Spatial Prediction of Landslide Hazard Using Fuzzy k-means and Dempster-Shafer Theory

Transactions in GIS

, 2005, 9(4): 455–474

© Blackwell Publishing Ltd. 2005. 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

Research Article

Spatial Prediction of Landslide Hazard Using Fuzzy

k

-means and Dempster-Shafer Theory

Pece V. Gorsevski

Department of Forest Resources University of Idaho

Paul E. Gessler

Department of Forest Resources University of Idaho

Piotr Jankowski

Department of Geography San Diego State University

Abstract

Landslide databases and input parameters used for modeling landslide hazardoften contain imprecisions and uncertainties inherent in the decision-making process.Dealing with imprecision and uncertainty requires techniques that go beyond classicallogic. In this paper, methods of fuzzy

k

-means classification were used to assign digitalterrain attributes to continuous landform classes whereas the Dempster-Shafer theoryof evidence was used to represent and manage imprecise information and to deal withuncertainties. The paper introduces the integration of the fuzzy

k

-means classificationmethod and the Dempster-Shafer theory of evidence to model landslide hazard inroaded and roadless areas illustrated through a case study in the Clearwater NationalForest in central Idaho, USA. Sample probabilistic maps of landslide hazard potentialand uncertainties are presented. The probabilistic maps are intended to help decision-making in effective forest management and planning.

1 Introduction

The dominant erosion and sedimentation processes are due to surface erosion or massfailure. Mass failure is a gravity-driven process that occurs when the shear strength ofa soil mass is overcome by the shear stress acting against it. Mass failure is triggeredby dynamic climatic (extreme rainfall), geodynamic (earthquake), volcanic or land-usechange events, and it is the most important cause of sedimentation in areas of steep

Address for correspondence:

Pece V. Gorsevski, Department of Forest Resources, College of NaturalResources, University of Idaho, Moscow, ID 83844-1133, USA. E-mail: [email protected]

456

P V Gorsevski, P Jankowski and P E Gessler

© Blackwell Publishing Ltd. 2005

slopes and unstable soils. Mass failures that are often referred as landslides cause damagescosting an estimated annual average of $1.5 billion in the US alone (Glade 1998). Effectsof mass failures include water quality, water quantity, and aquatic habitat impacts.

Human activities, such as poorly designed land use practices (i.e. road-building)accelerate the process of landslides (Chung et al. 1995, Burton and Bathurst 1998).Environmental effects from improper road-building and poor road maintenance can increasethe risk of erosion and landslide failure in steep mountainous regions. The environmentalconsequences from erosion and landslide failures are particularly severe after infrequentstorms and floods. The literature has shown (Dyrness 1967, McClelland et al. 1997)that the probability of landslide occurrence in forestlands increases whenever roads andassociated activities exist (i.e. logging, recreation). Consequently, the knowledge thatroad building affects natural resources has attracted public attention and demands rig-orous hazard assessment to support improved management practices. Independently, theeconomic impact of land sliding has led forest managers to exercise protective measuresto reduce landslide hazard. The success of protective measures, however, depends on theability to predict landslide hazard. Hence, robust methods and techniques of assessinglandslide hazard are vital for future planning and management of natural resources.

A number of methods and techniques were proposed to assess landslide hazard.Quantitative techniques include: stability ranking based on criteria such as slope, parentmaterial, and elevation (McClelland et al. 1997), the weighted linear combinationmethod which involves a pairwise comparison to create weights for predictor variables(Ayalew et al. 2004), statistical models linking environmental attributes using spatialcorrelation (Carrara 1983; Carrara et al. 1991; Chung et al. 1995; Chung and Fabbri1999; Dhakal et al. 2000; Dai and Lee 2002; Gorsevski 2002; Gorsevski et al.

2000,2003, 2004; Gorsevski and Gessler 2003), and process models that combine the infiniteslope equation and hydrological components (Montgomery and Dietrich 1994, Wu andSidle 1995, Gorsevski 2002). Many of these techniques were implemented with GISsoftware in order to present the results of landslide hazard assessment on maps (Carrara1983, Hammond et al. 1992, Montgomery and Dietrich 1994, Wu and Sidle 1995,Okimura and Ichikawa 1985, Carrara et al. 1995, Gorsevski 2002).

Gorsevski (2002) and Gorsevski et al. (2003, 2004) developed and tested an imple-mentation of a quantitative technique that integrates a fuzzy

k

-means classification anda Bayesian approach in the study site used by McClelland et al. (1997) to predict roadrelated (RR) landslide hazard (spatial locations within roaded areas) and non-roadrelated (NRR) landslide hazard (spatial locations within non-roaded areas). The fuzzy

k

-means classification was used to deal with uncertainties in terms of vagueness andincompleteness of known parameters associated with class overlap as well as to assigndigital terrain attributes into continuous landform classes. The continuous landformclasses were then combined in a GIS with known landslide locations for developing aprobabilistic model of landslide potential using the Bayesian approach. The Bayesianapproach or Bayes Theorem (Aspinal 1992, Aspinal and Veitch 1993, Skidmore et al.1996, Malczewski 1999) is a method used for decision-making under uncertainty. Thetheorem formula combines subjective probability (of being true or false) with condi-tional probability (of being true or false). Subjective probability is an expression of thedegree of belief in an event occurring based on a person’s experience, judgment, preju-dice, optimism, etc. (Malczewski 1999). Conditional probability is the knowledge aboutthe likelihood of the hypothesis to be true given a piece of evidence. However, whileBayesian probability theory is a well-structured statistical decision-making theory, it is

Fuzzy k-means and Dempster-Shafer theory

457


weak in representing and mapping imprecise information because it requires preciseprobability judgments (Sii et al. 2002). For example, the accuracy of landslideidentification from aerial photographs can vary depending upon the scale of the airphotos, the required mapping detail, the landslide size, the photo quality, the season,the forest cover density and height, and the skill of the interpreter. Rib and Liang (1978)recommended that an overall identification accuracy of 80 to 85 percent is realistic usingaerial photography, which introduces imprecision in the landslide databases.

To overcome the weakness of imprecision, uncertainty, and vague information inmodeling landslide hazard and consequently in the decision-making about reducinglandslide hazard, we introduce another modeling approach that integrates a fuzzy

k

-means classification with the Dempster-Shafer (D-S) theory of evidence (Dempster 1967,Shafer 1976). The D-S theory has been used in a number of practical applications in awide range of disciplines such as artificial intelligence (Shafer 1996), risk management(Sii et al. 2002), mineral exploration (Tangestani and Moore 2002), GIS (Eastman2001), and remote sensing (Le Hegarat-Mascle et al. 2000, Mertikas and Zervakis2001). D-S theory, which is based on probability theory, introduces uncertainties inmodeling and allows probability judgments to capture the imprecise and vague informa-tion in modeling evidence. This theory is an extension of the Bayes Theorem, yet it ismore flexible in the sense that it waives the need for complete knowledge of prior orconditional probabilities before modeling can take place. Additionally the D-S theoryintroduces the representation of ignorance.

The integrated approach (Fuzzy/D-S) is illustrated using a case study of the Clear-water National Forest (CNF) in central Idaho where the following three tasks were set:(1) to evaluate sets of different combinations of evidence sources to be used in thedevelopment of predictive D-S models for RR and NRR landslide hazard; (2) to evaluateif spatial prediction of landslide hazard using the integrated Fuzzy/D-S approach isbetter than spatial prediction using the Fuzzy/Bayesian approach; and (3) to evaluate ifpredictive D-S models of landslide hazard for RR and NRR landslides are different. Thefirst task will help answer a question whether different combinations of evidence sourcesare necessary in the development of predictive D-S models, the second task will deter-mine if the proposed approach using the D-S theory yields better spatial prediction thanthe Bayesian approach, and the third task will answer the question whether the devel-opment of two independent models for RR and NRR landslide hazard is necessary.

2 Modeling Theory

2.1 Fuzzy k-Means

The fuzzy

k

-means algorithm is described in greater detail in Bezdek (1981), McBratneyand deGruijter (1992), Burrough and McDonnell (1998), Burrough et al. (2000, 2001), andGorsevski et al. (2003), while in this paper the concept of fuzzy

k

-means clustering isbriefly summarized. The fuzzy

k

-means clustering approach is analogous to traditionalcluster analysis that uses a repetitive procedure by selecting a set of random cluster pointsof

n

objects and building

c

clusters around each seed. Given the cluster-allocation, thecenter

C

of each

c

th cluster for the

j

th attribute

X

is calculated as the weighted average:

(1)C Xcj ij iji

n

iji

n

( ) ( )== =∑ ∑µ µφ φ

1 1

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458

P V Gorsevski, P Jankowski and P E Gessler


where

µ

is the membership of the

i

th object to the

j

th

class and

φ

determines the amountof fuzziness or cluster overlap and is called the fuzzy exponent. For example, when

φ

= 1no overlap is allowed and there is no fuzziness (a “hard class” is generated), for large

φ

there is complete overlap and the clusters are identical.Reallocation of objects among the classes follows until a stable solution is reached,

meaning that the objects in each cluster are similar to one another while those in differ-ent clusters are not similar to one another. In ordinary

k

-means the membership

µ

ofthe

i

th object to the

j

th cluster is determined by:

(2)

where

d

is the distance function which is used to measure the similarity or dissimilaritybetween two individual observations and then later the similarity or dissimilaritybetween two clusters. Distances between attributes for correlated variables on the sameor different scale are calculated using Mahalanobis metric:

(3)

where

d

2

ij

is the square of the distance between an individual

i

and a class center

j

;

x

ic

isan attribute for individual

i

and the class

c

;

C

cj

denotes the centroid of class

c

forattribute

j

; and

Σ

−

1

is the pooled within-classes variance-covariance matrix. Other dis-tances such as a Euclidian metric is used for uncorrelated variables on the same scale,while a diagonal metric is used for uncorrelated variables on different scales.

Once a stable solution is reached the confusion index (

CI

), the optimal number ofclasses, or the degree of fuzziness can be determined. The

CI

is a measure of the degreeof class overlap in attribute space (Burrough and McDonnell 1998). The concept of a“confusion index” is a measure of how well each individual observation has beenclassified. The

CI

represents the ratio of the first sub-dominant membership (

MF

max2

)value to the dominant membership (

MF

max

) value for each observation. For example, ahigh

CI

value means that an observation likely belongs to two or more classes, and alow

CI

value suggests the observation belongs to one class. The

CI

is defined as:

CI

=

1

−

(

MF

max

−

MF

max2

) (4)

The user’s knowledge of the data is often used in choosing the optimal number ofclasses or the degree of fuzziness, or it can be based on repeated classification for a rangeof numbers of classes and evaluated using two validity functions: the fuzzy performanceindex (

FPI

), and the normalized classification entropy (

MPE

) (Modified PartitionEntropy). The fuzzy performance index (

FPI

) is computed as:

(5)

where

F

is the partition coefficient:

(6)

The Modified Partition Entropy (

MPE

) is computed as:

(7)

µ φ φij ij ik

k

c

d d [( ) ] [( ) ]/( ) /( )= − − − −

=∑2 1 1 2 1 1

1

( ) ( ) ( )d x C x Cij ic cj ic cj2 1= − −−∑

FPIF c

c

( / )( / )

= −−−

11

1 1

Fn

ijj

c

i

n

( )===∑∑1 2

11

µ

MPEH

c

log=

Fuzzy k-means and Dempster-Shafer theory

459


where

H

is the entropy function:

(8)

The fuzzy performance index function estimates the degree of fuzziness generated by aspecified number of classes, while the normalized classification entropy estimates thedegree of disorganization created by a specified number of classes (Minasny andMcBratney 2000). After

FPI

and

MPE

are calculated the optimum number of continu-ous and structured classes can be established on the basis of minimizing these twomeasures (McBratney and Moore 1985).

2.2 Dempster-Shafer Theory

Dempster (1967) originated the concept of lower and upper probability induced bymulti-valued mapping that was extended by Shafer (1976) into the current D-S theoryof evidence. Unlike the Bayesian theory that deals with the stochastic nature of datameasurements based on specific models of distribution and Bayesian logic of hypothesistesting, the D-S theory of evidence is based on approximate reasoning where imprecisionand uncertainties are introduced into the decision-making process by considering prob-ability intervals with lower and upper bounds (Mertikas and Zervakis 2001).

D-S theory starts with the definition of all possible hypotheses,

Θ

, called a frame ofdiscernment that a variable can take. It is equivalent to the sample space in probabilityterms. The frame of discernment corresponds to a hierarchical structure and includes allpossible (mutually exclusive) combinations between a set of hypotheses. The set of allhypotheses represented by the frame of discernment is 2Θ. For example, if the frame ofdiscernment includes three basic hypotheses then the number of hypotheses = 23 = 8 or:Θ = {A, B, C}, where the possible combinations are [A], [B], [C], [A, B], [A, C], [B, C],[A, B, C]. The eighth subset is represented by an empty set Ø, which corresponds to ahypothesis that is known to be false. The first three are called singleton hypothesesbecause each subset contains only one basic element, while the rest are called non-singleton hypotheses because each subset contains more than one basic element.Furthermore, each of these subsets is called a focal element. Based on the evidence,a numeric value with an interval between [0, 1] is assigned to each focal element. Thevalue 0 indicates no belief in a proposition, the value 1 indicates total belief, and valuesbetween [0, 1] indicate partial beliefs. Uncertainty is quantified as the amount of evi-dence not assigned to any particular subset, which is computed as one minus the sumof supports of all subsets. Therefore, the D-S theory allows gathering evidence not justabout individual hypotheses but about all possible subsets of Θ.

An exact belief value called a basic probability assignment (BPA) or a mass func-tion, m, is assigned (a number from 0 to 1) to each focal element A of Θ such that thesum of all BPA or masses will add up to one at all times. Thus, m distributes belief overall the subsets of Θ. For instance, if a piece of evidence increases the degree of belief ina focal element A, then the unassigned amount of belief gets assigned back to Θ insteadof to the complement of A, as would occur in Bayesian probability theory. If no levelof belief has been assigned to a focal element, the value of the belief function for thatfocal element is zero. Methods of calculating BPAs for a given hypothesis use a subjectivejudgment or empirical data (Dempster 1967). All belief values from the BPA functionmust satisfy Equation 9:

Hn

ij ijj

c

i

n

log( )= −==∑∑1

11

µ µ

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460 P V Gorsevski, P Jankowski and P E Gessler


(9)

where m(A) is the BPA of a set A in Θ, where Θ is a set of all possible outcomes or theframe of discernment.

Then, from the mass functions, belief (Bel) and plausibility (Pls) functions are derived.The belief function denotes the lower bound for an (unknown) probability function,whereas the plausibility function denotes the upper bound for an (unknown) probabilityfunction. The difference between the plausibility (Pls) and the belief (Bel) function rep-resents a measure of uncertainty. The belief function measures the amount of belief inthe hypotheses on the basis of observed evidence. It represents the total support for thehypothesis that is drawn from the BPAs for all subsets of that hypothesis (i.e. belief in[A, B] will be calculated as the sum of the BPAs for [A, B], [A], and [B]) and it is defined as:

(10)

The plausibility represents the maximum level of belief possible or the degree to whicha hypothesis cannot be disbelieved, given the amount of evidence negating the hypothesis.Specifically, the plausibility is obtained by subtracting the BPAs associated with all subsetsof the complement of the hypothesis (A) from 1. The plausibility function is defined as:

(11)

When two masses m1 and m2 for Θ are obtained as a result of two pieces of inde-pendent information, they can be combined using the Dempster’s rule of combinationto yield new BPAs (m1 ⊕ m2). This combination of m1 and m2 is defined as:

(12)

where the combination operator “⊕” is called “orthogonal summation”, A ≠ Ø, and thedenominator which represents a normalization factor (one minus the BPAs associatedwith empty intersection) is determined by summing the products of the BPAs of all setswhere the intersection is null. When the normalization factor equals 0 in such a situation,the two items of evidence are not combinable. The order of applying the orthogonalsummation does not affect the final results since Dempster’s rule of combination iscommutative and associative.

3 Study Area and Modeling Approach

The modeling approach to estimating landslide hazard probability in Idaho’s CNF wastested and compared against another approach described in Gorsevski et al. (2003). Thestudy area and the modeling approach are discussed in the following sections.

3.1 Study Area

The study area is within the CNF, located on the western slopes of the Rocky Moun-tains in north central Idaho (115°46′W, 46°07′N, 114°19′W, 47°00′N), USA. The CNF

m A m( ) ( ) = =⊂∑ 1 0

Αφ

Θ

and

Bel A m B( ) ( )=⊂∑

Β Α

Pls A m B( ) ( )=∩ =∑

Β Α φ

( )( )

( ) ( )

( ) ( )m m A

m B m C

m B m CC A

C

1 2

1 2

1 21⊕ =

−∩ =

∩ =

∑∑

Β

Β φ

Fuzzy k-means and Dempster-Shafer theory 461


is located west of the Montana state border and is bounded on three sides by four otherNational Forests; the Lolo National Forest in Montana; the Bitterroot National Forestin Montana and Idaho; the Nez Perce National Forest in Idaho; and the PanhandleNational Forest in Idaho. The CNF map is shown in Figure 1. The training area of111.8 km2 used for classification includes the Papoose, Badger and Squaw Creek water-sheds (Figure 1) located northwest of Lowell, Idaho in the Lochsa Basin of the CNF.The training area was chosen because of the high landslide density initiated by landslideevents in November 1995 and February 1996 and the similarity of the topographicattribute distributions with the overall CNF. The highly dissected mountainous topog-raphy of the training area is typical for Idaho’s Clearwater River Basin. Elevation in thetraining area watershed ranges from 966 to 2,154 m and slopes vary between 0 and 45degrees. The climate is characterized by dry and warm summers, and cool wet winters.Precipitation averages about 1,320 mm annually, which changes significantly across theelevation gradient. Most of the annual average precipitation falls as snow during winter

Figure 1 Distribution of landslides over the Clearwater National Forest drainage during thewinter 1995/96 storm events



and spring, while peak stream discharge occurs in late spring and early summer. Thehighly variable steep soils are well drained and are primarily derived from parent mate-rials such as granitics, metamorphic rocks, quartzites, and basalts or surface erosion anddeposition. Vegetation includes Grand fir (Abies grandis), Douglas fir (Pseudotsugamenziesii), Subalpine fir (Abies lasiocarpa), Western red cedar (Thuja plicata), Westernwhite pine (Pinus Monticola), and various other shrubs and grasses that have shortgrowing seasons, particularly at the higher elevations.

Historically, the CNF in Idaho has experienced periodic floods and landslide events.Major floods occurred in 1919, 1933, 1948, 1964, 1968, and 1974. All of these floodswere documented through streamflow records (McClelland et al. 1997). Although theapproximate frequency of major landslide events is known for the last century, accuratemapping of individual landslides through time does not exist.

3.2 Methods

Landslides were assessed through aerial reconnaissance flights and field inventory in July1996. Aerial photography was acquired at a scale of 1:15,840 (4 inches = 1 mile)followed by photo interpretation between October 1996 and February 1997 (McClel-land et al. 1997). The landslides interpreted from aerial photos were classified into roadrelated (RR) and non-road related (RR). A total of 865 landslides were recorded, with55% RR and 45% NRR landslides. The presence or absence of a landslide was repre-sented as a (30-m) grid coverage with values of 1 for presence and 0 for absence. Theinitiation area of each landslide (i.e. the area where the main scarp of the landslideoccurred) was interpreted as the point representing the presence of a landslide. The RRlandslides, which are associated with forest roads, were coded separately from the NRRlandslides. This enabled the development of independent quantitative models for eachdataset (RR, NRR).

Derivation and justification of the six fuzzy k-means classes and the maximum“Maxcl” class (most dominant membership value) that are used here for the study areaare described in detail in Gorsevski et al. (2003); only the main points are summarizedhere. The FuzME Program (Minasny and McBratney 2000) was applied to a trainingarea within the CNF using six topographic attributes (elevation, slope, profile curvature,tangent curvature, compound topographic index (CTI), and solar radiation). The train-ing area was representative of a broad area that closely matched the overall probabilitydensity functions. The fuzzy clustering was applied to the training area and was used todetermine the optimal number of classes required to classify the area, to decide theoptimal values of the performance parameters, and to calculate class centroids for inter-pretation of similarities and differences between classes. The fuzzy k-means algorithmwithout extragrades (observations that cannot be assigned to a hard class because theybelong equally little to any of the classes) was implemented using Mahalanobis distance,for a total of six classes, and a fuzzy exponent of 1.4 (Gorsevski et al. 2003). A custombuilt program using the Arc/Info GRID module was used to extrapolate the landformclassification of identifiable fuzzy k-means elements from the training area to the entirestudy area. Along with the derived fuzzy k-means classes, the maximum “Maxcl” class(most dominant membership value) was calculated.

In the framework of D-S theory of evidence the BPAs or the evidential support werecalculated from the fuzzy k-means classes on the basis of statistical frequency distribution.The occurrence of frequency distribution of presence and absence of landslide locations



and categorized membership values of fuzzy k-means classes (i.e. 0–0.1, 0.1–0.2, 0.2–0.3, 0.3–0.4, 0.4–0.5, 0.5–0.6, 0.6–0.7, 0.7–0.8, 0.8–0.9, 0.9–1) was determined forall classes and BPAs were derived for all pieces of evidence. The derived pieces of evidencewere intended to support two hypotheses: [presence of landslides] and [absence oflandslides] for both datasets (RR, NRR) separately. For example, the support for thefirst hypothesis represented the association between the presence of landslides andcategorized fuzzy k-means classes, while the support for the second hypothesisrepresented the association between the absence of landslides and categorized fuzzyk-means classes. The final integration of the Fuzzy/D-S approach was accomplished byaggregating the supporting evidence for the RR and NRR models and applying theDempster’s rule using the belief decision support module in the IDRISI GIS (Eastman2001) software.

4 Results and Discussion

The class centroids from the fuzzy k-means analysis given in Table 1 can be used tointerpret similarities and dissimilarities between classes. The centroid is the averagepoint in the multi-dimensional space and represents, in a sense, the center of gravity forthe respective cluster. For example, class “b” has the highest CTI, and the lowest (neg-ative) profile and tangent curvature values suggesting that this class represents conver-gent areas with high soil moisture. The high solar radiation of class “a” and the highprofile and tangent curvature indicates convex areas. Thus, from the cluster centers inTable 1, the analyst can interpret landscape pattern differences captured by the fuzzyclasses. The fuzzy k-means approach has distinguished potentially useful classes definingmultivariate patterns and clusters within the area of interest based on the input variables.

Figure 2 shows the topographical settings of these classes to complement the clustercenters. For example, class “a” represents mid elevation, gentle convex slopes with highsolar radiation, class “b” locations are mid elevation, concave drainages, class “c” ishigh elevation, high solar radiation (southerly slopes) locations, class “d” is low eleva-tion, steep low solar insolation locations (northerly slopes), class “e” is low elevation,steep high solar insolation locations (southerly slopes), and class “f ” is high elevation,low solar insolation (northerly slopes).

Except for the requirement that the evidence sources are independent in the D-Stheory there is not a strict limit to the knowledge backgrounds, the scale of measure-ments and distribution of the values of the evidence sources (Peddle 1995). In this paper

Table 1 Clusters centers for six classes

Input data Ca Cb Cc Cd Ce Cf

Elevation (meters) 1446 1413 1784 1363 1399 1607Slope (percent) 33 44 49 67 59 50CTI (no units) 9.3 11.6 9.4 9.0 8.9 9.2Solar Radiation (KWH/m2) 1109 944 1261 510 1101 691Profile Curvature (radians/100m) 0.10 −0.45 0.04 0.02 0.03 0.01Tangent Curvature (radians/100m) 0.31 −1.35 0.05 0.16 0.15 0.04



we treat the derived evidence sources from the fuzzy k-means classes as independentsources of evidence. In the next section we demonstrate the results from the methodo-logical approach that is akin to a stepwise regression approach, which takes one pieceof evidence at a time, and its effects are measured by the goodness-of-fit functionbetween individual models and independent test data.

4.1 Comparison of Predictor Datasets

Tables 2, 3, 4, and 5 show the results from the D-S modeling approach derived from differentcombinations of evidence sources. While results were generated from all possible combina-tions of evidence sources, in this paper we present the results that were important forthe comparison between the Fuzzy/D-S approach and the Fuzzy/Bayesian approach andresults that yielded a better goodness-of-fit from the evidence sources used. The goodness-of-fit between the models and the independent RR and NRR landslide test data was usedto measure the fit from different combinations of evidence sources tested.

Figure 2 Drapes of fuzzy k-means classification of the training area with six classes



The tables (2, 3, 4, and 5) show cross-tabulation of the independent test data forthe RR and NRR landslides against the results from the D-S model. The tables are alsoorganized by showing the lower probability bounds associated with the “Belief” func-tion as well as the upper probability bounds associated with the “Plausibility” function.The results presented in Table 2 were guided by the chi-square (χ 2) tests based oncalculating conditional probabilities from relative frequencies of datasets to be modeled(presence or absence of NRR or RR landslides) and categorized evidence sources derived

Table 2 Proportion of presence/absence associated with probabilities for NRR and RRlandslide hazard derived from the significant classes that were guided by the chi-square ( χ 2)tests. The NRR landslide hazard was derived from the following five classes “b”, “d”, “e”,“Maxcl”, and “Mincl” while the RR landslide hazard was derived from the following fourclasses “a”, “d”, “e”, and “Maxcl”

Non-road related Road related

Probability Presence Absence Presence Absence

Belief 0–0.2 3.2% 8.5% 1.4% 34.3%0.2–0.4 93.7% 88.0% 82.0% 58.7%0.4–0.6 3.2% 3.5% 16.7% 7.0%0.6–0.8 0.0% 0.0% 0.0% 0.0%0.8–1 0.0% 0.0% 0.0% 0.0%

Plausibility 0–0.2 0.0% 0.1% 0.0% 12.8%0.2–0.4 57.9% 87.1% 32.9% 60.1%0.4–0.6 18.4% 5.1% 30.2% 12.5%0.6–0.8 23.7% 7.7% 32.4% 13.2%0.8–1 0.0% 0.0% 4.5% 1.4%

Table 3 Proportion of presence/absence associated with probabilities for NRR and RRlandslide hazard derived from the following five classes “a”, “b”, “c”, “d”, and “e”



Belief 0–0.2 0.5% 18.5% 0.0% 10.0%0.2–0.4 3.7% 13.7% 2.7% 16.7%0.4–0.6 13.2% 27.8% 10.8% 32.7%0.6–0.8 30.5% 26.5% 39.6% 25.9%0.8–1 52.1% 13.5% 46.9% 14.7%

Plausibility 0–0.2 0.0% 0.0% 0.0% 0.0%0.2–0.4 2.1% 28.1% 0.0% 22.8%0.4–0.6 10.5% 24.5% 9.0% 30.3%0.6–0.8 22.6% 28.8% 33.8% 29.9%0.8–1 64.7% 18.6% 57.2% 17.0%



from the fuzzy k-means classes (Gorsevski et al. 2003). Significant sources of evidencefor the development of the NRR landslide model included the following: the maximumand minimum class (based on most and least dominant class membership); “b” – midelevation, concave drainages; “d” – low elevation, steep low solar insolation locations(northerly slopes); and “e ” – low elevation, steep high solar insolation locations (south-erly slopes). Significant sources of evidence for the development of the RR landslidemodel included the following: the maximum class (based on the most dominant classmembership); “a” – mid elevation, gentle convex slopes with high solar radiation; “d” –low elevation, steep low solar insolation locations (northerly slopes); and “e” – lowelevation, steep high solar radiation slopes (southerly slopes). The results in Table 2 fromthese sets of evidence sources appear to be weak because there is no clear distinction

Table 4 Proportion of presence/absence associated with probabilities for NRR and RRlandslide hazard derived from the following six classes “a”, “b”, “c”, “d”, “e”, and “f”



Belief 0–0.2 0.0% 18.9% 0.4% 21.2%0.2–0.4 6.8% 27.6% 5.4% 25.6%0.4–0.6 14.7% 26.3% 23.9% 25.7%0.6–0.8 40.0% 19.6% 48.7% 20.6%0.8–1 38.4% 7.6% 21.6% 6.9%

Plausibility 0–0.2 0.0% 17.3% 0.0% 17.4%0.2–0.4 4.2% 25.8% 3.5% 26.5%0.4–0.6 14.2% 26.5% 22.8% 25.4%0.6–0.8 37.4% 21.4% 47.1% 21.9%0.8–1 44.2% 8.9% 26.6% 8.9%

Table 5 Proportion of presence/absence associated with probabilities for NRR and RR landslidehazard derived from the following seven classes “a”, “b”, “c”, “d”, “e”, “f ” and “Maxcl”



Belief

0–0.2 0.0% 17.6% 0.5% 19.1%0.2–0.4 4.7% 20.7% 3.7% 17.9%0.4–0.6 20.0% 35.0% 29.0% 32.0%0.6–0.8 46.8% 21.1% 46.7% 21.7%0.8–1 28.4% 5.7% 20.1% 9.3%

Plausibility

0–0.2 0.0% 16.5% 0.0% 16.0%0.2–0.4 2.6% 19.1% 3.0% 18.3%0.4–0.6 20.5% 35.7% 28.5% 33.7%0.6–0.8 46.3% 22.2% 47.7% 22.6%0.8–1 30.5% 6.5% 20.8% 9.4%



between high and low hazard areas. For example, the table shows that the RR and NRRlandslide hazards associated with the “Belief” and the “Plausibility” functions are clusteredaround probabilities of 0.2–0.4 where the high percentage of correctly identified landslides(presence) are associated with a large percentage of areas that were stable (absence).However, the same sets of sources of evidence were used by Gorsevski et al. (2003) combinedwith the Bayes’ theorem where final results showed satisfactory outcomes in the develop-ment of RR and NRR landslide hazard (Table 6). Further discussion of the comparisonof the D-S predictive models with Bayes’ predictive models follows in the next section.

Table 3 shows the goodness-of-fit derived from the total of five sources of evidenceincluding “a”, “b”, “c”, “d”, and “e”, Table 4 shows the goodness-of-fit derived fromthe total of six sources of evidence including “a”, “b”, “c”, “d”, “e”, and “f”, andTable 5 shows the goodness-of-fit derived from the total of seven sources of evidenceincluding “a”, “b”, “c”, “d”, “e”, “f” and “Maxcl”. Tables 3, 4, and 5 show betteroverall fits with better discrimination between high and low hazard areas than Table 2.The trend in all tables appears to be that the goodness-of-fit coupled with the NRRlandslide hazard prediction is better than the goodness-of-fit coupled with RR landslidehazard prediction. For example, the high probabilities (0.8–1) for the belief function forpresence of NRR landslide hazard and corresponding absence is 52.1% versus 13.5%in Table 3, 38.4% versus 7.6% in Table 4, and 28.4% versus 5.7% in Table 5, whilethe high probabilities (0.8–1) for the belief function for the presence of RR landslidehazard and corresponding absence are 46.9% versus 14.7% in Table 3, 21.6% versus6.9% in Table 4, and 20.1% versus 9.3% in Table 5. The percentages for the plausibil-ity function of high probability belief (0.8–1) for presence of NRR landslide hazard andcorresponding absence are slightly higher in all tables.

4.2 Comparison of D-S and Bayes Predictive Models

Table 6 was derived by using the Fuzzy/Bayesian approach (Gorsevski et al. 2003) andshows cross-tabulation of goodness-of-fit between the models and the independent testdata for presence and absence associated with the probabilities for both RR and NRRlandslides. The results from Table 6 suggest that the NRR Bayesian model provides abetter overall fit with better discrimination between high and low hazard areas than theRR model. Bayesian prediction for NRR landslides seems to have a higher level ofcertainty than the prediction for RR landslides, which corresponds with the results from

Table 6 Proportion of presence/absence associated with probabilities for NRR and RRlandslide hazard using Fuzzy/Bayesian approach

Probability


Presence Absence Presence Absence

0–0.2 11.1% 60.8% 8.6% 45.6%0.2–0.4 5.3% 6.1% 8.1% 11.4%0.4–0.6 4.2% 4.5% 20.7% 19.6%0.6–0.8 17.4% 10.3% 26.1% 11.0%0.8–1 62.1% 18.4% 36.5% 12.4%



the Fuzzy/D-S approach described in the previous section. However, the difference isthat a clear distinction of high and low hazard areas is more apparent when the Fuzzy/Bayesian approach is used. For example, in Table 6 for the NRR landslides a lowprobability hazard (0–0.2) is associated with 60.8% of total cells that did not experiencelandslides (absence), while a high probability hazard (0.8–1) is associated with 62.1%of the total cells that experience landslides (presence). Such a clear distinction of high andlow hazard areas is not the case with the Fuzzy/D-S approach. However, it is interestingto note, especially in Table 3 where the Fuzzy/D-S approach was used, that a highprobability hazard (0.8–1) for plausibility correctly classifies 64.7% of cells that expe-rienced landslides versus 8.6% of cells that did not experience landslides. This is verysimilar to the results in Table 6 where 62.1% of presence cells corresponded to 18.4%of absence cells. Results in Table 3 for the high probabilities predictions also suggestthat RR landslide hazard was better predicted by using the Fuzzy/D-S approach.

Table 4 shows that the belief function for the NRR landslides is associated with atotal of 6.8% presence cells versus 46.5% absence cells for low probabilities (0–0.4) anda total of 78.4% presence cells versus 27.2% absence cells for high probabilities (0.6–1). The plausibility function is associated with a total of 4.2% presence cells versus43.1% absence cells for low probabilities (0–0.4) and a total of 81.6% presence cellsversus 30.3% absence cells for high probabilities (0.6–1). As a comparison in Table 6there is a total of 16.4% presence cells versus 66.9% absence cells for low probabilities(0–0.4) and there is a total of 79.5% presence cells versus 28.7% absence cells for highprobabilities (0.6–1). Similar conclusions can be reached from Table 5.

Figure 3 illustrates the spatial implementation of the D-S belief function for theNRR landslides derived from the six sources of evidence including “a”, “b”, “c”, “d”,“e”, and “f” and it is associated with Table 4. For the [presence of landslides] hypoth-esis for the NRR landslide hazard the map in Figure 3 represents the lower boundaryof our commitment to the hypothesis. Similarly, the spatial implementation of the D-Splausibility function can be used to represent the upper boundary of our commitmentto the hypothesis. The range between the lower and upper boundary is represented inFigure 4, which denotes the belief interval or the uncertainty in the [presence of land-slides] hypothesis. Although it is difficult to pinpoint the exact evidence sources contrib-uting to high belief intervals in Figure 4, closer examination of the figure and meantopographic attribute values of categorized probabilities from the D-S “Belief interval”(Table 7) reveals that concave drainages are potentially the areas with the highest uncer-tainty. Examination of other belief interval maps derived from different combinations

Table 7 Mean values of topographic attributes derived from categorized probabilities fromthe D-S “Belief interval” (Figure 4)

Attributes/Probabilities 0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5

Elevation (meters) 1509 1650 1459 1497 1544Slope (percent) 41 32 35 25 50CTI (no units) 9.8 11.9 13.9 14 8.8Solar Radiation (KWH/m2) 968 1037 947 1083 875Profile Curvature (radians/100m) –0.01 –0.04 0.055 –0.06 0.44Tangent Curvature (radians/100m) –0.01 –0.09 –0.34 0.154 1.47



of sources of evidence showed similar spatial patterns as Figure 4 indicating that con-cave drainages are potentially the areas with the highest uncertainty. These are the areaswhere new evidence is necessary in order to reduce the uncertainties.

The advantage of the D-S theory compared to the Bayesian theory can be seen inthe capability of the former to handle uncertainties and establish valuable informationfor devising a data gathering strategy to reduce the uncertainties from predicated models(Eastman 2001). Furthermore, since evidence is not without uncertainties the D-S theoryis better in handling uncertainties that involve ignorance. For example, an absence of alandslide in the database does not necessarily mean that landslides were not present inthose locations but it may suggest that the photo interpreter failed to identify the pres-ence of a landslide for various reasons. Therefore in the D-S theory an ignorance valuecan be used to represent the lack of evidence (complete ignorance is represented by 0)while in the Bayesian theory the lack of evidence for a hypothesis represents evidenceagainst that hypothesis (Eastman 2001). In the Bayesian theory only singleton hypo-theses are recognized that are assumed to be mutually exclusive and exhaustive (i.e. their

Figure 3 D-S “Belief” map. Probabilities of NRR landslide hazard (the goodness-of-fit forthis output is shown in Table 4)This figure appears in colour in the electronic version of this article and in the plate sectionat the back of the printed journal.



probabilities must sum to 1.0) whereas in the D-S theory a single piece of evidencesupports more than one hypothesis including singleton as well as non-singleton hypotheses.

4.3 Comparison of RR and NRR Predictive Models

There are differences between the predictive D-S models of landslide hazard for NRRand RR landslides. Figure 5 shows the differences of probabilities of the occurrenceassociated with the belief function (lower boundary) for NRR versus RR landslides,which was derived from the same sources of evidence as the previous figures. The legendin the figure suggests that the high hazard for NRR landslides is associated with positivevalues, while high hazard for RR landslides is associated with negative values. Values closeto zero represent areas of agreement between both landslide hazards. Concurrent examina-tion of the results of landslide hazard models for NRR and RR depicted in Figure 5 withthe results from Table 3 suggests that the NRR model provides a better overall fit than theRR model. Prediction of NRR landslides seems to have a higher level of certainty thanthe prediction for RR landslides perhaps because NRR landslides appear to occur in a more

Figure 4 D-S “Belief interval” or uncertainty map. Range between belief function (lowerboundary) and plausibility (upper boundary) This figure appears in colour in the electronic version of this article and in the plate sectionat the back of the printed journal



tightly constricted set of environments characterized by specific environmental attributes,all of which conforms to the results reported by Gorsevski et al. (2003). In addition, NRRlandslide hazard appears more likely associated with steeper concave drainages, whilethe RR landslide hazard is associated with steep as well as gentle slopes. Consequently,two independent models, one for RR and the other for NRR landslide hazard may benecessary in order to capture the landscape processes, driving landslide occurrences.

5 Conclusions

A D-S modeling approach for predicting potential NRR and RR landslide hazard wasdescribed and applied to a study area in the Clearwater National Forest in Idaho. Themodeling approach was based on integration of terrain attribute data through fuzzy k-means classes, computing BPA for evidence sources from these classes, and combiningthe evidence sources by using the Dempster’s rule. The Fuzzy/D-S approach presentedin this paper confirmed findings reported by Gorsevski et al. (2003) about the use of the

Figure 5 Differences of probabilities of the occurrence associated with the “belief function”for NRR versus RR landslides This figure appears in colour in the electronic version of this article and in the plate sectionat the back of the printed journal



Fuzzy/Bayesian methodology to predict landslide hazard. In this study, uncertaintiesassociated with the results were presented in the decision-making process through con-sideration of intervals with lower and upper bounds rather than explicit probabilityvalues as with the Fuzzy/Bayesian methodology.

The evaluation of task one demonstrated that different evidence sources calculatedfrom fuzzy k-means classes for RR and NRR landslide hazard models yielded differentresults. It appears that the best results for the [presence of landslides] hypothesis werederived from six evidence sources including “a”, “b”, “c”, “d”, “e”, and “f ” for bothNRR and RR landslide hazard while the NRR landslide hazard goodness-of-fit wasbetter than the RR landslide hazard goodness-of-fit for all different combinations ofevidence sources.

The second task demonstrated that a Fuzzy/D-S approach could achieve similarresults as the Fuzzy/Bayesian approach. However, when the Bayes theorem is imple-mented, prior probabilities may be unrealistically assigned to each piece of evidencesuch that the sum of all assigned probabilities equals one. With D-S theory, ignorancevalues can be used to represent the lack of information while uncertainties in thedecision-making process are introduced through lower and upper probability intervals.Therefore, the advantage of the D-S theory over the Bayesian probability theory is itsability to handle problems under uncertainty and to represent the degree of uncertaintyassociated with each hypothesis. As a result, a robust data gathering strategy can bedevised that is most effective in reducing the uncertainty especially in areas where highbelief intervals exist.

The third task confirmed that predictive D-S models of landslide hazard for RR andNRR landslides are different. The test data further demonstrated that the goodness-of-fit for the NRR landslides was better than for RR landslides, and that the predictionwith higher levels of certainty was possible for NRR landslides. This justifies the needfor development of two independent models, one for RR and another for NRR, thatmay be used to address different questions for future planning and management ofnatural resources in predicting landslide hazard.

In summary, the example application of the D-S theory demonstrated the potentialof this approach in generating valuable site-specific information for decision-makers.We believe that this approach has a potential to support decision-makers in identifyinglandslide hazard areas and providing the means of dealing with uncertainty at a reason-able cost to the users.

Acknowledgements

The work was supported though data provided by the Clearwater National Forest andthe Rocky Mountain Research Station in Moscow, Idaho.

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Spatial Prediction of Landslide Hazard Using Fuzzy k-means and Dempster-Shafer Theory