Fuzzy signal detection theory: basic postulates and formulas for analyzing human and machine performance.

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.