The Theory of Signal Detectability: ROC Curves and Their Character

The first problem in the theory of signal detectability deals with the decision between two alternative responses, corresponding to two possible classes of causes of an observation. When the goal of a decision process is to achieve the highest quality of terminal decision, the Receiver Operating Cha...

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Bibliographic Details
Main Author: Birdsall, Theodore G
Format: Report
Language:English
Subjects:
Acoustic Detection and Detectors
Active & Passive Radar Detection & Equipment
ALGEBRA
COMMUNICATION AND RADIO SYSTEMS
CONVEX SETS
CURVE FITTING
Cybernetics
DECISION MAKING
DECISION THEORY
DETECTION
EXPONENTIAL FUNCTIONS
INFORMATION THEORY
INTEGRAL TRANSFORMS
INTERCEPTION PROBABILITIES
MATHEMATICAL MODELS
PROBABILITY DENSITY FUNCTIONS
RADAR RECEIVERS
RADAR SIGNALS
RADIO RECEIVERS
RECEIVER OPERATING CHARACTERISTIC CURVES
SONAR SIGNALS
Statistics and Probability
THEOREMS
TRANSFORMATIONS(MATHEMATICS)
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Summary:The first problem in the theory of signal detectability deals with the decision between two alternative responses, corresponding to two possible classes of causes of an observation. When the goal of a decision process is to achieve the highest quality of terminal decision, the Receiver Operating Characteristic curve (ROC curve) contains all of the information necessary for the evaluation of the decision process. This present work introduces the ROC character, which is isomorphic to the ROC curve. The formal development is based on two key facts. The first is the fundamental theorem: if L(X) is the likelihood ratio of an observation, then the likelihood ratio of L is L itself. The second is the main theorem on ROC characters: each ROC character is isomorphic to a univariate probability distribution that possesses a moment generating function. The character convolution theorem and the character addition theorem follow directly from these. Families of ROC curves are developed from the main theorem on ROC characters. The normal, binormal, Q- table, power, and several discrete families of ROC curve have appeared in the literature. The new families include the Pearson type III, Fisher-Tippett doubly exponential, H-type, Poisson, and the regular conics. Additional families are generated from these by use of the metastatic relation, and the convolution and addition theorems.