evidential reasoning for assessing environmental impact

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EVIDENTIAL REASONING FOR ASSESSINGENVIRONMENTAL IMPACTYANG-CHI CHANG Research Assistant a , JEFF R. WRIGHT Professor of Civil Engineering b &BERNARD A. ENGEL ca School of Civil Engineering, Purdue University ,b Purdue University , West Lafayette, In., 47907-1284c Purdue University ,Published online: 04 Sep 2007.

To cite this article: YANG-CHI CHANG Research Assistant , JEFF R. WRIGHT Professor of Civil Engineering & BERNARD A.ENGEL (1996) EVIDENTIAL REASONING FOR ASSESSING ENVIRONMENTAL IMPACT, Civil Engineering Systems, 14:1, 55-78, DOI:10.1080/02630259608970210

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EVIDENTIAL REASONING FOR ASSESSING ENVIRONMENTAL IMPACT

YANG-CHI CHANG1, JEFF R. WRIGHT2 and BERNARD A. ENGEL3

Researclz Assistant, School of Civil Eng~neering, Purdtre University. 2*Professor of Civil Engineering, Purdue University, W e s l Lafayette,

111. 47907-1 284, Professor of Agricultural and Biological Engineerir~g, Purdue Uniuersity

(Receiveif 5 11fay 1994; Revised 2 Augusl 1995; in f i n d form 3 April 1996)

Many problems in the areas of environmental monitoring and assessment, site selection and planning, land use management, and natural resources allocation are characterized by the need to consider subjective judgement and uncertain data. Many of these problems are also spatial in nature. This research proposes a formal methodology for integrating subjective inferential reasoning anrl gcographic information systems (GIs) into a decision support system for use in these problem domains. The rationale for inferential spatial models, and the structure and function of a spatial modeling environment based on the Dempstcr-Shafer theory of evidence are presented.

Keywords: Dempster-Schafer theory; spatial decision support; geographic information systems (GIs); environmental assessment; land management; inferential reasoning; statistical methods

INTRODUCTION

Advances in information systems technology have spawned tremen- dous interest in the development of large scale databases for managing spatial information. These geographic information systems (GIs) have evolved rapidly over the past decade motivated by major database development efforts ranging in scale from very local, to global. The rationale for developing these large databases is to accurately and

*Corresponding author.

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56 YANG-CHI CHANG et al.

reliably represent spatial data to better facilitate public-sector decision making. Consequently, GIs has been embraced by decision makers responsible for managing spatially distributed resources, or for operat- ing systems within spatial domains (Laurini and Thompson, 1992). Potential applications in public-sector engineering disciplines include such things as watershed monitoring and management, facilities location and servicing, site selection and preparation, hydrologic and environmental modeling, network and transportation planning and management, and geotechnical engineering and analysis.

While reliable databases are an essential component of a comprehensive environmental decision support system, the integration of proven analytical modeling and analysis methodologies is too often a secondary consideration. The integration of GIS database technology and conventional engineering modeling methods is not sufficiently advanced t o allow even routine engineering analysis. Until a higher level of model-GIs integration is developed, the benefits of GIS will remain limited to simple data manipulation and display functions.

A domain where the thoughtful integration of spatial information and modeling technologies has received particular attention is the area of land suitability analysis; the assessment of available land resources to determine their potential for supporting specific activities. The analysis of land suitability plays an important role in many planning activities such as site selection, large-scale public or commercial development, military training, and environmental impact assessment. One approach to the integration of spatial database management systems (DBMS) and suitability indexing methods is shown schematically as Figure 1. In this context G I s is a database management tool that manipulates and extracts information from a spatial database as one of several possible inputs to a land suitability indexing model. For example, suppose that a user wishes to conduct a simple land suitability indexing activity using a weighted attribute analysis. The parameters of interest might be terrain slope, soil type, vegetative covcr and antecedent soil moisture. The model might be a simple assignment of weights based on the value of each parameter attribute for the region of interest. In this case, the CIS might provide precise information about the slope, soils and vegetation attributes, with soil moisture information being provided by other sources such

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EVIDENTIAL REASONING

User

II

FIGURE 1 General framework of a spatial decision support system for land suitability indexing.

Database

\ ,' A

as monitoring instrumentation or subjective judgement. The indexing procedure would draw on these data sources, compute the appropriate indices, and display the results of the analyses.

This indexing framework is particularly appropriate for the design of spatial decision support systems (SDSS) because of its modular nature; the user interface, systems models, and database may be de-

v

Spatial

signed and constructed as separate modules that can be integrated for specific applications. Collectively, these elements can be used to develop specific applications dictated by the needs of the user, and could accommodate, in addition to traditional weighting models, inany

A Other Models

Bayesian Model

1 (weighting Model

Non-Spatial

types of indexing procedures such as Bayesian methods and distributed parameter models, thereby providing land managers a flexible framework within which to work.

User Interface

This paper presents an alternate spatial indexing scheme based on Dempster-Shafer evidence theory, and demonstrates how this methodology may be integrated into the spatial indexing framework just presented. Dempster-Shafer theory is a statistical inference procedure that

Suitability - Indexing 7 Software

DatalSolution &

Map Display

deals with weights of evidence in terms of belief functions, which, in contrast to traditional probabilistic methods, may be more consistent

Procedure

Data

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with human subjective judgement. This is particularly relevant for indexing environmental impacts, where an experienced expert is required to assign a subjective judgement according to uncertain or incomplete evidence. Consequently Dempster-Shafer theory may provide a more defensible theoretical basis for implementation of land suitability analysis for a given application than might probabilistic or rule-based indexing procedures.

UNCERTAINTY IN L A N D SUITABILITY ANALYSIS

Assessment of the ability of a land resource to support a specific activity is frequently conducted using an indexing procedure that inte- grates quantitative as well as subjective factors. The selection of an indexing (or weighting) procedure is itself subjective, reflecting the experience or expertise of thc individual who assigns the weights (Wheeler, 1988). This approach to indexing has been explored in a number of interesting applications. Hopkins (1977) provided a de- tailed classification of indexing methods for generating si~itability maps. Cell indexing (or weighting models) identifies scvcral element attributes (factors) that contribute to the suitability of land for certain activity or impact. These factors may then be combined in a systematic fashion to produce a final suitability profile for that activity.

In some instances, sufficient understanding of fundamental physical system relationships upon which to base an indexing formulae is not available. An alternate way of assessing suitability is through the use of probability, which measures weight in terms of ratios (frequencies). Buehler and Wright (1991) constructed a system that employs both historic data (expressed as conditional probabilities for indexing related land attributes), and a Bayesian inference process to generate suitability indices (probabilities) over the observcd area. The present research re- places that Bayesian indexing procedure within the structure suggested in Figure 1, with one based on the Dempster-Shafer model of evidence.

Bayesian Suitability Indexing

Several approaches to land suitability indexing are based on prior observation and extrapolation. The work by Buchler and Wright (1991)

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EVIDENTIAL REASONING 59

resulted in the formal integration of GIS technology and Bayesian reasoning in developing Binfer (for Bayesian Inferencing); an applications development environment for land suitability analysis. The indexing procedure employed by Hinfer is the well-known Bayes Theorem reflected by:

where Hi is the set of possible outcome states (values for possible inferred attributes or impacts) and E j is the set of all possible condition states (values of attributes considered important in the determination of impact). The prior probability P(H,) and the conditional probability P(EjlHi) must be obtained before calculating the posterior probability P(Hi(Ej), and can be either derived from historical records if available or from a human's - usually a domain expert's - subjective judgement.

Two major assumptions inherent in Bayesian analysis are: 1) the elcments of set E, must be statistically independent, and 2) the set E must be comprehensive in explaining the set of outcomes H. Only if those two assumptions are satisfied will Bayesian theory be a valid methodology for a given application. Though these assumptions may restrict the use of the Bayesian model in a particular application, thcre are several advantages of using this approach for suitability indexing. Bayesian theory has been developed under the formal, axiomatic framework of probability theory for rationally quantifying uncertainty and predicting the likelihood of events (Wcst and Har- rison 1989). In addition to theoretical verifiability, Bayes theory provides a simple rule for manipulating uncertain information (DeGroot, 1989).

Bayesian models are sufficiently flexible that both spatial attribute data (e.g. topography, soil type, vegetation. etc.) and contextual information (e.g. temperature, soil moisture, etc.) can be included. Under the assumption that long-term historical data are available, explicit knowledge about cause-eflect can be captured implicitly as frequencies, avoiding the problem of subjectivity. In addition, probabilities are easily updated' with additional observations thus improving the statistical base and (presumably) the accuracy of prediction.

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While Bayesian statistics within this spatial context may be appropriate for many indexing applications, there remain several shortcom- ings of this approach for many others.

1. The expression of uncertain subjectivity with the use of Bayesian probability has long been criticized by its inability to deal sufficiently with ignorance; the distinction between disbelief and the lack of belief cannot be incorporated explicitly into a Bayesian approach because the relation P ( A ) + P(A) = 1 is assumed (Ishizuka et al., 1982; Gordon and Shortliffe, 1984; and Tanimoto, 1987). When a person assigns a belief to event A with probability P(A), it does not mean that one supports event A's negation 2 with probability 1 - P(A) . However Bayesian probability is required to do so.

2. Using the Bayesian model, there is no measure of the quality of human's subjective probability based on relevant evidence.

3. When evidence provides no information, the Bayesian method re- quires the assumption of equally probable outcomes (Shafer, 1987; Stoms, 1987; and Gordon and Shortliffe, 1984).

4. Bayesian statistics presumes that human experts can consistently assign subjective probability with great precision. The Bayesian model requries that two parameters be estimated: 1) an a priori probability of hypothesis P(H), and 2) a conditional probability P(E IH). Under such a second-degree-of-freedom system, any expec- tation of consistency in human judgement is probably unreason- able (Borden, 1987; and Stoms, 1987).

5. Traditional Bayesian methods tend to deal with single hypotheses and sources of evidence. However in the real world, evidence will arise from multiple sources and be of different types. A conditional probability relationship may exist with more than one hypothesis (Borden, 1987). In these situations it may be inappropriate to ex- pect traditional probabilistic approaches to deal adequately with estimation problems (Duda et al., 1976).

6. Bayesian methods are not appropriate for handling uncertainty due to imprecision, which is probably more hycriptive of subjective judgement (Borden, 1987; Stoms, 1987; and Shafer, 1987). The Dempster-Shafer theory of evidence evolved in response to the

need for methods to better address some of these limitations of classical

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EVIDENTIAL REASONlNG 6 1

probability (Bayesian theory) in modeling situations requiring qualitative descriptions of human judgement, particularly in instances where conflicts exist among different sources of information.

Dempster-Shafer Evidence Theory

The use of Bayesian probability may be inappropriate for measuring belief because uncertainty of belief does not appear to adhere to the laws of probability (Tanimoto, 1987). In 1967, Dempster proposed a new concept of lower and upper limits on probability. Shafer (1976) refined the theory with respect to a belief function based on plausibility, thus improving the measure of subjective uncertainty. The Dempster-Shafcr approach deviates radically from the Baycsian approach on two points. First, the Dempster-Shafer model overcomes the limitation of single evidence or single hypothesis inherent in the Bayesian model by attaching probabilities "...not to individual possible states of the world, but, more ambiguously, to subsets of these states so that when one sceks to compute an uncertainty measure for a particular outcome, one is led in general to a pair of numbers: a 'belief (or lower probability or minimal amount of probability assigned to the outcome) and a 'plausibility' (an upper probability or maximal amount of probability assigned to the outcome)" (Dempster and Kong, 1988). Second, the Dempsler-Shafer approach addresses the main deficiency of Bayesian probability: ignorance. According to Dempster-Shafer theory, the degree of belief assigned to support each hypothesis in a sample space may sum to a number less than one (Gordon and Shortliffe, 1985). Thus Dempster-Shafer theory may be a better qualitative description of human judgement.

To adapt this approach to spatial reasoning models, several important concepts and a terminology are introduced, Frame of discernment (denoted O) is synonymous to sample space in probability theory. A frame of discernment can be viewed as a set or collection of different outcomes (the elements of that set). A focal element A is a subset of frame of discernment O. 'A' can be a singleton subsct or contain multiple sub-elements; O = {A ,B) , where A = {a,,a,} and B = {b,), for example. The basic probability number m(A) is understood to be the measure of belief for subset A of O. A measure of belief assigned to one subset of a focal element is also assigned to every other subset of

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62 YANG-CHI CHANG er al.

that element. In the above example, if m(A) = 0.3, then m(a,) = 0.3 and m(u,) =0.3. In Dempster-Shafer theory, the basic probability assignment m:2@+[0,1] holds whenever:

and

An example is useful in explaining the details of basic probability assignment. Suppose a baby begins crying at midnight. The parents of the child assume that the baby's discomfort may be attributable to three possible factors (or hypotheses):

1. X, =the baby is hungry 2. X , = the baby is sick 3. X, = the baby needs changing

According to past experience, the parents might assign the following degree of belief (basic probability number) to support certain reason (s): m(X,) = 0.6, m(X,, X,) = 0.3, m(O) = 0.1. By assigning belief in this manner, the parents would be suggesting: 1) the measure of belief that the baby is hungry is 0.6 (m(X,) = 0.6), 2) the measure of belief that the baby is sick is 0.3 and the measure of bclief that the baby needs changing is 0.3 ( m ( X , , X , ) = 0.3), and 3) the measure of belief that the baby's crying is insufficient to indicate the nature of the problem is 0.1 (HI(@) =0.1). The value assigned to m(O) is thus the strength of the implication that the evidence does not mean anything. The basic probability assignment is similar to the testimony of an expert who gives reliable testimony 1 - ?n(@) and the unreliable testimony m(O), which is also called the discount on an expert's credibility (Cohcn 1986). The discount allows one to quantify uncertainty about a judgement without specifically assigning probability to each possible outcome.

The m(A) is the measure oC belief that one assigns to A, but it is not the total belief (belief function) assigned to A. To obtain the belief

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function, one must sum all the basic probability numbers m(B) where B represents the subsets of A:

The bclief function, also called a lower probability by Dempster, does not fully describe one's belief about the focal elcment A because 13el(A) does not convey all information of one's suspicion of A (i.e. to what extent one believes its negation A), Here Shafer introduces another measure called degree c?f doubt:

In turn, the degree of doubt allows the specification of a mesure of plausibility; the extent to which one fails to doubt A, or the extent to which one finds A credible or plausible (Shafer, 1976):

In terms of the basic probability number, plausibility can be written as follows:

= 1 - c -m (B) B c A

From (4) and (7), Pl(A) 2 Bel(A). Given several belicf functions over the same frame of discernment,

but based on different evidence, Dempster's rule of combintion en- ables us to compute their orthogonal sum; a new belief function based on the combined evidence. A new basic probability number of a pro- .. - - ..

position ~ - f r c % ~ two independeni sets of'eviacnce a'ana P.is iI'erived by -

the following equations:

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YANG-CHI CHANG et 01.

Here @ is the null set and A and B are possible subsets of the frame of discernment. m,(A) is the basic probability (number) that evidence cc implies hypothesis A and similarly for mp(B). K is used to normalize the number of orthogonal sum.

Consider our example of the crying baby, and let cl represent our previous evidence; that the baby is crying. At first, the parents only consider the baby's crying as evidence of a problem, but then recall that the baby had one bottle of milk an hour earlier. Based on this new evidence, the parents assign another set of basic probability numbers: mo(X, ) = 0.1, mp(X , , X,) = 0.5, and m p ( 0 ) = 0.4 (P reflects the new evidence; that the baby had milk recently). By applying the rule of combination, the final basic probability number assigned to X I is computed as follows:

The resulting probability number is lower because the new evidence weakens the hypothesis that the baby is hungry. The basic probability numbers for the other propositions can be updated using the same procedure.

Dempster's rule of combination can be denoted @ and is used repeatedly (the order in which evidence is combined is not critical) until all evidence has been considered. For the general case having

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four pieces of evidence, denoted a, 8, y, and a, respectively, the final probability number is computed sequentially:

In the following section, an example is presented suggesting how this methodology might be applied to problems of spatial analysis.

SPATIAL ANALYSIS USING THE DEMPSTER-SHAFER MODEL

Problems for which the Dempster-Shafer spatial reasoning model is appropriate (and possibly superior to Bayesian methods) for perform- ing land suitability indexing under conditions of uncertainty are those for which (1) variables important for determining suitability may not be fully independent, or (2) the outcome space is not fully known or understood. Consider a hypothetical example of a regional environmental authority interested in providing guidance to farmers relative to the use of a new pesticide with special attention to the impact of its use on water quality in surface waters. Historical data do not exist relating application strategies for this pesticide and affects on receiving waters, but experts agree that the likely factors that determine impact are: 1) slope of the land to which the chemical is applied, 2) distance between the application site and the receiving water, 3) amount of pesticide applied, 4) soil type, and 5) magnitude of recent rainfall. While all of these determinants have a spatial dimension (vary in space over the region of interest), the last parameter-recent rainfall- also varies significantly over time. From a data management perspec- tive, and particularly significant for our purposes, the first four factors could be stored as map layers in a GIs, while the rainfall factor would be considered a contextual observation to be provided by a .user on a case-by-case basis at run time.

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Returning to our example, the environmental authority is interested in developing a decision support system that would be used repeatedly by the individual responsible for developing a strategy for application of pesticides, and who may not have special expertise about the resulting impact on surface waters. This individual might be an extension agent who visits individual farmers providing information and advice. The decision support system would be used by this individual to estimate the likely impact of an application of this chemical on a specific area and at a specific point in time (that is, for different recent rainfall levels).

An individual is available who is knowledgeable about the adverse impacts on water quality from applying this particular chemical on land characterized by the above-listed five determining factors, and has articulated a specific belief function relative to likely impact. Three different degree-of-effects categories have been specified by this domain expert varying from L(area of least impact on water quality), M(area of moderate impact on water quality), to H(area of greatest impact on water quality). Attribute values have also been categorized by the expert, and basic probability numbers are assigned as presented in Table I. For example, the slope attribute value across the region being modeled has been reclassified as falling into one of three categories: flat (less than 5 degrees), moderate (6 to 10 degrees), and steep (greater than 10 degrees). Without considering other evidence, our expert feels that the application of pesticide to areas having a slope of 5 degrees or less will result in low impact on water quality with basic probability number .8, medium or high impact with basic probability '

number 0.1; the remaining basic probability number 0.1 reflects the strength of the assertion that a slope of 5 degrees does not contribute to the determination of impact. The completed table of determining factors, categorical attribute values for those factors, and associated probability numbers comprise the inference model for this hypothetical application.

Using Dernpster's rules of combination, seven probability numbers can be computed-one for each impact category or categories combination. Table I1 presents these belief functions (lower probabilities) when this model is used to infer likely impact resulting from the application of pesticide to an area having the following attribute values:

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TABLE I Database for assessing surface water quality

Slope Basic Probability Number()

0-5 degree ( f la t ) m,(L)=0.8m1(M,H)=0.1 m,(O)=O.l 6- 10 degree (moderate) m, (L,M,H) =0.7 m,(O) = 0.3 > 10 degree (steep) m,(H)=0.75 m,(L,M)=0.15 m,(O)=O.l

Distance to Stream Basic Probability Number()

0-300 m. (close) m,(H)=0.8m2(L,M)=O.1 m,(O)=O.I > 300 m. (far) m2(L)=0.8 m,(M,H)=O.l m,(O)=O.I

Amount of Chemical Applied Basic Probability Number()

> 1.50 Iblacre m3(H)=0.8 m3(L,M) = 0.1 m,(O) =O.l 1.01-1.50 Ib/acre m,(M) = 0.6 m3(L, H ) = 0.3 m 3 ( 0 ) = 0.1 0.51-1.00 Ib/arce m,(L) = 0.5 m,(M, H) = 0.4 m 3 ( 0 ) = 0.1 0.25-0.5 Ib/acre m3(L) = 0.9 m3(0 ) = 0.1

Soils Texture Basic Probablity Number()

loam type1 m,(L) = 0.6 m4(0 ) = 0.4 loam type2 m4(M) = 0.6 m4(0 ) = 0.4 clay m,(H) = 0.6 m4(0 ) = 0.4

Rainfall Basic Probability Number()

<O.lV(dry) m,(H)=O.l m,(L,M)-0.8m,(O)=O.l 0.1"-1.5" (damp) m,(L) = 0.3 m5(M,H) = 0.65 m,(O) = 0.05 > 1.5" (wet) m,(H)=0.8 m,(L,M)=O.I m,(O)=0.1

TABLE 11 Calculation o f lower probability

Impact Belief (Lower Probability)

Low m(L) = 0.494 Moderate m(M) = 0.2 1 High m(H) = 0.281 Moderate & High m(H) + m(M) + m(M, H ) = 0.492 Low & Moderate m(L)+ m ( M ) f m(L, M) =0.714 Low & High m(L) + m(H) + m(L,H) = 0.778 Ignorance (0) m(L) + m(M) + m(H) + m(M,H) + m(L, M ) +

m(L, H) + m(O) = 1.0

Slope = (flat) Distance to stream =,(close) Amount of chemical applied = (1.01 - 1.5 Ibs per acre)

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Soil texture = (clay) Rainfall = (dry)

The computation of these values is explained in detail in Appen- dix 11. This procedure would be used to determine lower probabilities for all combinations of attribute values deemed relevant for this scenario.

Plausibility as a function of Shafer's degree of doubt can be computed for the same model using Equation 7. The upper probabilities (plausibility numbers) for the same set of attribute values are presented in Table 111.

In this manner, evidence about any particular land use might be combined to generate impact profiles in the form of probability maps based on the Dempster rule of combining inferential evidence. Appro- priate long-term spatial attributes (such as slope, proximity to stream, etc.) would be provided as GIs map layers, and short-term or contextual atrribute values (recent rainfall, etc.) would be provided by the user at run-time.

MODELING CONSIDERATIONS

The Dempster-Schafer inferential reasoning shell (named DSinfer following the previous convention) has been tested on several representa- tive data sets including the hypothetical agricultural management scenario presented above. We conclude with a review of those experi-

TABLE 111 Calculation of upper probability

Impact Belief (Upper Probability)

Low m(L) + m(L, M ) + m ( L , H ) + m ( O ) = 0.508 Moderate m ( M ) + m ( M , H ) + m ( L , M ) + m(O) = 0.222 High m ( H ) + m ( M , H ) 4- m ( L , H ) + m ( O ) = 0.293 Moderate & High m ( M ) + m ( H ) + m ( M , H ) + m(L, M ) + m ( L , H ) +

m ( O ) = 0.506 Low & Moderate m ( L ) + m ( M ) + m ( L , M ) + m ( M , H ) + m ( L , H ) +

m(O) = 0.719 Low & High m ( L ) + m ( H ) + m ( L , H ) + m ( L , M ) -1- m ( M , H ) +

m ( O ) = 0.79 Ignorance(@) m ( L ) + m ( M ) + m(H) + m ( M , H ) + m ( L , M ) +

nl (L ,H)+m(@)= 1.0

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ences, and recommendations for future research and development of this technology.

Belief Assignment

Belief assignment is a key issue in the use of this methodology. Different styles of belief assignment may result in different inference outcomes in terms of the upper and lower probabilities (credibility intervals). As discussed in the previous section, the belief function (lower probability) is viewed as the minimum level of belief allocated to support the hypothesis, while the plausibility (upper probability) is the maximum amount of belief that one can attribute to the hypothesis. Any probability falling between these two ends might be considered credible.

The determination of upper and lower bounds on the length of the credible interval is subject to an expert's uncertainty. Generally, if an expert is more uncertain in assessing one event over another (which means the basic probability number estimated by that expert is lower), then the credible interval resulting from that estimation is likely to be larger. Two examples are presented to demonstrate this phenomenon. Table IV shows the belief assignment that might be expected from a very confident expert (Expert A) for a particular application and body of evidence. Table V presents the belief assignment from a less confident expert (Expert B). For this simple example, we assume that there are three categories in map slope and four categories in the vegetation cover map. Therefore twelve iterations of individual inference (each

TABLE IV Belief Assignment from Expert A

Slope Basic Probability Number() - - - -

0-5 degree m , ( L ) = 0.9 m , ( M , H ) =0.07 m , ( @ ) =0.03 6 - 10 degree m,(L , M)=0.85ml(M,H)=O.lm,(@)=0.05 > 10 degree m , ( M , H ) = 0 . 8 m, (L)=0 .1 m, (@)=O. l

Vegeration Cover Basic Probability Number()

irrigated m 3 ( M ) = 0.7 m, (L ,H)= 0.2 m,(@) =0.1 range.land. m,(L)-0.8-m,(-MjH.).iO..l -m,(@).=.O.l forest m,(L) = 0.9 m, (O) = 0.1 disturbed m,(H) ~ 0 . 8 m,(L ,M) = 0.15 m 3 ( 0 ) = 0.05

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TABLE V Belief Assignment from Expert B

Slope Basic Probability Number()

0-5 degree m,(L)=0.5m,(M,H)=0.3ml(0)=0.2 6- 10 degree m,(L, M ) = 0 . 4 m , ( M , H ) = 0 . 4 m , ( O ) = O . 2 > 10 degree m, ( M , H ) = 0.4 m, (L) = 0.2 m , ( O ) = 0.4

Vegetarion Couer Basic Probability Number()

irrigated m,(M) = 0.4 m3(L,H) = 0.3 m,(O) =0.3 range land m3(L) = 0.4 m3(M,H) = 0.3 m,(O) = 0.3 forest m,(L) = 0.6 m,(O) = 0.4 disturbed m3(H) = 0.5 m,(L ,M) = 0.3 m,(O) = 0.2

indicated as a test number in the figures) are required. The upper and lower bounds of each test are spatially marked with a probability scaled from 0% to 100%. The length of credible interval is also meas- ured on a scale from 0 to I to reveal the uncertainty of the expert.

After twelve iterations of the reasoning process, the result of likely impact 'H' (high) for this particular set of attribute attainment combinations is generated and plotted as Figure 2 and Figure 3 for expert A and expert B, respectively. With the exception of test number 8, the credible interval of each iteration in Figure 3 is wider than that in

Credible Interval

Test Number

FIGURE 2 Inference results for Expert A; confident expert.

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Credible Interval

Test Number

FIGURE 3 Inference results for Expert B; less confident expert.

Figure 2. Test number 2 of Figure 2 shows a very small interval (al- most zero) which is about the same result as Bayesian analysis (no interval). This result also verifies the axiom that the Bayesian approach is a special case of Dempster-Shafer theory when the ignorance is zero. Similar analyses could be performed for likely impacts 'C (low) and 'M' (moderate) for such an example.

Practical Considerations in Model Use

While experience with this modeling approach sugg&sts that the use of Dempster-Shafer theory for assigning land suitability indices is a reasonable alternative to Bayesian models, there are three major issues that should be considered prior to model development and use. First, the model designer (and user) should give careful consideration to the interpretation of upper and lower probabilities resulting from the inferencing process in order to help decision maker take full advantage of the inference result. The definition of lower probability (belief function) and upper probability (plausibility) is well addressed in the litera- ture, but precisely how to make an effective decision based on these two probabilities is probably situational, and not generalizable across applications, maybe even within a single domain. or the example of

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the environmental authority used earlier, the resulting suitability map would classify "high impact" by upper and lower probability numbers; a range, not a single number. The interpretation of this impact category thus may be different for different decision makers or in different instances. The thoery of model use lags far behind the available theory of model development in this regard.

Second, more experience with real problems is required in order to evaluate the appropriateness of Dempster-Shafer spatial analysis for a given application. Comprehensive evaluation of the methodology will require real experts assigning belief measures to real problem data, and in support of a real, and critical decision. Even then, the expert is likely to have to ignore what they might know about probability theory from past experiences or education. Therefore identification of an "expert" willing to articulate their feelings relative to belief in a particular problem domain is likely to be difficult, or at least require time and training.

Lastly, while the methodology presented in this paper is computa- tionally very efficient, problems having combinatorially large sets of attributes may result in an excessively large computational burden. This has traditionally been a criticism of Dempster-Shafer evidential models as pointed out by Gordon et al. (1986); "the [Dempster-Shafer] theory is widely assumed to be impractical for computer-based implementation due to an evidence-combination scheme that assures computational complexity with exponential-time requirements." The issue of computational complexity in this domain is somewhat a function of advances in computer technology, and will be resolved only after a sufficient number of such models have been built and tested. (Other criticisms have been offered from the advocates of Bayesian methods who challenge the validity of the underlying Dempster-Shafer theory (Cheeseman, 1985; and Pearl, 1989). From a more objective point of view, Shafer (1989) argues that both approaches should be thought of as languages for expressing probability.)

CONCLUSION

The object of this research was not to demonstrate the superiority of the Dempster-Shafer theory over Bayesian statistics in a spatial

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context and applied to the domain of environmental monitoring and assessment. Rather, the intent was to suggest an approach to environmental assessment of land resources that can take advantage of spatial data, and that can be used to address uncertain, or poorly understood problems. The Dempster-Shafer theory of inferential reasoning has proven to be a reasonable alternative to Bayesian models when data attributes may not be independent, or when the outcome space is not fully known or understood.

Of particular value in a decision making context is the seamless integration of georeferenced data through this process. Consider the example of surface water quality as discussed in the previous section. The results from these analyses are probability maps for likely impact, which can be used directly by decision makers for identifying poten- tially dangerous areas for pesticide application. Furthermore the probability maps could be used in conjunction with other sources of information to perform better, at least more extensive sensitivity analysis. In particular, applications that are repeated frequently (in time or space) and problem domains that are complex enough to require expertise not ordinarily available to decision makers, may benefit from using this approach.

References

[I] Borden, A. (1987) "Computer, Know Thine Enemy", AI Expert, July, pp. 48-54. [2] Buehler, K. A. and Jeff R. Wright (1991) "Bayesian reasoning shell for land man-

agement. I. Structure and function", Journal of Computing in Civil Engineering, 5(3), 267-282.

[3] Buehler, K. A. and Jeff R. Wright (1991) "Bayesian reasoning shell for land management. 11. Implementation", Jotrrnal of Computing in Civil Engineering, 5(3), 283-299.

[4] Chesseman, P. (1985) "In Defence of Probability", Proceedings of Ninth Interna- tional Joint Conference on Artificial Intelligence, pp. 1002-1009.

[5] Cohen, M. S. (1986) "An Expert System Framework for Non-monotonic Reason- ing About Probabilistic Assumptions", Uncertainty in Artifical Inrelligence, L.N. Kanal and J.F. Lemmer eds., New York, Elsevier Science Publishers, B.V.

[6] DeGroot, M. H. (1986) Probability and Statistics. Addison-Wesley, Reading, Mass.

[7] Dempster, A. P. (1967) "Upper and lower probabilities induced by a multivalued mapping", Ann Math. Statist., 38, 325-247.

[8] Dempster, A. P. and Kong, A. (1988) "Uncertain Evidence and Artificial Analy- sis", Journal of Statistical Planning and lnference, 20, 355-368.

[9] -Duds, 'R. O., Peter E. Hart and Nils J. Nilsson (1976) "Subjective Bayesian Methods for Rule-based lnference Systems", National Computer Conference, pp. 1075-1082.

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1101 Engel, B. A., Jones, D. D., Wright, J. R. and Benbdallah, S. (1991) "Selection of an Expert System Development Shell", A1 Application, 5(1), 15-22.

[ I I] Gordon, J. and ShortlifTe, E. H. (1985) "A Method for Managing Evidential Rea- soning in a Hierarchical Hypothesis Space", Arr$cial Intelligence, 26, 323-357.

[ l2 j Hopkins, L. D. (1977) "Methods for Generating Land Suitability Maps: A Com- parative Evaluation", Journal of the American Institute of Planners, 43(4), 386-400.

[I31 Laurini, R. and Thompson, D. (1992) Fundamentals of Spatial lnformation Sys- tems, Academic Press, Dan Diego.

1141 Pearl, J. (1989) "Bayesian and Belief-Functions Formalisms for Evidential Reason- ing: A Conceptual Analysis", Readings in Uncertain Reasoning, Morgan Kauf- mann Publishers, San Mateo, 540-574.

1151 Shafer, G. (1976) A Mathematical Theory of Evidence, Princeton University Press, Princeton.

[16] Shafer, G. (1987) "Probability Judgement in Artifical Intelligence and Expert Sys- tems", Statistical Science, 2(1), 3-44.

[I71 Stoms, D. (1987) "Reasoning with Uncertainty in Intelligent Geographic Informa- tion Systems", G1S387-San Francisco, 2, 693- 700.

[I81 Tanimoto, S. L. (1987) The Elements of Artificial Intelligence, Computer Science Press, Rockville.

[I91 West, M. and Harrison, J. (1989) Bayesian Forecasting and Dynamic Models. Springer, New York.

[20] Wheeler, D. J. (1988) "A Look At Model-Building with Geographic lnformation System", GIS/LIS'88 Proceedings, 2, 580-589.

APPENDIX I - NOTATION

The following symbols are used in this paper: A = focal element

a i = ith sub-element of focal element A; B = focal element;

b i = ith sub-element of focal element B; Bel(A) = belief function of focal element A; Dou(A) = degree of doubt of focal element A;

E j = set of all possible condition states; H = area of greatest impact on water quality; Hi = set of possible outcome states; K - ' = normalization factor of orthogonal sum; L = area of least impact on water quality; M = area of moderate impact on water quality; m(A) = basic probability number of focal element A; map(A) = basic probability number based on two independent

pieces of evidence: ct and 8; Pl(A) = plausibility of focal element A

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Xi = the th factor (piece of evidence); @ = the empty set 0 = frame of discernment; and 0 = Dempster's rule of combination.

APPENDIX I1 - EXAMPLE COMPUTATION OF EVIDENCE COMBINATION

The following calculations demostrate one complete iteration or rule combination necessary to find the belief function (lower probability) and plausibility (upper probability).

The Computational Procedure

First, the subset 'flat' of slope, 'close' of distance to stream, '1.01-1.50 Ib/acre' of amount of chemical applied ,'clay' of soils texture and 'dry' of rainfall are considered. The K factor is calculated using Equation 9,

which is used to normalize the inferred attribute, obtained from Equa- tion (8):

m12(L) = K,, x Cml (L) x m,(L, M) + m, (L) x m,(@)] = 0.44

m,,(M) = K,, x [m,(M,,H x m,(L, M)] = 0.03

m,,(M,H) = K,, x [m,(M,H) x m,(@)] = 0.03

m,.,(M,H) = K,, x [m,(M,H) x m,[O)] = 0.03

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76 YANG-CHI CHANG e t a / .

Next the evidence of 1.01-1.50 lb/acre of chemical application is considered by applying the same equations,

and the evidence from attributes soils texture (clay) is considered:

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Finally the short term event rainfall (dry) is considered in the same manner:

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evidential reasoning for assessing environmental impact

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