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Fuzzy Signal Detection Theory: Basic Postulates andFormulas for Analyzing Human and Machine Performance

Raja Parasuraman and Anthony J. Masalonis, Catholic University of America, Wash-ington D.C., and Peter A. Hancock, University of Minnesota, Minneapolis, Minnesota

Signal detection theory (SDT) assumes a division of objective truths or "states ofthe world" into the nonoverlapping categories of signal and noise. The definitionof a signal in many real settings, however, varies with context and over time. Inthe terminology of fuzzy logic, a real-world signal has a value that falls in a rangebetween unequivocal presence and unequivocal absence. The definition of aresponse can also be nonbinary. Accordingly the methods of fuzzy logic can becombined with SDT, yielding fuzzy SDT. We describe the basic postulates offuzzy SDT and provide formulas for fuzzy analysis of detection performance,based on four steps: (a) selection of mapping functions for signal and response,(b) use of mixed-implication functions to assign degrees of membership in hits,false alarms, misses, and correct rejections; (c) computation of fuzzy hit, falsealarm, miss, and correct rejection rates; and (d) computation of fuzzy sensitivityand bias measures. Fuzzy SDT can considerably extend the range and utility ofSDT by handling the contextual and temporal variability of most real-world sig-nals. Actual or potential applications of fuzzy SDT include evaluation of the per-formance of human, machine, and human-machine detectors in real systems.

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end in doubts; but if he will be content to beginwith doubts he shall end in certainties.

- Francis Bacon, The Advancementof Learning (1605, bk. 1, v. 8)

INTRODUCTION

A . A = 0. This seemingly irrefutable math-ematical expression asserts that a statementand its opposite can never coexist. Somethingeither is or is not. A tumor is either cancerousor benign: a new car is either reliable or faulty;a politician is either honest or dishonest, andso on. Confidence in the truth of this mathe-matical expression and of these representativestatements pervades both academic and every-day thinking.

A little reflection reveals that although thelogical expression is often true. it need not al-ways be true. After all, tumors, cars, and politi-

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white. To cope with this possibility, Zadeh(1965) developed fuzzy logic, sometimes sim-ply called fuzzy (Kosko, 1993, 1997). Fuzzylogic allows the possibility that the intersectionof A and A is nonzero. Is it or isn't it? Thetruth lies somewhere in between. To the extentthat an event is somewhere in between, forcingits categorization into nonoverlapping sets ofblack and white can result in the loss of usefulinformation and less sensitive analysis. Rather,if we follow Bacon's admonition to "begin withdoubts" and express those doubts mathemati-cally in fuzzy terms, then the analyses that fol-low may very well "end in certainties."

If we are doubtful whether a given event isa member of a particular category or not, howcan we reach meaningful decisions and takeappropriate actions based on our knowledge ofthe properties of that category? To return toour examples, the tumor needs to be operated

Address correspondence to Raja Parasuraman, Cognitive Science Laboratory, Catholic University of America, NVashington,DC 20064; [email protected]. HUMAN FACTORS, Vol. 42, No. 4. Winter 2000. pp. 636-659. Copyright O 2000,Human Factors and Ergonomics Society. All rights reserved.

Parasuraman, R., Masalonis, A.J., & Hancock, P.A. (2000). Fuzzy signal detection theory: Basic postulates and formulas for analyzing human and machine performance. Human Factors, 42 (4), 636-659.

FUZZY SIGNAL DETECTION THEORY

on. the faulty car returned to the dealer, thepolitician voted out of office. Is a fuzzy charac-terization of events a recipe for indecisivenessand inaction? Not necessarily.

As we demonstrate in this paper, fuzzy logiccan be combined with a well-known methodol-ogy for analyzing decision making: signal detec-tion theory (SDT). The result, which we termfrzzy SDL allows for a broader and potentiallymore powerful analysis of decision-making per-formance than do conventional methods. More-over, as we shall show, use of fuzzy SDT canavoid the possibility of erroneous conclusionsthat may arise from the application of standardSDT to situations in which signal definition isfuzzy.

SDT was initially developed to quantify theperformance of electronic receivers for detectingnoisy radio signals (Peterson, Birdsall, & Fox.1954). It was later extended to describe humandetection of threshold-level sigrnals (Tanner &Swets, 1954). Green and Swets ( 1 966) describ-ed the modified theory in a landmark bookthat led to widespread application of SDT (inpsychology and related disciplines) to a varietyof perceptual and cognitive tasks involvingdecision making (Swets & Pickett, 1982).

SDT has been shown to provide indepen-dent measures of the bias and the accuracy ofdecision outcomes, thereby granting it manyadvantages over competing theories and com-putational methods. such as high-thresholdtheory (MacMillan & Creelman, 1991). Anoth-er advantage is that SDT can be used to analyzehumran, machine, or joint human-machine per-formance, thus providing a common metric todescribe diverse aspects of detection perfor-mance in many application domains (Parasur-aman, 1985; Parasuraman & WVisdom, 1985:Sheridan & Ferrell, 1974; Sorkin & Woods, 1985:Swets, 1996).

This well-established theoretical approach isbased on a division of objective truths, or statesof the world, into one of two nonoverlappingcategories: signal or noise. SDT also typically(but not always) assumes binary responses madeby the human or machine observer; for exam-ple, "yes, a signal is present," or `no, a signal isnot present." Traditional SDT requires the map-ping of environmental events or sources of evi-dence into two categorical states of the world.

As our opening examples of various events andstates indicate, such mapping is often fuzzyrather than exact. The question 'What is a sig-nal?" does not usually arise in laboratory stud-ies, The exception is when the experimenter isinterested in examining the effects of uncer-tainty regarding some aspect of the signal, suchas its duration, starting time, frequency, and soon. WVe discuss the general problem of uncer-tain signal detection (Tanner & Birdsall, 1958;Green & Swets, 1966) in relation to fuzzySDT in a later section of this article.

In the laboratory, the signal is whatever theexperimenter defines it to be. In a perceptualexperiment on visual discrimination, for exam-ple, the signal may be defined as a line orientedat 90°, whereas lines with orientations ofO_80c are defined as nonsignals or noise. Aresearcher studying recognition memory maydefine a signal as a face that has been shown tothe participant during a prior study period andother previously unseen faces as noise.

In most such laboratory studies of percep-tion, memory, and cognition, the categorizationof a physical event as signal or noise is fixed. Incontrast, in most real settings the definition of asignal is often context-dependent and varieswith several factors. Real-world signals arefuzzy. For example, a sonar operator trying todetect the electronic signature of an approach-ing submarine on a visual display terminal hasto look for a line with a luminance, contrast, andspatial frequency that will depend on the sub-marine speed, ocean currents. presence of othernearby objects, and so on. The signal will alsovary over time as the submarine approaches.

Even when the formal or legal definition ofa signal is clearly specified in terms of somemeasurable physical event, people may treatthe event variably in different contexts. Forexample, the legal definition of a conflict in theflight paths of two aircraft being monitored byair traffic control (ATC) is fixed. According toFederal Aviation Administration regulations, a"signal" in ATC occurs when two aircraft comewithin 5 nautical miles (nmi) horizontally and1 000 ft vertically of each other.

The acceptable minimum aircraft separationvalues (vertical and lateral) actually depend onseveral variables, including the altitude of theaircraft and the environment (cruise, approach

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Winter 2000 - Human Factors

for landing, etc.). For the sake of simplicity, how-ever, in this paper we use the 5-nmi/l000-ftminimum standard that prevails in low-altitude(below 29,000 ft) cruise flight.

However, the separation distances that theair traffic controller will consider a signal re-quiring action will often exceed these mini-mum values. The controller's definition ofsignal will vary depending upon the complexityof the traffic, the nature of the ATC sectorbeing controlled, and other factors. In one con-text, say, with many climbing and descendingaircraft, two planes separated by 8 nmi may tosome extent represent a signal requiring urgentaction because of the potentially high likeli-hood that the planes will come closer while thecontroller is distracted by other aircraft. Inanother situation, say, level cruise flight, twoaircraft that approach to within 6 nmi may notparticularly perturb the controller.

The definition of a signal in many real-world situations is therefore variable or fuzzy.We introduce the concepts of contextual andtemporal variability as key characteristics ofsignals in real settings and propose that fuzzymethods are particularly well suited to dealwith these sources of variability. These consid-erations suggest that fuzzy logic could be usedto extend traditional SDT analysis to situationsin which membership of events to signal andnoise sets is not strictly dichotomous.

There has been some limited work incorpo-rating both fuzzy logic and SDT. Fuzzy meth-ods have been used to create decision-makingalgorithms whose results have then been ana-lyzed with traditional SDT or other decision-theoretical analysis; such combinations havebeen used in the domain of personnel psychol-ogy (Alliger, Feinzig, & Janak, 1993; Craiger &Coovert, 1994). In contrast, the present workconsiders fuzzy logic and SDT simultaneouslyas opposed to successively, and develops gener-al methods that can be used across a wide vari-ety of applied and basic domains. Furthermore,our methods allow for an analysis of the influ-ence of the fuzziness inherent in the real worldat all stages, from signal to response. We de-velop the basic postulates of fuzzy SDT andderive formulas for the fuzzy analysis of detec-tion performance. We also discuss the implica-tions of fuzzy SDT analysis for understanding

human and machine detection performance.Before describing the formulas for fuzzy SDTanalysis, we provide brief overviews of fuzzylogic and conventional SDT in the next two sec-tions. Readers familiar with these topics maywish to skip either or both of these sections.

FUZZY LOGIC: A REVIEW

Fuzzy logic represents an alternative meth-od to traditional set theorv for assignment ofmembership of events to sets (Zadeh, 1965).Fuzzy methods have been used in a variety ofbasic and applied areas (see Kosko, 1993, fornumerous examples from many scientific do-mains). Among the areas that are particularlyrelevant to the present paper are applicationsof fuzzy logic in the following domains; (a)cognitive psychology (Brainerd & Reyna, 1993;Campbell & Massaro, 1997; Ellison & Mas-saro, 1997; Heiser & Groenen, 1997; Massaro,1l988, 1998), (b) human factors/ergonomics andhuman-computer interaction (e.g., Genaidy etal., 1998; Karwowski & Mital, 1986; Kreifeldt& Rao, 1986; Lehto & Sorock, 1996; Moray,King, Turksen, & Waterton, 1987; Moray, Krus-chelnicky, Eisen, Money, & Turksen, 1988),and (c) biomedical engineering and neuroscience(Baumgartner, Vinddischberger, & Moser, 1998;Bellazzi, Silviero, Stefanelli, & De Nicolao,1995; Boston, 1997; Lowe, t larrison, & Jones,1999; Phelps & Hutson, 1995). Despite the greatnumber of these applications, we must reiterateone point: The application of fuzzy logic in thepreviously cited works has been either to de-tect a discrete (nonfuzzy) signal or to analyze asystem or result that was derived using non-fuzzy methods. In contrast, in the present paperwe present a general model for analysis offuzzy signals and fuzzy responses.

Fuzzy is best thought of as an extension oftraditional set theory, in which elements eitherdo or do not belong to a given set. Because aclear boundary exists between set members andnonmembers, traditional sets are often referredto as crisp sets. Suppose we wish to define a setto describe the range of temperatures (t) that anormal human being would consider comfort-able. An example of a crisp set is

C(t) = [55, 851.

638

(1)



which would mean that the numbers 55 and85 (temperatures in degrees Fahrenheit), andall real numbers between these two, are mem-bers of the set C, and all numbers below 55 orabove 85 are nonmembers. If we plot C(t) ver-sus t, the result would appear as shown in thesolid line in Figure 1. Hiowever, it would seemmore appropriate to distinguish between levelsof comfort rather than to assign every temper-ature to either the comfortable or uncom-fortable sets. We could instead develop afunction that permitted a temperature's mem-bership in the set comfortable to be some-where between yes and no, or between 0 andl. The mapping function could be derived in anumber of ways. For example, one could use ascale from 0 (completely uncomfortable) to 1(completely comfortable) and have a sampleof people rate several temperature values onthis scale. The average rating for each temper-ature and the resulting function might bedefined as follows:

0C() = { (t - 50)/20,

(90 - t)/200

t < 5050 < t < 70 ).70 < t < 90t > 90

(2a)(2b)(2c)(2d)

This function, which is shown in the dottedlines in Figure 1, provides a more realistic as-sessment of comfort versus temperature andbetter captures the variability in comfort level.The act of assigning nonbinary membershipdegrees to a previously binary definition can becalled fuzzification.

SIGNAL DETECTION THEORY: A REVIEW

SDT assumes two possible states of theworld: signal (s), in which the event of interestis present, and noise (n), in which it is absent(Green & Swets, 1966). At any given time oneof these states of the world occurs. The detec-tion system (human or machine, or some com-bination) makes a yes (Y) or a no 'N) judgment,

CrispMapping

FuzzyMapping

II

II

PI I

I %I I

I \I x

I %

I II %I x

I II I

I II I

0 10 20 30 40 50 60 70 80 90 100

t (temperature, OF)Figure 1. Crisp and fuzzy plots of comnfort, C, as a function of temperature, t.

0.9

0.8

0.7

C(t)0o6

0.5 F

04 F

0.3

0.2

0.1

0

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indicating whether or not it is believed that thesignal is present or absent. Across many suchoccurrences or trials, the two possible states ofthe world and the two possible decisions resultin four possible outcomes, each with an associ-ated probability (P):

Hit; signal present, yes judginent,P(Yls) = Hit rate (HR) (3a)

Miss; signal present, no judgment,P(Nis) = Miss rate (MR) (3b)

False Alarm (FA); signal absent,yes judgment, P(Yln) = FA rate (FAR) (3c)

Correct Rejection (CR); signal absent, nojudgment, P(Nln) = CR rate (CRR). (3d)

Onlv two of the four probabilities are need-ed for complete characterization of the perfor-mance outcomes, because HR + MR = 1, andCRR + FAIR = 1. The convention is to use theprobability of a Hlit, or I-lit Rate (HR) and theprobability of a False Alarm, or FA Rate (FAR),to describe the decision outcomes. The hit andfalse alarm probabilities can then be used tocompute various measures of the performanceof the detection system. In general, it is neces-sary to distinguish the sensitivity or bias-freeaccuracy of the detection system from the cri-terion or decision threshold associated withthe choice of judgments or responses. In SDT,sensitivity is indexed by the parameter d' andthe criterion by the parameter 3 (Green &Swets, 1966; MacMillan & Creelman, 199 1).

BASIC ELEMENTS OF FUZZY SDT

Mapping Functions for Signaland Response

The basis for the notion of fuzzy SDT is thatan event or trial can belong to the set signalwith some degree between 0 and 1. In addition,the response can belong to the set response witha degree between 0 and 1. Throughout thispaper the parameter s will refer to the degree towhich an event is a signal, The parameter r willrefer to the degree to which a yes (signal pre-sent) response was made. In fuzzy SDT, either sor r or both must be continuous variables in therange L0, 1]. One of the two may be binary; thatis, E {0, 1 1. I-lowever, s and r cannot both be

binary, for this would then reduce fuzzy SDT tocrisp SDT. This paper primarily deals with casesin which s and r are both continuous, or s iscontinuous but r is binary.

In assigning degrees of membership to thesignal (s) and response (r) sets, it is necessary toevaluate all possible states of the world (SWs)and each possible response value (RV), and todetermine what values will be assigned to each.A mapping function is required to derive a sig-nal value, s, based on some variable or set ofvariables that describe the SW. Many suchmapping functions are possible for the signal.

As noted by Moray et al. (1987), in manyapplications of fuzzy logic the mapping func-tions that are chosen are plausible but never-theless appear somewhat arbitrary (but seeTsoukalas and Uhrig, 1997). This is particular-ly true if a mapping function has to be devel-oped for some perceived characteristic, such asthe emotional quality of a face (Ellison & Mas-saro, 1 997) or the heaviness of a lifted object(Genaidy et al., 1998). This problem is some-what less severe in fuzzy SDT, because themapping function for the signal can generallybe based on objective physical variables corre-sponding to the state of the world. Given thatthese variables can be measured, the mappingfunction can be specified.

Of course the type and complexity of themapping function will vary with the applica-tion. For instance in the ATC example describedearlier, a function could map the distance sepa-rating two aircraft, call it a, onto s. The map-ping function could also be considerably morecomplex than this. by taking into account notonly a. but other variables as well.

The mapping function for the responsecould be based simply on a confidence ratingof signal presence: for example, "80% confi-dent a signal occurred." Such mapping is anextension of traditional SDT methodology(Green & Swets, 1966). Often in standard SDT,in order to derive multiple receiver operatingcharacteristic (ROC) points, judgments of con-fidence are permitted to be incorporated intothe response (e.g., yes-sure, yes-not sure, no-not sure, no-sure). Despite the nonbinarynature of the judgments, the traditional analyt-ic procedure for such rating data has still beencategorical rather than fuzzy. Typically each

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confidence rating is mapped cumulatively intodichotomous yes-no categories (i.e., first ananalysis is conducted with yes-sure as a yesand all other responses as a no, next the yes-sure and yes-not sure are analyzed as yes re-sponses with the others being no, and so on).Fuzzy SDT differs fundamentally from crispSDT in that membership assignments to theyes and no categories remain fuzzy rather thanbeing forced into crisp categories. Finally. themapping function for r can also be based onreported signal severity, criticality, or intensity.

In SDT, the value of a random variable, theevidence variable, represents the strength ofthe signal as perceived by the operator (Green& Swets, 1966). As the evidence variable isgenerally continuous, it is amenable to fuzzifi-cation. For some applications, in fact, the r val-ues can be assigned based on the evidencevariable. This strategy is appropriate if signaldetection performance is being measured withregard to how decisions are made based onwhat is perceived. However, if one is interestedin decision-making performance and patternswith respect to some "true" or objective stateof the world, then the variable mapped into rmust be some measure of the actual objectivestate of the system about which decisions arebeing made.

Mapping functions: Discrete states of theuworld and responses. The following formulascan be used whenever there are a finite numberof discrete states of the world (see Tsoukalas &Uhrig, 1997):

s =s5 for SW= 1 to ns (4a)r rR for RV= to nR (4b)

in which SW = each possible value of the stateof the world, RV = each possible responsevalue that could be made, ns= number of pos-sible discrete SW values, and nR = number ofpossible discrete RV values. One case in whichthis mapping approach would be used is whendiscrete ratings of confidence level of responseare taken (yes-sure, yes-unsure, and so on).

Mapping functions: Continuous states of theworld and responses. In other situations theoriginal variables representing SW and RVmay be continuous. In these cases. SW or RVcan take on an infinite number of values, and a

function should be constructed mapping theSW or RV values to s or r values, respectively,in the range between 0 and 1:

s (SW) = fSW)r (RV) = g(RV)

(5a)(5b)

in which SW = original value of the variablerepresenting state of the world. on an intervalscale, RV = original value of the variable repre-senting response value, on an interval scale,and f(.), g(.) = functions defined according tothe needs of the analysis being conducted andwhose range must be [0, I] or included within[0, 1].

Some features of these mapping functionsshould be noted. First, the function itself neednot be continuous, depending on contingenciesin the domain being analyzed. For example ifsome legal cutoff point exists in the SW, abreak in the function's continuity can occur atthat point (e.g., a jump from r = .5 to r = .9).Also, in fuzzy SDT it is not necessary to choosebetween Equations 4a-4b and Equations 5a-5b.For example it is permissible to have s be basedon a continuous interval value (use Equation5a) and r on a categorical value (use Equation4b), or vice versa.

Example mapping ftunction. Consider con-flict detection in air traffic control, in whichthe signal (presence of an aircraft conflict) iscoded according to the separation distance, a,between two aircraft potentially headed for aconflict (horizontal separation less than 5nmi). The function mapping the SW variable.a, to s can be determined according to the fol-lowing rationale. As the separation distance, a,decreases, the event becomes more signal-likeand conversely, becomes less signal-like as aincreases. As mentioned previously, the legalFAA definition of a conflict in ATC is 5 nmihorizontal separation. Clearly, though, worsethings can happen besides a violation of thelegal minimum. For this reason a collision (a =0) is defined as s = I and a marginal violationof the 5-nmi criterion (a = 5) as s = .9. Amonotonic decreasing function can be con-structed that allows for an increasingly sharpdrop-off of s as a increases beyond the 5-nmicutoff and yields relatively similar (low) valuesof s at all high values of a.

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Table I gives some values of a and s, andFigure 2 plots the mapping function. The crispmapping function that would be used in stan-dard SDT is also shown. For a > 5 nmi, thecrisp function assigns no value to signal. Thiscould result in some loss of information andpotential insensitivity in analyzing detectionperformance in comparison with fuzzy SDTbecause it is well known that air traffic con-trollers typically attend to, and sometimes evenact on, pairs of aircraft that will be greaterthan 5 nmi apart.

If there are many or infinite SW values, or atheoretical basis exists for using a specificmapping function, or both, a continuous map-ping function (Equation 5a) should be used.For example, a could be mapped to s using thesigmoid function, s = 1/[1 t (a/k)n], in whichk is a constant and the exponent n can be cho-sen depending upon the desired sharpness ofthe mapping function. The function in Figure 2can be approximated with a sigmoid with k =10 and n = 5. Other sigmoid functions couldalso be used, such as the "squashing" functionsthat are commonly used in connectionist(neural network) models (Rumelhart, Hinton,& Williams, 1986).

Exploration of Methods for CalculatingFuzzy Set Membership

Let us retrace the steps of the analysis so far.First we allowed the definition of both signal, s,

and response, r, to be continuous rather thanbinary. Next, we discussed how mapping func-tions can be selected for deriving values of s andr. An obvious next step is to determine to whichof the four traditional SDT categories (Hit,Miss, FA, and CR) a given s and r pair shouldbe assigned. To which category should the Is, r}pair be assigned and how? The solution is toallow a given event pair to belong with somedegree to more than one of the four categories.In traditional SDT, each event falls completelyinto one of the four outcome categories, basedon two dichotomies (SW - signal vs. noise; RV- yes vs. no). In fuzzy SDT, on the other hand,each event can fall into more than one of thefour categories used in traditional SDT.

In evaluating methods for determining anevent's degree of membership in each of thefour outcome categories, we applied two criteriato develop an appropriate set of functions. First,the functions had to reduce to those of crispSDT when s and r were made binary, becausefuzzy SDT is an extension of the crisp case (oralternatively, crisp SDT is a special case of fuzzySDT). A second criterion was face validity - if aset of functions gave results that were illogicalfor any or all inputs, they were rejected.

An important component of our judgmentof face validity was the expectation that perfor-mance will be considered better the closer thatr is to s. For example, if s = .7, the optimumdegree of response r should also be .7. In the

TABLE 1: Example of a Mapping Function in an Air Traffic Control Application

Separation Signal (s) Separation Signal (s) Separation Signai (s)a (nmi) a (nmi) a (nmi)

0.0 1.00 4.1 0.92 9.0 0.651.0 0.98 4.6 0.91 10.0 0.501.3 0.97 4.9 0.90 11.2 0.331.9 0.96 5.0 0.90 12.0 0.252.0 0.96 5.8 0.88 12.3 0.232.2 0.96 6.0 0.87 13.4 0.152.4 0.95 6.9 0.83 14.5 0.122.9 0.94 7.0 0.83 15.6 0.093.0 0.94 7.1 0.82 16.1 0.083.1 0.94 7.3 0.81 16.2 0.083.3 0.93 7.4 0.80 17.8 0.043.7 0.93 7.7 0.77 19.0 0.023.9 0.92 7.9 0.76 19.3 0.014.0 0.92 8.8 0.67 20.2 0.00

Note. An Aircraft-to-Aircraft "Conflict" or Signal (s) is Defined in Terms of the Separation Distance (a) Between Aircraft.

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Crisp Mapping

1

0.9

0.8

0.7

0.6

S 05

0.4

0.3

0.2

0.1*

0

Fuzzy Mapping

0 5 10 15 20 25

State of the World: a (separation distance in nm)Figure 2. Sigmoid mapping function relating signal value, s, to each possible value of aircraft separation dis-tance, a. as might occur in an air traffic control case. The rectangular function represents the standard orcrisp SDT mapping of s to a.

same vein, if r > s, the ideal set of functionswill assign at least some degree of FA member-ship to the event because the response degreeis higher than ideal and the Miss degree forsuch an event should be 0. The oppositeshould hold true when r < s. The Miss degreemembership should be nonzero and that forFA should be 0.

The considerations just discussed are basedon the assumption that both s and r are definedon scales that are mappable to each other. Inother words, whatever the application domain,the method of assigning memberships to s andr should be done such that an s of 0 calls forabsolutely no response (r = 0), and an s of .3calls for whatever degree of response has beendefined as r = .3, and so on. There is arguablysome danger in the assumption that such amapping is alwavs possible, because theassignment methods for defining s and r in afuzzy system can be subjective. Ilowever, this

does not mean that there will be no defensiblebasis for membership assignments. This pointis well recognized in the literature on fuzzylogic: "Membership functions may representan individual's (subjective) notion of a vagueclass.. .Membership functions may also bedetermined on the basis of statistical data orthrough the aid of neural networks.. .mem-bership functions are primarily subjective in na-ture; this does not mean that thev are assignedarbitrarily, but rather on the basis of applica-tion-specific criteria" (Tsoukalas & Uhrig,1997, p. 15).

It should be remembered that assignmentsin crisp analyses are frequently arbitrary as well.Take the aforementioned heating example. If aheater operates on a simple crisp thermostatrule in which the heat will be on below a certainroom temperature and off above that tempera-ture, the designer has defined (perhaps arbitrar-ily, or perhaps because of engineering constraints

643


644 Winter 2000 - Human Factors

not related to the end goal of human comfort)what heating strength (e.g. furnace tempera-ture, fuel consumption) corresponds to on.This setting cannot always be easily changed bythe user of a home or office heating system.Therefore the arbitrary nature of signal/responseassignments is not peculiar to fuzzy SDT. Aswith crisp SDT, some fuzzy SDT s and r as-signments will be more arbitrary than others,and the s assignments will be mappable to the rassignments more so in some cases than in oth-ers. For example in aircraft conflict detection,both signal and response degree might be as-signed based on the same function translatingdistance between aircraft into the range [0, I].In other words, the true state of the world mightbe defined in terms of the closest approach dis-tance between two aircraft, and the responsemight also be a judgment, by a human or anautomated tool, of that distance. This mappingstrategy might be considered better than bas-ing s on distance and r on probability. However,sometimes the nature of the problem will besuch that we must use whatever data best re-flects the question we are trying to answerabout human or machine performance, even atthe expense of the best possible mapping. Con-tinuing with the ATC example, the most impor-tant performance measure is likely to be thehuman controller's or machine's degree of con-fidence that a legal separation violation willoccur, rather than the exact distance withinwhich it is believed the planes will approach.

One cautionary note should be considered.lf in a fuzzy SDT application there is any sus-picion that the selected s-to-r mapping mightbe flawed (e.g., whatever has been defined as r= .9 is not likely to be the desired response towhatever is defined as s = .9). care should betaken in comparing derived outcome parame-ters of sensitivity, hit rate, and so on. to othersystems. The mapping could still be used, how-ever, to compare the performance of different

detectors within the sarne system or to evalu-ate the performance of the same detector inthat system under different conditions.

BASIC FORMULAS OF FUZZY SDT

We turn now to a recommended set of meth-ods for carrying out a fuzzy SDT analysis. Onces and r are mapped onto [0, 1] using appropri-ate mapping functions, the next step is to deriveevent membership in the four outcome cate-gories, Hit, Miss, FA, and CR. In this sectionwe present the methods that met the twin cri-teria of matching crisp SDT results and facevalidity. We examined several other methods,all of which were found to be invalid for onereason or another. (A short report describingthese rejected methods is available from theauthors on request.)

Mixed-implication Functions forFuzzy SDT

In standard SDT, the function mapping val-ues of s and r to the four outcome categoriescan be specified simply in a truth table format.The truth table, based on logical implicationfunctions (e.g., "If SW = signal and response =yes, then Hit"), is shown in Table 2. Each eventis mapped exclusively to only one of the fouroutcome categories, with a value of 1, whereasthe remaining categories have entries of 0. Incontrast as mentioned previously, in fuzzy SDT,events will claim nonzero membership in morethan one outcome category.

Traditional crisp implication functions havebeen adapted for fuzzy logic (e.g., Klir & Yuan,1995; Tsoukalas & Uhrig, 1997). We examinedimplication functions for fuzzy SDT based onmaximum and minimum values, as well as amultiplication function. All had certain unde-sirable features precluding their use in assign-ing degrees of category membership for use infuzzy SDT. However, a mixed set of functions,

TABLE 2: Truth Table for Standard (Crisp) SDT

Signal (s) Response (r) Hit FA Miss CR

o 0 0 0 01

O 1 0 1 0 0

1 0 0 0 1 01 1 1 0 0 0

Winter 2000 - Human Factors644


FUZ SINA DEETO TER 4

combining both the maximum and minimumfunctions, possesses properties suitable forapplication of fuzzy SDT. The set involves acombination of different implication functions,using maximum functions for the error cate-gories (Miss and FA) and minimum functionsfor the correct response categories (Hit andCR). The membership values for the four deci-sion outcomes, Hit, Miss, FA. and CR, aredefined by the following functions:

Hit:miss;False alarm;Correct rejection;

H = min (s. r) (6a)M = max (s - r. 0) (6b)FA = max (r- s, 0) (6c)CR =min (1 -s, I -r).(6d)

Table 3 shows a fuzzy SDT truth table forseveral selected values of s and r. As an exam-ple of the results of this function set, supposethat s = .8 and r = .9. That is, the state of theworld strongly, but not absolutely, points to asignal, and the observer strongly responds thata signal is present. Applying equations 6a-6d,the resulting category memberships are H = .8,M = 0, FA = . 1, and CR = .1. Hence the out-come strongly points to a hit, but unlike crispSDT, there is also some membership in the FAcategory, representing the fact that the responsewas stronger than what was called for by the

signal. The CR category is also nonzero, reflect-ing the small membership of the event in thenoise category and the fact that an unequivocalyes response was not made. Note also that ifboth s and r are forced to be binary (e.g., 0 or1!. then equations 6a-6d will result in a rever-sion to the crisp truth table shown in Table 2.

To get an intuitive appreciation of Equations6a-6d, consider the minimum (Hit, CR) andmaximum (Miss, FA) functions separately. Theminimum functions can be thought of as re-flecting the degree of overlap (or intersection)of the fuzzy sets of signal and response (Hits)and of that of nonsignal and nonresponse(CR). For hits, for example, one can think ofhit membership as reflecting the degree towhich the response set overlaps the signal set.If the overlap is perfect, then the Hit member-ship takes on the value of the signal (and theresponse) membership. If the overlap is lessthan complete, Hit membership is lower andtakes on the value of the signal or response,whichever is smaller.

The same reasoning applies to CR member-ship. For Miss and FA, the function reflects therationale that the degree of overresponding orunderresponding (with respect to the signal)defines the degree of FA and Miss member-ship, respectively. The rationale for the use of

TABLE 3: Example Truth Table for Fuzzy SDT

Signal (s) Response (r) Hit FA Miss CR

0.8 0.9 0.8 0.1A 0 0.10.1 0.2 0.1 0.1 0 0.80., 0.1 0.1 0 0 0.90.1 0.9 0.1 0.8 0 0.10.2 0.2 0.2 0 0 0.80.2 0.1 0.1 0 0.1 0.80.2 0.3 0.2 0.1 0 0.70.3 0.2 0.2 0 0.2 0.70.3 0.5 0.3 0.2 0 0.50.3 0.9 0.3 0.6 0 0.10.5 0.2 0.2 0 0.3 0.50.5 0.5 0.5 0 0 0.50.5 0.9 0.5 0.4 0 0.,0.75 0.1 0.1 0 0.65 0.250.75 0.75 0.75 0 0 0.250.75 0.8 0.75 0.05 0 0.20.9 0.1 0.1 0 0.8 0.10.9 0.9 0.9 0 0 0.10.9 0.8 0.8 0 0.1 0.1

Fuzzy SiGNAL DETECTION THEORY A45


v Wi 0

the maximum is as follows. If the response iscloser to a full yes (r = 1) than the signal is,then the event membership should fall in theFA category to some nonzero degree but shouldnot have any membership in the Miss category.This follows because r > s. On the other hand,if the response is closer to a full no than thesignal is (r < s), then the event membershipshould fall in the Miss category to some nonze-ro degree but should not belong with anydegree to the FA category.

Table 3 reveals some interesting features offuzzy SDT membership degrees. First, there isalways at least one zero value among the cate-gories. (In contrast, as Table 2 shows, in crispSDT a given event is always associated withthree zero and one unit membership values).In fact, zero values occur only in the Miss cate-gory, the FA category, or both, for any givenevent, and a zero must occur in at least one ofthe two. This outcome follows from the facevalidity assumption described earlier.

Note that when r = s, both Miss and FA havea value of zero, because the response was ex-actly the correct strength. For example, when r= s = .9, the event is mostly a hit (.9), partly aCR (.1). and not at all a Miss or FA (both 0).Also notice that the values of the four categoriesalways sum to 1. This is another sensible anddesirable result, because in crisp SDT the fourcategories are mutually exclusive. ln our tem-perature example it is reasonable for a givenvalue of t to belong, say, to the category cold tothe extent of .8 and to cool with .3, for a totalvalue greater than 1, as these two descriptorscan overlap. But it is not reasonable for thememberships of the Hit. Miss, FA, and CRoutcomes to sum to greater than I. Put anotherway, although fuzzy SDT permits A r) A to benonzero, the sum of the four mutually exclu-sive outcome categories should be 1, becauseeven in fuzzv SDT the four categories representthe full universe of possible outcomes; togetherthey encompass the "whole truth."

Relating Membership Functionsfor Decision Outcomes toStates of the World (SW)

First, the mapping functions map SW to thesignal s and RV to the response r. Next. mem-bership values for decision outcomes are com-

puted by using the mixed-implication functionswith the values for s and r. It is then relativelystraightforward to examine the relationshipbetween the membership functions for thedecision outcomes and the SW Given an ana-lytical mapping function, a family of curves forthe different outcomes can be generated byvarying the response variable r

Figures 3A and 3B plot membership func-tions for Hits, Misses, FAs, and CRs as a func-tion of the SW variable a. The parameter is theresponse r. (Discrete r values of 0, .3, .5, .7,and 1 are shown for illustrative purposes only.In actuality a continuous family of curves wouldbe generated. as r takes on any value between 0and 1.) These outcome categories were generat-ed using the sigmnoid mapping function for air-craft separation distances shown in Figure 2 (s= 1/[1+(a/10)5 ]). Because the a-to-s mappingis nonlinear, so are the functions describing theoutcome categories. These four functions takeon a shape that reflects the shape of the originalfunction mapping the SW to s. but the value ofr puts a limit on the maximum membership inI-fit and FA, whereas (1 - r) represents the max-imum possible membership for iMiss or CR.

The membership functions show that ingeneral, the membership values for Hlit and FAtrade off against each other, as do those for CRand Miss. Consider Figure 3A. For any givenvalue of r (except r = 0), the membership valuefor IFlit decreases with an increase in a (greateraircraft separation), whereas that for FAincreases. This makes intuitive sense becauseas a increases, the event (aircraft separation)becomes less signal-like and more noise-like,so that the outcome of a response, irrespectiveof its strength, will become less Hit-like andmore FA-like. Also as Figure 3A shows, forany given value of a, both Hit and FA member-ships increase with increased r. This is againreasonable because for any given signal value,as response strength increases, the outcomeshould become both more Hit-like and moreFA-like. These patterns are similar for the CRand Miss memberships, except that the varia-tions with a and r are inverted (see Figure 3B).

Figure 3 also shows that membership valuesfor Hit and Miss do not sum to 1; that is, I-I +M X 1. By the same reasoning, CR + FA X 1.Recall, however, that the sum of the member-

Winter 2000 - Human FactorsAAA



A!

10.9

0.8

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State of the World (a)

Miss

L

r

CR

0

0IN

.3

, _ _ _ _ s~

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.5

_ s~~~~I -

N%.5

I . 0 . - 1.0 r

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1617 1819 20 r


Figure 3. Fuzzy Hit. Miss, FA, and CR membership values with sigmnoid mapping of s to a.

ship values of all four outcomes is always 1;that is, H + M + CR + FA = 1.

Figure 4 gives the outcomes if the SW-to-smapping is linear. These figures are includedFr illustrative purposes only and most likelywould not reflect a suitable SW-to-s mappingfor air traffic control (but they might be appro-priate for another application). Both Figures 3and 4 show the correspondence between the

original mapping function's shape and that ofthe outcome category functions, as well asshowing that the limits set by r and (I - r, areindependent of the shape of the original map-ping function.

Computation of Fuzzy Hit and FalseAlarm Rates

We now turn to a discussion of how to

0.9

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FA

A

- o .7

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- .3

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Miss

0,9.

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%N .3

N

N .5S.7

IN.7 N-

N><II

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

.0

r


Figure 4. Fuzzy Hit, Miss. FA, and CR membership values with linear mapping of s to a.

compute some of the standard SDT measureswhen the signal and response are defined fuzzi-ly. The rates for each of the four decision out-comes (FHR, MR, FAR, CRR) can be calculatedby summing the membership values in eachcategory over all trials and dividing by the sumacross trials of membership values in signal (s)or not-signal (1-s). Formally, all four decisionoutcomes can be computed as follows,

IIR = I(Hj)/ E(si) for i = 1 to NMR = E(AMj)/ E(si) for i = I to NFAR = Y(FAi)/ E( l - s;) for i = I to NCRR = Y(CRi)I X(1 - sj) for i = I to N

(7a)(7b)(7c)(7d)

in which i is the trial number, N is the total num-ber of trials and Hi is the degree of Hit for triali, M; is the degree of Miss for trial i, and so on.

The E(1 - s,) term, although formally com-

Hit

1.0

LA1>

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0.6

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0.4

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0.2

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648

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FuzzY SIGNAL DETECTION THEORY 649

puted by adding (1 - s;) for each trial i, can bemore efficiently calculated as N - X(s) for i = Ito N. By substituting Equations 6a-6d for themembership values of H;, M;. and so on, Equa-tions 7a-7d can be rewritten as:

HR = 1(min(si, ri))/E(si) for i = 1 to N (8a)MR = Y;(max(s;- ri. O0))

X(s) for i= I to N (8b)F.R = 1(max(ri- si,. O))

(Il-s) for i = I to N (8c)CRR = X(min(I - si, I - ri))/

O l-s;) for i = I to N. (8d)

Either Equations 7a-7d in conjunction withEquations 6a-6d, or Equations 8a-8d alone,can be used in any application in which s ismanipulated experimentally (or varies natural-ly) over several trials and separate detection re-sponses are made on each trial. Note that theredundancy between Hit and Miss rates (i.e.,HR + MR = 1), and between FA and CR rates(FAR + CRR = 1), holds true for both crispand fuzzy SDT. (This redundancy should notbe confused with the lack of redundancy be-tween membership values on individual trialsfor Hit and Miss and between values for CRand FA.)

When we use fuzzy SDT Equations 6a-6dand 7a-7d, or Equations 8a-8d to aggregateperformance across cases or trials, the methodfollows directly from that used for crisp SDT.The total Hit and Miss memberships for all tri-als are summed and divided by the sum of thesignal memberships for all trials. For the FAand CR rates, which are dependent on thedegree to which each trial did not contain asignal, the sum of memberships should bedivided by the sum of the extent to which eachtrial was not a signal (for each trial i, this isequal to [I - si]!). Applying Equations 7a-7d tothe data set presented in Table I yields the fol-lowing. We obtain X(s,) = 9.05 and E(l - si) =19 - 9.05 = 9.95. just as in the crisp case, theMiss rate will be ( I - HR), and the CR rate is(1 - FAR). The sums of the four outcome cate-gories are 7.00, 2.05, 2.35, and 7.60 for Hit,Miss, FA. and CR, respectively. The degrees ofmembership for each category can now be com-puted by entering these values into Equations7a-7d.

HR = 7.00/9.05 = .773MR = 2.05/9.05 = .227FAR = 2.35/9.95 = .236CRR = 7.60/9.95 = .764

The same calculations could be done direct-ly from the individual s and r values using Equa-tions 8a-8d. With these calculations completed,standard SDT parameters (sensitivity, bias) canbe computed just as with crisp SDT. In thenext section we continue the analysis begun inthis section with our sample data set, and dis-cuss an alternative method for assessing sensi-tivity in fuzzy SDT.

Sensitivity and Criterion in Fuzzy SDT

In traditional SDT the parametric estimateof operator sensitivity. d', represents the stan-dardized distance between the normal curveapproximating the signal distribution and theone approximating the noise distribution. Inother words, a curve is approximated for thefrequency of each level of the evidence variablegiven that there is a signal, and another curveis approximated given that there is noise. Thed' measure is the horizontal distance, in unitsof standard deviations, between the curves.Another way of saying this is that the place-ment of the observed Hlit rate on a normal dis-tribution, minus the placement of the observedFA rate on a normal distribution, will be equiv-alent to the distance between the signal andnoise distributions and will provide an esti-mate of the detector's sensitivity. The d' para-meter is most easily calculated by determiningthe difference between the standardized Hitand FA rates using the following formula:

d = Z(HR) - Z(FAR). (9)

Because the fuzziness of the signal has al-ready been captured in the definitions of s, andr, and from them, fuzzy HR, FAR, MR, andCRR, the traditional d' formula can be used infuzzy SDT analysis. WVe illustrate the calcula-tion of d' using the sample data from Table 3.For these data, FHR is .773 and FAR is .236.Taking the normal deviates of these values andusing equation 9 gives d' = 1.47.

Another way to assess d' when s is fuzzy is

649FLIZZY SIGNAL DETECTION THEORY



through integration. In traditional SDT, d' isthe horizontal distance, in units of standarddeviations, between the means of the Gaussianprobability density distributions for signal andnoise. In crisp SDT the noise curve is essential-ly the curve for s = 0, whereas the signal curveis for s = 1. In fuzzy SDT the value of s can fallanywhere between 0 and 1; therefore, anotherappropriate way to assess sensitivity is to deter-mine how far, in standard deviation units, allthe possible curves sit from each other. In otherwords, theoretically each of the infinite curvesbetween s = 0 and s = 1 would be approxi-mated. Using the s = 0 curve as a reference, thecontribution of each of the infinite curves to d'would be weighted by the distance of its s valuefrom 0, and the integral would be calculated:

d = fIZ(x value at s = q) - Z(x value at s = 0)] dq,o (10)

in which x is the value of the evidence variableor SW. If s had only a finite set of values, thenthe distances of each of the existing s curvesfrom s = 0 would be computed.

d' = ', [Z(x value at s = qi) -

Z(x value at s = 0)] for i = 1 to N, ( 1.)

in which x is the value of the evidence variableor SW; q, represents each of the discrete valuesof s, and N is the total number of discrete svalues.

The traditional SDT criterion measure, ,can also be calculated using fuzzily derived HIRand FAR. WAre define ,B in the standard way as

/3= Y(HR)/Y(FAR), (12)

in which Y(.) represents the ordinate of thenormal distribution. For the fuzzy HR andFAR derived in our sample data set. /3= 0.98.

Fuzzy SDT Correlation AnalysisA completely different method for assessing

the accuracy of detection performance when s,r, or both, are fuzzy is correlation. The com-puted correlation coefficient between signaldegree and response degree can be an indica-tion of the accuracy of detecting a fuzzy signal.

A high correlation would indicate a linear rela-tionship between r and s. Correlational analysishas some advantages. If the mapping from SWto s or from RV to r is erroneous for any rea-son, then accuracy as computed by fuzzy SDTmay be reduced compared with other methods,whereas correlation would still capture the lin-ear nature of the relationship.

The correlational analysis can best bedescribed with an example. Consider the sam-ple s and r data shown in Table 4. The respond-ing system tends to generate r values towardthe middle range of the spectrum, even for moreextreme values of s. The fuzzy SDT implica-tion functions (Equations 6a-6d) would assignrelatively high Miss and FA values for some ofthe events (e.g., s = .1 and r= .35 would resultin an FA membership of .25). However, becausethe relationship between s and r is almost per-fectly linear, as shown in Figure 5, the compu-ted correlation is very high (greater than .99).

This dissociation between the results of thevarious fuzzy SDT analysis methods can havedifferent consequences, depending upon theapplication. It might be that the implicationfunction is a better measure of performance, ifindeed it was desirable for s to closely match r.Alternatively it could be that the mappingstrategy used to convert RV to r was too proneto assigning middle values and did not makeenough use of the extreme (near I or 0) r val-ues to be reflective of the question of interest.

Correlation analysis can also pose some dan-ger, similar to that discussed for the implicationfunctions. For example, it too will penalize thedetector for overly crisp definitions of signaland response, because a restricted range on oneor both variables can deflate a correlation coef-ficient. Fortunately, methods exist for correctingthis problem when it occurs. For example, if theresponse is always defined in a binary fashion,point-biserial correlations can be used as amore appropriate measure of correlation.

Fuzzy Signals, Crisp ResponsesFuzzy SDT analysis can also be extended to

cases in which the signal is fuzzy but theresponse is discrete or binary; for example, yesor no. This is particularly useful when consid-ering the analysis of actions that are contingenton a particular decision choice. For example, a

650



TABLE 4: Sample s and r Data to Illustrate Differential Results of ImplicationFunctions Versus Correlation in Measuring Sensitivity in Fuzzy SDT

s r

0 0.30.1 0.350.2 0.40.3 0.450.4 0.4750.5 0.50.6 0.5250.7 0.550.8 0.60.9 0.651.0 0.7

participant in a memory study may decide witha response r = .8 that he has been shown a par-ticular face before by the experimenter. If theexperimenter then requires the participant topick the face seen before from a group of dis-tractors, the participant must take a discreteaction consistent with his decision choice.

1-

0.9

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0.7

Ca&c

c.ccGLr.

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0

Outside the laboratory, too, human andmachine decision choices often need to betranslated into discrete actions. For example,an automated fault management system in anuclear power plant may make a judgment of r= .9 (e.g., probability, confidence, or severity)that a fault is present. This fault then needs to

0

a

.

0

0

S

e

.

Figure 5. Piot of sample s and r values illustrating linear relationship.

0 o.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

s (signal)

651



be translated into a binary action, that is, toturn off the plant or not. In general, an overtresponse is more difficult to "fuzzify" than thedecision about the state of the world uponwhich the overt action is based.

In a recent comparative analysis of theinformation-processing performance of hu-mans and automation, Parasuraman, Sheridan,and Wickens (2000) also distinguished deci-sion making from action selection as separatestages of any detection process. Although theyallowed action levels to be continuous, theysuggested that in most cases actions will becategorical, even when decision making hasmultiple levels.

Table 5 shows the result of applying theimplication functions (Equations 6a-6d) tosample data with r = 0 or 1. A binary responseto a fuzzy signal always results in the event'shaving nonzero membership in exactly two ofthe categories. If the binary response is yes (r =l), then Equations 6a through 6d reduce to:

H=sM=OFA = I -sCR = 0.

(1 3a)(13b)(1 3c)(I 3d)

Note that the available membership degree ofI is assigned partly to Hiit in the amount of sand the rest, I - s, to FA. If the binary responseis no (r = 0). then

H = 0M = sFA = 0CR= I -s.

(14a)(14b)(14c)(1 4d)

In this case, the available membership degreeof 1 is assigned partly to Miss in the amount ofs and the rest, 1 - s, to CR.

Equations 13 and 14 can be used to com-pute detection statistics for the special case offuzzy signals with crisp responses. Some addi-tional special cases should also be noted. Forexample, if r values are allowed to vary alongthe full range [0, 13 and the observer simplychooses to make binary responses, then theHit, Miss, FA, and CR rates, as computed usingEquations 13 and 14, are appropriate. Whenaggregated, they will give measures of accuracyand bias that can be validly compared withother observers or systems. However, if thenature of the response is such that its very def-inition constrains the possible responses toonly 0 or 1, then fuzzy SDT analysis could un-derestimate accuracy. This occurs because theobserver is penalized by being forced to usebinary responses when the signal is fuzzy, andthe penalty increases as s becomes fuzzier (iscloser to .5).

To see this, suppose s = .6, but r can be only0 or 1. The permissible response closest to theactual s value is 1, which results in IH = .6, M= 0, FA = .4. and CR = 0, or a sensitivity thatis lower than could be achieved with a fuzzyresponse (e.g., r = .6). This outcome rendersinvalid any comparisons of fuzzy SDT parame-ters between systems or situations in which r isfree to vary across the full range [0, 1] andthose in which r is only allowed to be 0 or 1.

However, the results from fuzzy SDT analy-sis of such a situation are, at worst. no lessdetrimental to calculated accuracy measuresthan the outcomes of a traditional forced-choice

TABLE 5: Sample Hit, Miss, FA, and CR Outcomes for Fuzzy SDT Analysis When the Response r Is Binary(0 or 1)

s r Hit Miss FA CR

0.1 1 0.1 0 0.9 00.3 1 0.3 0 0.7 00.5 1 0.5 0 0.5 00.7 1 0.7 0 0.3 00.9 1 0.9 0 0.1 00.1 0 0 0.1 0 0.90.3 0 0 0.3 0 0.70.5 0 0 0.5 0 0.50.7 0 0 0.7 0 0.30.9 0 0 0.9 0 0.1

652


FuzzY SIGNAL DETECTION THEORY

paradigm in which both signal and response areconstrained to binary values. Furthermore. al-lowing the definition of s to be fuzzy will improveestimates of bias by capturing the variability insignal strength, making it possible to assess, forexample, the extent to which stronger signalsare more likely to generate a response.

The only potential problem remaining withthis type of situation is as follows: When thedomain involves fuzzy s and binary r, attemptsto compare different operators or different sit-uations within the same system may be con-founded by the 'luck of the draw." That is, ifoperator P happens to get a larger number ofthe "more fuzzy" signals (e.g., s values of .4, .5,.61 than operator Q, then it will be more diffi-cult for P to achieve high HR and CRR valuesoverall.

Two solutions exist for this situation. First,one might reconsider whether there might besome way to fuzzify r, such as using confidenceor probability ratings, even if a binary action istaken. Second, if this is not possible or desir-able, the s value can be rounded to 0 or 1, andtraditional crisp SDT analysis conducted. Notethat rounded does not have to mean that an sof .5 or more rounds to 1, whereas an s of lessthan .5 rounds to 0. The cutoff can be placedanvwhere that it makes sense to do so, accord-ing to the needs of the particular situation.

It is also possible to conduct an analysis inwhich trials with middle values of s are ex-cluded from analysis, high s values are classedas signal-present, and low s values as signal-absent. Altematively, the cutoff can be placedat one point and the traditional SDT analysisconducted, then the cutoff can be moved andanother traditional SDT analysis conducted,and so on. Such a moving-cutoff strategy isanalogous to the use of confidence intervals intraditional SDT - or for some analyses, notmerely analogous but equivalent, because con-fidence interval ratings can be converted to r.

DISCUSSION

Summary of Fuzzy SDT Analysis

The fuzzy SDT analysis developed in thispaper involves four main steps: (a) selectionand application of mapping functions for states

of the world and responses; (b) use of mixed-implication functions to assign degrees of mem-bership in the decision outcomes of hits, misses,false alarms, and correct rejections; (c) compu-tation of fuzzy hit, miss, false alarm, and correctrejection rates; and (d) computation of fuzzysensitivity and bias measures.

The first step is the selection of mappingfunctions. The signal mapping function mapsvariables that describe the states of the worldinto the fuzzy signal set with membershipstrength s. WVe described several ways in whichsuch functions could be selected. Mappingfunctions will vary with the application andcan be relatively simple, through the use of asingle variable (as in the ATC example: seeFigure 2), or could be more complex by beingbased on many variables.

Mapping functions can also be discrete orcontinuous and can be derived empirically orbased on some theoretical premise. Amongcontinuous analytical functions, the sigmoidmay best capture the variability inherent inmany applications. Selecting the response map-ping function is relatively straightforward andcan be based on judgments of confidence, onthe reported strength, severity, or criticality ofthe signal, or both.

The second step is the use of mixed-implicationfunctions for assigning degrees of membershipin the conventional SDT outcomes of hits, miss-es, false alarms, and correct rejections. We usedtwo criteria for choosing appropriate functions:comparability with crisp SDT and face validity.The implication functions we propose, whichcombine minimum and maximum functions,meet both criteria and seem suitable for a widerange of applications. However, we make noclaim that these are the best functions or thatother functions may not be more suited to par-ticular applications. Can the implication func-tions be optimized? Perhaps. One possibilitywould be to use Monte Carlo methods or sim-ulation with a data set with well-defined statesof the world and outcomes, and choose a set offunctions that best matches the defined out-comes.

The third step in fuzzv SDT analysis in-volves computing the mean fuzzy hit and falsealarm rates by weighting the membership de-grees in these outcome categories by the average

653



degrees of membership in the signal (s) andnoise (1 - s) sets. The fourth step, computa-tion of sensitivity and bias measures, then fol-lows in exactly the same manner as it would incrisp SDT.

Crisp and Fuzzy SDTSDT could arguably be viewed as one of the

most robust and useful quantitative theories inpsychology. The familiar adage, "there is noth-ing so practical as a good theory," is also par-ticularly well suited to SDT, given the varietyand range of the practical applications of SDTin psychology, human factors, and other fields(Swets, 1996). In the present paper we haveproposed that combining fuzzy logic and SDTcan further enhance the range, power, and util-ity of SDT. Nevertheless, it is instructive toexamine the points of convergence of crisp andfuzzy SDT.

The approach to fuzzy SDT we have out-lined in this paper shares some features withthe uncertain signal problem. This refers to thecase in which there is uncertainty regardingsome dimension of the signal - for example, itsduration, starting time, frequency, location, orother characteristics - so that the signal isknown statistically (SKS). The case of no un-certainty is referred to as a signal known exact-ly (SKE: Tanner & Birdsall, 1958). A numberof researchers have compared detection perfor-mance for SKS and SKE for a number of dif-ferent dimensions of signal uncertainty, thegeneral finding being that uncertainty reducesdetectability (Egan, Greenberg, & Schulman,1961; Green & Swets, 1966; Swets, 1984). SKScould be linked to the mapping function con-cept of fuzzy SDT. For example, one could de-velop a function that mapped all sources ofuncertainty regarding the signal onto the signalstrength parameter, s.

In this sense, fuzzy SDT, and the SKS case inSDT can be considered to be related. In anoth-er respect, however, the two are distinct, partic-ularly in nonlaboratory conditions in whichthere is uncontrolled variability in the signal.In the SKS case, the mapping of SW to the sig-nal is nevertheless crisp, and the uncertainty isintroduced in its presentation to the observer.For example, an SKS experiment in the labora-tory could involve detection of a 1 000-Hz tone

presented at random times (Egan et al., 1961).Although the time of presentation of the signalis uncertain, the definition of the signal, in termsof the SW-to-signal correspondence, is crisp.In contrast, in the fuzzy SDT cases we consid-er, the SW-to-signal correspondence is uncertain.An example might be a sonar operator listen-ing to sonar returns consisting of multiple fre-quencies: A signal indicating the presence of asubmarine is not defined crisply as a 1000-Hztone but as an auditory stimulus whose fre-quencies vary with context and over time.

Previous Related Work on Fuzzy LogicFuzzy logic has been applied to a wide array

of problems in psychology (e.g., Massaro, 1998)and human factors (e.g., Moray et al., 1987).For the most part, however, fuzzy logic has notpreviously been explicitly combined with SDTto develop a general model of fuzzy detectionperformance, as presented in this paper. Never-theless, there has been some relevant priorwork; for example, a study of word recognitionby Meng and Li (1990). Although this studydid use fuzzy logic and SDT, it did not presenta general model of fuzzy SDT as we do.

There has also been some relevant work onmachine classification of noisy signals. Boston(1997) carried out an analysis of the classifica-tion of event-related brain potentials (ERPs).H-le used fuzzy logic to define the degree towhich the values of a potential signal on twodifferent ERP features reflected a true ERP sig-nal. He contrasted the judgments regarding sig-nal presence resulting from fuzzy analysis tothe performance of a Bayesian detection algo-rithm. But the end result of his analysis involvedrounding out the decision to relatively crispoutcomes: signal present, signal absent, or un-certain.

Although these studies did consider fuzzylogic in the context of particular signal detec-tion problems, they stopped short of developingthe general case of fuzzy SDT, which retainsthe continuous definition of signal, response,or both, resulting from contextual and tempo-ral variability. The methods we presented inthis paper allow one to carry the fuzzinessthrough all the stages of analysis, rather thanreverting back to categorical characterizationsof the outcome. Of course, there are prior exam-

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ples of the use of nonbinary responses to im-prove the precision of SDT analysis, such asthe use of confidence levels surrounding a yes-no judgment (Green & Swets, 1966), exploringthe costs and benefits when a decision criterionis set at different levels (Lehto, Papastavrou, &Giffen, 1998; Parasuraman, Hlancock, & Olofin-boba, 1997), and Balakrishnan's (1998) methodfor computing response bias by comparing thetypes of events (signal vs. noise) correspondingto each self-rated degree of confidence.

Although these and other methods have ac-knowledged the existence of continua in SW.in the response of a decision maker, or in both,SDT analyses and the resulting interpretationsof sensitivity, bias, and payoffs have generallyassumed crisp states of the world and crisp re-sponses.

The fuzzy SDT model we have outlined isalso conceptually related to fuzzy logic modelsof perception and cognition that have beendeveloped by cognitive psychologists. Probablythe best known of such models is the fuzzylogical model of perception (FLMP) of Mas-saro (1988; Massaro & Friedman, 1990). TheFLMP model assumes that perceptual tasksinvolve three stages: feature evaluation, inte-gration, and decision making. Fuzzy set mem-berships are used to derive the relationshipsbetween feature values and their integrationprior to the decision stage. In this respect, themodel is similar to Bayesian models of infor-mation integration. although the use of fuzzymembership values (or fuzzy truth values) isunique to the model and distinct from the sub-jective probabilities that are used in Bayesiananalysis. Given that SDT originated as a per-ceptual theory. links between fuzzy SDT andthe FLMP model are to be expected. Moreover,Massaro has advocated the use of "graded" and"expanded" factorial designs in testing modelsof perception and cognition. This method re-fers to the use of all possible combinations ofgraded values of stimulus features, as well asthe presentation of stimuli possessing each fea-ture value in isolation (Massaro & Hary, 1986).Massaro and Friedman also recommended test-ing the FLMP model by using ratings ratherthan categorical responses.

In some respects, therefore, there are linksbetween the FLMP model and fuzzy SDT. The

two are nevertheless quite distinct. FLMP in-volves the integration of perceived informationderived from multiple stimulus features thatmatch stimulus prototypes in a fuzzy manner.It uses an integration procedure functionallyequivalent to Bayesian analysis. Fuzzy SDT,although it can be based on what is perceived(evidence variable), is generally concerned withthe analysis of objectively measured stimuli(states of the world) that map onto the signal setin a fuzzy manner. It uses mixed-implicationfunctions to derive fuzzy hit and FA rates, fol-lowed by conventional SDT analyses.

Our intention in proposing fuzzy SDT is noticonoclastic; rather, it is to allow the extensionof SDT when it would be useful to do so. Asregression is the global case of analysis of vari-ance, so fuzzy SDT is the global case of SDT,improving the precision of signal detectionanalyses by retaining the information providedby the middle ground. rather than by roundingit into oblivion. Of course, situations still existin which the simplicity of the variables at handmakes the crisp SDT method more parsimo-nious without a significant loss of information.We conclude, however, that fuzzy SDT is par-ticularly well suited to evaluation of detectionperformance when there is significant variabil-ity in the state of the world that defines the sig-nal to be detected.

Contextual and Temporal Variability inSignals

There can be both contextual and temporalvariability in signal definition. Contextual varia-bility refers to the dependence of signal defini-tion on situation-specific factors (e.g., see thepreviously discussed example of signals in ATCdepending on the type of flights, whether levelcruise or climbing/descending). An invariantcrisp definition of a signal may result in erro-neous conclusions when these factors are pre-sent. To the extent that the factors can be encodedinto a mapping function as we describe in StepI of our fuzzy SDT analysis, this form of anal-ysis can capture the contextual variability bet-ter than crisp SDT.

Temporal variability refers to the variationin signal strength over time. In many real set-tings, the signal is not a discrete event in timebut represents an event that unfolds over time.

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In industrial process control, for example, thevalue of a critical system variable (e.g., core tem-perature in a power plant) will fluctuate overtime but may gradually drift toward a danger-ously high value if an emergency condition isdeveloping (Moray, 1986). In medicine, to takeanother example, detection of emergency con-ditions during surgery requires analysis of tem-poral patterns and trends in the values of vitalpatient signs (Lowe et al., 1999). In fuzzy SDTterminology, the value of s may start low andincrease over time, or start high and decreaseover time.

This type of signal variability over time hasnot been explicitly modeled in psychologicalstudies. This is a curious omission because psy-chologists have developed sequential samplingmodels of the perceived strength of a discretesignal (e.g., Laming, 1968; Link & Heath, 1975).In most such models, the perceived signalstrength grows with time following the presen-tation of the discrete stimulus until some thresh-old for response is reached. In models of thediscrimination between two stimuli, SI and S2,the evidence makes a "random walk" betweenS1 and S2 until the threshold for one alterna-tive is crossed.

Sequential sampling and random-walk mod-els have been used to examine the trade-offbetween detection accuracy and speed of re-sponse. The reason that these and related mod-els have not been extended to analysis of actualrather than perceived temporal variability insignal strength is that most laboratory studiesof detection and discrimination involve dis-crete rather than continuous signals.

Models of temporal variabilitv of the statesof the world associated with a signal could bedeveloped or empirically defined. The influenceof temporal variability could then be examinedby extending the mapping function in a partic-ular application by time, T. If the signal map-ping function is s = f(SXVJ, then an extendedmapping function would be of the form f(SW,D). We propose that the ability to incorporatecontextual and temporal variability in signaldefinition into the analysis of detection perfor-mance is one of the major advantages of fuzzySDT. Conventional SDT can be thought of asproviding a snapshot of detection performanceat: a particular time. Although several such snap-

shots could be obtained in successive crispSDT analyses at discrete points in time, fuzzySDT provides a natural way for analyzing con-tinuous signals by incorporating temporal vari-ability into the mapping function.

ApplicationsMany domains in which SDT has been ap-

plied are amenable to fuzzy SDT analysis. FuzzySDT should prove useful in basic studies ofperception, memory, and cognition. In manylaboratory studies, the strength of a signal isvaried as an independent variable. For exam-ple, in perceptual experiments, participantsmay be presented with stimuli of different size,luminance, or orientation. Depending on thequestion being addressed. it may be appropri-ate to map each level of an independent vari-able involving signal strength to an s value andconduct fuzzy SDT analysis on detection ratesacross conditions, weighted by the s value ofthe given condition. This would provide aclearer picture of how sensitivity and bias areaffected by the other independent variablesbeing studied in the experiment.

Fuzzy SDT is also particularly well suited toapplied studies and may better capture theshades of gray inherent in almost any real-world domain. In some cases dichotomizationof the signal, the response, or both is neces-sary. As discussed previously, crisp responses(yes or no) are often necessary when translat-ing decision choices into action. However, aslong as the contextual and temporal variabilityof the state of the world can be captured (e.g.,signal fuzzification only), the methods outlinedherein can enrich the performance analysis ofany decision-making system. Furthermore, aswith crisp SDT, fuzzy SDT can be applied tothe analysis of the detection performance ofhumans, machines, or joint human-machine"teams" (Parasuraman, 1987; Sheridan &Ferrell, 1974; Sorkin & Woods, 1985; Swets,1996).

Some potential real-world applications offuzzy SDT are briefly discussed here. Considerfor example, the evaluation of collision warn-ings for automobiles and conflict warning sys-tems for aircraft (Hlancock & Parasuraman,1992; Parasuraman et al., 1997). Farber andPaley (1993) discussed the idea of "acceptable"

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false alarms (FA), those being alerts that acti-vate when the threshold for a collision is almostviolated, thus reminding the user what thealert looks/sounds like and assuring him or herthe system is still working. Traditional SDTwould class this kind of alert as an FA, where-as fuzzy SDT would class it primarily as a Hit,with a small degree of FA. which would moreclosely reflect the fact that the event is notwholly undesirable. The fuzzy SDT analysis issimilar to the concept of graded or likelihoodalartns proposed by Sorkin, Kantowitz, andKantowitz (1988) but provides a more generalbasis for determining the alarm degree that isdisplayed to the human operator.

Similar examples can be drawn from qualitycontrol and process control. In these applica-tions a warning system may indicate when thestate of a plant or a product has deviated fromthe norm by more than some cutoff value (Cale,Paglione, Ryian, Timoteo, & Oaks, 1998). IHow-ever, in evaluating the performance of thedetector, one would probably wish to know theextent to which false alarms (in crisp SDT par-lance) were generated by events which werenear to that cutoff point and would wish thesame information for so-called misses. FuzzySDT would capture the extent to which falsealarms, for example, had some degree of "Flit"by virtue of the event being close to the cutofffor a crisp signal, whereas crisp SDT wouldclassifv all alerts generated by events beyondthe cutoff equally as false alarms.

An example from industrial psychology servesto illustrate the potential diversity of fuzzySDT applications. Crisp SDT analysis has beenused to evaluate personnel selection tests byranking individuals who score above or belowa cutoff score on a selection test of ability orknowledge, and determining whether they rateas high or low performers according to a sub-sequent assessment test or performance ap-praisal. However, a crisp SDT analysis of thistype of data ignores gradations within high andlow performance as well as ignoring how farabove or below the cutoff score the employeescored. The use of fuzzy methods by Alliger etal. (1993) for evaluating employment inter-view data represented a step in this direction.Additional fuzzy mappings of employment orperformance-related variables in order to con-

duct a full fuzzy SDT analysis would furtherenrich the techniques those authors have sug-gested.

CONCLUSIONS

In this paper the basic postulates of fuzzySDT were presented and simple but powerfulformulas were identified for conducting a fuzzySDT analysis of detection performance. Exam-ples were also provided of applications in basicand applied work. Of course, fuzzy SDT willultimately stand or fall, to some degree, basedon the results it generates in future applicationsin psychology, human factors, engineering, andother domains. We believe its membership inthe set {stand} will be much greater than itsmembership in the set {falll.

ACKNOWLEDGMENTS

This work was supported by Grant NAG-2-1096 from NASA Ames Research Center,Moffett Field, California. Kevin Corker wasthe technical monitor. We thank Neville Moray,Tom Sheridan, and Robert Sorkin for helpfulcomments on a previous version of this paper.

APPENDIX

Symbol DefinitionsSW State of the World; objective truth

specifying a priori some physical states Signal event in standard signal

detection theory (SDT): also, degreeof membership of state of the world(SW) in the fuzzy set signal in fuzzySDT

M Noise event in standard SDTY Yes, or response of a detection system

that a signa' has occurredN No, or response of a detection system

that a signal has not occurredr Degree of membership of a response in

the Yes response set in fuzzy SDTRV Response Value; equivalent to rP(01s) Probability of a Yes response given that

a signal occurred; also known as thehit rate (HR)

P' Yln) Probability of a Yes response given that

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a signal did not occur; also known asthe false alarm (FA) rate (FAR)

P(Nls) Probability of a No response given thata signal occurred; also known as themiss rate (MR)

P(Nin) Probability of a No response given thata signal did not occur; also known asthe correct rejection (CR) rate (CRR)

d' Sensitivity of a detection system instandard SDT

p3 Criterion or decision threshold ofdetection svstem

ns Number of possible discrete SW valuesnR Number of possible discrete RV valuesf(SW) Mlapping function for SWg(RV) Mapping function for RVII Degree of membership of a decision

outcome in the fuzzy set HlltM Degree of membership of a decision

outcome in the fuzzy set MissFA Degree of membership of a decision

outcome in the fuzzy set FACR Degree of membership of a decision

outcome in the fuzzy set CR

Fuzzy Implication Functions

Hit: min (s, r)Miss: max (s - r, 0)False alarm: max (r - s, 0)Correct rejection: min (1 - s, I - r)

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Raja Parasuraman is a professor of psychology anddirector of the Cognitive Science Laboratory at theCatholic University of America. Washington, D.C.He received a Ph.D. in psychology from the Univer-sity of Aston in 1976.

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Date received: November 2. 1999Date accepted: Alarch 18, 2000

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