Map Measurement and
Transformation
Longley et al., ch. 13
What is spatial analysis?
• Methods for working with spatial data
– to detect patterns, anomalies
– to find answers to questions
– to test or confirm theories
• deductive reasoning
– to generate new theories and generalizations
• inductive reasoning
• "a set of methods whose results change
when the locations of the objects being
analyzed change"
What is Spatial Analysis (cont.)
• Methods for adding value to data
– in doing scientific research
– in trying to convince others
• Turning raw data into useful information
• A collaboration between human and
machine
– Human directs, makes interpretations and
inferences
– Machine does tedious, complex stuff
Early Spatial Analysis
• John Snow, 1854
• Cholera via polluted water, not air
• Broad Street Pump
John Snow’s Map
Updating Snow:
Openshaw 1965-’98
• Geographic Analysis
Machine
• Search datasets for
event clusters
– cases: pop at risk
• Geographical
correlates for:
– Cancer
– Floods
– Nuclear attack
– Crime
Objectives of Spatial Analysis
• Queries and reasoning
• Measurements
– Aspects of geographic data, length, area, etc.
• Transformations
– New data, raster to vector, geometric rules
• Descriptive summaries
– Essence of data in a few parameters
• Optimization - ideal locations, routes
• Hypothesis testing – from a sample to entire
population
Answering Queries
• A GIS can present several distinct views
• Each view can be used to answer simple
queries
– ArcCatalog
– ArcMap
Views to Help w/Queries
• hierarchy of devices, folders, datasets, files
• Map, table, metadata
Views to Help w/Queries
• ArcMap - map view
•
Views to Help w/Queries
ArcMap - table view linked to map
•
Views to Help w/Queries
ArcMap - histogram and scatterplot views
Exploratory Data Analysis ( EDA )
• Interactive methods to explore spatial data
• Use of linked views
• Finding anomalies, outliers
• In images, finding particular features
• Data mining large masses of data
– e.g., credit card companies
– anomalous behavior in space and time
SQL in EDA
• Structured or Standard query language
• SELECT FROM counties WHERE median value > 100,000
Result is HIGHLIGHTed
Spatial Reasoning with GIS
• GIS would be easier to use if it could "think"
and "talk" more like humans
– or if there could be smooth transitions between
our vague world and its precise world
– Google Maps
• In our vague world, terms like “near”, far”,
“south of”, etc. are context-specific
– From Santa Barbara: LA is far from SB
– From London: LA is right next to SB
Measurement with GIS
• Often difficult to make by hand from maps
– measuring the length of a complex feature
– measuring area
– how did we measure area before GIS?
• Distance and length
– calculation from metric coordinates
– straight-line distance on a plane
Measuring the length of a feature
vs.
Distance
• Simplest distance calculation in GIS
• d = sqrt [(x1-x2)2+(y1-y2)2 ]
• But does it work for latitude and longitude?
Spherical (not spheroidal) geometry
• Note: a and b are distinct from A (alpha) and B (beta).
• 1. Find distances a and b in degrees from the pole P.
• 2. Find angle P by arithmetic comparison of longitudes.
– (If angle P is greater than 180 degrees subtract angle P from 360
degrees.)
– Subtract result from 180 degrees to find angle y.
– 3. Solve for 1/2 ( a - b ) and 1/2 ( a + b ) as follows:
tan 1/2 ( a - b ) = - { [ sin 1/2 ( a - b ) ] / [ sin 1/2 ( a + b ) ] } tan 1/2 y
tan 1/2 ( a + b ) = - { [ cos 1/2 ( a - b ) ] / [ cos 1/2 ( a + b ) ] } tan 1/2 y
• 4. Find c as follows:
– tan 1/2 c = { [ sin 1/2 ( a + b ) ] x [ tan 1/2 ( a - b ) ] } / sin 1/2 ( a - b )
• 5. Find angles A and B as follows:
– A = 180 - [ ( 1/2 a + b ) + ( 1/2 a - b ) ]
– B = 180 - [ ( 1/2 a + b ) - ( 1/2 a - b ) ]
Distance
• GIS usually uses spherical calculations
• From (lat1,long1) to (lat2,long2)
• R is the radius of the Earth
d = R cos-1 [sin lat1 sin lat2 + cos lat1 cos lat2 cos (long1 - long2)]
What R to use?
• Quadratic mean radius
– best approximation of Earth's average transverse meridional arcradius
and radius.
– Ellipsoid's average great ellipse.
– 6 372 795.48 m (≈3,959.871 mi; ≈3,441.034 nm).
• Authalic mean radius
– "equal area" mean radius
– 6 371 005.08 m (≈3,958.759 mi; ≈3,440.067 nm).
– Square root of the average (latitudinally cosine corrected) geometric
mean of the meridional and transverse equatorial (i.e., perpendicular),
arcradii of all surface points on the spheroid
• Volumic radius
– Less utilized, volumic radius
– radius of a sphere of equal volume:
– 6 370 998.69 m (≈3,958.755 mi; ≈3,440.064 nm).
• (Source Wikipedia)
Length
• add the lengths of polyline or polygon
segments
• Two types of distortions
(1) if segments are straight,
length will be
underestimated
in general
Length
• Two types of distortions
(2) line in 2-D GIS on a plane considerably
shorter than 3-D
Area of land parcel based on area of horiz.
projection, not true surface area
Area
• How do we measure area of a polygon?
• Proceed in clockwise direction around the
polygon
• For each segment:
– drop perpendiculars to the x axis
– this constructs a trapezium
– compute the area of the trapezium
– difference in x times average of y
– keep a cumulative sum of areas
Area (cont.)
• Green, orange, blue trapezia
• Areas = differences in x times averages of y
• Subtract 4th to get area of polygon
Area by formula
(x1,y1)= (x5,y5)
(x4,y4) (x2,y2)
(x3,y3)
Applying the Algorithm to a Coverage
• For each polygon
• For each arc:
– proceed segment by segment from
FNODE to TNODE
– add trapezia areas to R polygon area
– subtract from L polygon area
• On completing all arcs, totals
are correct area
Algorithm
• Area of poly - a “numerical recipe”
• a set of rules executed in sequence
to solve a problem
– “islands” must all be
scanned clockwise
– “holes” must be scanned
anticlockwise
– holes have negative
area
– Polygons can have
outliers
Shape
• How can we measure the shape of an area?
• Compact shapes have a small perimeter for a
given area (P/A)
• Compare perimeter to the perimeter of a circle of
the same area [A = P R2
• So R = sqrt(A/ P )
• shape = perimeter / sqrt (A/ P)
• Many other measures
What Use are Shape
Measures?
• “Gerrymandering”
– creating oddly shaped districts to manipulate
the vote
– named for Elbridge Gerry, governer of MA and
signatory of the Declaration of Independence
– today GIS is used to design districts
After 1990 Census
Example: Landscape Metrics
Slope and Aspect
• measured from an elevation or bathymetry
raster
– compare elevations of points in a 3x3 (Moore)
neighborhood
– slope and aspect at one point estimated from
elevations of it and surrounding 8 points
• number points row by row, from top left from 1 to 9
1 2 3
4 5 6
7 8 9
Slope and Aspect
Slope Calculation
• b = (z3 + 2z6 + z9 - z1 - 2z4 - z7) / 8r
• c = (z1 + 2z2 + z3 - z7 - 2z8 - z9) / 8r
– b denotes slope in the x direction
– c denotes slope in the y direction
– r is the spacing of points (30 m)
• find the slope that fits best to the 9 elevations
• minimizes the total of squared differences
between point elevation and the fitted slope
• weighting four closer neighbors higher
• tan (slope) = sqrt (b2 + c2)
Slope Definitions
• Slope defined as an angle
• … or rise over horizontal run
• … or rise over actual run
• Or in percent
• various methods
– important to know how your favorite GIS
calculates slope
– Different algorithms create different
slopes/aspects
Slope Definitions (cont.)
Aspect
• tan (aspect) = b/c
• Angle between vertical and direction of
steepest slope
• Measured clockwise
• Add 180 to aspect if c is positive, 360 to
aspect if c is negative and b is positive
Transformations
• Buffering (Point, Line, Area)
• Point-in-polygon
• Polygon Overlay
• Spatial Interpolation
– Theissen polygons
– Inverse-distance weighting
– Kriging
– Density estimation
Basic Approach
Map New map
Transformation
Example
