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

## Article excerpt

Signal detection theory (SDT) assumes a division of objective truths or "states of the world" into the nonoverlapping categories of signal and noise. The definition of a signal in many real settings, however, varies with context and over time. In the terminology of fuzzy logic, a real-world signal has a value that falls in a range between unequivocal presence and unequivocal absence. The definition of a response can also be nonbinary. Accordingly the methods of fuzzy logic can be combined with SDT, yielding fuzzy SDT. We describe the basic postulates of fuzzy 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, false alarm, miss, and correct rejection rates; and (d) computation of fuzzy sensitivity and bias measures. Fuzzy SDT can considerably extend the range and utility of SDT by handling the contextual and temporal variability of most real-world signals. Actual or potential applications of fuzzy SDT include evaluation of the performance of human, machine, and human-machine detectors in real systems.

If a man will begin with certainties, he shall end in doubts; but if he will be content to begin with doubts he shall end in certainties.

Francis Bacon, The Advancement of Learning (1605, bk. 1, v. 8)

INTRODUCTION

A [cap] A = 0. This seemingly irrefutable mathematical expression asserts that a statement and its opposite can never coexist. Something either is or is not. A tumor is either cancerous or benign; a new car is either reliable or faulty; a politician is either honest or dishonest, and so on. Confidence in the truth of this mathematical expression and of these representative statements pervades both academic and everyday thinking.

A little reflection reveals that although the logical expression is often true, it need not always be true. After all, tumors, cars, and politicians come in all shades, not just black and white. To cope with this possibility, Zadeh (1965) developed fuzzy logic, sometimes simply called fuzzy (Kosko, 1993, 1997). Fuzzy logic allows the possibility that the intersection of A and A is nonzero. Is it or isn't it? The truth lies somewhere in between. To the extent that an event is somewhere in between, forcing its categorization into nonoverlapping sets of black and white can result in the loss of useful information and less sensitive analysis. Rather, if we follow Bacon's admonition to "begin with doubts" and express those doubts mathematically in fuzzy terms, then the analyses that follow may very well "end in certainties."

If we are doubtful whether a given event is a member of a particular category or not, how can we reach meaningful decisions and take appropriate actions based on our knowledge of the properties of that category? To return to our examples, the tumor needs to be operated on, the faulty car returned to the dealer, the politician voted out of office. Is a fuzzy characterization of events a recipe for indecisiveness and inaction? Not necessarily.

As we demonstrate in this paper, fuzzy logic can be combined with a well-known methodology for analyzing decision making: signal detection theory (SDT). The result, which we term fuzzy SDT, allows for a broader and potentially more powerful analysis of decision-making performance than do conventional methods. Moreover, as we shall show, use of fuzzy SDT can avoid the possibility of erroneous conclusions that may arise from the application of standard SDT to situations in which signal definition is fuzzy.

SDT was initially developed to quantify the performance of electronic receivers for detecting noisy radio signals (Peterson, Birdsall, & Fox, 1954). It was later extended to describe human detection of threshold-level signals (Tanner & Swets, 1954). …

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