Lev Ginzburg

Different methods are needed to propagate ignorance and variability

There are two kinds of uncertainty. One kind arises as variability resulting from heterogeneity or stochasticity. The other arises as partial ignorance resulting from systematic measurement error or subjective (epistemic) uncertainty. As most researchers recognize, variability and ignorance should be treated separately in risk analyses. Although a second-order Monte Carlo simulation is commonly employed for this task, this approach often requires unjustified assumptions which may be inappropriate in some circumstances. We argue that the two kinds of uncertainty should be propagated through mathematical expressions with different calculation methods. Basically, interval analysis should be used to propagate ignorance, and probability theory should be used to propagate variability. We demonstrate how using an inappropriate method can yield erroneous results. We also show how ignorance and variability can be represented simultaneously and manipulated in a coherent analysis that does not confound the two forms of uncertainty and distinguishes what is known from what is assumed.

Dependence in probabilistic modeling, Dempster-Shafer theory, and probability bounds analysis.

This report summarizes methods to incorporate information (or lack of information) about inter-variable dependence into risk assessments that use Dempster-Shafer theory or probability bounds analysis to address epistemic and aleatory uncertainty. The report reviews techniques for simulating correlated variates for a given correlation measure and dependence model, computation of bounds on distribution functions under a specified dependence model, formulation of parametric and empirical dependence models, and bounding approaches that can be used when information about the intervariable dependence is incomplete. The report also reviews several of the most pervasive and dangerous myths among risk analysts about dependence in probabilistic models.

Extreme event risk analysis for age-structured populations

Abstract Since population explosions and extinctions often have decidedly undesirable environmental or economic consequences, we would like to be able to estimate the chances that a particular species will crash or bloom. Although mathematical demography has employed the Leslie matrix for population projection for several decades, it is quite diffficult to make these estimations analytically when stochasticity and density dependence are included in the age-structured model. We describe microcomputer software, ramas , that was designed to overcome this problem. It can represent a great deal of the complexity that is possible in age-structured dynamics of single populations and yet is extremely user-friendly. This methodology permits a comprehensive analysis of the population-level consequences of environmental impacts in terms of the risks of demographically extreme events.

Using the phase shift for assessing the causation of population cycles

Abstract The occurrence of an appreciable phase shift between the predator and prey curves is a characteristic signature of predator–prey cycles. The magnitude of the phase shift can be used for assessing whether the direct effect of predation is likely to cause the observed cyclic dynamics. The statistical estimation of the phase shift requires having simultaneous, long-term data series for predator and prey in the same locality, a condition that is almost never satisfied by long-term data of most cyclic populations. A method for estimating the magnitude of the phase shift between the predator and prey trajectories using only single species data. This method is applied to two cases in which predation has been suggested to be the ecological cause of the cycles: the lynx–hare cycles in boreal Canada and the gyrfalcon–ptarmigan cycles in northern Iceland. The estimates of the phase shift indicate that while the lynx–hare cycles are unlikely to be due by the direct effect of lynx predation, those of gyrfalcon–ptarmigan are consistent with predator–prey causation.

Judgment under uncertainty: Evolution may not favor a probabilistic calculus

The environment in which humans evolved is strongly and positively autocorrelated in space and time. Probabilistic judgments based on the assumption of independence may not yield evolutionarily adaptive behavior. A number of “faults” of human reasoning are not faulty under fuzzy arithmetic, a nonprobabilistic calculus of reasoning under uncertainty that may be closer to that underlying human decision making.

On the use of intervals in scientific computing: what is the best transition from linear to quadratic approximation?

In many problems from science and engineering, the measurements are reasonably accurate, so we can use linearization (= sensitivity analysis) to describe the effect of measurement errors on the result of data processing. n nIn many practical cases, the measurement accuracy is not so good, so, to get a good estimate of the resulting error, we need to take quadratic terms into consideration – i.e., in effect, approximate the original algorithm by a quadratic function. The problem of estimating the range of a quadratic function is NP-hard, so, in the general case, we can only hope for a good heuristic. n nTraditional heuristic is similar to straightforward interval computations: we replace each operation with numbers with the corresponding operation of interval arithmetic (or of the arithmetic that takes partial probabilistic information into consideration). Alternatively, we can first diagonalize the quadratic matrix – and then apply the same approach to the result of diagonalization. n nWhich heuristic is better? We show that sometimes, the traditional heuristic is better; sometimes, the new approach is better; asymptotically, which heuristic is better depends on how fast, when sorted in decreasing order, the eigenvalues decrease.