Detection theory
Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called stimulus in living organisms, signal in machines) and random patterns that distract from the information (called noise, consisting of background stimuli and random activity of the detection machine and of the nervous system of the operator).
In the field of electronics, signal recovery is the separation of such patterns from a disguising background.[1]
According to the theory, there are a number of determiners of how a detecting system will detect a signal, and where its threshold levels will be. The theory can explain how changing the threshold will affect the ability to discern, often exposing how adapted the system is to the task, purpose or goal at which it is aimed. When the detecting system is a human being, characteristics such as experience, expectations, physiological state (e.g.
fatigue) and other factors can affect the threshold applied. For instance, a sentry in wartime might be likely to detect fainter stimuli than the same sentry in peacetime due to a lower criterion, however they might also be more likely to treat innocuous stimuli as a threat.
Much of the early work in detection theory was done by radar researchers.[2] By 1954, the theory was fully developed on the theoretical side as described by Peterson, Birdsall and Fox[3] and the foundation for the psychological theory was made by Wilson P. Tanner, David M. Green, and John A. Swets, also in 1954.[4]
Detection theory was used in 1966 by John A. Swets and David M. Green for psychophysics.[5] Green and Swets criticized the traditional methods of psychophysics for their inability to discriminate between the real sensitivity of subjects and their (potential) response biases.[6]
Detection theory has applications in many fields such as diagnostics of any kind, quality control, telecommunications, and psychology. The concept is similar to the signal-to-noise ratio used in the sciences and confusion matrices used in artificial intelligence. It is also usable in alarm management, where it is important to separate important events from background noise.
Mathematics[edit]
P(H1|y) > P(H2|y) / MAP testing[edit]
In the case of making a decision between two hypotheses, H1, absent, and H2, present, in the event of a particular observation, y, a classical approach is to choose H1 when p(H1|y) > p(H2|y) and H2 in the reverse case.[14] In the event that the two a posteriori probabilities are equal, one might choose to default to a single choice (either always choose H1 or always choose H2), or might randomly select either H1 or H2. The a priori probabilities of H1 and H2 can guide this choice, e.g. by always choosing the hypothesis with the higher a priori probability.
When taking this approach, usually what one knows are the conditional probabilities, p(y|H1) and p(y|H2), and the a priori probabilities and . In this case,
,
where p(y) is the total probability of event y,
.
H2 is chosen in case
and H1 otherwise.
Often, the ratio is called and is called , the likelihood ratio.
Using this terminology, H2 is chosen in case . This is called MAP testing, where MAP stands for "maximum a posteriori").
Taking this approach minimizes the expected number of errors one will make.
Bayes criterion[edit]
In some cases, it is far more important to respond appropriately to H1 than it is to respond appropriately to H2. For example, if an alarm goes off, indicating H1 (an incoming bomber is carrying a nuclear weapon), it is much more important to shoot down the bomber if H1 = TRUE, than it is to avoid sending a fighter squadron to inspect a false alarm (i.e., H1 = FALSE, H2 = TRUE) (assuming a large supply of fighter squadrons). The Bayes criterion is an approach suitable for such cases.[14]
Here a utility is associated with each of four situations: