Computer intensive statistical bootstrap

Although this approach is still used, it is undesirable for statistical reasons error calculations underestimate the true uncertainty associated with the equations (17, 21). A better approach is to use the equations developed for one set of lakes to infer chemistry values from counts of taxa from a second set of lakes (i.e., cross-validation). The extra time and effort required to develop the additional data for the test set is a major limitation to this approach. Computer-intensive techniques, such as jackknifing or bootstrapping, can produce error estimates from the original training set (53), without having to collect data for additional lakes. 

Uncertainties inherent to the risk assessment process can be quantitatively described using, for example, statistical distributions, fuzzy numbers, or intervals. Corresponding methods are available for propagating these kinds of uncertainties through the process of risk estimation, including Monte Carlo simulation, fuzzy arithmetic, and interval analysis. Computationally intensive methods (e.g., the bootstrap) that work directly from the data to characterize and propagate uncertainties can also be applied in ERA. Implementation of these methods for incorporating uncertainty can lead to risk estimates that are consistent with a probabilistic definition of risk. 

Bootstrapping involves the repetitive drawing of random samples with replacement from the observed population and computing statistics. A complete bootstrap in an observed population with eight variables would require the calculation of bootstrap statistics for 8 = 16,777,216 samples, quite a computer-intensive process. Therefore bootstrap samples are usually Umited to hundreds or thousands of drawings. 

Sometimes, the distribution of the statistic must be derived under asymptotic or best case conditions, which assume an infinite number of observations, like the sampling distribution for a regression parameter which assumes a normal distribution. However, the asymptotic assumption of normality is not always valid. Further, sometimes the distribution of the statistic may not be known at all. For example, what is the sampling distribution for the ratio of the largest to smallest value in some distribution Parametric theory is not entirely forthcoming with an answer. The bootstrap and jackknife, which are two types of computer intensive analysis methods, could be used to assess the precision of a sample-derived statistic when its sampling distribution is unknown or when asymptotic theory may not be appropriate. 

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