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

.

.

.

.

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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

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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