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K—Uncertainty in Stock Assessment Methods and Models

Interval Analysis

In much of fisheries science, the assignment of quantitative values to parameters is inexact. Consequently, a measurement is never known certainly, and scientists tend to report measured values as the best estimate and the possible error around the estimate. This is sometimes expressed with the ''plus or minus" convention or as an interval that is believed to contain the actual value. Interval analysis, described in detail by Moore (1966, 1978) and by Alefield and Herzberger (1983), is considered useful for this type of uncertainty projection.

Briefly, interval analysis permits one to circumscribe estimates with bounds, such as a = [al, a2] where al < a2, and to use range arithmetic to propagate uncertainty. A simple example is provided by illustrating the addition of intervals. Suppose we know that M, the instantaneous natural mortality rate, is between 0.2 and 0.4 and F, the instantaneous fishing mortality rate, is between 0.5 and 0.8, and we want an estimate of Z, the total instantaneous mortality rate, as the sum of these two intervals. Interval addition proceeds as follows:

By substitution, we obtain

From the above, we know that the value for Z must be in the interval [0.7, 1.2]. Rules for various other operators, such as subtraction, multiplication, and division, are relatively simple. However, interval analysis is not easy to use in complex calculations. Fortunately, software capable of handling all computational details is available (Ferson and Kuhn, 1994).

Although interval arithmetic is relatively easy to explain and to apply and seems to work regardless of the source of uncertainty, there are some shortcomings. For example, ranges can grow rapidly in some applications, so that the results may lead to overly conservative fishery management advice. In addition, there is an apparent paradox because no exact values are specified, but the end points (bounds) are presumed to be exact. In summary, interval analysis may be appropriate in situations where no repetitive sampling has occurred and no probability is defined.

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--> K— Uncertainty in Stock Assessment Methods and Models Interval Analysis In much of fisheries science, the assignment of quantitative values to parameters is inexact. Consequently, a measurement is never known certainly, and scientists tend to report measured values as the best estimate and the possible error around the estimate. This is sometimes expressed with the ''plus or minus" convention or as an interval that is believed to contain the actual value. Interval analysis, described in detail by Moore (1966, 1978) and by Alefield and Herzberger (1983), is considered useful for this type of uncertainty projection. Briefly, interval analysis permits one to circumscribe estimates with bounds, such as a = [al, a2] where al < a2, and to use range arithmetic to propagate uncertainty. A simple example is provided by illustrating the addition of intervals. Suppose we know that M, the instantaneous natural mortality rate, is between 0.2 and 0.4 and F, the instantaneous fishing mortality rate, is between 0.5 and 0.8, and we want an estimate of Z, the total instantaneous mortality rate, as the sum of these two intervals. Interval addition proceeds as follows: By substitution, we obtain From the above, we know that the value for Z must be in the interval [0.7, 1.2]. Rules for various other operators, such as subtraction, multiplication, and division, are relatively simple. However, interval analysis is not easy to use in complex calculations. Fortunately, software capable of handling all computational details is available (Ferson and Kuhn, 1994). Although interval arithmetic is relatively easy to explain and to apply and seems to work regardless of the source of uncertainty, there are some shortcomings. For example, ranges can grow rapidly in some applications, so that the results may lead to overly conservative fishery management advice. In addition, there is an apparent paradox because no exact values are specified, but the end points (bounds) are presumed to be exact. In summary, interval analysis may be appropriate in situations where no repetitive sampling has occurred and no probability is defined.

OCR for page 175