adaptive sampling - helsingin yliopisto - matematiikan ja ... · adaptive sampling: an adaptive...
TRANSCRIPT
![Page 1: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/1.jpg)
Adaptive SamplingSteve Thompson
Simon Fraser University
16-17 June 2011
BANOCOSS 2011
Adaptive Sampling – p. 1/??
![Page 2: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/2.jpg)
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.
Adaptive Sampling – p. 2/??
![Page 3: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/3.jpg)
Sketch
1. Adaptive sampling ideas and examples
2. Design and inference considerations
3. Spatial, network, temporal settings
Adaptive Sampling – p. 3/??
![Page 4: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/4.jpg)
Rare, clustered population
Adaptive Sampling – p. 4/??
![Page 5: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/5.jpg)
Random sample of 40 units
Adaptive Sampling – p. 5/??
![Page 6: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/6.jpg)
Same population
Adaptive Sampling – p. 6/??
![Page 7: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/7.jpg)
Initial sample of 20 units
Adaptive Sampling – p. 7/??
![Page 8: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/8.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 8/??
![Page 9: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/9.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 8/??
![Page 10: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/10.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 8/??
![Page 11: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/11.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 8/??
![Page 12: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/12.jpg)
Changed population!
Adaptive Sampling – p. 9/??
![Page 13: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/13.jpg)
Initial sample of 20 units
Adaptive Sampling – p. 10/??
![Page 14: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/14.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 11/??
![Page 15: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/15.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 11/??
![Page 16: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/16.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 11/??
![Page 17: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/17.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 11/??
![Page 18: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/18.jpg)
Adaptive cluster sample
Adaptive Sampling – p. 11/??
![Page 19: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/19.jpg)
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.)
Adaptive Sampling – p. 12/??
![Page 20: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/20.jpg)
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).
Adaptive Sampling – p. 13/??
![Page 21: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/21.jpg)
Trawl survey, Kodiak Island
L. WatsonAdaptive Sampling – p. 14/??
![Page 22: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/22.jpg)
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.
Adaptive Sampling – p. 15/??
![Page 23: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/23.jpg)
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.
minE[
(Z − Z)2 | s1,ys1
]
(Zacks 1969, Thompson and Seber 1996, Chao and Thompson 2000)
Adaptive Sampling – p. 16/??
![Page 24: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/24.jpg)
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
mins1
E[(τ − τ)2 | s0, ys0 ]︸ ︷︷ ︸
using current data
≤ mins
E(τ − τ)2
︸ ︷︷ ︸
not using
Adaptive Sampling – p. 17/??
![Page 25: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/25.jpg)
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 Sampling – p. 18/??
![Page 26: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/26.jpg)
Adaptive cluster sampling estimation
y is not unbiased for µ
Unbiased estimate has form
∑ yiαi
αi = network intersection probability,
or Rao-Blackwell form.
Adaptive Sampling – p. 19/??
![Page 27: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/27.jpg)
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.
Adaptive Sampling – p. 20/??
![Page 28: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/28.jpg)
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 {
Adaptive Sampling – p. 21/??
![Page 29: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/29.jpg)
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.
Adaptive Sampling – p. 22/??
![Page 30: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/30.jpg)
“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 – p. 23/??
![Page 31: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/31.jpg)
Adaptive sampling in networks
Adaptive Sampling – p. 24/??
![Page 32: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/32.jpg)
Studies of hidden populations
HIV/AIDS at-risk study
M. Miller
Adaptive Sampling – p. 25/??
![Page 33: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/33.jpg)
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).
Adaptive Sampling – p. 26/??
![Page 34: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/34.jpg)
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
Adaptive Sampling – p. 27/??
![Page 35: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/35.jpg)
Snowball and Random Walk Designs
1. Snowball designs and inference
2. Random walk designs and inference
Adaptive Sampling – p. 28/??
![Page 36: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/36.jpg)
Example network population
population graph
Adaptive Sampling – p. 29/??
![Page 37: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/37.jpg)
Random sample
sample
Adaptive Sampling – p. 30/??
![Page 38: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/38.jpg)
Snowball sample
sample
Adaptive Sampling – p. 31/??
![Page 39: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/39.jpg)
Snowball sample
sample
Adaptive Sampling – p. 31/??
![Page 40: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/40.jpg)
One-wave snowball selection probabilities
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
One−wave selection probabilities
Adaptive Sampling – p. 32/??
![Page 41: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/41.jpg)
The population again
population graph
Adaptive Sampling – p. 33/??
![Page 42: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/42.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 43: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/43.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 44: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/44.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 45: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/45.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 46: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/46.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 47: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/47.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 48: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/48.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 49: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/49.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 50: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/50.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 51: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/51.jpg)
Random walk sample
walk
Adaptive Sampling – p. 34/??
![Page 52: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/52.jpg)
Random walk limit selection probabilities
Limit random walk probabilities
Adaptive Sampling – p. 35/??
![Page 53: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/53.jpg)
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 .
Adaptive Sampling – p. 36/??
![Page 54: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/54.jpg)
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:
µ =
∑
s yi/di∑
s 1/di
Adaptive Sampling – p. 37/??
![Page 55: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/55.jpg)
Targeted random walk designs
1. Uniform random walk
2. More general targetting
Adaptive Sampling – p. 38/??
![Page 56: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/56.jpg)
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−∑
j 6=i
Pij
where
αij = min
{πjqjiπiqij
, 1
}
Adaptive Sampling – p. 39/??
![Page 57: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/57.jpg)
TARGETED RANDOM WALK DESIGNS
1. Random walk as a Markov chain
2. Random, uniform, and targeted walks
Adaptive Sampling – p. 40/??
![Page 58: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/58.jpg)
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 .
Adaptive Sampling – p. 41/??
![Page 59: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/59.jpg)
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−∑
j 6=i
Pij
where
αij = min
{πjqjiπiqij
, 1
}
Adaptive Sampling – p. 42/??
![Page 60: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/60.jpg)
Uniform targeted walk design
0.0 0.2 0.4 0.6 0.8 1.0
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
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
0.0
0.2
0.4
0.6
0.8
1.0
1
‘random walk’ sample
2
3
4
5
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1
uniform walk sample
2
3 4
5
Adaptive Sampling – p. 43/??
![Page 61: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/61.jpg)
Random, uniform, and targeted walks
0 10 20 30 40 50
0.10
0.20
0.30
0.40
random walk, no jumps
wave
expe
cted
nod
e va
lue
0 10 20 30 40 50
0.10
0.20
0.30
0.40
random walk (with jumps)
wave
expe
cted
nod
e va
lue
0 10 20 30 40 50
0.10
0.20
0.30
0.40
uniform walk
wave
expe
cted
nod
e va
lue
0 10 20 30 40 50
0.10
0.20
0.30
0.40
value 2/1 walk
wave
expe
cted
nod
e va
lue
0 10 20 30 40 50
0.10
0.20
0.30
0.40
degree+1 walk
expe
cted
nod
e va
lue
0 10 20 30 40 50
0.10
0.20
0.30
0.40
degree walk
expe
cted
nod
e va
lue
Adaptive Sampling – p. 44/??
![Page 62: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/62.jpg)
Adaptive web sampling
1. How they work
2. Variations
3. Inference methods
Adaptive Sampling – p. 45/??
![Page 63: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/63.jpg)
Adaptive web sampling
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 )
Adaptive Sampling – p. 46/??
![Page 64: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/64.jpg)
Adaptive web sampling
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 )
Adaptive Sampling – p. 46/??
![Page 65: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/65.jpg)
Population graph
population graph
Adaptive Sampling – p. 47/??
![Page 66: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/66.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 67: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/67.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 68: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/68.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 69: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/69.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 70: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/70.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 71: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/71.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 72: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/72.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 73: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/73.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 74: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/74.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 75: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/75.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 76: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/76.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 77: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/77.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 78: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/78.jpg)
Adaptive web design
weighted links
Adaptive Sampling – p. 48/??
![Page 79: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/79.jpg)
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
Adaptive Sampling – p. 49/??
![Page 80: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/80.jpg)
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
Adaptive Sampling – p. 49/??
![Page 81: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/81.jpg)
Design-unbiased estimators
• Start with some preliminary unbiased estimator µ0, suchas the initial sample mean , an unequal probabilityestimator , or conditional probability estimator
Adaptive Sampling – p. 50/??
![Page 82: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/82.jpg)
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) =∑
paths
µ0(s)p(s | d)
Adaptive Sampling – p. 50/??
![Page 83: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/83.jpg)
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) =∑
paths
µ0(s)p(s | d)
d is the minimal sufficient statistic
Adaptive Sampling – p. 50/??
![Page 84: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/84.jpg)
Estimator based on initial sample mean
• Start with unbiased estimator of µ based on the initialsample s0.
For example,µ01 = y0
or
µ01 =1
N
∑
i∈s0
yiπi
Adaptive Sampling – p. 51/??
![Page 85: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/85.jpg)
Estimator based on initial sample mean
• Start with unbiased estimator of µ based on the initialsample s0.
For example,µ01 = y0
or
µ01 =1
N
∑
i∈s0
yiπi
• Improve it using the Rao-Blackwell method:
µ1 = E(µ01|dr) =∑
{s:r(s)=s}
µ01(s)p(s | dr)
Adaptive Sampling – p. 51/??
![Page 86: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/86.jpg)
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
Nn
[
n0τs0 +
n∑
i=n0+1
zi
]
The improved estimator is
µ2 = E(µ02|dr) =∑
{s:r(s)=s}
µ02(s)p(s | dr)
Adaptive Sampling – p. 52/??
![Page 87: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/87.jpg)
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
N =1
n
[
n0N0 +
n∑
i=n0+1
Ni
]
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)
Adaptive Sampling – p. 53/??
![Page 88: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/88.jpg)
Composite conditional mean of ratios
An alternate way to use the ratios of unbiased estimators in a compositeestimator is
µ04 =1
n
[
n0z0
N0
+
n∑
i=n0+1
zi
Ni
]
The improved version of this estimator is
µ4 = E(µ04|dr) =∑
{s:r(s)=s}
µ04(s)p(s | dr)
Adaptive Sampling – p. 54/??
![Page 89: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/89.jpg)
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.
Adaptive Sampling – p. 55/??
![Page 90: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/90.jpg)
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.
• n = 9 has 362,880 reordings.
Adaptive Sampling – p. 55/??
![Page 91: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/91.jpg)
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.
• n = 9 has 362,880 reordings.
• n = 10 has 3.6 million.
Adaptive Sampling – p. 55/??
![Page 92: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/92.jpg)
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.
• n = 9 has 362,880 reordings.
• n = 10 has 3.6 million.
• n = 20 has 2.4 quintillion (1018), as in “million, billion, trillion,quadrillion, quintillion,...”
Adaptive Sampling – p. 55/??
![Page 93: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/93.jpg)
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
}
Adaptive Sampling – p. 56/??
![Page 94: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/94.jpg)
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
}
Adaptive Sampling – p. 57/??
![Page 95: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/95.jpg)
Spatial adaptive web sampling
spatial population
Adaptive Sampling – p. 58/??
![Page 96: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/96.jpg)
Network structure of spatial population
population graph
Adaptive Sampling – p. 59/??
![Page 97: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/97.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 98: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/98.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 99: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/99.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 100: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/100.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 101: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/101.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 102: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/102.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 103: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/103.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 104: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/104.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 105: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/105.jpg)
Adaptive web sample
sample
Adaptive Sampling – p. 60/??
![Page 106: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/106.jpg)
The resulting spatial sample
spatial population
Adaptive Sampling – p. 61/??
![Page 107: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/107.jpg)
Active set design variations
spatial population
population graph
active set sample active set sample
Adaptive Sampling – p. 62/??
![Page 108: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/108.jpg)
Blue-winged teal population
spatial population
3 5
24 14
2 3 2
10 103
13639 1
14 122
1772 00000000
00000000
00000
000000
00000000
population graph
Adaptive Sampling – p. 63/??
![Page 109: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/109.jpg)
Two samples, n=20. Top:n0 = 13. Bottom: n0 = 1.
sample
sample
Adaptive Sampling – p. 64/??
![Page 110: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/110.jpg)
MSE of estimators depending onn0, with n = 20
5 10 15 20
0.6
0.8
1.0
1.2
1.4
1.6
1.8
n0
mse
µ1µ2µ3µ4
Adaptive Sampling – p. 65/??
![Page 111: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/111.jpg)
Model-based inference with network designs
(Work with Ove Frank, Mosuk Chow, Mike Kwanisai, andothers).
Adaptive Sampling – p. 66/??
![Page 112: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/112.jpg)
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.
Adaptive Sampling – p. 67/??
![Page 113: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/113.jpg)
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.
Adaptive Sampling – p. 68/??
![Page 114: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/114.jpg)
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)
Adaptive Sampling – p. 69/??
![Page 115: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/115.jpg)
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.
Adaptive Sampling – p. 70/??
![Page 116: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/116.jpg)
Bayes predictive inference; Actual pattern
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
Adaptive Sampling – p. 71/??
![Page 117: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/117.jpg)
Sample and observed values
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample and observed values
Adaptive Sampling – p. 72/??
![Page 118: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/118.jpg)
Realization from posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
inferred possible pattern, given data
Adaptive Sampling – p. 73/??
![Page 119: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/119.jpg)
Realization from posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
inferred possible pattern, given data
Adaptive Sampling – p. 74/??
![Page 120: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/120.jpg)
Realization from posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
inferred possible pattern, given data
Adaptive Sampling – p. 75/??
![Page 121: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/121.jpg)
Realization from posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
inferred possible pattern, given data
Adaptive Sampling – p. 76/??
![Page 122: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/122.jpg)
Realization from posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
inferred possible pattern, given data
Adaptive Sampling – p. 77/??
![Page 123: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/123.jpg)
Median of posterior distribution
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
median of possible patterns, given data
Adaptive Sampling – p. 78/??
![Page 124: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/124.jpg)
Systematic sample, 16 sites
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
exp
((−
(x/d
elta
)^2
))
spatial covariance
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample data
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional median
Adaptive Sampling – p. 79/??
![Page 125: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/125.jpg)
Systematic sample, 4 sites
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
exp
((−
(x/d
elta
)^2
))
spatial covariance
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample data
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional median
Adaptive Sampling – p. 80/??
![Page 126: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/126.jpg)
Random sample, 16 sites
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
exp
((−
(x/d
elta
)^2
))
spatial covariance
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample data
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional median
Adaptive Sampling – p. 81/??
![Page 127: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/127.jpg)
Random sample, 16 sites
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
x
exp
((−
(x/d
elta
)^2
))
spatial covariance
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample data
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional realization
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
conditional median
Adaptive Sampling – p. 82/??
![Page 128: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/128.jpg)
MCMC data augmentation steps
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
actual pattern in region
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1[ss]
co
ord
2[s
s]
sample data
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 1
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 2
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 3
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 4
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 5
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 6
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.2
0.4
0.6
0.8
1.0
coord1
co
ord
2
mcmc augmented data 7
Adaptive Sampling – p. 83/??
![Page 129: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/129.jpg)
Design and estimation comparisons
population graph
Adaptive Sampling – p. 84/??
![Page 130: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/130.jpg)
Random walk n=20, initial pp-degree
sample mean, random walk
rwmean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
mean of draws, random walk
rwnommean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
gen ratio est, random walk
rwnaive
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
gen ratio of draws, random walk
rwnaivenom
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Adaptive Sampling – p. 85/??
![Page 131: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/131.jpg)
5 random walks, n=4 each, pp-deg starts
sample mean, random walk
rwmean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
mean of draws, random walk
rwnommean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
2.5
gen ratio est, random walk
rwnaive
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
gen ratio of draws, random walk
rwnaivenom
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
Adaptive Sampling – p. 86/??
![Page 132: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/132.jpg)
random walk, n=20 , equal probability start
sample mean, random walk
rwmean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
mean of draws, random walk
rwnommean
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
gen ratio est, random walk
rwnaive
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
gen ratio of draws, random walk
rwnaivenom
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
6
Adaptive Sampling – p. 87/??
![Page 133: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/133.jpg)
AWS, n0=1, n=20, random links, jump=.1
generalized ratio estimate 1
dgre1
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
generalized ratio estimate 2
dgre2
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Adaptive Sampling – p. 88/??
![Page 134: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/134.jpg)
AWS, n0=10, n=20, random links, jump=.1
generalized ratio estimate 1
dgre1
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
generalized ratio estimate 2
dgre2
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
4
Adaptive Sampling – p. 89/??
![Page 135: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/135.jpg)
Design and model based estimators, AWS n0=10, n=20
sample mean
ybar
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
rb initial mean
rbw0vec
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
rb norepl est
rbnoreplvec
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
rb nr alt
rbnraltvec
De
nsity
0.0 0.2 0.4 0.6 0.8 1.00
.01
.02
.03
.0
rb nr alt0
rbnraltvec0
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
bayes predictor
bayes.predtvec
De
nsity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
Adaptive Sampling – p. 90/??
![Page 136: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/136.jpg)
Designs and Estimators
grhh grht gre est1 est3 gre est1 est3 bayes
0.0
00
.02
0.0
40
.06
0.0
80
.10
0.1
20
.14
DESIGN
Random WalkAWS n0=1, n=20AWS n0=10, n=20
Adaptive Sampling – p. 91/??
![Page 137: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/137.jpg)
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)
Adaptive Sampling – p. 92/??
![Page 138: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/138.jpg)
Colorado springs study population
population graph
Adaptive Sampling – p. 93/??
![Page 139: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/139.jpg)
Sample of 80 individuals
Initial n0 = 10, final n = 20, m = 4 independent selections.
samplesamplesamplesample
Adaptive Sampling – p. 94/??
![Page 140: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/140.jpg)
Design and Model based inferences
sample mean
s.mean
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
mle
mle.theta
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
bayes estimator
bayes.t
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
bayes predictor
pred.tD
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
design unbiased
dunbiased
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
design consistent
dconst
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
(Sex worker data) Adaptive Sampling – p. 95/??
![Page 141: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/141.jpg)
HIV Behavioral Monitoring Design Study
population graph
Adaptive Sampling – p. 96/??
![Page 142: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/142.jpg)
Design-based and Bayes estimators
sample mean
ybar
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
0.5
1.0
1.5
2.0
rb initial mean
rbw0vec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
rb norepl est
rbnoreplvec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
rb nr alt
rbnraltvec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
rb nr alt0
rbnraltvec0
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
bayes predictor
bayes.predtvec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
0.0
1.0
2.0
3.0
Adaptive Sampling – p. 97/??
![Page 143: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/143.jpg)
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)
Adaptive Sampling – p. 98/??
![Page 144: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/144.jpg)
Colorado springs study population
population graph
Adaptive Sampling – p. 99/??
![Page 145: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/145.jpg)
Sample of 80 individuals
Initial n0 = 10, final n = 20, m = 4 independent selections.
samplesamplesamplesample
Adaptive Sampling – p. 100/??
![Page 146: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/146.jpg)
Estimating idu use, random links design
initial mean
y0
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
02
4rb initial mean
rbw0vec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
02
4
norepl est
norepl.est
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
1.0
2.0
rb norepl est
rbnoreplvec
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
1.0
2.0
norepl alt
noreplalt
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
0.0
1.5
3.0
rb nr alt
rbnraltvecD
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0 1.2
02
4
norepl alt0
noreplalt0
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
02
4
rb nr alt0
rbnraltvec0
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0 1.2
02
4
Adaptive Sampling – p. 101/??
![Page 147: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/147.jpg)
Estimating idu use, weighted links design
initial mean
y0
Den
sity
0.0 0.5 1.0 1.5
02
4rb initial mean
rbw0vec
Den
sity
0.0 0.5 1.0 1.5
02
4
norepl est
norepl.est
Den
sity
0.0 0.5 1.0 1.5
0.0
1.0
2.0
rb norepl est
rbnoreplvec
Den
sity
0.0 0.5 1.0 1.5
0.0
1.0
2.0
norepl alt
noreplalt
Den
sity
0.0 0.5 1.0 1.5
01
23
rb nr alt
rbnraltvecD
ensi
ty
0.0 0.5 1.0 1.5
01
23
4
norepl alt0
noreplalt0
Den
sity
0.0 0.5 1.0 1.5
02
46
rb nr alt0
rbnraltvec0
Den
sity
0.0 0.5 1.0 1.5
02
46
Adaptive Sampling – p. 102/??
![Page 148: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/148.jpg)
Degree distribution, HIV/AIDS study
0 5 10 15 20
0.0
0.1
0.2
0.3
0.4
degree distribution
degree
freq
uenc
y
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−6
−5
−4
−3
−2
−1
log(degree)
log(
freq
)
0 5 10 15 20
0.0
0.1
0.2
0.3
0.4
sample degree distribution
degree
freq
uenc
y
0.0 0.5 1.0 1.5 2.0 2.5 3.0
−6
−5
−4
−3
−2
−1
log(degree)
log(
freq
)
Adaptive Sampling – p. 103/??
![Page 149: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/149.jpg)
Estimating mean degree
average degree in sample
degree
De
nsity
0 2 4 6 8
0.0
0.1
0.2
0.3
0.4
estimate of mean degree
rbw0vec
De
nsity
0 2 4 6 80
.00
.10
.20
.30
.40
.50
.6
Adaptive Sampling – p. 104/??
![Page 150: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/150.jpg)
Design and Model based inferences
sample mean
s.mean
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
mle
mle.theta
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
bayes estimator
bayes.t
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
bayes predictor
pred.tD
ensi
ty
0.0 0.2 0.4 0.6 0.8 1.0
01
23
45
67
design unbiased
dunbiased
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
design consistent
dconst
Den
sity
0.0 0.2 0.4 0.6 0.8 1.0
02
46
(Sex worker data) Adaptive Sampling – p. 105/??
![Page 151: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/151.jpg)
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?
Adaptive Sampling – p. 106/??
![Page 152: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/152.jpg)
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.
Adaptive Sampling – p. 107/??
![Page 153: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/153.jpg)
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
Adaptive Sampling – p. 108/??
![Page 154: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/154.jpg)
Purpose:
• Rapid health response - earlier diagnosis and treatment
Adaptive Sampling – p. 109/??
![Page 155: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/155.jpg)
Purpose:
• Rapid health response - earlier diagnosis and treatment
• Environmental remediation
Adaptive Sampling – p. 109/??
![Page 156: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/156.jpg)
“Streets and avenues design”
Adaptive Sampling – p. 110/??
![Page 157: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/157.jpg)
Array of rectangular (square) paths
Adaptive Sampling – p. 111/??
![Page 158: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/158.jpg)
Sample size, moving units
4 6 8 10 12 14 16
0.70
0.75
0.80
0.85
0.90
0.95
1.00
n
prob
det
ect
streets
squareslines
one line
Adaptive Sampling – p. 112/??
![Page 159: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/159.jpg)
Sample size, fixed units
10 20 30 40 50 60
0.2
0.4
0.6
0.8
1.0
n
prob
det
ect
fixed
moving
Adaptive Sampling – p. 113/??
![Page 160: Adaptive Sampling - Helsingin yliopisto - Matematiikan ja ... · Adaptive sampling: An adaptive sampling design is one in which the selection of units to include in the sample depends](https://reader031.vdocuments.mx/reader031/viewer/2022030615/5ae00a117f8b9a1c248cba8f/html5/thumbnails/160.jpg)
Adaptive designs in space-time-network settings
Adaptive Sampling – p. 114/??