Adaptive sampling is a new sampling design in which sampling regions, defined as "units", are selected based on values of the variables of interest observed during a sampling survey. For example, a region containing endangered species or land contamination would be defined as a unit. Positive identifications meeting a specified criteria within a region are then used for future input. In a conventional sampling design, the selection for a sampling unit does not depend on previous observations made during an initial survey; entire sampling units are selected before any physical sampling in the field ever takes place. Therefore, conventional sampling guarantees that the calculated statistics will be unbiased. To use the adaptive sampling technique, however, different estimators must be implemented to guarantee unbiasedness. (Thompson, 1992)
According to Thompson (1992), there are two advantages to adaptive sampling. The primary advantage is the ability to incorporate population characteristics to obtain more precise estimates of population density. For a given sample size and cost, more valuable information can be obtained than is possible with conventional designs. For example, populations of plants and animals, minerals, and fossil fuels tend to exhibit natural aggregation tendencies due to schooling, flocking, or environmental patchiness. Since the location and shape of these natural aggregations cannot often be predicted, adaptive sampling may provide a way to dramatically increase the effectiveness of the sampling project. A secondary advantage of adaptive sampling is an increase in the yield of important observations (e.g. the number of endangered species observed), which can result in higher quality estimates of parameters such as the mean and variance.
Thompson defines a "neighborhood" as all the surrounding units (or regions) within a given dimension. For example, the figure below represents a one dimensional case. The initial unit is illustrated in blue, the two black units to the right and left of the blue unit are said to be in the "neighborhood" of the blue unit. The two black units belong in the neighborhood of the blue unit whether or not they meet a given criteria.
For a two dimensional case the four black units are in the neighborhood of the blue unit.
Finally, for a three dimensional case the six black units are in the neighborhood of the blue unit.
Adaptive sampling introduces biases into conventional estimators so new unbiased estimators are needed. According to Thompson (1992), if additional units are added to the sampling design wherever high positive identifications are observed, the sample mean will over-estimate the population mean. A method of obtaining unbiased estimators is to make use of new observations in addition to the observations initially selected. (Thompson, 1992)
Adaptive Cluster Sampling
Adaptive cluster sampling was conceived as a response to the problem of sampling clustered, yet rare, populations. An initial random sample of units is selected and whenever the variable of interest satisfies a condition, additional neighboring units are added to the sample. For example, the conditions may be a given set of criteria, or a ranking system using the largest, second largest, and/or third largest, etc., order observation. The following two examples are one-dimensional case scenarios, one with a specified criteria and the other with a ranking criteria. (Thompson, 1996)
Example Using a Specified Criteria
Here is an example of a small population in a one dimensional case. The population consists of 10 units whose y-values are depicted below. The actual population mean is 26.2 units with a variance of 626.6 units^2.
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2 |
20 |
8 |
60 |
48 |
1 |
9 |
15 |
71 |
28 |
The neighborhood of each unit includes all adjacent units. The criteria, "C", for taking additional samples is defined as C = {y: y >= 10 ppm}.
1. An initial random sample yields 3 y-values; these results are shown in red.
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60 |
48 |
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71 |
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2. The units that conform to the criteria are y-values 60 and 71 ppm. These are shown in blue.
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60 |
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71 |
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3. Now, sample to the right and to the left of each unit meeting the criteria. The new units are shown in red.
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60 |
48 |
1 |
9 |
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71 |
28 |
4. Those units meeting the criteria, C = {y: y >= 10}, are shown in green.
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60 |
48 |
1 |
9 |
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71 |
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5. Sampling continues to the left and right of each new unit. These values are shown in red.
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60 |
48 |
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9 |
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71 |
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6. Again, the units meeting the criteria are shown in green.
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60 |
48 |
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7. This is the full table with all of the collected samples.
The estimators for the above example are as follows:
So,
The probability that unit "i" is included in the sample is:
(Note: Partial inclusion probability is assumed and therefore edge units are ignored. Boundary units are included in a network; they do not become edge units if they fall on a boundary).
So, N = 10.
where m1 = 1, since there is only one unit in A1.
So, the unbiased estimator, 
is:
where K = total number of distinct networks in the population, and yk* = sum of y-values for the kth network.
An estimator of can
be described as:
Example Using a Ranking System
Here is an example of a small population in a one dimensional case. The population consists of 10 units whose y-values are depicted below. In this example, sampling is only carried out in the vicinity of the largest order statistic of the initial sample.
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8 |
99 |
55 |
12 |
9 |
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35 |
28 |
1. An initial random sample yields 3 y-values; these results are shown in red. Therefore, their n1 = 3. The neighborhood of each unit consists of itself plus any other adjacent units.
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99 |
55 |
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35 |
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2. The initial sample maximum y(1) is 99.
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99 |
55 |
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35 |
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3. Therefore, units in the neighborhood of 99 are added to the sample.
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99 |
55 |
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35 |
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4. The value 55 exceeds the second largest initial sample value y(2) = 35, the unit 12 is added.
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99 |
55 |
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5. Since the y-value 12 does not exceed y(2) = 35, no new units are added to the sample, and 99 and 55 form a network.
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The estimators for this example are:
Samples 8 and 12 do not intersect the initial sample, so they
receive no weight in the calculation of 
.
The initial sample variance is:
So,
The variance of 
is:
So, 
The variance of 
is:
Systematic and Strip Adaptive Cluster Sampling: 2-Dimensional Cases
Two adaptive sampling designs that are two dimensional cases are the systematic and strip adaptive cluster sampling designs. For both designs, the initial sample is selected in terms of primary units; subsequent additions to the sample are in terms of secondary units. In strip adaptive cluster sampling, an area of potential contamination is cut into strips selected randomly. These strips form the primary units; the square plots form the secondary units. As an example, wherever blue whales are found, the areas to the left and the right of the sightings are sampled. The sampling procedure is repeated up, down, left, and right for the next positive identifications. Sampling stops when no further whales are found based on the original aggregation. Please refer to animation number one below for a graphical illustration. (Thompson, 1992)
Animation number one (and the following animation number two) depict an initial grid of blue dots representing the ultimate number of blue whales present in a given unit (region). The whales that are positively identified in a given unit turn the color red.
Strip Adaptive Cluster Sampling
In systematic adaptive cluster sampling, systematic units are randomly selected based on two sets of diagonally placed secondary square units. Samples are taken up, down, left, and right wherever the blue whales are initially sighted. Sampling stops when there are no more positive identifications. Please refer to animation number two for a graphical illustration of systematic adaptive sampling. (Thompson, 1992)
Systematic Adaptive Cluster Sampling
References
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Student Authors: Mary Rust, mrust@vt.edu , and Katie McArthur, kmcarthu@vt.edu
Faculty Advisor: Daniel Gallagher, dang@vt.edu
Copyright © 1997 Daniel Gallagher
Last Modified: 07-02-1998