Detection theory

The signal detection theory was developed by John A. Swets and David M. Green and first presented in 1966 in her book, Signal Detection Theory and Psychophysics. It analyzes the detection of difficult to detect signals and provides a measure of the quality of the human or system responsible for this detection. The first impetus for this research delivered in December 1941, the Japanese air attack on Pearl Harbor, which had been discovered according to official figures neither the radar nor of the radio monitoring of U.S. defense. Hence the name of receiver operator characteristic (see below) comes from.

Original experiment

Green and Swets played their subjects many noise samples before that contained only noise to some extent, and partly also a barely audible tone. The subjects indicated whether they heard a sound or noise. As with most previous psychophysical experiments showed that its performance not only on the signal - to-noise ratio ( ie how much the sound stood out from noise) and the actual detection performance depended, but was influenced by many additional factors, including motivation, vigilance / fatigue, distraction by interference, information such as the percentage of the noise samples sounds contained, etc. Green and Swets particular interest, however, was the response bias exhibited by the subjects when they were unsure: Some opted more often for " yes, I have a sound can be heard " (so-called liberal criterion), while others are in doubt rather" no, I have not heard a sound, " replied ( conservative criterion). The actual detection performance can be determined only if the response bias is eliminated, as the following example shows: Two medical students are each 20 X-ray survey, of which, what they do not know, 10 show a tumor. A student wants to see anything possible and decides at 13 shots for the diagnosis of " tumor ". Of these, 9 are correctly identified and 4 wrong. Student B on the other hand wants to be sure and decides at 7 shots for "tumor". Of these, 6 right and 1 wrong. Both words have the same number of correct diagnoses ( 9-4 = 6-1 ), except that Student A a more liberal response criterion than B and thus has a higher number of misdiagnoses.

Metrics

The signal may be present or not, and the test subject (or detection system ) may report a detection or not, so that the four combinations are possible:

To calculate the Degree of sensitivity d '( English pronunciation dee prime ), we first determined the relative frequencies of hits and false alarms, leading to these values ​​by a z-transform and forms calculate the difference between: d '= z (hits) - z ( false alarm )

Another, less commonly used measure is the response bias (also called tilt reaction ): c = -0.5 * (z ( false alarm ) z (hits) ).

Example: A young drug search dog has a relative frequency of hits by 89% ( estimate of its hit probability 0.89) and a relative frequency of false alarms by 59% ( estimate of its false alarm probability 0.59). From the corresponding z- values ​​z ( 0.89) = 1.23 and z ( 0.59) = 0.23, the calculated sensitivity of d '= 1 and a response bias of -0.73. After several years 'professional experience' the dog has a hit rate of 96% and only in 39 % of cases suggests a false alarm. Therefore, its detection performance improved d ' 2.03, while his response bias c has remained the same with -0.74 (which is to be expected of a dog too).

Theoretical assumptions

While in fact, originally the noise was meant in headphones ' receiver operator "with" noise ", we group today under this term all those internal and external influences together that can move the diagnostician to" yes " to say, although no signal is available. The probability that the noise causes a false alarm is assumed to be normally distributed (in the picture the upper bell curve ). Against this background noise, which is always present, the signal must " prevail ", ie it is the noise added to it. Thus, the probability distribution is shifted to the right (in the picture the lower bell curve ). With heavy tasks (low signal-to- noise ratio), the curves are flat and wide and overlap strongly (as shown here), with easy tasks they are steep and narrow and overlap only slightly.

Whether the " operator" is now actually saying "yes", depends on his response criterion. In the picture is the threshold more to the right (more "No" - as " yes" answers ), so he drives a more conservative strategy.

Applications

The signal detection theory can come from any type of diagnosis for the application; some of their areas of application are:

• Empirical science (see type 1 error, type 2 error )
• Medicine (including assessing radiographs, PET and fMRI scans, laboratory tests, etc. )
• Quality Management
• Air traffic control
• Baggage check ( eg airports )
• Access control to sports stadiums, discos, etc.

In such real-life situations is a further significant influence on the detection performance to bear, namely how large the benefit of hits and especially how dangerous the consequences of Verpassern.

ROC

A frequently used graphical representation of these metrics is the Isosensitivitäts or ROC curve, ROC stands for receiver operator characteristic ( the two medical students from the example above are isosensitiv ). To the relative frequency of hits is plotted versus the relative frequency of false alarms. All diagnosticians with the same detection capability are therefore on the same ROC curve: Whoever only advises ( hit rate = miss- rate = 0.5, d = 0 ) lies on a straight line of slope 1 Good (more hits than false alarms, d ' > 1 ) and very good diagnostician (a lot more hits than false alarms, d ' > 2) are more or less strongly convex curves. The ROC curves are thus independent of the response bias c. If our two medical students to completely change their response tendency, encouraged by a new boss, for example, they would still remain on its ROC curve. This can be changed only through training. Are the conservative respondents ( few hits and a few false alarms ) in the left (lower) range, the liberal respondents ( many hits and many false alarms ) in the right ( upper ) region on each ROC curve.