Characterization of Uncertainty in Environmental and Biological Models Employed in Risk Assessment

Bernard R. Parresol

USDA Forest Service Southern Research Station

1. Types of Uncertainty

Uncertainties in models can be classified as natural, model, and data uncertainties.  Environmental and biological systems are inherently stochastic.  Some variables are random in principle, while other variables that are precisely measurable are modeled as random quantities as a practical matter due to the cost and/or effort involved with continuous measurement.  Some quantities vary over time, over space, or across individuals in a population; this is termed variability.  The differences in uncertainty and variability are relevant in decision making.  The knowledge of the frequency distribution for variability can guide the identification of significant subpopulations which merit more focused study.  In contrast, the knowledge of uncertainty can aid in determining areas where additional research or alternative measurement techniques are needed to reduce uncertainty.  Mathematical models are simplified representations of the phenomena being studied.  The structure of mathematical models used to represent biological systems is often a key source of uncertainty.  Different sources of model uncertainties can be summarized as follows.  (1) Model structure: uncertainty arises when there are alternative sets of assumptions for developing a model.  (2) Model detail: often, models are simplified for purposes of tractability.  (3) Extrapolation: models that are valid for one portion of input space may be completely inappropriate for making predictions in other regions of the parameter space.  And (4) model resolution: selection of a spatial and/or temporal grid size often involves uncertainty.  Uncertainties in data stem from a variety of sources.  Some uncertainties arise from measurement errors such as random errors in analytic devices or systematic biases from imprecise calibration.  Other potential sources of uncertainties include misclassification, estimation of parameters through a small sample, and estimation of parameters through non-representative samples.  Uncertainty associated with model formulation and application can also be classified as reducible and irreducible.  Reducible uncertainty can be lowered by better inventory methods, improved instrumentation, and improvements in model formulation.

2. Approaches for Representation of Uncertainty

Various approaches for representing uncertainty can be summarized as follows.  (1) Classical set theory: uncertainty is expressed by sets of mutually exclusive alternatives in situations where one alternative is desired.  This includes diagnostic, predictive and retrodictive uncertainties.  (2) Probability theory: uncertainty is expressed in terms of a measure on subsets of a universal set of alternatives (events).  The uncertainty measure is a function that assigns a number between 0 and 1 to each subset of the universal set.  This number, called probability of the subset, expresses the likelihood that the desired unique alternative is in this subset.  (3) Fuzzy set theory: fuzzy sets, similar to classical sets, are capable of expressing nonspecificity.  In addition, they are also capable of expressing vagueness.  Vagueness is different from nonspecificity in the sense that vagueness emerges from imprecision of definitions.  In fuzzy sets, membership is a matter of degree.  (4) Fuzzy measure theory: this theory considers a number of special classes of measures, each of which is characterized by a special property.  Some of the measures used in this theory are plausibility and belief measures, and the classical probability measures.  Fuzzy measure theory and fuzzy set theory are notably different.  In fuzzy set theory, the conditions for the membership of an element into a set are vague, whereas in fuzzy measure theory, the conditions are precise, but the information about an element is insufficient to determine whether it satisfies those conditions.  (5) Rough set theory: a rough set is an imprecise representation of a crisp set in terms of two subsets, a lower approximation and an upper approximation.

3. Sensitivity/Uncertainty Analysis

The aim of sensitivity analysis is to estimate the rate of change in the output of a model with respect to changes in model inputs.  Such knowledge is important for evaluating the applicability of the model, determining parameters for which it is important to have more accurate values, and understanding the behavior of the system being modeled.  Conventional methods for sensitivity analysis and uncertainty propagation can be broadly classified into three categories: (1) sensitivity testing, (2) analytical methods, and (3) sampling based methods.  Sensitivity testing involves studying model response for a set of changes in model formulation, and for selected model parameter combinations. Some of the widely used analytical methods for sensitivity/uncertainty are: (a) differential analysis methods, (b) Green’s function method, (c) spectral based stochastic finite element method, and (d) coupled and decoupled direct methods.  Sampling based methods involve establishing a relationship between inputs and outputs using the model results at the sample points. Some of the widely used sampling based sensitivity/uncertainty analysis methods are: (a) Monte Carlo and Latin Hypercube Sampling methods, (b) Fourier Amplitude Sensitivity Test (FAST) (c) reliability based methods, and (d) response surface methods (RSM).  A recent state-of-the-art RSM called the Stochastic Response Surface Method has been used in evaluating airshed models looking at reactive plumes (such as might occur from a prescribed burn or wildfire) and water quality models for groundwater systems.  Both are of interest in forest threat assessment, especially at the forest/urban interface.

Statistical Methods Session - Wednesday Afternoon

corresponding author:

Bernard R. Parresol
USDA Forest Service
Southern Research Station
200 WT Weaver Boulevard
Asheville, NC 28804
828-259-0500
bparresol@fs.fed.us