A Comparative Analysis of Favorability Mappings1

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Natural Resources Research (NRR) pp1067-nrr-477452 December 11, 2003 18:23 Style file version Nov. 07, 2000

Natural Resources Research, Vol. 12, No. 4, December 2003 ( C© 2003)

A Comparative Analysis of Favorability Mappings byWeights of Evidence, Probabilistic Neural Networks,Discriminant Analysis, and Logistic Regression

DeVerle Harris,1 Lukas Zurcher,1,4 Michael Stanley,2 Josef Marlow,2 and Guocheng Pan3

Received 11 November 2002; accepted 27 February 2003

This study compares the performance of favorability mappings by weights of evidence (WOE),probabilistic neural networks (PNN), logistic regression (LR), and discriminant analysis (DA).Comparisons are made by an objective measure of performance that is based on statisticaldecision theory. The study further emphasizes out-of-sample inference, and quantifies theextent to which outcome is influenced by optimum variable discretization with classificationand regression trees (CARTS).

Favorability mapping methodologies are evaluated systematically across three case studieswith contrasting scale and geologic information:

Case Study Carlin Alamos Nevadasediment-hosted intrusion-related intrusion-related

gold copper copperScale deposit district regionalCell size small (0.01 km2) medium (1 km2) large (7 km2)Information level high moderate lowGeovariables complex simple simpleVariable interdependency moderate low highAsymmetry in frequency of modest considerable severebarren and mineralized cells

Estimated favorabilities for all cells then are represented by computed percent correctclassification, and expected loss of optimum decision.

The deposit-scale Carlin study reveals that the performances of the various methods fromlowest to highest expected decision loss are: PNN, nonparametric DA, binary PNN (WOEvariables), LR, and WOE. Moreover, the study indicates that approximately 40% of theincrease in expected decision loss using WOE instead of PNN is the result of information lossfrom variable discretization. The remaining increases in losses using WOE are the result of itslesser inferential power than PNN.

The district-scale Alamos study shows that the lowest expected decision loss is not byPNN, but by canonical DA. CARTS discretization improves greatly the performance of WOE.However, PNN and DA perform better than WOE.

Unlike findings from the Alamos and Carlin studies, results from the regional-scale Nevadastudy indicate that decision losses by LR and DA are lower than those by WOE or PNN.

1 Department of Geosciences, University of Arizona, Tucson,Arizona, USA.

2 Resource Science, Inc., Tucson, Arizona, USA.3 GeoSight, Inc., Highlands Ranch, Colorado, USA.

4 To whom correspondence should be addressed: Department ofGeosciences, 1040 E. Fourth Street, Tucson, Arizona 85721,phone: (520) 626-4962, Fax: (520) 621-2672; e-mail: [email protected].

2411520-7439/03/1200-0241/1 C© 2003 International Association for Mathematical Geology

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242 Harris, Zurcher, Stanley, Marlow, and Pan

Moreover, decision losses by CARTS-based canonical DA are noticeably the lowest of all,including those by LR and DA using the original variables.

KEY WORDS: Decision theory; mineral potential prediction; method performance.

INTRODUCTION

Favorability Mapping Defined

The favorability mapping paradigm consists of(1) training a method on the relationship of geovari-ables (lithology, structure, geophysics, geochemistry,etc.) to mineral occurrence on an explored area, (2)employing the trained method to infer favorabilityof unexplored cells (subdivisions of an unexploredarea) from their geovariables, and (3) selectingthe most favorable unexplored cells for follow-onexploration.

Data Integration by Weights of Evidence:A Favorability Mapping

Some geologists use Weights of Evidence (WOE)to generate a numerical index of mineral occurrencefavorability from data on geovariables (Agterbergand others, 1993; Mihalasky and Bonham-Carter,2001; Bonham-Carter and others, 1998). Among thereasons for this use of WOE is the congruence of twodevelopments: (1) the increased use of GeographicInformation Systems (GIS) to store, retrieve, pro-cess, display, and analyze geological information(Bonham-Carter, 1994), and (2) the perception ofWOE as a data integration tool. The phrase data inte-gration refers to the combining of several GIS layersinto a single layer that depicts the spatial distributionof a constructed index of a specific geological featureor state. When that index reflects geological favor-ability for the occurrence of a mineral deposit, theGIS layer is referred to as a favorability map. Theavailability of an ARCVIEWTM “plug in” for WOE(Kemp, Bonham-Carter, and Raines, 1999) enhancesthe appeal of WOE for data integration, as also doesthe capability of WOE to accommodate an unevenspatial distribution of information, that is some cellsfor which the states of one or more geological con-ditions are unknown—see Singer and Kouda (1999)for a more extensive discussion of these and otherfactors.

Perspective for this Study

The starting point for this study is the recognitionthat the integration of GIS layers by WOE is the cre-ation of a new layer showing the spatial distribution ofa Bayesian probability for mineralization. Thus, al-though WOE usually is described in intuitive terms ofevidence for and evidence against, it is, nevertheless,simply a user friendly way of estimating a Bayesianprobability, which is the probability for deposit oc-currence, given the geology: P(Deposit Occurrence/Geology). This view of WOE prompts examination ofalternative methodologies that accomplish the sameobjective: the mapping of geological favorability formineralization. Hereafter, the expression geologicalfavorability for mineralization is shortened to favor-ability. Accordingly, this study compares favorabilitymappings by selected mathematical methods, all ofwhich yield a Bayesian probability either directly orindirectly:

• Weights of Evidence (WOE)• Probabilistic Neural Networks (PNN)• Logistic Regression (LR)• Discriminant Analysis (DA)

Relevant Previous Work

Some of the research reported in this paperis a follow-on of the investigation by Harris andPan (1999) of favorability mapping for Carlin gold.In order of decreasing classification accuracy, thetested methods ranked as follows: Probabilistic Neu-ral Networks (PNN), Nonparametric DiscriminantAnalysis (DA), Parametric (Fisher) DiscriminantAnalysis, and Logistic Regression (LR). Singer andKouda (1999) compare PNN and WOE on the ChiselLake-Anderson Lake area, Manitoba, Canada, whichhad been analyzed previously by Wright andBonham-Carter (1996). They determined that whenone half of the data are used for training and theother half for validation, PNN had a lower error ratethan WOE. They also concluded that PNN, similarto LR, avoids some of the problems caused by inter-dependent (correlated) variables in WOE: The study

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A Comparative Analysis of Favorability Mappings 243

reported in this paper is an extension of the worksof Harris and Pan (1999) and of Singer and Kouda(1999) in that it compares favorability mapping byWOE with mappings by PNN and multivariate statis-tical methods. This study differs from other studies inthree ways: (1) the measure by which performances ofthe methods are compared, (2) the method by whichvariables are discretized to binary form, and (3) theselection of a subset of variables to be used by allfavorability mapping models by an external method-ology, that is, a methodology other than those beingcompared. Explanations of these procedures are pro-vided in subsequent sections.

CONCEPTS AND ANALYTICAL STRUCTUREFOR JUDGING PERFORMANCE

The Use of Case Studiesand Out-of-Sample Inference

This investigation employs case studies instead ofsimulation studies as a method for comparing favor-ability mapping methodologies. Moreover, the studyemphasizes out-of-sample inference, not just the fit ofa model to its training data. As used here, the phraseout-of-sample inference refers to using an estimated(trained) model to infer the favorability of areas thatwere not used to estimate model parameters. Suchan approach is consistent with a mineral explorationperspective, as it imitates the training of a model ona well-explored area for use on unexplored areas toselect potentially mineralized cells. However, in thisstudy, the areas selected for out-of-sample inferencewere well-explored, thereby providing a method forjudging relative performances of the different mod-els; such areas are referred to as validation areas(cells).

The study design, although well motivated,presents a number of challenging problems. Chiefamong them is the ambiguity that arises from incom-plete exploration in training or validation areas of thecase studies. A second difficulty arises from the in-terdependency of much natural geological informa-tion that is converted to geovariables. Interdepen-dency of geovariables affects model estimation andperformance. A third difficulty arises from the limita-tion of WOE to binary or ternary variables. A fourthdifficulty is the judging of performance by compar-ing model outputs, which are probabilities, with ac-tual occurrences, which are realizations of a stochasticprocess.

Features of the Case Studies

The three case studies represent three differ-ent scales of exploration: regional (Nevada), min-ing district (Alamos), and individual mineral deposit(Carlin). Accordingly, exploration completeness, cellsize, and specificity of geovariables differ greatlyacross these three scales of exploration:

Carlin gold mineralization: High-level information,small cell size (10,000 m2, or 0.01 km2), small ge-ographic area with relatively homogeneous geol-ogy, a single deposit type, complete exploration,complex geovariables, low to moderate variableinterdependency, moderate asymmetry in numbersof barren and mineralized cells.

Alamos District intrusion-related mineralization:Moderate-level information, cell size of 1 km2, in-complete exploration, simple geovariables, low in-terdependency of variables, multiple deposit types,considerable asymmetry in number of mineralizedand barren cells.

Nevada (north-central) intrusion-related mineraliza-tion: Low-level information, relatively large cellsize (7 km2, compared with 1 km2 for Alamosand 0.01 km2 for Carlin), incomplete exploration,simple geovariables, high geovariable interdepen-dency, low homogeneity of geology, multiple de-posit types, and severe asymmetry in numbers ofmineralized and barren cells.

Each of these case studies followed the sameparadigm:

• Divide the cells of the case study randomly intotwo subsets: Training set and validation set;• train the method (estimate model parameters)

on the training set only;• use the trained model to estimate favorability

for all cells (training cells + validation cells);• compute the percent correct classification for

all cells; and• compute expected decision loss of optimum

decision.

The number of randomly selected cells in the train-ing set was constrained by the capacity of the PNNsoftware. Moreover, the fraction of those randomlyselected cells that were mineralized was determinedtotally by the random sampling process.

It is important to emphasize that only one of thesecase studies, Carlin Gold, satisfies the requirement forunqualified comparison: All cells in the training andvalidation areas have been drill-tested and determined

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to be either barren or mineralized. Thus, although twoadditional case studies are employed to examine per-formances on different scales and complexity of geol-ogy and mineralization, only the Carlin study permitsunqualified judgment.

Misclassified Cells: A Difficult Problem

The most problematic of the factors affecting per-formance is the misclassification of mineralized cellsin the training set as barren, reflecting the fact thatsuch cells have deposits, but they have not yet beendiscovered. The presence of these misclassified cellsin the training set presents models with contradictoryor ambiguous information about geology and min-eral occurrence. To the extent that the geological vari-ables do capture important differences in the geologyof mineralized and barren cells, misclassification de-creases the capability of the models to identify thesedifferences and utilize them in classification of un-known cells. The presence of misclassified cells in thebarren group causes the trained models to not fit theirtraining data as well as they would otherwise, and theirperformances on the validation set are similarly con-founded. Although misclassification of training cellsdiminishes the performance of all methods, its effectmay not be equal across methods.

Measuring Performance

Although all investigated methods were selectedbecause they provide a Bayesian probability for min-eralization, output is treated only as a scaled [0,1]index. Accordingly, a method’s performance is de-scribed by the percentage of mineralized cells or bar-ren cells correctly classified when a specific indexvalue is used as the classification criterion (see Fig. 1).

Figure 1. Illustration of classification of mineralized and barrencells.

Probability bias is not important in this measure ofperformance, as the Bayesian probability is treatedonly as a decision index. What is important is howaccurately the index discriminates mineralized frombarren cells at selected cutoff probabilities (decisioncriteria). Moreover, as incorrect exploration deci-sions have economic consequences, each methodol-ogy is described by expected loss when its decisionindex is used optimally to select cells for further ex-ploration. The mechanics of performance valuationare explained further and illustrated in a subsequentsection entitled “A Decision-Theory Approach toValuation.”

Variable Redundancy, Discretization, and CARTS

Inasmuch as most geological data sets are redun-dant to some degree, comparison of model perfor-mances requires a ground-rule on model estimationand variable selection. The rule adopted in this studyis that all models are required to use the same variables.It is acknowledged that this rule obviates the selectionof variables that optimize each model’s performanceindividually, as such selections could differ across themethods. However, invoking this ground-rule permitsthe attribution of performances to the model, and notto the differences in geological information that arisefrom different sets of variables. When a data set hasgreat redundancy, a second analysis is made on a sub-set of relatively independent variables selected by anexternal method, CARTS (Classification and Regres-sion Trees), because the performance of each modelmay be affected differently by data redundancy. Then,all models are retrained and evaluated on the same re-duced set of variables. CARTS (Steinberg and Colla,1995; Breiman and others, 1984) is a software thatgrows classification trees by binary recursive parti-tioning of the variables.

The selection of an external methodology to re-duce the number of variables was made to removethe advantage that one method would gain over an-other if its reduced set were employed by all methods.The choice of CARTS as that external methodology isbased upon the capability it provides for achieving asecond objective: Enhanced discretization of continu-ous geovariables. Discretization is required by WOE,as it is the only tested methodology that requires bi-nary or ternary variables. Although a simple pres-ence/absence is the most frequently used discretiza-tion, it may not be the best. In order to present WOEin its best light, CARTS is used on two of the case

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studies not only to reduce the number of variablesbut also to identify discretizations that enhance theinformation content of the selected variables.

A DECISION-THEORY APPROACHTO PERFORMANCE VALUATION

The Need for an Objective Performance Measure

Two methods may differ so greatly by percent ofcells correctly classified that judgment about relativeperformance of the two methods can be made visu-ally. Apart from such polar cases, however, judgingrelative performance solely by visual inspection ofgraphs or tables may be difficult. One method mayclassify mineralized cells well but not the barren cells;a second method may classify barren cells well but notmineralized cells; and a third method may classify bothonly moderately well. Judgment about relative perfor-mance may require consideration of additional infor-mation as well as an objective measure. For mineralexploration decisions, the most important additionalinformation is the economic consequences of incor-rect classifications of barren and mineralized cells, forthese consequences may differ. For example, in ex-ploration for porphyry copper, the economic loss ofincorrectly rejecting a mineralized cell may be con-siderably greater than the loss from incorrectly re-taining a cell that is barren. When losses are asym-metric, the measure of relative performance by whichmethods are compared should accommodate thatasymmetry.

The performance measure selected for this studyis the expected loss of optimum decision when theconditional losses of acceptance and rejection are nor-malized on acceptance loss. Rejection loss is definedto be the opportunity lost when a mineralized cell isincorrectly rejected, indicating that it is judged to bebarren. Acceptance loss is the cost of additional ex-ploration required to determine that a cell retainedas mineralized is actually barren. Suppose, for exam-ple, that rejection and acceptance dollar losses were$1,000,000 and $50,000, respectively; then standard-ized rejection and acceptance losses would be 20 and1, respectively. Thus, the total expected loss of op-timum decision is a weighted sum of the two nor-malized losses, with the weights being the expectednumber of validation cells incorrectly rejected and ac-cepted, respectively. Clearly, the expected loss of op-timum decision can be computed only if performanceof a methodology has been described by percent

correct classification for a range of cutoff probabil-ities for mineralization, as described in the followingsection.

Classification Performance on the Validation Set

Correct classification, which indicates correctlyretaining mineralized cells and correctly rejecting bar-ren cells, varies with the cutoff probability used forclassification. Thus, for a cutoff probability, p, the per-cent of mineralized cells having a probability of atleast p is computed. Similarly, the percent of barrencells having a probability less than p is computed. Byrepeating this operation for several cutoff probabili-ties, a graph can be constructed that depicts the per-formance of a methodology as an exploration decisiontool (see Fig. 1). The following section describes howpercent correct classification and conditional losses ofmisclassifications are combined to give the expectedloss of optimum decision, which is the performancemeasure used in this study.

Expected Loss of Optimum Decision

Suppose that for an area containing NM mineral-ized cells and NB barren cells, percent correct classifi-cation has been determined for each at several cutoffprobabilities ranging from 0 to 1. Let m(p) be the per-cent of NM mineralized cells that are correctly classi-fied given cutoff probability for mineralization, p, andb(p) the percent of the NB barren cells that are cor-rectly classified. Define L as the ratio of unit rejectionloss to unit acceptance loss. Then, the expected deci-sion loss, denoted as E(p; L), is a function of p, giventhe normalized rejection loss for a given L (ratio ofrejection loss to acceptance loss):

E(p; L) = NM{[100−m(p)]/100}L+NB{[100− b(p)]/100}{1} (1)

Consider the first part of Equation (1). The term[100 − m(p)]/100 describes the fraction of the NMmineralized cells incorrectly classified as barren. Thus,the expression NM{[100 − m(p)]/100}L is the mag-nitude of opportunity loss as a result of misclassifi-cation of mineralized cells as barren, given a cutoffprobability (decision criterion) p. The second part ofEquation (1) describes loss resulting from misclas-sification of barren cells as mineralized using deci-sion criterion p. The sum of the two parts is the ex-pected decision loss for decision criterion p. Table 1

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Table 1. Expected Decision Loss

Fraction classified correctly Rejection loss as a multiple of acceptance loss (L)Cutoff probability for

mineralization (p) Mineralized Barren 1 10 25 50 100

0.0 1 0 14612.0 14612.0 14612.0 14612.0 14612.00.1 0.9448 0.8716 1973.9 2853.8 4320.2 6764.1 11652.10.2 0.9428 0.8944 1644.3 2556.0 4075.6 6608.1 11673.10.3 0.9338 0.9116 1408.9 2464.1 4222.7 7153.7 13015.70.4 0.9196 0.9297 1169.6 2451.1 4586.9 8146.6 15266.10.5 0.893 0.9447 997.5 2703.0 5545.5 10282.9 19757.70.6 0.8596 0.9597 837.5 3075.3 6805.1 13021.3 25453.70.7 0.7917 0.9754 728.4 4048.4 9581.9 18804.4 37249.40.8 0.6858 0.9884 725.9 5734.0 14080.7 27991.9 55814.30.9 0.4977 0.9973 929.0 8935.2 22278.8 44518.1 88996.81.0 0.2635 1 1304.3 13043.4 32608.5 65217.1 130434.2

Optimum Cutoff Probability 0.8 0.4 0.2 0.2 0.1Minimum Expected Loss [C(L)] 725.9 2451.1 4075.6 6608.1 11652.1

demonstrates the computation of expected decisionloss. Let NM = 1771 and NB = 14612, L = 1 and p =0.8. L= 1 implies that rejection and acceptance lossesare equal. Inspection of Table 1 shows that for p= 0.8,m(0.8)/100 = 0.6858 and b(0.8)/100 = 0.9884, whichare the decimal fractions of mineralized and barrencells, respectively, that are classified correctly whenp = 0.8 is used as the decision criterion. Substitutingappropriately in the equation for expected decisionloss, we have:

E(0.8; 1) = 1771(1− 0.6858)(1)

+ 14612(1− 0.9884)(1) = 725.9

This expected loss (725.9) is shown in the fourth col-umn on the row for p=0.8. Similarly, calculated valuesfor all other cutoff probabilities at L = 1 also areshown in the fourth column. Note that of these losses,that for p = 0.8 is the smallest. Thus, if the relevantratio of rejection loss to acceptance loss is 1, the op-timum decision rule is to retain all cells with proba-bilities of at least 0.8 and reject all cells with proba-bilities smaller than 0.8. Changing L and performingthese same calculations gives the decision losses forvalues of L of 10, 25, 50 and 100, as shown in the othercolumns of Table 1. The bottom 2 rows of this tableshow the optimum decision rule for each value of Land associated expected losses C(L):

C(L) = Minp{E(p : L)}= Expected loss of optimum decision. (2)

Figure 2 shows the plot of C(L) against L for thisillustrative example.

Notice that when L = 50, the optimum decisioncriterion is p = 0.2, with an expected loss of 6608.1.Thus, when rejection loss is 50 times the acceptanceloss, our expected losses are minimized by loweringthe cutoff probability to 0.2 so as to retain more of themineralized cells, even though doing so increases thecost of incorrectly retaining more barren cells. Natu-rally, the larger L is, the smaller the optimum cutoffprobability, as shown in Figure 3.

Suppressing intermediate computations, eachfavorability mapping method will be described bythe graph of C(L) plotted against L. Thus, for a givenarea and relevant L, the preferred methodology forfavorability mapping will have the smallest C(L).One methodology may have lower values of C(L)for small values of L, whereas C(L) for anothermethodology may be lower for high values of L, orvice-a-versa. A methodology is dominant when it has

Figure 2. Illustration of Decision Loss relationship.

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Figure 3. Effect of Rejection Loss multiple on optimum cutoffprobability.

the smallest C(L) over the entire range of relevant Lvalues.

THE CARLIN STUDY

Perspective

The Carlin sediment-hosted gold deposit islocated at −116.32◦ and 40.91◦ in north–centralNevada. This case study is an important one becauseexploration is considered to be complete to some rea-sonable depth: all cells have been tested for mineral-ization by the drill. Given the small cell size (100 mby 100 m) and Carlin-type mineralization, the likeli-hood of misclassifying a mineralized cell as barren isconsidered to be negligible. The Carlin case study isspecial for reasons other than exploration complete-ness. Of the three case studies, Carlin has the smallestnumber of geovariables (8), presenting less data re-dundancy. Moreover, the eight geovariables are theproducts of prior data processing and enhancement,making them rich in information related to Carlin goldoccurrence. A final benefit of using Carlin as a casestudy is that it already had been intensely investigatedin the comparison of PNN with multivariate statisticalmethods for favorability mapping (Harris and Pan,1999; Pan and Harris, 2000). The present study ex-tends that investigation to include WOE and binaryPNN with CARTS discretization.

Generation of Variables

Each of the cells is described by the pres-ence/absence of gold, as confirmed by drilling, and

by measurements on eight geological variables:

tm TM linear scoretrend regional structural trend scorevmag vertical magnetic componentresis resistivity ratiotc total radiometric countau soil Au in ppbas soil As in ppmhg soil Hg in ppb

Tm is a quantified structural score from a set of lin-eaments interpreted from Thematic Mapping images.The quantification is based upon a set of structuralmeasurements generated from a moving window. De-tails of this quantification procedure are given in Pan(1989). Trend is a quantified structural score basedupon regional structures interpreted by geologists.The quantification procedure is the same as for tm.Vmag is the first vertical derivative of the total mag-netic field. Resis is the ratio of the resistivity value at900 Hz frequency to the resistivity value at 7200 Hzfrequency. Tc is the total count of K, Th, and U from anairborne radiometric survey. Au, as, and hg are inter-polated, nontransformed, values of gold, arsenic, andmercury, respectively, from a soil sample geochemicalsurvey.

Training and Validation Sets

The Carlin case study consists of 16,383 cells eachof 10,000 m2 area (100 m by 100 m), all of which havebeen drill-tested for mineralization. In order to testthe predictive power of the models, cells were dividedrandomly into two sets: training and validation. Afterthe models were trained (estimated) on the trainingset, they were used to estimate probability for miner-alization for the validation cells as well as the trainingcells.

Random sampling of the 16,383 cells for 3,277training cells created the training set, which consti-tutes about 20% of all cells. Of the 3,277 training cells,321 are mineralized, the remaining 2,956 are barren.After training, the model was used to estimate prob-ability for mineralization for all 16,383 cells, of which13,106 had not been used in training. Of the 16,383cells, 1,771 are mineralized; the remaining 14,612 areknown to be barren. Thus, the favorability mappingmodels are judged by how well they perform on theentire set (16,383) of cells, 80% of which were notused in training of the models.

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Discretization of Variables and Analysis by WOE

The first hurdle to performing a WOE analysisis the discretization of the original variables to bi-nary variables. The procedure employed in the Carlinstudy examined each variable separately by splittingits range of values into several subclasses and iden-tifying those having the greatest relative frequencyof mineralization. That produced the following dis-cretization:

trend 1 for presence of trend, −1 for absenceof trend

vmag 1 for 1.8 < vmag <3.4, −1 otherwiseresis 1 for resis > 2.3, −1 otherwisetc 1 for tc> 11.755, −1 otherwiseau 1 for 20 < au < 30 or

au > 100, −1 otherwiseas 1 for as > 19, −1 otherwisehg 1 for 80 < hg < 95 or

hg > 100, −1 otherwise

Tm was dropped, as there were no preferred subdi-visions. Dropping of tm probably had little effect onrelative performance of WOE, as a subsequent analy-sis by CARTS (Classification And Regression Trees)revealed that tm was next to last in importance in thebuilding of a classification tree.

WOE analysis of the 3,277 training cells revealedthat the most important variables are the geochemicalscores, as they have the highest contrasts:

trend 0.174vmag 0.762resis 0.594tc 0.771au 1.665as 1.116hg 2.188

The WOE model based upon all 7 variables wasapplied to all 16,383 cells. The results of this analysisare shown in Figure 4.

For comparison purposes, the classification per-formance of PNN is shown in Figure 5. Figures 4 and5 reveal a performance by PNN that surpasses con-siderably that of WOE. For all cutoff probabilitiesgreater than 0, PNN correctly classifies larger per-centages of both mineralized and barren cells thandoes WOE. Moreover, favorability mapping by WOEis inferior to the mappings by LR and DA reportedby Harris and Pan (1999), who also determined fa-vorability mappings by LR and DA to be inferior toPNN.

Figure 4. Classification of all Carlin Gold cells by WOE.

An important and intriguing question is: Howmuch of the superior performance by PNN over WOEis the result of the loss of information in discretizingthe variables in order to use WOE? In an attempt toexplore that question, PNN was trained on the samebinary variables used in the WOE analysis. The resultsof applying that model to the validation set are shownin Figure 6.

Comparison of Figures 5 and 6 reveals a classi-fication performance by the binary PNN—using theWOE binary variables—that is noticeably inferior tothat of the PNN that used the original variables. Thebinary PNN classifies the mineralized cells well, but itdoes not do nearly as well on the barren cells as doesthe PNN using the original variables.

A strict comparison of the binary PNN withWOE requires consideration of the conditional lossesof incorrect rejection of mineralized and incorrect ac-ceptance of barren cells. Figure 7 depicts expecteddecision losses for selected normalized rejection lossmultiples from 1 to 100, given optimum decision rules,

Figure 5. Classification of all Carlin Gold cells by PNN usingcontinuous variables.

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Figure 6. Classification of all Carlin Gold cells by PNN using binaryvariables.

that is cutoff probabilities for each method. As shownin this figure, WOE has the highest decision losses bya considerable margin.

Of particular interest is the graph of expecteddecision loss for the binary PNN, because that modelused the same binary variables used by WOE. Espe-cially noteworthy are the considerably lower expecteddecision losses for the binary PNN than for WOE.Comparison of expected decision losses of PNN onthe original variables, PNN on the binary WOE vari-ables, and WOE strongly suggest that: Approximately40% of the increase in expected decision loss usingWOE instead of PNN is the result of information lossfrom discretizing the geovariables, given rejection lossmultiples of 20 or less. That percentage increases toabout 60% as rejection losses multiples increase to 100.The remaining increases (60% to 40%) in losses foreach rejection loss multiple are because of the greaterinferential power of PNN over WOE.

Inspection of Figure 7 further reveals that formost of the range of rejection loss multiples, expected

Figure 7. Comparison of Decision Loss for Carlin by logisticregression, nonparametric discriminant analysis, WOE, and PNN.

decision loss by the binary PNN is lower than that forLR (logistic regression) using the original variables.Generally, for the Carlin case study, Figure 7 showsthat the favorability mapping methods, ranked fromlowest to highest expected decision loss are:

• Probabilistic Neural Networks (PNN)• Nonparametric Discriminant Analysis (DA)• Binary PNN (WOE variables)• Logistic Regression (LR)• Weights of Evidence (WOE)

THE ALAMOS STUDY

Perspective

The Alamos district is located about −108.93◦

and 27.05◦ in southern Sonora, Mexico. The Alamoscase study differs from Carlin in ways that have thepotential to affect favorability mapping. The most im-portant of these is incomplete exploration, meaningthat some cells that are currently classified as barrenmay have undiscovered mineral deposits. Alamos dif-fers from Carlin in other ways, besides incompleteexploration: (1) larger cell size (1 km2 compared with0.01 km2), (2) simpler geovariables, (3) a narrowerspectrum of geoinformation, (4) multiple Cu deposittypes (skarn, porphyry, and polymetallic veins), and(5) less information enhancement.


The study area contains thirty seven prospects.Two, or possibly three prospects, are related to a por-phyry stock, seven are associated with skarn, and therest are polymetallic vein systems.

From old to young, explanatory variables in-clude metasediments (TJ seds), intermediate vol-canics (Kande), limestone (Kls), batholith (KTgd),felsic volcanics (Trhy), bimodal volcanics (Tvolcs),conglomerate (Tbaucarit), alluvium (Qal), and frac-tures (Frac). The variables were quantified by theiraerial extent within a given unit area (1 km2) cell,based on a digitized version of the map by Vazquez-Perez (1975). Because of their small outcrop propor-tion, porphyry intrusions were left out. Fractures werequantified based on selected influence buffers. In theinitial analysis, a buffer of 300 meters was used toquantify fractures (Frac300), as that buffer size max-imized WOE’s contrast. A buffer of 500 meters was

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assigned to each mineral occurrence to increase theinfluence ratio between barren and mineralized cells.


The 1,939 cells of the Alamos area were sampledrandomly for a training set of 970 cells, of which 41were mineralized. The remaining set of 969 cells con-stituted the validation set, of which 55 were mineral-ized. Then, the trained model was used to compute theconditional probability for mineralization for each ofthe 1,939 (training plus validation) cells.

Analyses

Initial Analysis. The initial analysis for Alamosused all 9 variables. Application of WOE requiredthat all continuous variables be discretized to binaryvariables. For the initial analysis, discretization wasa simple presence or absence. Figure 8 depicts theexpected decision loss for the various methods. Thefigure shows decision loss patterns similar to those ofthe Carlin study in that losses by WOE are greaterthan losses by PNN or DA.

Of course, some of the WOE loss could be theresult of loss of information by the simple pres-ence/absence discretization. To investigate the magni-tude of that loss, a second PNN was performed on thesame binary variables used by WOE. As in the Carlincase, expected decision loss by the Binary PNN is con-siderably less than it is for WOE, which demonstratesthe greater inferential power of PNN. Unlike theCarlin study, however, the lowest expected decisionloss is not by PNN, but by canonical discriminant anal-ysis (DA), based upon separate group covariance ma-

Figure 8. Comparison of Decision Loss for Alamos by PNN, WOE,and discriminant analysis.

trices. Interestingly, this DA achieves the lowest de-cision loss even though it is parametric, being basedupon the assumption that discriminant scores are nor-mally distributed.

A Second Analysis—An Investigation of En-hanced Discretization. The second analysis investi-gates the impact of (1) a more rigorous discretiza-tion than simple presence or absence, and (2) fewergeovariables. The finding of the large decision loss byWOE begs the question: How well would WOE per-form when it is applied under conditions that enhanceits performance? Such conditions would include re-duction of variables and a discretization scheme thatenhances the relationship between the resulting bi-nary geovariables and mineral occurrence.

Although there are various methods for dis-cretization (Pan and Harris, 2000), this study em-ployed CARTS. The criterion used to rank candidatesplits was the heterogeneity of a tree node, which inthis study was described by the Gini coefficient. Anal-ysis by CARTS determined the optimum tree to have5 nodes, when prior probability is an average of rela-tive frequency from the data and equal priors, andwhen rejection loss is 3 times acceptance loss. Be-cause a tree with 7 nodes had only a slightly higherdecision cost, it was selected to ensure that the favor-ability mapping methods were not overly constrainedby the methodology of CARTS, and to accommodatedifferences in how the various methods employed thegeovariables. For a 7-node classification tree, the mostimportant variables were, in decreasing order, Kande,KTgd, Trhy, TJseds, and Frac200. The cut points se-lected for discretization were early splits on the fivevariables: Kande > 0.00478; KTgd > 0.0237; Trhy >0.0796; Frac200 > 0.111; and TJseds > 0.158.

The new binary variables improved greatly theperformance by WOE. Figure 9 shows a great

Figure 9. Comparison of Decision Loss for Alamos by WOE:original binary and CARTS binary variables.

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Figure 10. Comparison of Decision Loss for Alamos by PNN,discriminant analysis, and WOE using CARTS binary variables.

reduction in decision loss when WOE is trained onthe same cells but with the new CARTS binary vari-ables. Decision loss by WOE on the CARTS variablesis closer to that of PNN and DA on the original con-tinuous variables than that of WOE on the originalbinary variables. This seems to suggest great bene-fits from information enhancement by the CARTSdiscretization and a concomitant decrease in interde-pendency of variables. The conditional independenceratio computed by WOE for the training set is 0.93,indicating an acceptable model in terms of variableautocorrelation. However, for the entire data set (val-idation+ training sets), the conditional independenceratio is considerably lower, 0.45.

Interestingly, further analysis reveals that bothPNN and DA perform better than WOE when ap-plied to the same CARTS binary variables. As shownin Figure 10, decision losses by PNN and DA are ap-proximately one half of that by WOE, with the lossby PNN being slightly lower than that for DA. Theselower losses are attributed to the greater inferentialpower of these methods over WOE. Again, the lowdecision loss by DA is especially noteworthy in thatthe CARTS binary variables depart greatly from nor-mality.

THE NEVADA STUDY

Perspective

The regional-scale Nevada study area is boundedby−118.25◦, 41.75◦ and−114.50◦, 38.75◦ in the centraland northeastern sector of the state. It is 103,908 km2

in size and consists of 14,844 cells, each 7 km2. Of thethree case studies, the Nevada study cell size is by farthe largest, being 7 times that of Alamos and 700 times

that of Carlin. A consequence of the larger cell size isdecreased specificity of geovariables in terms of theirrelationship to mineral occurrence. The larger num-ber of geovariables (15) describe a broad spectrumof information, for example magnetics, radiometrics,and geochemistry, in addition to lithologies and struc-ture. Everything else being equal, a large number ofgeovariables commonly results in increased variableinterdependency.

Known mineralization in the Nevada study areais not nearly as abundant as in the Carlin and Alamosareas. For example, mineralized cells constitute onlyabout 1.4% of the cells in the Nevada study area. Thisis a small percentage when compared with Carlin,for which 10.8% are mineralized, and with Alamos,for which about 5% are mineralized. Thus, as statisti-cal populations, the two classes of cells (mineralizedand barren) are asymmetric in number. This asymme-try could represent lower mineralization rates or in-creased post-mineralization cover. As with Alamos,mineralized cells in the Nevada study area includeseveral types of intrusion-related Cu deposits.


Conforming to the field descriptor “Genlith” ofthe digital version of the geologic map of Nevada(Turner, Bawiec, and Ambroziak, 1991), variableswere delineated based on lithology and age associ-ations. They include:

Uptec Upper Tertiary sedimentary, volcanicand unconsolidated materials

Lowtev Lower Tertiary volcanic unitsLowtep Lower Tertiary plutonic unitsMzvol Mesozoic volcanic unitsMzplut Mesozoic plutonic unitsLmzsed Lower Mesozoic foreland and coastal

sedimentary unitsUpzseq Upper Paleozoic sequences generalized

from: Antler sequence,Carbonate-detrital belt,Foreland/Shelf Carbonate sequence,and Siliceous and Volcanic assemblage

Lpzcarb Lower Paleozoic carbonate assemblageLpztrans Lower Paleozoic transitional

assemblageLpzclast Lower Paleozoic siliclastic assemblageThrusts Thrust fault influence buffers of 6 kms

based upon WOE contrastmaximization.

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Rad Radiometrics (Phillips, Duval, andAmbroziak, 1993) generalized fromthe frequency distribution to includehigh (Eu ≥ 2) and low (Eu < 2) valuesfrom an original cell size of 2.4× 2.4kms

Mag Total magnetic intensity (Kucks, 1999a)generalized from the frequencydistribution to include high (Gammas≥ 0) and low (Gammas < 0) valuesfrom an original cell size of2.4× 2.4 kms

Grav Bouguer gravity (Kucks, 1999b)generalized from the frequencydistribution to include high (Milligals≥ −220) and low (Milligals < −220)values from an original cell size of4.8× 4.8 kms

Geochem Principal components score generatedfrom analysis of 1553 NURE(Grossman, 1998) base metal streamsediment geochemical data. Thehigh-low contour map is based on thefrequency distribution of theln(Cu)-bearing factor.

Mineral deposit sites used for this analysis in-clude 207 copper-bearing intrusion-related mines andprospects extracted from MRDS (Frank, 1999) basedon the presence of Cu as a major commodity: replace-ment, skarn, disseminated, or porphyry copper.


The 14,844 cells were sampled randomly for atraining set of 1,929 cells, of which 31 were mineral-ized. The remaining 12,915 cells were withheld as avalidation set, of which 176 were mineralized. Thus,the training set represents only about 13% of the totalcells. Approximately 1.61% and 1.36% of the trainingset and validation set, respectively, are mineralized.As in the other case study areas, the trained methodswere used to estimate the probability for mineraliza-tion for each of the 14,844 cells, of which about 87%had not been seen by the models in training.

Analyses

Initial Analysis. The initial analysis employed all15 variables. Both WOE and Binary PNN employed asimple presence and absence discretization. As shown

Figure 11. Comparison of Decision Loss for Nevada by PNN, WOE,logistic regression, and discriminant analysis.

by Figure 11, relative performances of PNN, WOE,and Binary PNN on this data set are markedly dif-ferent from those on Carlin and Alamos. The moststriking feature of Figure 11 is that decision losses byLR and DA are much lower than those by WOE orPNN. Moreover, unlike Alamos and Carlin, the lossesby LR and DA are about the same, with Logistic Re-gression being slightly smaller. In the other studiesDA consistently outperformed LR.

A second remarkable feature of Figure 11 is thatlosses by Binary PNN are less than those of PNN onthe original variables. Decision losses by PNN, WOE,and binary PNN are approximately equal for rejec-tion loss multiples up to 20. The binary PNN gives thelowest expected decision loss for rejection loss multi-ples greater than 20. Interestingly, decision losses byPNN exceed those of WOE and binary PNN for mod-erate rejection loss multiples, for example multiplesbetween 20 and 70, but WOE incurs the largest lossesfor multiples greater than 70. Such a marked change inrelative performances begs explanation, given the rel-ative performances on Carlin and Alamos. This dataset suggests that:

(a) As quantified, the geovariables express littleinformation that is relevant to mineral occur-rence;

(b) redundant or conflicting information con-fuses training of models; or

(c) some of the cells in the training set thatare currently classified as barren are mis-classified, meaning that they have undiscov-ered mineral deposits.

Because there is no way to determine how manybarren cells have undiscovered deposits, the investi-gation turned to examination of geovariables for adiscretization that would enhance their information

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content, and to a selection that would reduce interde-pendency.

A Second Analysis—Enhanced Discretizationwith CARTS. Using CARTS to build a classificationtree revealed a problematic data set. For example,the optimum classification tree, given equal priors, arejection cost multiplier of 1 (indicating equal losses),and splitting by the Gini Coefficient employed only 1variable, Uptec (Upper Tertiary cover!). Moreover,for priors taken only from the data, CARTS wouldnot even build a tree. Varying the rejection costmultiplier changed classification accuracy, but hadlittle effect on the complexity of the tree or theimportance of the variables. Thus, as with Alamos,a decision tree a little larger than the optimum wasselected to identify variables to use and their cutpoints (splits) for discretization. From that analysis,only six of the 15 variables were selected, with thefollowing cut points for discretization: Uptec at0.90034, Thrusts at 0.36100, Mzvol at 0.00013, Radat 0.93890, Geochem at 0.03750, and Lmzsed at0.20600.

PNN and WOE were trained on the CARTS bi-nary variables and then used to estimate a probabilityfor mineralization for each of the 14,844 cells. A simi-lar analysis was made for a PNN using the continuousforms of the CARTS variables. Figure 12 shows thatdecision losses for all three methods are considerablyless than with the initial set of variables. However, themost dramatic result is the great decrease in WOE’sdecision losses brought about by using the CARTSvariables: from about 12000 to approximately 1200.Moreover, decision losses by CARTS-based WOE areconsiderably lower than those by either the CARTS-based binary or continuous PNNs. The WOE con-ditional independence ratio for the CARTS binaryvariables on the training set is 0.64 but for the

Figure 12. Decision Loss for Nevada based upon CARTS variables:WOE and PNN (continuous and binary variables).

Figure 13. Decision Loss for Nevada by WOE, logistic regression,and discriminant analysis using CARTS variables.

entire (training + validation) set, that ratio is ex-tremely low, 0.08, indicating severe violation of theconditional independency assumption even with theCARTS variables.

The performance of WOE on the CARTS vari-ables contrasts markedly with the relative perfor-mances of PNN and WOE in the Alamos and Carlinstudies. It is important to note, however, that eventhough losses by the CARTS-based WOE are re-duced, they are considerably larger than the decisionlosses by LR and DA for all 15 original variables (com-pare Figs. 11 and 12).

Using the same CARTS binary variables, LRand canonical DA were trained and used to estimatea probability for mineralization for all 14,844 cells.Decision losses by CARTS-based LR are somewhatlower than for the CARTS-based WOE, but not aslow as LR on the original variables. However, deci-sion losses by CARTS-based canonical DA are notice-ably the lowest of all, being lower than those of theCARTS-based WOE or LR and lower than those bythe LR and DA using the original variables (compareFigs. 11 and 13).

DISCUSSION AND CONCLUSIONS

The Carlin case study provides the rare opportu-nity to examine relative performances of favorabilitymapping methodologies for a single deposit typewhen exploration is complete. This circumstance isimportant when comparing mapping methodologies.Conclusions about relative performance of mappingmethods must be qualified whenever exploration isincomplete because of possible conflicting or ambigu-ous information presented for training when mineral-ized cells are misidentified as belonging to the barren

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set. Moreover, everything else being equal, the pres-ence of multiple deposit types further compounds thetraining of mapping models and the analysis of out-of-sample performance. This study determined the per-formance of WOE to be inferior to the other methods,which included PNN, DA, and LR. Furthermore, thisstudy also demonstrated that only part of this inferiorperformance by WOE is the result of the loss of infor-mation when the variables are discretized to satisfydata requirement by WOE.

The relative inferential power of PNN andWOE was examined by training PNN on the samediscretized variables used by WOE. Interestingly,expected decision losses by this binary PNN areconsiderably lower than those by WOE, therebydemonstrating that part of the noticeably inferior per-formance by WOE reflects a less powerful use ofinformation than by PNN. Moreover, decision lossesby the binary PNN are less than those by FisherDA and LR based upon the original variables.Favorability methods ranked by decision loss onthe Carlin case study from lowest to highestare:

• Probabilistic Neural Networks (original vari-ables)• Nonparametric Discriminant Analysis (origi-

nal variables)• Binary Probabilistic Neural Networks (WOE

variables)• Fisher Discriminant Analysis (original vari-

ables)• Logistic Regression (original variables)• Weights of Evidence (binary variables)

Although informative, the given ranking fails to notethe large disparity of performance by the poorest,WOE, and the best, PNN. In that regard, expecteddecision losses by WOE are three to six times thelosses by PNN, except for small multiples of rejectionloss. The only possible conclusion to be drawn fromthe Carlin study is: The routine use of WOE for fa-vorability mapping incurs a high opportunity cost, be-cause other methods, for example PNN and DA, pro-vide more accurate mappings.

The Carlin study is not typical of applicationsof favorability mapping. Seldom has each cell in thetraining area been explored to the point that itsmineral state is known with near certainty. Likewise,seldom is the mineralized state of every cell in a valida-tion set known with near certainty. Besides these, theCarlin study offered other favorable circumstances,such as a single deposit type, relatively homogeneous

geology and highly specific geological descriptionsthat relate to mineralization.

Considerable exploration is conducted at the dis-trict level (Alamos) or large size area (Nevada) undercircumstances less favorable than those of Carlin. Thecase studies of Alamos and Nevada explore favora-bility mappings under some of these less favorablecircumstances. Acknowledging that all results mustbe qualified because of incomplete exploration andabstracting from the details, the following broad, ten-tative conclusions can be drawn:

• The performance of WOE may be improvedconsiderably by replacing a simple pres-ence/absence discretization by one that en-hances information relevant to mineral occur-rence.• The performance of all methods on problem-

atic data sets can be improved by deletingor combining redundant, interdependent vari-ables.• Even when variables are discretized to en-

hance relevant information and redundancy isreduced, decision losses by WOE are greaterthan those by either PNN or canonical DA.In other words, one of the other alternatives(PNN and DA) incurs lower decision lossesthan WOE.• However, WOE is useful as a data exploratory

tool because it deals effectively with cells withmissing information. Moreover, the method isintuitive to the explorationist, because its ap-plication simulates the superposition of Mylarmaps over a light table.

The Alamos and Nevada case studies stronglysuggest that canonical DA is a more powerful fa-vorability mapping method than either WOE or LR.These case studies purposefully elected to test the per-formance of canonical DA instead of nonparametricDA, even though the study by Harris and Pan (1999)determined nonparametric DA to compete well withPNN. A priori considerations would dictate the useof nonparametric DA because it assumes no specificdistributional properties of the geological variables.Even so, this study examined canonical DA becausenonparametric DA is more difficult to use, as it in-volves search parameters. Accordingly, a question ad-dressed by these case studies is: How important isthe assumption made in classical discriminant anal-ysis that the geological variables must be normallydistributed? The approach of this study is a pragmaticone: If that assumption is important to favorability

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mapping, it will show up in the relative classifica-tion performances and expected decision losses ofcanonical DA. Interestingly, by both classification per-centages and expected loss of optimal decision, theassumption of normality does not seem to be an im-portant constraint to the application of canonical DAto favorability mapping. Even when the geologicalvariables are binary, having distinctly nonnormal dis-tributions, performances of canonical DA are superiorto LR, WOE, and in some situations PNN. Perhaps,this is because with canonical DA, probabilities arebased upon discriminant scores, not directly upon thegeological variables. As pointed out by Cooley andLohnes (1962) and later by Harris (1984), becausethe canonical discriminant score is a weighted sumof variables, it tends to be more normally distributedthan do the initial variables.

Finally, these three case studies collectively sug-gest that if WOE is to be used to map favorability,then good procedure also would be to use PNN andcanonical DA, and compare their performances on avalidation set prior to using any of them to infer favor-ability on an unexplored area. Good practice wouldalso include information enhancement and reductionof variable interdependency.

ACKNOWLEDGMENTS

Special thanks are extended to Don Singer fromthe USGS Mineral Resources Team. This manuscriptbenefited extensively from his critical review. We alsoare grateful to the USGS’s Tucson Field Office forallowing us the use of their GIS Lab.

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A Comparative Analysis of Favorability Mappings1

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