signal-detection theory
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Detection theory or signal detection theory is a means to measure the ability to differentiate between information-bearing patterns (called
stimulus A stimulus is something that causes a physiological response. It may refer to: *Stimulation **Stimulus (physiology), something external that influences an activity **Stimulus (psychology), a concept in behaviorism and perception *Stimulus (economi ...
in living organisms,
signal In signal processing, a signal is a function that conveys information about a phenomenon. Any quantity that can vary over space or time can be used as a signal to share messages between observers. The '' IEEE Transactions on Signal Processing' ...
in machines) and random patterns that distract from the information (called
noise Noise is unwanted sound considered unpleasant, loud or disruptive to hearing. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrations through a medium, such as air or water. The difference aris ...
, consisting of background stimuli and random activity of the detection machine and of the nervous system of the operator). In the field of
electronics The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification ...
, signal recovery is the separation of such patterns from a disguising background. 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 Radar is a detection system that uses radio waves to determine the distance ('' ranging''), angle, and radial velocity of objects relative to the site. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, we ...
researchers. By 1954, the theory was fully developed on the theoretical side as described by Peterson, Birdsall and Fox and the foundation for the psychological theory was made by Wilson P. Tanner, David M. Green, and John A. Swets, also in 1954. Detection theory was used in 1966 by John A. Swets and David M. Green for psychophysics. Green and Swets criticized the traditional methods of psychophysics for their inability to discriminate between the real sensitivity of subjects and their (potential)
response bias Response bias is a general term for a wide range of tendencies for participants to respond inaccurately or falsely to questions. These biases are prevalent in research involving participant self-report, such as structured interviews or surveys. ...
es.Green, D.M., Swets J.A. (1966) ''Signal Detection Theory and Psychophysics''. New York: Wiley. () Detection theory has applications in many fields such as diagnostics of any kind, quality control,
telecommunications Telecommunication is the transmission of information by various types of technologies over wire, radio, optical, or other electromagnetic systems. It has its origin in the desire of humans for communication over a distance greater than that fe ...
, and
psychology Psychology is the scientific study of mind and behavior. Psychology includes the study of conscious and unconscious phenomena, including feelings and thoughts. It is an academic discipline of immense scope, crossing the boundaries between ...
. The concept is similar to the signal-to-noise ratio used in the sciences and confusion matrices used in
artificial intelligence Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speech r ...
. It is also usable in
alarm management Alarm management is the application of human factors and ergonomics along with instrumentation engineering and systems thinking to manage the design of an alarm system to increase its usability. Most often the major usability problem is that t ...
, where it is important to separate important events from
background noise Background noise or ambient noise is any sound other than the sound being monitored (primary sound). Background noise is a form of noise pollution or interference. Background noise is an important concept in setting noise levels. Background n ...
.


Psychology

Signal detection theory (SDT) is used when psychologists want to measure the way we make decisions under conditions of uncertainty, such as how we would perceive distances in foggy conditions or during eyewitness identification. SDT assumes that the decision maker is not a passive receiver of information, but an active decision-maker who makes difficult perceptual judgments under conditions of uncertainty. In foggy circumstances, we are forced to decide how far away from us an object is, based solely upon visual stimulus which is impaired by the fog. Since the brightness of the object, such as a traffic light, is used by the brain to discriminate the distance of an object, and the fog reduces the brightness of objects, we perceive the object to be much farther away than it actually is (see also
decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
). According to SDT, during eyewitness identifications, witnesses base their decision as to whether a suspect is the culprit or not based on their perceived level of familiarity with the suspect. To apply signal detection theory to a data set where stimuli were either present or absent, and the observer categorized each trial as having the stimulus present or absent, the trials are sorted into one of four categories: : Based on the proportions of these types of trials, numerical estimates of sensitivity can be obtained with statistics like the sensitivity index ''d and A', and response bias can be estimated with statistics like c and β. Signal detection theory can also be applied to memory experiments, where items are presented on a study list for later testing. A test list is created by combining these 'old' items with novel, 'new' items that did not appear on the study list. On each test trial the subject will respond 'yes, this was on the study list' or 'no, this was not on the study list'. Items presented on the study list are called Targets, and new items are called Distractors. Saying 'Yes' to a target constitutes a Hit, while saying 'Yes' to a distractor constitutes a False Alarm. :


Applications

Signal Detection Theory has wide application, both in humans and
animals Animals are multicellular, eukaryotic organisms in the biological kingdom Animalia. With few exceptions, animals consume organic material, breathe oxygen, are able to move, can reproduce sexually, and go through an ontogenetic stage in ...
. Topics include
memory Memory is the faculty of the mind by which data or information is encoded, stored, and retrieved when needed. It is the retention of information over time for the purpose of influencing future action. If past events could not be remembered ...
, stimulus characteristics of schedules of reinforcement, etc.


Sensitivity or discriminability

Conceptually, sensitivity refers to how hard or easy it is to detect that a target stimulus is present from background events. For example, in a recognition memory paradigm, having longer to study to-be-remembered words makes it easier to recognize previously seen or heard words. In contrast, having to remember 30 words rather than 5 makes the discrimination harder. One of the most commonly used statistics for computing sensitivity is the so-called
sensitivity index The sensitivity index or discriminability index or detectability index is a dimensionless statistic used in signal detection theory. A higher index indicates that the signal can be more readily detected. Definition The discriminability index is ...
or ''d. There are also
non-parametric Nonparametric statistics is the branch of statistics that is not based solely on parametrized families of probability distributions (common examples of parameters are the mean and variance). Nonparametric statistics is based on either being distri ...
measures, such as the area under the ROC-curve.


Bias

Bias is the extent to which one response is more probable than another. That is, a receiver may be more likely to respond that a stimulus is present or more likely to respond that a stimulus is not present. Bias is independent of sensitivity. For example, if there is a penalty for either false alarms or misses, this may influence bias. If the stimulus is a bomber, then a miss (failing to detect the plane) may increase deaths, so a liberal bias is likely. In contrast, crying wolf (a false alarm) too often may make people less likely to respond, grounds for a conservative bias.


Compressed sensing

Another field which is closely related to signal detection theory is called ''
compressed sensing Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a signal, by finding solutions to underdetermined linear systems. This ...
'' (or compressive sensing). The objective of compressed sensing is to recover high dimensional but with low complexity entities from only a few measurements. Thus, one of the most important applications of compressed sensing is in the recovery of high dimensional signals which are known to be sparse (or nearly sparse) with only a few linear measurements. The number of measurements needed in the recovery of signals is by far smaller than what Nyquist sampling theorem requires provided that the signal is sparse, meaning that it only contains a few non-zero elements. There are different methods of signal recovery in compressed sensing including '' basis pursuit'' , ''expander recovery algorithm', CoSaMP'' and also ''fast'' ''non-iterative algorithm''.Lotfi, M.; Vidyasagar, M." A Fast Noniterative Algorithm for Compressive Sensing Using Binary Measurement Matrices". In all of the recovery methods mentioned above, choosing an appropriate measurement matrix using probabilistic constructions or deterministic constructions, is of great importance. In other words, measurement matrices must satisfy certain specific conditions such as '' RIP'' (Restricted Isometry Property) or '' Null-Space property'' in order to achieve robust sparse recovery.


Mathematics


P(H1, y) > P(H2, y) / MAP testing

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.Schonhoff, T.A. and Giordano, A.A. (2006) ''Detection and Estimation Theory and Its Applications''. New Jersey: Pearson Education () 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 ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
'' 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 ("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...
'' probabilities p(H1) = \pi_1 and p(H2) = \pi_2. In this case, p(H1, y) = \frac , p(H2, y) = \frac where ''p(y)'' is the total probability of event ''y'', p(y, H1) \cdot \pi_1 + p(y, H2) \cdot \pi_2 . ''H2'' is chosen in case \frac \ge \frac \Rightarrow \frac \ge \frac and ''H1'' otherwise. Often, the ratio \frac is called \tau_ and \frac is called L(y), the ''
likelihood ratio The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
''. Using this terminology, ''H2'' is chosen in case L(y) \ge \tau_. 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

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 A nuclear weapon is an explosive device that derives its destructive force from nuclear reactions, either fission (fission bomb) or a combination of fission and fusion reactions ( thermonuclear bomb), producing a nuclear explosion. Both bom ...
), 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 A false alarm, also called a nuisance alarm, is the deceptive or erroneous report of an emergency, causing unnecessary panic and/or bringing resources (such as emergency services) to a place where they are not needed. False alarms may occur with ...
(i.e., H1 = FALSE, H2 = TRUE) (assuming a large supply of fighter squadrons). The Bayes criterion is an approach suitable for such cases. Here a
utility As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosoph ...
is associated with each of four situations: * U_: One responds with behavior appropriate to H1 and H1 is true: fighters destroy bomber, incurring fuel, maintenance, and weapons costs, take risk of some being shot down; * U_: One responds with behavior appropriate to H1 and H2 is true: fighters sent out, incurring fuel and maintenance costs, bomber location remains unknown; * U_: One responds with behavior appropriate to H2 and H1 is true: city destroyed; * U_: One responds with behavior appropriate to H2 and H2 is true: fighters stay home, bomber location remains unknown; As is shown below, what is important are the differences, U_ - U_ and U_ - U_. Similarly, there are four probabilities, P_, P_, etc., for each of the cases (which are dependent on one's decision strategy). The Bayes criterion approach is to maximize the expected utility: E\ = P_ \cdot U_ + P_ \cdot U_ + P_ \cdot U_ + P_ \cdot U_ E\ = P_ \cdot U_ + (1-P_) \cdot U_ + P_ \cdot U_ + (1-P_) \cdot U_ E\ = U_ + U_ + P_ \cdot (U_ - U_) - P_ \cdot (U_ - U_) Effectively, one may maximize the sum, U' = P_ \cdot (U_ - U_) - P_ \cdot (U_ - U_) , and make the following substitutions: P_ = \pi_1 \cdot \int_p(y, H1)\, dy P_ = \pi_2 \cdot \int_p(y, H2)\, dy where \pi_1 and \pi_2 are the ''a priori'' probabilities, P(H1) and P(H2), and R_1 is the region of observation events, ''y'', that are responded to as though ''H1'' is true. \Rightarrow U' = \int_ \left \ \, dy U' and thus U are maximized by extending R_1 over the region where \pi_1 \cdot (U_ - U_) \cdot p(y, H1) - \pi_2 \cdot (U_ - U_) \cdot p(y, H2) > 0 This is accomplished by deciding H2 in case \pi_2 \cdot (U_ - U_) \cdot p(y, H2) \ge \pi_1 \cdot (U_ - U_) \cdot p(y, H1) \Rightarrow L(y) \equiv \frac \ge \frac \equiv \tau_B and H1 otherwise, where ''L(y)'' is the so-defined ''
likelihood ratio The likelihood function (often simply called the likelihood) represents the probability of random variable realizations conditional on particular values of the statistical parameters. Thus, when evaluated on a given sample, the likelihood functi ...
''.


Normal distribution models

Das and Geisler extended the results of signal detection theory for normally distributed stimuli, and derived methods of computing the error rate and
confusion matrix In the field of machine learning and specifically the problem of statistical classification, a confusion matrix, also known as an error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a su ...
for ideal observers and non-ideal observers for detecting and categorizing univariate and multivariate normal signals from two or more categories.


See also

*
Binary classification Binary classification is the task of classifying the elements of a set into two groups (each called ''class'') on the basis of a classification rule. Typical binary classification problems include: * Medical testing to determine if a patient has c ...
* Constant false alarm rate *
Decision theory Decision theory (or the theory of choice; not to be confused with choice theory) is a branch of applied probability theory concerned with the theory of making decisions based on assigning probabilities to various factors and assigning numerical ...
* Demodulation *
Detector (radio) In radio, a detector is a device or circuit that extracts information from a modulated radio frequency current or voltage. The term dates from the first three decades of radio (1888-1918). Unlike modern radio stations which transmit sound (an audi ...
* Estimation theory *
Just-noticeable difference In the branch of experimental psychology focused on sense, sensation, and perception, which is called psychophysics, a just-noticeable difference or JND is the amount something must be changed in order for a difference to be noticeable, detectable ...
*
Likelihood-ratio test In statistics, the likelihood-ratio test assesses the goodness of fit of two competing statistical models based on the ratio of their likelihoods, specifically one found by maximization over the entire parameter space and another found after im ...
* Modulation *
Neyman–Pearson lemma In statistics, the Neyman–Pearson lemma was introduced by Jerzy Neyman and Egon Pearson in a paper in 1933. The Neyman-Pearson lemma is part of the Neyman-Pearson theory of statistical testing, which introduced concepts like errors of the seco ...
*
Psychometric function Psychometrics is a field of study within psychology concerned with the theory and technique of measurement. Psychometrics generally refers to specialized fields within psychology and education devoted to testing, measurement, assessment, and ...
*
Receiver operating characteristic A receiver operating characteristic curve, or ROC curve, is a graphical plot that illustrates the diagnostic ability of a binary classifier system as its discrimination threshold is varied. The method was originally developed for operators of m ...
* Statistical hypothesis testing *
Statistical signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...
* Two-alternative forced choice *
Type I and type II errors In statistical hypothesis testing, a type I error is the mistaken rejection of an actually true null hypothesis (also known as a "false positive" finding or conclusion; example: "an innocent person is convicted"), while a type II error is the fa ...


References

* Coren, S., Ward, L.M., Enns, J. T. (1994) ''Sensation and Perception''. (4th Ed.) Toronto: Harcourt Brace. * Kay, SM. ''Fundamentals of Statistical Signal Processing: Detection Theory'' () * McNichol, D. (1972) ''A Primer of Signal Detection Theory''. London: George Allen & Unwin. * Van Trees HL. ''Detection, Estimation, and Modulation Theory, Part 1'' (
website
* Wickens, Thomas D., (2002) ''Elementary Signal Detection Theory''. New York: Oxford University Press. ()


External links




An application of SDT to safety

Signal Detection Theory
by Garrett Neske,
The Wolfram Demonstrations Project The Wolfram Demonstrations Project is an organized, open-source collection of small (or medium-size) interactive programs called Demonstrations, which are meant to visually and interactively represent ideas from a range of fields. It is hos ...

Lecture by Steven Pinker
{{DEFAULTSORT:Detection Theory Signal processing Telecommunication theory Psychophysics Mathematical psychology