Signal Detection Theory
Signal Detection Theory (SDT) is a framework from psychophysics that models how observers detect faint signals in noise. It separates sensitivity (d' — ability to distinguish signal from noise) from response bias (criterion c — tendency to say yes or no). Four outcomes: hit (signal present, detected), miss (present, not detected), false alarm (absent, said yes), correct rejection (absent, said no). Originally from radar engineering, SDT is fundamental in psychology, medicine, and machine learning.
The Four Outcomes of Signal Detection
Every detection decision falls into four categories. Signal present and detected: Hit. Signal present but missed: Miss. No signal but observer says yes: False Alarm. No signal and correctly says no: Correct Rejection. These form a 2×2 matrix describing detection performance. The hit rate and false alarm rate together determine the ROC curve — a powerful tool for evaluating any detection system, from radiologists reading X-rays to spam filters classifying email.
Key Formulas
Sensitivity (d') and Response Bias (c)
SDT's genius is separating two independent factors. Sensitivity (d-prime) measures how well the observer distinguishes signal from noise — the standardized distance between noise-only and signal-plus-noise distributions. A radiologist with d' = 3.0 is much better at spotting tumors than one with d' = 1.0. Response bias (criterion c) measures willingness to say 'signal present' — a cautious radiologist requires strong evidence (high criterion, fewer false alarms but more misses). Both metrics are independent: changing bias doesn't change sensitivity.
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Open CalculatorROC Curves: Visualizing the Tradeoff
The ROC curve plots hit rate versus false alarm rate as criterion varies. Perfect detection (d' = ∞) reaches the upper-left corner. Random guessing (d' = 0) falls on the diagonal. Higher sensitivity pushes the curve upper-left. Area under the ROC (AUC) summarizes performance: 1.0 is perfect, 0.5 is chance. ROC analysis originated in WWII radar engineering, was adopted by psychology in the 1950s, then medicine (diagnostic testing) and machine learning (classifier evaluation).
SDT in Psychology: Perception, Memory, and Decisions
SDT models any yes/no judgment under uncertainty. Pain detection: is that sensation painful or just pressure? Recognition memory: have I seen this face before? Eyewitness identification: is this the perpetrator? In each case, the signal is embedded in noise, and SDT quantifies both discrimination ability and response tendency. Understanding that hits and false alarms are linked through the criterion helps design better experimental and forensic procedures.
SDT in Engineering and Machine Learning
SDT applies identically to electronic and algorithmic detection. A radar receiver decides if a return pulse is target or noise. A spam filter decides if an email is legitimate or junk. A medical AI decides if a scan shows a tumor. In each case, the threshold determines the hit/false alarm tradeoff. ROC curves and AUC from SDT are now standard metrics for evaluating binary classifiers in any domain — the mathematical framework is universal.
Related Topics in signal processing techniques
Understanding signal detection theory connects to several related concepts: detection theory, signal detection theory psychology, signal detection theory example, and signal detection theory psychology definition. Each builds on the mathematical foundations covered in this guide.
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