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
A signal is both the process and the result of transmission of data over some media accomplished by embedding some variation. Signals are important in multiple subject fields including signal processing, information theory and biology.
In ...
in machines) and random patterns that distract from the information (called
noise
Noise is sound, chiefly unwanted, unintentional, or harmful sound considered unpleasant, loud, or disruptive to mental or hearing faculties. From a physics standpoint, there is no distinction between noise and desired sound, as both are vibrat ...
, consisting of background stimuli and random activity of the detection machine and of the nervous system of the operator).
In the field of
electronics
Electronics is a scientific and engineering discipline that studies and applies the principles of physics to design, create, and operate devices that manipulate electrons and other Electric charge, electrically charged particles. It is a subfield ...
, 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 system that uses radio waves to determine the distance ('' ranging''), direction ( azimuth and elevation angles), and radial velocity of objects relative to the site. It is a radiodetermination method used to detect and track ...
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
Psychophysics is the field of psychology which quantitatively investigates the relationship between physical stimulus (physiology), stimuli and the sensation (psychology), sensations and perceptions they produce. Psychophysics has been described ...
. 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.
[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
Quality control (QC) is a process by which entities review the quality of all factors involved in production. ISO 9000 defines quality control as "a part of quality management focused on fulfilling quality requirements".
This approach plac ...
,
telecommunications
Telecommunication, often used in its plural form or abbreviated as telecom, is the transmission of information over a distance using electronic means, typically through cables, radio waves, or other communication technologies. These means of ...
, and
psychology
Psychology is the scientific study of mind and behavior. Its subject matter includes the behavior of humans and nonhumans, both consciousness, conscious and Unconscious mind, unconscious phenomena, and mental processes such as thoughts, feel ...
. The concept is similar to the
signal-to-noise ratio
Signal-to-noise ratio (SNR or S/N) is a measure used in science and engineering that compares the level of a desired signal to the level of background noise. SNR is defined as the ratio of signal power to noise power, often expressed in deci ...
used in the sciences and
confusion matrices used in
artificial intelligence
Artificial intelligence (AI) is the capability of computer, computational systems to perform tasks typically associated with human intelligence, such as learning, reasoning, problem-solving, perception, and decision-making. It is a field of re ...
. 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 ...
, 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 no ...
.
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 rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
). 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.
β is the measure of response bias.
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, have myocytes and are able to move, can reproduce sexually, and grow from a ...
. 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 remembe ...
, 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 or ''d. There are also
non-parametric measures, such as the area under the
ROC-curve.
Bias
Bias is the extent to which one response is more probable than another, averaging across stimulus-present and stimulus-absent cases. That is, a receiver may be more likely overall to respond that a stimulus is present or more likely overall to respond that a stimulus is not present. Bias is independent of sensitivity. Bias can be desirable if false alarms and misses lead to different costs. For example, if the stimulus is a bomber, then a miss (failing to detect the bomber) may be more costly than a false alarm (reporting a bomber when there is not one), making a liberal response bias desirable. In contrast, giving false alarms too often (
crying wolf) may make people less likely to respond, a problem that can be reduced by a conservative response 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 (electronics), signal by finding solutions to Underdetermined s ...
'' (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
A hypothesis (: hypotheses) is a proposed explanation for a phenomenon. A scientific method, scientific hypothesis must be based on observations and make a testable and reproducible prediction about reality, in a process beginning with an educ ...
, ''H1'', absent, and ''H2'', present, in the event of a particular
observation
Observation in the natural sciences is an act or instance of noticing or perceiving and the acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the percep ...
, ''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
('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 experience. knowledge is independent from any experience. Examples include ...
''
probabilities
Probability is a branch of mathematics and statistics concerning Event (probability theory), events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probab ...
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 (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
'' 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 (epistemology), justification, or argument by their reliance on experience. knowledge is independent from any ...
'' 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
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 or atomic bomb) or a combination of fission and fusion reactions (thermonuclear weapon), producing a nuclear exp ...
), 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.
[
Here a ]utility
In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings.
* In a normative context, utility refers to a goal or objective that we wish ...
is associated with each of four situations:
* : 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;
* : One responds with behavior appropriate to H1 and H2 is true: fighters sent out, incurring fuel and maintenance costs, bomber location remains unknown;
* : One responds with behavior appropriate to H2 and H1 is true: city destroyed;
* : 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, and .
Similarly, there are four probabilities, , , etc., for each of the cases (which are dependent on one's decision strategy).
The Bayes criterion approach is to maximize the expected utility:
Effectively, one may maximize the sum,
,
and make the following substitutions:
where and are the ''a priori'' probabilities, and , and is the region of observation events, ''y'', that are responded to as though ''H1'' is true.
and thus are maximized by extending over the region where
This is accomplished by deciding H2 in case
and H1 otherwise, where ''L(y)'' is the so-defined '' likelihood ratio''.
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 error matrix, is a specific table layout that allows visualization of the performance of an algorithm, typically a super ...
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 one of two groups (each called ''class''). Typical binary classification problems include:
* Medical testing to determine if a patient has a certain disease or not;
* Qual ...
* Constant false alarm rate
* Decision theory
Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
* Demodulation
Demodulation is the process of extracting the original information-bearing signal from a carrier wave. A demodulator is an electronic circuit (or computer program in a software-defined radio) that is used to recover the information content fro ...
* 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 ...
* Estimation theory
Estimation theory is a branch of statistics that deals with estimating the values of Statistical parameter, parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such ...
* 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, detectabl ...
* Likelihood-ratio test
* Modulation
Signal modulation is the process of varying one or more properties of a periodic waveform in electronics and telecommunication for the purpose of transmitting information.
The process encodes information in form of the modulation or message ...
* Neyman–Pearson lemma
* Psychometric function
* Receiver operating characteristic
A receiver operating characteristic curve, or ROC curve, is a graph of a function, graphical plot that illustrates the performance of a binary classifier model (can be used for multi class classification as well) at varying threshold values. ROC ...
* Statistical hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...
* Statistical signal processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as sound, images, potential fields, seismic signals, altimetry processing, and scientific measurements. Signal ...
* Two-alternative forced choice
* Type I and type II errors
Type I error, or a false positive, is the erroneous rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the erroneous failure in bringing about appropriate rejection of a false null hy ...
References
Bibliography
* 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 open-source collection of interactive programmes called Demonstrations. It is hosted by Wolfram Research. At its launch, it contained 1300 demonstrations but has grown to over 10,000. The site won a Pa ...
Lecture by Steven Pinker
{{DEFAULTSORT:Detection Theory
Signal processing
Telecommunication theory
Psychophysics
Mathematical psychology