Sampling Design, Spatial Allocation, and Proposed Analyses
Don Stevens
Department of Statistics
Oregon State University
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
Sampling Environmental Populations
• Systematic sample has substantial disadvantages – Well known problems with periodic response – Less well recognized problem: patch-like
response– Inflexible point density doesn’t accommodate
• Adjustment for frame errors
• Sampling through time
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
RTS Design
• Does not resolve systematic sample difficulties with – variable probability (point density)– unreliable frame material– Sampling through time
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.
x
B x s 0 f s x =
F(x)
s
y1
y2
yi
yM
.
.
.
.
.
.
x1 x2 . . . xi . . . xM
quadrant-recursive, hierarchical random map
systematic sample
s f=1–
x
F x s s dB x
=
x F1–
= y
x f s =
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).
for k 1 2 k = { , , ..., }, k MS s s s
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.
1 2 k{ , , ..., }s s s
Voronoi Polygons
GRTS SampleUniform Sample
Sample Size
Effi
cie
ncy
of G
RT
S D
esi
gn
0 10 20 30 40 50 60
02
46
81
0
At n = 8, efficiency is 2.4
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
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.
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
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
( , 1) ( ( , ))y s t f y s t
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.
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
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”
Proposed Analyses
• Spatial displays– Contour plots– Perspective plots– Hexagon mosaic plots– Multivariate displays
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