*4.10. Adaptive Sampling for Pollution 'Hot Spots'*

The population mean concentration of the chemical pollutant will be estimated by identifying hotspots. Some clusters may be overlooked if basic random sampling is used. The sample mean, while unbiased as a population mean estimate, will have a substantial variance. In this circumstance, adaptive sampling is a viable alternative. In this case, the sampling procedure's direction at any stage is influenced at least in part by the information gathered in prior samplings.

The sampling procedure is as follows. Take a random sample of a certain size from the study area. Return and sample every unit adjacent to the contaminated unit if any of the selected units reveals contamination. If any neighboring units exhibit contamination, sample their neighbors, and so on, until each detected cluster has a clean boundary.

The total sample size is unknown in advance, however, the accuracy of the outcome will overcome this disadvantage. However, if the resulting data are evaluated naively, this strategy will produce erroneous estimates of population parameters. To avoid this, the authors of [147] outlined a sampling theory, i.e., employed a useful strategy for selecting the initial sample in clusters and stratifying those samples. Then, using modified Horvitz–Thompson or Hansen–Hurwitz estimators, unbiased estimators of the unknown population's mean can be obtained. These estimators, such as the mean of the initial sample, are unbiased, but they do not always have the lowest variance. The Rao–Blackwell theorem can be used to improve them.

## *4.11. Trend Analysis*

Analysis of trends in environmental science leads to adjustments for autoregressive effects or other spatial-temporal correlations in the data. This is another important area of environmental trend analysis (see [148]). Any data that posses time-dependency will lead to auto-correlation and then to time series analysis.

## *4.12. Ecological Modeling*

In building stochastic models of vertebrate populations, statistics have an useful interaction with fisheries and wildlife sciences. Analyzing the survival of the northern spotted owl after it experiences habitat loss and employing the well-known Leslie–Lefkovitch model suggested by [149] is an example. The model uses information about survival and fecundity in a matrix framework to predict future age structure based on past age structure information. After statistical analysis, it is found that the characteristic root was significantly less than zero, suggesting a decline in female owl populations due to habitat loss. However, other parameters, including vitality rates, do not show any negative trend. Here, they use an appropriate variance model, which is critical in stochastic modeling.

If single sampling is considered, the variance estimate computed will be misleading. However, if a number of sampling occasions are considered, then the process variance will give a better estimate.
