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.
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Transcription
Voiceover: In this video, I'm going to be talking about something known as Signal Detection Theory. Signal Detection Theory, basically, looks to see how we make decisions, so decision making, under conditions of uncertainty, so with uncertainty. Let me give you an example of what Signal Detection Theory is trying to do. I want you to look at the screen and tell me if there is any change. It was pretty obvious that I put up this bright green circle. Okay? Keep looking at the screen and think to yourself and try and notice any changes. Okay? This time around, I put up this fainter green dot as compared to this bright green dot. The bright green dot is a fairly strong signal and this faint green dot is a fairly weak signal. Signal Detection Theory is, basically, trying to figure out at what point is a signal strong enough that we are able to notice it, in the first place, and also in order to… Signal Detection Theory is, basically, trying to decide at what point are we able to detect a signal, and it had its origins in radar. Back when radar was being developed, they had to figure out a way to determine whether a strong signal is a ship or a large whale or a school of fish, and that's where it had its origins. Signal Detection Theory also plays a role in psychology and, in psychology, imagine that we show a list of words to an individual, and then we show them a second list of words, and we ask them to recall which words from the second list were on the first list. The decision that they have to make is to decide which word on the second list was also present on the first list, and the uncertainty is their ability to memorize all the words on the first list, so they're not sure, 100%, whether a word is exactly the same as the one on the first list, or very similar, and I can give you a real-world example of Signal Detection Theory. Imagine that you're driving to work or school and you're waiting at a traffic light. It's a foggy day, and you have to decide when to start driving, so you have to decide when the light turns green and you have to start driving. Now it's really hard to see the green light, so it might be, kind of, faint, kind of, like this green light, and you have to decide, at what point in time, how strong does the signal have to be in order for you to say, "Yes, "the light is definitely green. Let me start driving." In that case, there are a few different options. Let me just draw a quick table here. There are a few different options. Either the signal is present, so the light is green, so signal can be present, or the signal can be absent, so the light is red, or it's not green. You can either say, "Yes, the light is green," or you can say, "No, the light is not green." There are a few different possibilities. If you say, "Yes, the light is definitely green," you're 100% sure, maybe it's something like this, then that would be a hit. However, if the light is present, or maybe it's really faint, and it's present, but you're not 100% sure whether it's green or not, you might say, "No," and, since the signal is present, and you're saying, "No," that's a miss, so it's incorrect. Another possibility would be the signal being absent, so maybe the signal's absent, but you say, "Yes," and that would be a false alarm, so false alarm. The final possibility is that the signal is absent and you say, "No," and that's correct, so that would be a correct rejection, a correct rejection. If the signal is really, really strong, so if it's this, you might always get it right. Whenever the signal is present, you'll always say, "Yes," and when the signal's gone, you'll always say, "No." In that case, it's pretty easy to decide whether a signal is present or not. On the other hand, if it's a fairly weak signal, maybe this faint green dot, you might get some false alarms. You might say, "Yes," when the signal's actually present, or you might say, "No," because you don't see this faint green dot, and you might get some misses. An easy signal, like the really bright green dot, would create more hits than misses, whereas a weak signal would create less hits than misses. Now, the strength of a signal, which is what we were just talking about, is a variable known as D Prime, so this is the strength of a signal. Another variable is C, and that is strategy. Let me just talk about this. Strategy would be… Let's look at the example of you driving to work and you're waiting at the traffic light. One strategy could be, if you see any light, you're going to say, "Yes," and start driving. Another strategy would be, if you see any green light, you're going to say, "Yes," and start driving, or, third strategy would be, if it's a green light and it's elevated up off the floor, and it was presented immediately after a red light, then you're going to start driving, so you have different strategies. There are two big strategies. You could either have a conservative strategy, so conservative, or you can have a liberal strategy. If you have a conservative strategy, you would always say, "No," unless you're 100% sure that the signal is present. You're always going to be saying, "No," unless you're 100% sure the signal is present, and the bad thing about that is that, even though you'll get all the correct rejections, you might also get some misses. On the other hand, you can have a liberal strategy, where you always say, "Yes," and, in that case, you'll always get all the hits; however, you might get a few false alarms. These are the two different strategies that you can use, so this would be strategy and this would be the strength, so D Prime and C.
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.[7][8] 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). 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:
Respond "Absent" Respond "Present" Stimulus Present Miss Hit Stimulus Absent Correct Rejection False Alarm
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',[9] and response bias can be estimated with statistics like c and β.[9] β is the measure of response bias.[10]
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.
Respond "No" Respond "Yes" Target Miss Hit Distractor Correct Rejection False Alarm
Applications
Signal Detection Theory has wide application, both in humans and animals. Topics include memory, 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.[6]
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 (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[11], CoSaMP[12] and also fast non-iterative algorithm.[13] 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.[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
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:
- : 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 [15] extended the results of signal detection theory for normally distributed stimuli, and derived methods of computing the error rate and confusion matrix 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
- Constant false alarm rate
- Decision theory
- Demodulation
- Detector (radio)
- Estimation theory
- Just-noticeable difference
- Likelihood-ratio test
- Modulation
- Neyman–Pearson lemma
- Psychometric function
- Receiver operating characteristic
- Statistical hypothesis testing
- Statistical signal processing
- Two-alternative forced choice
- Type I and type II errors
References
- ^ T. H. Wilmshurst (1990). Signal Recovery from Noise in Electronic Instrumentation (2nd ed.). CRC Press. pp. 11 ff. ISBN 978-0-7503-0058-2.
- ^ Marcum, J. I. (1947). "A Statistical Theory of Target Detection by Pulsed Radar". The Research Memorandum: 90. Retrieved 2009-06-28.
- ^ Peterson, W.; Birdsall, T.; Fox, W. (September 1954). "The theory of signal detectability". Transactions of the IRE Professional Group on Information Theory. 4 (4): 171–212. doi:10.1109/TIT.1954.1057460.
- ^ Tanner, Wilson P.; Swets, John A. (1954). "A decision-making theory of visual detection". Psychological Review. 61 (6): 401–409. doi:10.1037/h0058700. PMID 13215690.
- ^ Swets, J.A. (ed.) (1964) Signal detection and recognition by human observers. New York: Wiley[page needed]
- ^ a b Green, D.M., Swets J.A. (1966) Signal Detection Theory and Psychophysics. New York: Wiley. (ISBN 0-471-32420-5)[page needed]
- ^ Clark, Steven E.; Benjamin, Aaron S.; Wixted, John T.; Mickes, Laura; Gronlund, Scott D. (2015). "Eyewitness Identification and the Accuracy of the Criminal Justice System". Policy Insights from the Behavioral and Brain Sciences. 2: 175–186. doi:10.1177/2372732215602267. hdl:11244/49353. S2CID 18529957.
- ^ Haw, Ryann Michelle (January 2005). "A theoretical analysis of eyewitness identification: Dual -process theory, signal detection theory and eyewitness confidence". ProQuest Etd Collection for Fiu: 1–98.
- ^ a b Stanislaw, Harold; Todorov, Natasha (March 1999). "Calculation of signal detection theory measures". Behavior Research Methods, Instruments, & Computers. 31 (1): 137–149. doi:10.3758/BF03207704. PMID 10495845.
- ^ "Signal Detection Theory". elvers.us. Retrieved 2023-07-14.
- ^ Jafarpour, Sina; Xu, Weiyu; Hassibi, Babak; Calderbank, Robert (September 2009). "Efficient and Robust Compressed Sensing Using Optimized Expander Graphs" (PDF). IEEE Transactions on Information Theory. 55 (9): 4299–4308. doi:10.1109/tit.2009.2025528. S2CID 15490427.
- ^ Needell, D.; Tropp, J.A. (2009). "CoSaMP: Iterative signal recovery from incomplete and inaccurate samples". Applied and Computational Harmonic Analysis. 26 (3): 301–321. arXiv:0803.2392. doi:10.1016/j.acha.2008.07.002. S2CID 1642637.
- ^ Lotfi, M.; Vidyasagar, M."A Fast Noniterative Algorithm for Compressive Sensing Using Binary Measurement Matrices".
- ^ a b Schonhoff, T.A. and Giordano, A.A. (2006) Detection and Estimation Theory and Its Applications. New Jersey: Pearson Education (ISBN 0-13-089499-0)
- ^ Das, Abhranil; Geisler, Wilson (2021). "A method to integrate and classify normal distributions". Journal of Vision. 21 (10): 1. arXiv:2012.14331. doi:10.1167/jov.21.10.1. PMC 8419883. PMID 34468706.
- 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 (ISBN 0-13-504135-X)
- McNichol, D. (1972) A Primer of Signal Detection Theory. London: George Allen & Unwin.
- Van Trees HL. Detection, Estimation, and Modulation Theory, Part 1 (ISBN 0-471-09517-6; website)
- Wickens, Thomas D., (2002) Elementary Signal Detection Theory. New York: Oxford University Press. (ISBN 0-19-509250-3)
External links
- A Description of Signal Detection Theory
- An application of SDT to safety
- Signal Detection Theory by Garrett Neske, The Wolfram Demonstrations Project
- Lecture by Steven Pinker
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