Some spatial modelling examples for discussion

Marian Scott

NERC September 2011

ECCOMAGS Project

• Funded by EC FPV – Nuclear Safety, • Partners

– BfS (Germany), CEA (France), DEMA/DTU (Denmark), GBA (Austria), HSK (Switzerland), IGM (Portugal), SSI (Sweden), SGU (Sweden), STUK (Finland), SURRC (UK)

• Airborne Gamma-ray Spectrometry (AGS)• To assess comparability of the European AGS

systems• To assess comparability of AGS with ground-

based systems.

Exercise Survey Areas

I JF G HEC DA B

X

Y

Z

Dumfries

StranraerWigtownMerse

Inch Farm

CastleKennedy

ECCOMAGS Exercise : Survey Areas

Solway Firth

Dumfries and Galloway Region

0 10 20km

• Three spatial scales of measurement:– Ground based soil sampling– In-situ monitoring– Aerial monitoring

• How comparable are the three sets of measurement? What issues do we need to consider (downscaling/upscaling)?

Ground sampling locations

Pre-Characterisation-sampling plan for in-situ measurement

• Expanding hexagonal sampling pattern showing the radial numbers and shell designations.

137Cs Map 6th June

Area X – 137Cs

Example: Mapping SO2 over Europe and through time

How has SO2 levels changed over time as a result of international agreements?

How is SO2 distributed over Europe?

A spatial model (over Europe), which may change over time- a spatio-temporal model

Example: River quality over a river basin

How have nitrate levels changed over time as a result of changes in management and diffuse pollution control?

How are nitrate levels distributed over river basin?

A spatio-temporal model

some commentsmodelling over space and time is challenging- it lies at

much of the controversies we have seen about climate change predictions

some simple approaches are to look at each spatial location over time, to look at each spatial pattern at each timepoint

A spatio-temporal model combines these approaches and may introduce an interaction, so that the change over time is not uniform over space

Spatial patterns of change- at river basin scale• Spatial patterns of change

may be important • interpolation over the

entire network from the stations is possible, but needs a spatial model-a river distance model (where river distance is defined as the shortest distance between two locations, along the river network) is useful

the circles represent the stations on the network, clearly not spatially representative Joint work with David O’Donnell, Mark Hallard

(SEPA), Adrian Bowman

Spatial patterns of change- at river scale- the river Dee

Joint work with Stephen Reid, David O’Donnell, Mark Hallard (SEPA), Adrian Bowman

euclidean

river

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Outline

• Variation

• General sampling principles

• Methods of sampling– Simple random sampling– Stratified sampling– Systematic sampling– How many samples (power calculations)

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network design-ECN

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Variation

• Natural variation in the attribute of interest, might be due to – feeding habits if measuring sheep, rainfall patterns if

measuring plants

• Also variation/ uncertainty due to analytical measurement techniques.

• Natural variation may well exceed the analytical uncertainty

• Expect therefore that if you measure a series of replicate samples, they will vary and if there is sufficient you may be able to define the distribution of the attribute of interest.

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Statistical sampling

The use of valid statistical sampling techniques increases the chance that a set of specimens (the sample, in the collective sense) is collected in a manner that is representative of the population.

Statistical sampling also allows a quantification of the precision with which inferences or conclusions can be drawn about the population.

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Statistical sampling

• the issue of representativeness is important because of the variability that is characteristic of environmental measurements.

• Because of variability within the population, its description from an individual sample is imprecise, but this precision can be described in quantitative terms and improved by the choice of sampling design and sampling intensity (Peterson and Calvin, 1986).

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Good books

• The general sampling textbooks by Cochran (1977) and Thompson (1992),

• the environmental statistics textbook by Gilbert (1987), and

• papers by Anderson-Sprecher et al. (1994), Crepin and Johnson (1993), Peterson and Calvin (1986), and Stehman and Overton (1994).

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Know what you are setting out to do before you start

 describing a characteristic of interest (usually the average),

describing the magnitude in variability of a characteristic, describing spatial patterns of a characteristic,mapping the

spatial distribution, quantifying contamination above a background or

specified intervention level  detecting temporal or spatial trends,   assessing human health or environmental impacts of

specific facilities, or of events such as accidental releases,

• assessing compliance with regulations

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Rules

• Rule 1: specify the objective • Rule 2: use your knowledge of the

environmental context

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Use your scientific knowledge

   the nature of the population such as the physical or biological material of interest, its spatial extent, its temporal stability, and other important characteristics,

  the expected behaviour and environmental properties of the compound of interest in the population members,

   the sampling unit (i.e., individual sample or specimen),

• the expected pattern and magnitude of variability in the observations .

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What is the population?

• The concept of the population is important. The population is the set of all items that could be sampled, such as all fish in a lake, all people living in the UK, all trees in a spatially defined forest, or all 20-g soil samples from a field. Appropriate specification of the population includes a description of its spatial extent and perhaps its temporal stability

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What is the sampling unit?

• The environmental context helps define the sampling unit. It is not practical to consider sampling units so small that their concentration cannot be easily measured. to consider extremely large sampling units, if they are too difficult to manipulate or process.

• A sampling unit is a unique element of the population that can be selected as an individual sample for collection and measurement.

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Sampling units

• In some cases, sampling units are discrete entities (i.e., animals, trees), but in others, the sampling unit might be investigator-defined, and arbitrarily sized.

• Statistical sampling leads to a description of the sampled members of the population and inference(s) and conclusion(s) about the population as a whole.

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RepresentativityAn essential concept is that the taking of a sufficient number of individual samples should provide a collective sample that is representative of all samples that could be taken and thus provides a true reflection of the population.

A representative collective sample should reflect the population not only in terms of the attribute of interest, but also in terms of any incidental factors that affect the attribute of interest.

Representativeness of environmental samples is difficult to demonstrate. Usually, representativeness is considered justified by the procedure used to select the samples.

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5 step approach

• Define the objectives and questions to be answered• Summarize the environmental context for the

quantities being measured.• Identify the population, including spatial and

temporal extent.• Select an appropriate sampling design.• Document the sampling design and its rationale.

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Methods

Simple random sampling

With simple random sampling, every sampling unit in the population has, in theory, an equal probability of being included in the sample. The resulting estimator based on such a sample will be unbiased, but it may not be efficient, in either the statistical or practical senses. Simple random sampling designs are easy to describe but may be difficult to achieve in practice.

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Population of N units-10 randomly selected

1 2 3 4 5 9

17

23 25

31 33

42 45

46 51 54

Random digits: 5,17,23, 25, 31, 33,42, 45,46,51

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Sample mean and variance

n

yy i

1

2

2

n

yys

i

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Sampling error

n

fsyVar

1)( 2

the sampling fraction f is usually very small and given by n/N.

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Methods

Stratified sampling

The population is divided into strata, each of which is likely to be more homogeneous than the entire population. In other words, the individual strata have characteristics that allow them to be distinguished from the other strata, and such characteristics are known to affect the measured attribute of interest. Some ordinary sampling method (e.g., a simple random sample or systematic sample) is used to estimate the properties of each stratum.

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Methods

Stratified samplingUsually, the proportion of sample observations taken in each stratum is similar to the stratum proportion of the population, but this is not a requirement. If good estimates are wanted for rare strata that have a small occurrence frequency in the population, then the number of samples taken from the rare strata can be increased. Stratified sampling is more complex and requires more prior knowledge than simple random sampling, and estimates of the population quantities can be biased if the stratum proportions are incorrectly specified.

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Methods

Systematic sampling

Systematic sampling is often more practical than random sampling because the procedures are relatively easy to implement in practice, but this approach may miss important features if the quantity being sampled varies with regular periodicity and the sampling scheme has similar periodicity.

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Methods

Systematic sampling

Systematic sampling is probably the most commonly used method for field sampling. It is generally unbiased as long as the starting point is randomly selected and the systematic rules are followed with care. Line transects and two dimensional grids are specific types of systematic samples that are described in more detail in the spatial section.

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Population of N (9x6) units-9 systematically selected

1 2 3 4 5 6

12 18

24

30 36

42

48 54

Systematic selection: 6,12,18,24,30,36,42,48

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Number of samples needs you to

• state the desired limits of precision for the population inference (how precisely does one want to know the average PCB concentration, or, what size of difference is needed to be detected and with what precision?),

• state the inherent population variability of the attribute of interest, and

• derive an equation which relates the number (n) of samples with the desired precision of the parameter estimator and the degree of significance (the chance of being wrong in the inference).

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Number of samples

What is the power?

Power is a probability, it is the probability that we correctly conclude that the null hypothesis should be rejected. The null would say there is no difference/no effect/no trend.

We want a high power

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Power Curves

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PCB

• estimate the mean concentration with an estimated standard error (e.s.e.) precision of 0.1 mg kg-1. The variation of PCB in salmon flesh is 3.192. Therefore, how many samples would be required? Since the e.s.e. of the sample mean is s/n, then one must solve for n, for example:

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Sample size-too big

2

...

ese

sn 1018

1.0

19.32

Thus this degree of improvement in precision, can only be achieved by increasing the number of samples taken to approximately 1000. This may well be impractical; therefore the only solution may be to accept a lower precision.

Example: 60Co activity in sediment of an estuary- rp.firth

Suppose wanted to estimate the inventory of 60Co in the sediments of an estuary whose boundaries have been clearly defined. We know that 60Co is particle reactive and we have a map of sediment type in the estuary.

How would we make use of this information?

Stratified sampling chosen

Example: tritium in a South Pacific atoll- rp.mururoa

Suppose wanted to map the distribution of tritium inside and atoll (tritium in sea water). This will require us to collect water samples. How should we do this?

Transect sampling chosen

Examples to try

• try the Mururoa and Rosyth examples again and look at the map that is produced and compare it with the truth- try various sampling strategies

• Library(rpanel)• rp.mururoa()• rp.firth()

Summary

• Spatial modelling is an important skill, since much environmental data are spatially registered

• the complexities include describing the spatial correlation structure and considering if there is a temporal component how this can be modelled.