The Generalized Random Tessellation Stratified Sampling Design for Selecting Spatially-Balanced Samples

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The Generalized Random Tessellation Stratified
Sampling Design for Selecting Spatially-Balanced
Samples

• Don L. Stevens, Jr.
• Department of Statistics
• Oregon State University

Monitoring Science Technology
Symposium September 20 - 24, 2004 Denver, Colorado
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This presentation was developed under STAR
Research Assistance Agreement No. CR82-9096-01
Program on Designs and Models for Aquatic
Resource Surveys awarded by the U.S.
Environmental Protection Agency to Oregon State
University. It has not been subjected to the
Agency's review and therefore does not
necessarily reflect the views of the Agency, and
no official endorsement should be inferred
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Historical Context

• GRTS design evolved from EMAP work on global
  tessellations in the early 1990s
• Scott Overton, Denis White, Jon Kimmerling
  developed EMAPs triangular grid hexagonal
  tessellation

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Historical Context

• EMAP began with a triangular grid hexagonal
  tessellation
• Expected to intensify grid as needed
• Triangular grid has several advantages
• More compact than square grid
• More subdivision factors
• Became clear that basic concept did not have
  enough flexibility to accommodate the
  characteristics of environmental resource sampling

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Environmental Resource Populations

• Point-like
• Finite population of discrete units, e.g., small-
  to medium-sized lakes
• Linear
• Width is very small relative to length, e.g.,
  streams or riparian vegetation belts
• Extensive
• Covers large area in a more or less continuous
  and connected fashion, e.g., a large estuary

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Environmental Resource Populations

• Tobler's First Law of Geography Things that are
  close together in space tend to have more similar
  properties than things that are far apart.
• OR
• Spatial correlation functions tend to decrease
  with distance

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Sampling Environmental Resource Populations

• Environmental Resource Populations exist in a
  spatial matrix
• Population elements close to one another tend to
  be more similar than widely separated elements
• Good sampling designs tend to spread out the
  sample points more or less regularly
• Simple random sampling tends to exhibit uneven
  spatial patterns

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Simple random sample of a domain with 3
subdomains
A B C 28
28 15
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Sampling Environmental Resource Populations

• Patterned response (gradients, patches, periodic
  responses)
• Variable inclusion probability
• 0, 1, and 2 dimensional populations (points,
  lines, areas)
• Pattern in population occurrence (density )
• Unreliable frame material
• Temporal panels often needed

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Environmental Resource Populations

• Ecological importance, environmental stressor
  levels, scientific interest, and political
  importance are not uniform over the extent of the
  resource

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Desirable Properties of Environmental Resource
Samples

• (1) Accommodate varying spatial sample intensity
•  
• (2) Spread the sample points evenly and regularly
  over the domain, subject to (1)
• (3) Allow augmentation of the sample
  after-the-fact, while maintaining (2)

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Desirable Properties of Environmental Resource
Samples

• (4) Accommodate varying population spatial
  density for finite linear populations, subject
  to (1) (2).
• (2) (4) Þ Sample spatial pattern should
  reflect the (finite or linear) population spatial
  pattern

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Sampling Environmental Resource Populations

• Systematic sample has substantial disadvantages
• Well known problems with periodic response
• Less well recognized problem patch-like response

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A B C 26
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A B C 32
20 16
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Sampling Environmental Resource Populations

• Systematic sample has substantial disadvantages
• Well known problems with periodic response
• Less well recognized problem patch-like response
• Difficult to apply to finite populations , e.g.,
  Lakes
• Limited flexibility to change sample point
  density
• Difficult to accommodate variable inclusion
  probability or sample adjustment for frame errors
  

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Sample point intensity can be changed using
nested grids
A B C 26
88 15
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RANDOM-TESSELLATION STRATIFIED (RTS) DESIGN

• Compromise between systematic SRS that
  resolves periodic/patchy response
• Cover the population domain with a grid
• Randomly located
• Regular (square or triangular)
• Spacing chosen to give required spatial
  resolution
• Tile the domain with equal-sized regular polygons
  containing the grid points
• Select one sample point at random from each
  tessellation polygon

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RANDOM-TESSELLATION STRATIFIED (RTS) DESIGN

• Solves some of systematic sample problems
• Non-zero pairwise inclusion probability
• Alignment with geographic features of population
• Lets points get close together with low
  probability

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RTS DESIGN

• Does not resolve systematic sample difficulties
  with
• variable probability
• finite linear populations
• pattern in population occurrence (density)
• unreliable frame material
• Limited ability to change density

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Generalized Random-Tessellation Stratified (GRTS)
Design

• Conceptual structure
• Population indexed by points contained within a
  region R
• Have inclusion probability p(s) defined on R
• Select a sample by picking points
• Finite points represent units
• p(s) is usual inclusion probability
• Linear points on the lines
• p(s) is a density sample points /unit length
• Extensive points are in region area
• p(s) is a density sample points/unit area

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GRTS Design Mechanics

• Map R into first quadrant of unit square, add a
  random offset
• Subdivide unit square into small grid cells
• At least small enough so that total inclusion
  probability for a cell (expected number of
  samples in the cell) is less than 1
• Total inclusion probability for cell is sum or
  integral of p(s) over the extent of the cell

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Population region image
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Population region image random offset
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GRTS Design Mechanics

• Order the cells so that some 2-dimensional
  proximity relationships are preserved
• Cant preserve everything, because a 1-1, onto,
  continuous map from unit square to unit interval
  is impossible
• Can get 1-1,onto, measureable, which is good
  enough
• GRTS uses a quadrant-recursive function, similar
  to the space filling curve developed by Guiseppe
  Peano in 1890.

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Assign each cell an address corresponding to the
order of subdivision The address of the shaded
quadrant is 0.213 Order the cells following the
address order
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GRTS DesignMechanics

• If we carry the process to the limit, letting the
  grid cell size ? 0, the result is a quadrant
  recursive function, that is, a function that maps
  the unit square onto the unit interval such that
  the image of every quadrant is an interval.
• Apply a restricted randomization that preserves
  quadrant recursiveness

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HIERARCHICAL RANDOMIZATION

• Each cell address is a base 4 fraction, that is,
  t  0.t1t2t3..., where each digit ti is either a
  0, 1, 2, or 3. A function hp is a hierarchical
  permutation if
• where is a possibly
  distinct permutation of 0,1,2,3 for each unique
  combination of digits
• t1, t2, ..., tn - 1.

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HIERARCHICAL RANDOMIZATION

• If the permutations that define hp() are chosen
  at random and independently from the set of all
  possible permutations, we call hp() a
  hierarchical randomization function, and the
  process of applying hp() hierarchical
  randomization.
• Compose the basic q-r map with a hierarchical
  randomization function

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GRTS DesignMechanics

• The result is a random order of the small grid
  cells such that
• All grid cells in the same quadrant have
  consecutive order positions
• But will be randomly ordered within those
  positions
• This holds for all quadrant levels
• This induces a random ordering of population
  elements

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GRTS DesignMechanics

• Assign each grid cell a length equal to its total
  inclusion probability
• String the lengths in the random order
• Result is a line with length equal to target
  sample size
• Take systematic sample along line (random start
  unit interval)
• Map back to population using inverse random qr
  function

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GRTS DesignMechanics

• Points will be in hierarchical random order
• Re-order into reverse hierarchical order gives
  some very useful features to the sample

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Reverse Hierarchical Order

• Illustrate for 2-levels of addressing

First 16 addresses as base 4-fractions 00 01 02
03 10 11 12 13 20 21 22 23 30
31 32 33
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Reverse Hierarchical Order

• Illustrate for 2-levels of addressing

First 16 addresses as base 4-fractions 00 01 02
03 10 11 12 13 20 21 22 23 30
31 32 33 Reversed digits 00 10 20 30
01 11 21 31 02 12 22 32 03 13 23
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Reverse Hierarchical Order

• Illustrate for 2-levels of addressing

First 16 addresses as base 4-numbers 00 01 02
03 10 11 12 13 20 21 22 23 30 31
32 33 Reversed digits 00 10 20 30 01
11 21 31 02 12 22 32 03 13 23
33 Reversed digits as base 10 numbers 0 4
8 12 1 5 9 13 2 6 10 14
3 7 11 15
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SPATIAL PROPERTIES OF REVERSE HIERARCHICAL
ORDERED GRTS SAMPLE

• The complete sample is nearly regular, capturing
  much of the potential efficiency of a systematic
  sample without the potential flaws
• Any subsample consisting of a consecutive
  subsequence is almost as regular as the full
  sample in particular, the subsequence
• 
  , is a spatially well-balanced sample.
• Any consecutive sequence subsample, restricted to
  the accessible domain, is a spatially
  well-balanced sample of the accessible domain.

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Inclusion probability density surface
Region is (0,1)x(0,0.8)
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SPATIAL PROPERTIES OF REVERSE HIERARCHICAL
ORDERED GRTS SAMPLE

• Assess spatial balance by variance of size of
  Voronoi polygons, compared to SRS sample of the
  same size.
• Voronoi polygons for a set of points
  The ith polygon is the collection of points
  in the domain that are closer to si than to any
  other sj in the set.
• Estimate variance by 1000 replications of a
  sample of size 256 in unit square

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SPATIAL PROPERTIES OF REVERSE HIERARCHICAL
ORDERED GRTS SAMPLE

• Compare regularity as points are added one at a
  time, following reverse hierarchical order under
  4 scenarios
• Complete, continuous domain
• Domains with holes excluding 20 , modeling
  non-response/access refusal
• 20 randomly-located square holes, constant size
• 20 randomly-located square holes, increasing
  linearly in size
• 10 randomly-located square holes, increasing
  exponentially in size

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20 point GRTS Sample
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Four 20-point GRTS Panels
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Five 20-point GRTS Panels
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Five 20-point GRTS Panels Special Study Area
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Finite Population Example
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Equi-probable GRTS Sample
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GRTS Sample Probability inversely proportional
to population density
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Equi-probable
Inverse density