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Adaptive SamplingSteve Thompson

thompson@stat.sfu.ca

Simon Fraser University

16-17 June 2011

BANOCOSS 2011

Adaptive Sampling – p. 1/??

Adaptive sampling:

An adaptive sampling design is one in which theselection of units to include in the sample depends onvalues of the variable of interest observed during thesurvey.

Sketch

1. Adaptive sampling ideas and examples

2. Design and inference considerations

3. Spatial, network, temporal settings

Rare, clustered population

Random sample of 40 units

Same population

Initial sample of 20 units

Adaptive cluster sample

Changed population!

Initial sample of 20 units

Types of sampling designs

The procedure by which we select the sample.

Conventional design : p(s)Procedure for selecting the sample does not depend onvalues of variables of interest observed during the survey.

Adaptive design : p(s | y)Procedure for selecting sample can depend on values ofvariables of interest.

(Design can also depend on auxiliary variables x.)

Approaches to inference from samples

Design based approach:The values of the variables of interest in the population arefixed, unknown constants.

y = (y1, . . . , yN )

Model based approach:The population values are random variables, which we tryto model.

Y1, . . . , YN have some joint probability distributionf(y1, . . . , yN | θ).

Trawl survey, Kodiak Island

L. WatsonAdaptive Sampling – p. 14/??

Optimal sampling strategies

Find the design p(s |y) and estimator Z of populationquantity Z to minimize the mean square error

E(Z − Z)2

subject to unbiasedness, E(Z) = E(Z)

The optimal strategy is in most cases an adaptive one.

Reasoning:

1. Stop part way through the survey and look at what hasbeen observed so far:

initial sample and values (s1, ys1)

2. Choose the rest of the sample s2 to minimize the meansquare error of the estimate given what has been observedso far.

(Z − Z)2 | s1,ys1

(Zacks 1969, Thompson and Seber 1996, Chao and Thompson 2000)

The idea:

Given what you’ve observed so far, choose subsequentsample units with good design conditional on that.

Say the sample is in two phases, s = (s0, s1).The data are d = (s, ys).

E[(τ − τ)2 | s0, ys0 ]︸︷︷︸

using current data

≤ mins

E(τ − τ)2

︸︷︷︸

not using

Practical, efficient designs

Theoretically optimal designs are hard to implement,computationally complex, and overly dependent on modelbased assumptions.

We seek instead practical, efficient, robust designs

Adaptive cluster sampling estimation

y is not unbiased for µ

Unbiased estimate has form

∑ yiαi

αi = network intersection probability,

or Rao-Blackwell form.

Sufficiency, completeness, Rao-Blackwell

sampling data = (s, ys)

sufficient statistic = set of distinct units, associated yvalues

Rao-Blackwell estimate =E[simple estimator | sufficient statistic]

Minimal sufficient statistic is not complete so more thanone possible estimator.

Bering Sea king crab surveyJohnson, Chao, Thompson and Stevens - Draft 10/28/2001

Figure 1.| The result of adaptive cluster sampling during 1995 in the Bering Sea. Initiallyselected primary units were 43, 56 and 86. Neighborhood pattern was north, south, eastand west. Mature female red king crabs (Paralithodes camtschaticus) were sampled witha condition of 18 in the southern stratum, and a condition of 60 in the northern stratum.Maturity was determined by presence of embryos or empty egg cases.{ 15 {

Likelihood function

Prob(data | parameters) = P(s,ys | θ)

L(θ; s,ys) =

p(s |y; θ)f(y; θ)dys

(design)(model)d(unobserved)

In general a likelihood function involves both the selectionmechanism (design) and the model and effective inferenceshould take into account both.

“Ignorable” design

If the design depends only on values that are observed andrecorded in the data, then the design disappears fromlikelihood-based estimates.

L(θ; s,ys) = p(s |ys; θ1)

f(y; θ2)dys

But to be ignorable for frequentist model-based inference,the design must be a conventional one p(s), depending onno y-values at all.

It can be argued that in most real situations, the design isignorable for data analysis only if the study used a knownprobability design.

Adaptive sampling in networks

Studies of hidden populations

HIV/AIDS at-risk study

M. Miller

Sampling in networks

Population of units or nodes: 1, 2, . . . , N

Node variables of interest : y1, y2, . . . , yN

Link-indicators or weights: wij , i, j = 1, . . . , N

(Variables of interest associated with pairs of nodes)

Sample : A subset or sequence s of units and pairs of units

from the population: s = (s(1), s(2))y is observed in s(1).w is observed in s(2).

Approaches to inference in network sampling

Design based approach:The values of the variables of interest in the population arefixed, unknown constants.

y = (y1, . . . , yN )

w = {wij}, i, j ∈ {1, . . . , N}

Probability enters only through the design

Model based approach:The population values are random variables, which we tryto model.

Y1, . . . , YN , W11, . . . ,WNN have some joint probabilitydistribution, described by a stochastic graph model

Snowball and Random Walk Designs

1. Snowball designs and inference

2. Random walk designs and inference

Example network population

population graph

Random sample

sample

Snowball sample

sample

Snowball sample

sample

One-wave snowball selection probabilities

0.0 0.2 0.4 0.6 0.8 1.0

One−wave selection probabilities

The population again

population graph

Random walk sample

Random walk limit selection probabilities

Limit random walk probabilities

Random walk as Markov chain

Wk is the node of the graph selected at kth wave.aij = 1 indicates a link from node i to node j.

{W0,W1,W2, . . . } is a Markov chain withP (Wk+1 = j |Wk = i) = aij/ai·

Q is the transition matrix of the chain,qij = P (Wk+1 = j |Wk = i).

The stationary probabilities (π1, . . . , πN ) satisfy πj =∑

πiqijfor j = 1, . . . , N .

Approach using limiting distribution of random walk

For random walk design with-replacement in asingle-component network and if the links are symmetric ,

then the limiting selection probability is proportional to theperson’s degree (di)

Generalized ratio estimator of mean for behavioralcharacteristic y:

s yi/di∑

s 1/di

Targeted random walk designs

1. Uniform random walk

2. More general targetting

Targeted walk designs

Let πi(y) denote the desired stationary selection probabilityfor the ith node as a function of its value or degree.

The transition probabilities for the targeted walk are

Pij = qijαij for i 6= j

Pii = 1−∑

αij = min

{πjqjiπiqij

TARGETED RANDOM WALK DESIGNS

1. Random walk as a Markov chain

2. Random, uniform, and targeted walks

Random walk as Markov chain

Wk is the node of the graph selected at kth wave.aij = 1 indicates a link from node i to node j.

{W0,W1,W2, . . . } is a Markov chain withP (Wk+1 = j |Wk = i) = aij/ai·

Q is the transition matrix of the chain,qij = P (Wk+1 = j |Wk = i).

The stationary probabilities (π1, . . . , πN ) satisfy πj =∑

πiqijfor j = 1, . . . , N .

Targeted walk design

Let πi(y) denote the desired stationary selection probabilityfor the ith node as a function of its value or degree.

The transition probabilities for the targeted walk are

Pij = qijαij for i 6= j

Pii = 1−∑

αij = min

{πjqjiπiqij

Uniform targeted walk design

0.0 0.2 0.4 0.6 0.8 1.0

population

0.0 0.2 0.4 0.6 0.8 1.0

Limit selection propabalities

0.0 0.2 0.4 0.6 0.8 1.0

‘random walk’ sample

0.0 0.2 0.4 0.6 0.8 1.0

uniform walk sample

Random, uniform, and targeted walks

0 10 20 30 40 50

random walk, no jumps

0 10 20 30 40 50

random walk (with jumps)

0 10 20 30 40 50

uniform walk

0 10 20 30 40 50

value 2/1 walk

0 10 20 30 40 50

degree+1 walk

0 10 20 30 40 50

degree walk

Adaptive web sampling

1. How they work

2. Variations

3. Inference methods

At any point in the sampling,

• the next unit or set of units is selected from a distributionthat depends on the values of variables of interest in anactive set of units already selected. (follow a link )

At any point in the sampling,

• the next unit or set of units is selected from a distributionthat depends on the values of variables of interest in anactive set of units already selected. (follow a link )

• With some probability, however, the selection may bemade from a distribution not dependent on thosevalues. (random jump )

Population graph

population graph

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Adaptive web design

weighted links

Inference

Estimation of a population characteristic such as apopulation mean, degree distribution, or other quantity,based on the sample data.

• Design-basedsimple preliminary estimatorimprove with Rao-Blackwell or resampling

Inference

Estimation of a population characteristic such as apopulation mean, degree distribution, or other quantity,based on the sample data.

• Design-basedsimple preliminary estimatorimprove with Rao-Blackwell or resampling

• Model-basedassume stochastic graph modelproduce realizations from predictive posterior

Design-unbiased estimators

• Start with some preliminary unbiased estimator µ0, suchas the initial sample mean , an unequal probabilityestimator , or conditional probability estimator

• Improve it using the Rao-Blackwell method:

µ = E(µ0|d) =∑

µ0(s)p(s | d)

µ = E(µ0|d) =∑

µ0(s)p(s | d)

d is the minimal sufficient statistic

Estimator based on initial sample mean

• Start with unbiased estimator of µ based on the initialsample s0.

For example,µ01 = y0

µ01 =1

i∈s0

Estimator based on initial sample mean

• Start with unbiased estimator of µ based on the initialsample s0.

For example,µ01 = y0

µ01 =1

i∈s0

µ1 = E(µ01|dr) =∑

{s:r(s)=s}

µ01(s)p(s | dr)

Estimators based on conditional probabilities

τs0 , an unbiased estimator of the population total based on the initialsample s0.

For the kth selection after the initial sample, zk =∑

i∈sckyi + yk/qaki

where qaki is the conditional probability of selecting person i given thecurrent active set ak.

An unbiased estimator of the population mean is

µ02 =1

n0τs0 +

i=n0+1

The improved estimator is

µ2 = E(µ02|dr) =∑

{s:r(s)=s}

µ02(s)p(s | dr)

Composite conditional generalized ratio

N0 an estimator of the population size N based on the initial sample, forexample, N0 =

k∈s0(1/πk).

After the initial sample, Nk = nck + 1/qaki, where nck is the size of thecurrent sample.

A composite estimator of N is

n0N0 +

i=n0+1

A generalized ratio estimator is then formed as the ratio of the twounbiased estimators: µ03 = Nµ02/N

The improved version of this estimator is

µ3 = E(µ03|dr) =∑

{s:r(s)=s}

µ03(s)p(s | dr)

Composite conditional mean of ratios

An alternate way to use the ratios of unbiased estimators in a compositeestimator is

µ04 =1

i=n0+1

The improved version of this estimator is

µ4 = E(µ04|dr) =∑

{s:r(s)=s}

µ04(s)p(s | dr)

Computational issue

µ = E(µ0|dr) =∑

{s:r(s)=s}

µ01(s)p(s)

The sum is over all possible sample paths giving dr.

Sample sequence s = (s0, in0+1, . . . , in) has selection probability

p(s) = p0qan0,i1 · · · qan−1in

For a nonreplacement design, n! reordings of the sample.

Computational issue

µ = E(µ0|dr) =∑

{s:r(s)=s}

µ01(s)p(s)

• n = 9 has 362,880 reordings.

Computational issue

µ = E(µ0|dr) =∑

{s:r(s)=s}

µ01(s)p(s)

• n = 10 has 3.6 million.

Computational issue

µ = E(µ0|dr) =∑

{s:r(s)=s}

µ01(s)p(s)

• n = 10 has 3.6 million.

• n = 20 has 2.4 quintillion (1018), as in “million, billion, trillion,quadrillion, quintillion,...”

Markov chain resampling estimators

Let x be a permutation of the sample s.

The object is to obtain a Markov chain x0, x1, x2, . . . having stationarydistribution p(x | dr).

1. A tentative permutation tk is produced by applying the originalsampling design to the data as if the sample were the wholepopulation.

2. With probability α, tk is accepted and xk = tk, while with probability1− α, tk is rejected and xk = xk−1, where

α = min

{p(tk)

p(xk−1)

pt(xk−1)

pt(tk), 1

Markov chain resampling estimators

Let x be a permutation of the sample s.

The object is to obtain a Markov chain x0, x1, x2, . . . having stationarydistribution p(x | dr).

1. A tentative permutation tk is produced by applying the originalsampling design to the data as if the sample were the wholepopulation.

2. With probability α, tk is accepted and xk = tk, while with probability1− α, tk is rejected and xk = xk−1, where

α = min

{p(tk)

p(xk−1)

pt(xk−1)

pt(tk), 1

Spatial adaptive web sampling

spatial population

Network structure of spatial population

population graph

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

Adaptive web sample

sample

The resulting spatial sample

spatial population

Active set design variations

spatial population

population graph

active set sample active set sample

Blue-winged teal population

spatial population

10 103

13639 1

14 122

1772 00000000

00000000

000000

00000000

population graph

Two samples, n=20. Top:n0 = 13. Bottom: n0 = 1.

sample

MSE of estimators depending onn0, with n = 20

5 10 15 20

µ1µ2µ3µ4

Model-based inference with network designs

(Work with Ove Frank, Mosuk Chow, Mike Kwanisai, andothers).

Bayes predictive inference

Inference about population characteristics based on theBayes predictive posterior distribution given the datad = (s,ys,ws)

P (ys,ws | d) =

P (ys,ws , θ, β | d) dθ dβ

Based on an assumed stochastic graph model f(y,w; θ, β),θ = node paramaters, β = link parameters.

Sampling from predictive posterior

The object is to produce many realizations of the entirepopulation from the posterior distribution given the sampledata.

This is the data augmentation step of a Markov chain MonteCarlo procedure.

Within Bayes: The likelihood function

The likelihood function depends on both the design used inobtaining the data and the model describing the population.

Prob(data | parameters) = P(s,ys,ws | θ, β)

L(θ, β; s,ys,ws) =

p(s |y,w; θ, β)f(y,w; θ, β)dysdws

(design)(model)d(unobserved)

MCMC for network Bayes inference

1. Using current values of θ and β, select a realization of(ys,ws) from P (ys,ws | θ, β, s,ys,ws).

2. Using the values (ys,ws) obtained in step (1) toaugment the data values (ys,ws), select new parametervalues (θ, β) from the posterior distribution of theparameters given the whole graph realizationπ(θ, β |ys,ys,ws,ws)

Repeat.

Bayes predictive inference; Actual pattern

0.0 0.2 0.4 0.6 0.8 1.0

coord1

actual pattern in region

Sample and observed values

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample and observed values

Realization from posterior distribution

0.0 0.2 0.4 0.6 0.8 1.0

coord1

inferred possible pattern, given data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

Median of posterior distribution

0.0 0.2 0.4 0.6 0.8 1.0

coord1

median of possible patterns, given data

Systematic sample, 16 sites

0.0 0.2 0.4 0.6 0.8 1.0

spatial covariance

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

conditional realization

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

conditional median

Systematic sample, 4 sites

0.0 0.2 0.4 0.6 0.8 1.0

spatial covariance

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

conditional median

Random sample, 16 sites

0.0 0.2 0.4 0.6 0.8 1.0

spatial covariance

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

conditional median

Random sample, 16 sites

0.0 0.2 0.4 0.6 0.8 1.0

spatial covariance

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

conditional median

MCMC data augmentation steps

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1[ss]

sample data

0.0 0.2 0.4 0.6 0.8 1.0

coord1

mcmc augmented data 1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

0.0 0.2 0.4 0.6 0.8 1.0

coord1

Design and estimation comparisons

population graph

Random walk n=20, initial pp-degree

sample mean, random walk

rwmean

0.0 0.2 0.4 0.6 0.8 1.0

mean of draws, random walk

rwnommean

0.0 0.2 0.4 0.6 0.8 1.0

gen ratio est, random walk

rwnaive

0.0 0.2 0.4 0.6 0.8 1.0

gen ratio of draws, random walk

rwnaivenom

0.0 0.2 0.4 0.6 0.8 1.0

5 random walks, n=4 each, pp-deg starts

rwmean

0.0 0.2 0.4 0.6 0.8 1.0

rwnommean

0.0 0.2 0.4 0.6 0.8 1.0

rwnaive

0.0 0.2 0.4 0.6 0.8 1.0

rwnaivenom

0.0 0.2 0.4 0.6 0.8 1.0

random walk, n=20 , equal probability start

rwmean

0.0 0.2 0.4 0.6 0.8 1.0

rwnommean

0.0 0.2 0.4 0.6 0.8 1.0

rwnaive

0.0 0.2 0.4 0.6 0.8 1.0

rwnaivenom

0.0 0.2 0.4 0.6 0.8 1.0

AWS, n0=1, n=20, random links, jump=.1

generalized ratio estimate 1

0.0 0.2 0.4 0.6 0.8 1.0

AWS, n0=10, n=20, random links, jump=.1

0.0 0.2 0.4 0.6 0.8 1.0

Design and model based estimators, AWS n0=10, n=20

sample mean

0.0 0.2 0.4 0.6 0.8 1.0

rb initial mean

rbw0vec

0.0 0.2 0.4 0.6 0.8 1.0

rb norepl est

rbnoreplvec

0.0 0.2 0.4 0.6 0.8 1.0

rb nr alt

rbnraltvec

0.0 0.2 0.4 0.6 0.8 1.00

rb nr alt0

rbnraltvec0

0.0 0.2 0.4 0.6 0.8 1.0

bayes predictor

bayes.predtvec

0.0 0.2 0.4 0.6 0.8 1.0

Designs and Estimators

grhh grht gre est1 est3 gre est1 est3 bayes

DESIGN

Random WalkAWS n0=1, n=20AWS n0=10, n=20

Empirical Example

HIV/AIDS at-risk hidden population: Colorado SpringsStudy on the heterosexual transmission of HIV/AIDS(Potterat et al. 1993, Rothenberg et al. 1995, Darrow et al.1999)

Colorado springs study population

population graph

Sample of 80 individuals

Initial n0 = 10, final n = 20, m = 4 independent selections.

samplesamplesamplesample

Design and Model based inferences

sample mean

s.mean

0.0 0.2 0.4 0.6 0.8 1.0

mle.theta

0.0 0.2 0.4 0.6 0.8 1.0

bayes estimator

bayes.t

0.0 0.2 0.4 0.6 0.8 1.0

bayes predictor

pred.tD

0.0 0.2 0.4 0.6 0.8 1.0

design unbiased

dunbiased

0.0 0.2 0.4 0.6 0.8 1.0

design consistent

dconst

0.0 0.2 0.4 0.6 0.8 1.0

(Sex worker data) Adaptive Sampling – p. 95/??

HIV Behavioral Monitoring Design Study

population graph

Design-based and Bayes estimators

sample mean

0.0 0.2 0.4 0.6 0.8 1.0

rb initial mean

rbw0vec

0.0 0.2 0.4 0.6 0.8 1.0

rb norepl est

rbnoreplvec

0.0 0.2 0.4 0.6 0.8 1.0

rb nr alt

rbnraltvec

0.0 0.2 0.4 0.6 0.8 1.0

rb nr alt0

rbnraltvec0

0.0 0.2 0.4 0.6 0.8 1.0

bayes predictor

bayes.predtvec

0.0 0.2 0.4 0.6 0.8 1.0

Empirical Example

HIV/AIDS at-risk hidden population: Colorado SpringsStudy on the heterosexual transmission of HIV/AIDS(Potterat et al. 1993, Rothenberg et al. 1995, Darrow et al.1999)

Colorado springs study population

population graph

Sample of 80 individuals

Initial n0 = 10, final n = 20, m = 4 independent selections.

samplesamplesamplesample

Estimating idu use, random links design

initial mean

0.0 0.2 0.4 0.6 0.8 1.0 1.2

4rb initial mean

rbw0vec

0.0 0.2 0.4 0.6 0.8 1.0 1.2

norepl est

norepl.est

0.0 0.2 0.4 0.6 0.8 1.0 1.2

rb norepl est

rbnoreplvec

0.0 0.2 0.4 0.6 0.8 1.0 1.2

norepl alt

noreplalt

0.0 0.2 0.4 0.6 0.8 1.0 1.2

rb nr alt

rbnraltvecD

0.0 0.2 0.4 0.6 0.8 1.0 1.2

norepl alt0

noreplalt0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

rb nr alt0

rbnraltvec0

0.0 0.2 0.4 0.6 0.8 1.0 1.2

Estimating idu use, weighted links design

initial mean

0.0 0.5 1.0 1.5

4rb initial mean

rbw0vec

0.0 0.5 1.0 1.5

norepl est

norepl.est

0.0 0.5 1.0 1.5

rb norepl est

rbnoreplvec

0.0 0.5 1.0 1.5

norepl alt

noreplalt

0.0 0.5 1.0 1.5

rb nr alt

rbnraltvecD

0.0 0.5 1.0 1.5

norepl alt0

noreplalt0

0.0 0.5 1.0 1.5

rb nr alt0

rbnraltvec0

0.0 0.5 1.0 1.5

Degree distribution, HIV/AIDS study

0 5 10 15 20

degree distribution

degree

0.0 0.5 1.0 1.5 2.0 2.5 3.0

log(degree)

0 5 10 15 20

sample degree distribution

degree

0.0 0.5 1.0 1.5 2.0 2.5 3.0

log(degree)

Estimating mean degree

average degree in sample

degree

0 2 4 6 8

estimate of mean degree

rbw0vec

0 2 4 6 80

Design and Model based inferences

sample mean

s.mean

0.0 0.2 0.4 0.6 0.8 1.0

mle.theta

0.0 0.2 0.4 0.6 0.8 1.0

bayes estimator

bayes.t

0.0 0.2 0.4 0.6 0.8 1.0

bayes predictor

pred.tD

0.0 0.2 0.4 0.6 0.8 1.0

design unbiased

dunbiased

0.0 0.2 0.4 0.6 0.8 1.0

design consistent

dconst

0.0 0.2 0.4 0.6 0.8 1.0

(Sex worker data) Adaptive Sampling – p. 105/??

Spatial-Temporal designs

For detecting releases of biological pathogens and otherairborne health hazards,

is it better to set out sensors in fixed positions or to havethem move in some pattern?

The more basic sampling question:

What is the best design for sampling a population that ischanging, when the sampling units themselves may moveas observations are collected.

Background

Early health warning of exposure to airborne biologicalpathogens

Bionet Project places fixed sensor units in selected cities

Builds on the Envionmental Protection Agency’s air qualitymonitoring program

Purpose:

• Rapid health response - earlier diagnosis and treatment

Purpose:

• Rapid health response - earlier diagnosis and treatment

• Environmental remediation

“Streets and avenues design”

Array of rectangular (square) paths

Sample size, moving units

4 6 8 10 12 14 16

streets

squareslines

one line

Sample size, fixed units

10 20 30 40 50 60

moving

Adaptive designs in space-time-network settings

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