Cutting Spatial Data Into Ribbons

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Cutting Spatial Data Into Ribbons

• Sampling Proximity Relations to Permit Some
  Spatial Analysis While Preventing Other Spatial
  Analysis
• Alan Saalfeld
• The Ohio State University

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NOT CALLEDCutting Spatial Data into Scraps

• Throwing Away All Adjacency Relations, Thereby
  Permitting No InterRegional Spatial Analysis
  Whatsoever!

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Preview

• Some discrete math
• Some affine geometry
• Some spatial relations
• Some existence results
• Some non-existence results
• Some transformations
• Some solutions in search of problems

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If only we lived in a 1-D world!

• Everything would be comparable.
• There would be a metric-compatible total order on
  point locations (no such order exists in 2-D or
  higher-D).
• Neighborhoods would always be intervals (a,b).
• After sorting the data, many of the processing
  tasks would have linear-time complexity.
• E.g., finding all intervals of a fixed size
  would be a linear-time (O(n)) operation.

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All right, youse guys, now I want you to line
up alphabetically, by height...
Yogi Berra finds a way to sort 2-D data
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Data Ribbons

• Decompose the dataset into square regions that
  form a fixed width ribbon.
• Each region has a predecessor and a successor
  region (of different shades).
• Each region shares a single square edge with its
  predecessor and shares a different single edge
  with its successor.
• Exactly ? of the 4-connected adjacencies in 2-D
  space are preserved.

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Hilbert-order Ribbons

• Every square data set may be broken into square
  regions and fitted with a fixed width,
  non-overlapping, covering ribbon.
• The ribbon order is quadrant-recursive the
  ribbon covers a (sub)quadrant before moving on to
  the next (sub)quadrant.
• Each ribbon has two possible subquadrant
  refinements.

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Statistical Behavior

• (Quad)tree hierarchical data structures have
  well-behaved variance/covariance structure
• ...covariance of values associated with any two
  subquadrants is equal to the variance of the
  first common ancestor (Cressie,1998).
• The Hilbert ribbon structure organizes data to
  facilitate covariance estimation of contiguous
  cells.

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Advantages of Hilbert Ribbons

• They are area-based, not point-based.
• They are regular, composed of same-size squares.
• They preserve exactly half the adjacencies.
• They fill area of entire subquadrants at a time.
• They follow the hierarchy of quad-trees.
• Opportunities for multi-resolution statistical
  analysis.
• Opportunities to increase or decrease resolution.

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Disadvantages of Hilbert Ribbons

• Their squares are regular in size and
  orientation.
• They cannot accommodate traditional geography.
• They may allow recovery of the underlying grid.
• Recovering that grid would disclose all
  locations!
• There are few degrees of freedom in Hilbert
  ribbon placement.
• Spatial statistical theory of ribbon structure
  may prove accessible, but it still needs
  development.

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Suppose that we begin with our space decomposed
into convex regions. We want to order data
elements in those regions to try to keep all data
from each region together on our file and to try
to keep data from adjacent regions in adjacent
data blocks. Wed also like to be able to analyze
some spatial properties from our ordered data
sets alone based on proximity properties.
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Regions and their planar dual graph, with dual
edges crossing region edges at midpoints
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Drawing of m regions and their planar dual graph
on m vertices. Each edge has a two-piece
bisecting dual edge. Together the edge halves and
dual edge halves decompose each n-sided convex
polygon into n quadrilaterals
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Finally we have the structure sufficiently in
place to be able to describe how to cut the
data to ribbons
When we finish our cutting, each quadrilateral
will be connected to exactly two of its four
neighboring quadrilaterals. The connections will
form a cyclic ribbon of quadrilaterals.
1. Cut along any (m-1) dual edges that form
a spanning tree of the dual graph. 2. Cut along
every edge in the original graph whose dual edge
has not been cut.
How many different decomposing ribbons exist?
One for each spanning tree of the planar dual
graph!!
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The Ribbon of Quadrilaterals

• On average, regions will have their set of
  interior quadrilaterals distributed over slightly
  less than 2 unbroken ribbon segments.
• The edges chosen for the spanning tree of the
  dual graph may be constrained to respect
  hierarchies of regions to a large extent.
• For example, a county-to-county link may be given
  a very low priority of being in the spanning tree
  if the neighboring counties are in different
  states.

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To limit the number of crossings of major
boundaries by the spanning tree of the dual
graph, dual edges that cross the boundary may be
assigned a much higher cost.
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Piecewise Linear Homeomorphisms

• PLH maps are described fully by their action on
  triangles
• PLH maps are described fully by their action on
  triangle vertices
• PLH maps agree on shared edges that are straight
  line segments

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Any polygon can be mapped to any other by a PLH
map

• Any n-sided polygon can be triangulated.
• Any triangulation of an n-sided polygon is also a
  triangulation of a similarly labeled regular
  n-gon.
• The composition of two PLH maps is a PLH map.

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Area-preserving transformations

• Lemma A PLH map, realized as a bijection of
  triangles, is area-preserving everywhere if and
  only if it sends triangles to triangles of the
  same area.
• Proof A necessary and sufficient condition for
  an affine function (x,y) ? (axbyc, dxeyf) to
  preserve area is for the determinant of its
  Jacobian, ae-bd, to be equal to 1 or (-1).

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Area-preserving transformations

• PLH maps may be found that are area-proportional
  everywhere.
• Proof by induction
• Trivial for n3.
• Construction for n4.
• First show for convex sets for ngt4.

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Area-preserving transformations

• The following are equivalent
• 1. Any two convex n-sided polygons of equal area
  are homeomorphic under an area-preserving PLH map
  that is linear on each corresponding edge pair.
• 2. Any convex n-sided polygon is homeomorphic
  under an area-preserving PLH map to a regular
  n-gon of the same area. The homeomorphism may be
  taken to be linear on each corresponding edge
  pair.

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Node Splitting and Area Splitting

• Every convex polygon possesses a splitter that
  divides the area in equal parts and also splits
  the vertices into equal groups.

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Area-preserving transformations

• Proof for convex sets
• Suppose 1. and 2. are true for all convex sets of
  size kltn, where ngt4.
• Find a splitter for a convex n-gon that splits it
  into two equal area (?n/2?2)-sided convex
  polygons. Note ?n/2?2ltn.
• Map each half into half of a regular n-gon.

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Area-preserving transformations

• A technical detail
• On 1 or 2 of the boundary edges, the ratio of the
  two pieces of the divided edge may differ for the
  convex (?n/2?2)-gon and for the half of the
  regular n-gon. We can always adjust the edge
  pieces with yet another area-preserving PLH map
  so that they recover the original ratio.

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Area-preserving transformations

• Proof for non-convex sets
• Suppose any non-convex k-gon for kltn has a PLH
  area-preserving, boundary-extending map to any
  same area convex k-gon, where ngt4.
• For the non-convex n-gon, triangulate and remove
  an ear. Ears always exist.
• Map the (n-1)-gon to a slab convex set with the
  edge from the missing ear going to the long edge
  of the slab.
• Map the ear to an ear-sized triangle attached
  to the slab along the long edge. (Construction
  makes it possible.).

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In Summary

• We saw how to cut data into ribbons to remove
  exactly half of the neighbor relations. We
  discussed a randomization procedure for relation
  sampling.
• We proved the existence of conforming meshes with
  prescribed PLH behavior on the boundary that
  scale all triangles uniformly. We illustrated
  transformations that preserve some spatial
  analysis capabilities.

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More Things to Do

• We will analyze how well randomization of the
  spanning tree generation will protect sensitive
  location information.
• We will develop theory for spatial analysis of
  ribboned data sets, including measures and bounds
  on the uncertainty that is due to our ribboning
  procedure.

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An Earlier Strategy

• 1. Transform the 2-D dataset into a 1-D dataset
  (using a proximity-preserving transformation).
• 2. Create contextual variables for the 1-D
  dataset.
• 3. Interpret the 1-D contextual variables in
  terms of their corresponding sampled 2-D
  contexts.
• 4. Assess information loss and uncertainty, and
  interpret them as data protection measures.

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Gamut of Spatial Detail

• Smoothed local summary statistics
• Aggregated measures Averages, Densities
• Measures of local interaction
• Regression, Spatial autocorrelation
• Raw data to permit any spatial analysis
• Exact coordinates of every data point