Sampling Design, Spatial Allocation, and Proposed Analyses

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Sampling Design, Spatial Allocation, and Proposed
Analyses

• Don Stevens
• Department of Statistics
• Oregon State University

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Sampling Environmental Populations

• Environmental 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 (SRS) tends to result in
  point patterns with voids and clusters of points

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Sampling Environmental Populations

• Systematic sample has substantial disadvantages
• Well known problems with periodic response
• Less well recognized problem patch-like response
• Inflexible point density doesnt accommodate
• Adjustment for frame errors
• Sampling through time

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Random-tessellation Stratified (RTS) Design

• Compromise between systematic SRS that
  resolves periodic/patchy response
• Cover the population domain with a randomly
  placed grid
• Select one sample point at random from each grid
  cell

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RTS Design

• Does not resolve systematic sample difficulties
  with
• variable probability (point density)
• unreliable frame material
• Sampling through time

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Generalized Random-tessellation Stratified (GRTS)
Design

• Design is based on a random function that maps
  the unit square into the unit interval.
• The random function is constructed so that it is
  1-1 and preserves some 2-dimensional proximity
  relationships in the 1-dimensional image.
• Accommodates variable sample point density,
  sample augmentation, and spatially-structured
  temporal samples.

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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
  (critical for sediment sample).

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Spatially Balanced 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.

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Sampling Through Time

• Detection of a signal that is small relative to
  noise magnitude requires replication
• Spatial replication (more samples per year)
  addresses spatial variation
• Need temporal replication (more years) to address
  temporal variation
• Detection of trend in slowly changing status
  requires many years

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Sampling Through Time

• Repeat sampling of same site eliminates a
  variance component if the site retains its
  identity through time.
• Design based on assumption that sediment does
  retain identity, but water does not.
• Both water and sediment samples have spatial
  balance through time, but sediment sample
  includes revisits at 1, 5, and 10 year intervals.

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Proposed Analyses

• Annual descriptive summaries
• Mean values, proportions, distributions,
  precision estimates based on annual data
• Mean concentration ? confidence limits
• Percent area in non-compliance ? confidence
  limits
• Histograms
• Distribution function plots ? confidence limits
• Subpopulation comparisons

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Proposed Analyses

• Composite estimation Annual status estimates
  that incorporate prior data
• Model that predicts current value at site s based
  on prior observation
• Composite estimator is weighted combination of
  mean of current observation and mean of predicted
  values based on prior observations
• Results in increased precision for annual
  estimates
• Can also be used to borrow strength from
  spatially proximate data

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Proposed Analyses

• Trend Analyses.
• Need to describe trend at the segment or Bay
  level.
• Usual approach trend in mean value.
• Also consider trend in spatial pattern, trend
  in population distribution, distribution of
  trend, and mean value of trend.
• Trend analyses will exploit repeat visit pattern
  for sediment samples.

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Proposed Analyses

• Space-Time Models
• Use random field approach to account for
  correlation through space and time
• Panel structure (repeat visits) in sediment
  sample is a good structure to estimate space-time
  correlation
• Long-term need 10 years to get sufficient data
  to estimate model parameters

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Proposed Analyses

• Bayesian Hierarchical Models
• Good way to incorporate ancillary information
  into status estimates
• E.g., loading estimates, flow data, metrological
  data
• Distribution of response is modeled as a function
  of parameters whose distribution in turn depends
  on ancillary data, hence, hierarchical

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Proposed Analyses

• Spatial displays
• Contour plots
• Perspective plots
• Hexagon mosaic plots
• Multivariate displays

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