.g., all of the census tracts of Los Angeles It is consequently difficult to think of such data as a random sample of anything not a random sample of all census tracts Tobler s Law guarantees that independence is problematic unless samples are drawn very far apart

80 Possible Approaches to Inference Treat the data as one of a very large number of possible spatial arrangements useful for testing for significant spatial patterns Discard data until cases are independent no one likes to discard data Use models that account directly for spatial dependence Be content with descriptions and avoid inference

81 Spatial Sampling Sample frames Probability of selection All geographic representations are samples Geographic data are only as good as the sampling scheme used to create them

82 Sample Designs Types of samples Random samples Stratified samples Clustered samples Weighting of observations

83 Examples of Sampling Designs