Spatial Analysis Methodologies and Spatial Processes

Outline

Spatial Analysis basics – Geometric Properties:
Calculating Distance, Area, Shape
Generalisation and Centroids
Spatial analysis concepts: 

• Topology
• Clip, Merge, Intersect

Spatial Analysis

“Spatial analysis or spatial statistics includes any of the formaltechniques which study entities using their topological, geometric, or geographic properties.” – Wikipedia (slaps wrist...!)
Spatial analysis is a set of methods whose results are not invariant under changes in the locations of the objects being analysed.
Spatial Analysis (simple or complex)can make what is implicit explicit.
We do not look at analysis of attribute now.
Geometric properties are associated the shape, size and relative positions of objects
Topological properties relate to the elements of an entities geometry that remain unchanged if you alter their shape, size etc.
Geographic properties relate to an entities place on the earth – the importance of location

Distance

Calculating Distance on a National Grid

Very easy using trigonometry: A2 + B2 = C2
Distance between:
Blue dot (x = 300,y = 800) Red dot (x = 500,y = 200)
Abs(300-500)2+ Abs(800-200) 2 = C2
2002 + 6002 = 400,000
Sqrt(400,000)= 632.5km

The importance of distance in spatial analysis...
We do many analysis and processes using distance such as Select, Clip, Proximity and Buffer, Interpolation, Overlay and Union and Intersect

Geometric properties: Area
Depends on distance
Very common 

calculation in GIS
Different to shape
Can be summarized as a bounding box (i.e. a box which contains the shape of interest)
Very important as a denominator for density, e.g. Population / Area = Population density
Calculating Area – a standard GIS method
Calculating area is straight forward when we know the coordinates of the vertices...

Shape – assessing the level of Gerrymandering
Shapes of electoral districts have become very irregular in some places as politicians have attempted to alter boundaries to capture more votes
Ansolabehere and Palmer (2015) use measures of shape compactness to assess the level of Gerrymandering in US congressional districts
http://www.vanderbilt.edu/csdi/events/ansolabehere_palmer_gerrymander.pdf

Geometric properties: Shape
Again, dependent on distance.
Various ways to measure irregular polygons:
Perimeter to area ratio. 𝐿𝑖2 𝐴𝑖

Compactness ratio. 𝑠=𝑃𝐴. 𝐴𝑖 𝐵𝑖. 𝐶𝑖 = 𝜋2

Geometric properties: Smoothing/ Generalisation
Overly complex (detailed – large in computer memory) shapes can be generalised.
Various algorithms will generalise shapes
Smoothing can be used to estimate shapes with poor spatial resolution

Topology

Konigsberg Bridge Problem - Can you find a walk through the city that would cross each bridge once and only once? Euler, 1736

Geography, shape, distance all irrelevant, only the relationship between the land masses and the bridges connecting them

Laid the foundations of Graph Theory

Topology

An important concept in spatial analysis. 4 main properties:
Dimensionality: the distinction between point, line, area, and volume, which are said to have topological dimensions of 0, 1, 2, and 3 respectively
Adjacency: including the touching of land parcels, counties, and nation-states
Connectivity: including junctions between streets, roads, railroads, and rivers
Containment: when a point lies inside rather than outside an area

Topological Analysis - Next To, Within, Intersects
Spatial queries are one of the core functions of any GIS system
They rely on topological relationships

Making intersection test more efficient
Take care of easy cases using coordinate comparisons
Only of f bounding boxes intersect

Sweepline Algorithm

Event queue

1. Start Segment (S2)
2. End Segment (E2) 
3. Intersection (I2,4)

Move sweepline to next event
Maintain vertical order of segments as line sweeps across

• Start Segment

Insert in list
Check above and below for intersection

• End Segment

Remove from list
Check newly adjacent segments for intersection

• Intersection

Reorder segments
Check above and below for intersection

Topological Analysis - Point In Polygon
Topological rules can be put to good use when carrying out some spatial analysis tasks
Counting the number of points contained within a polygon is one of the most common of these

Spatial Autocorrelation - Tobler’s First Law of Geography

"Everything is related to everything else, but near things are more related than distant things.” See Tobler, 1970.

Formalising Tobler’s First Law: Spatial Dependence/ Autocorrelation
Points closer to the red point are more likely to have characteristics which are similar
Sunlight,slope, vegetation, soil ph etc.

Conclusions
These concepts are at the heart of spatial analysis software and processes.
Many aspects are now routine in GIS.
Effective spatial analysis requires an intelligent user, not just powerful computer
Try and avoid complacency.