Editing Receiver operating characteristic (section)

==History==
The ROC curve was first used during [[World War II]] for the analysis of [[radar|radar signals]] before it was employed in [[signal detection theory]].<ref name="green66">{{cite book |first1=David M. |last1=Green |first2=John A. |last2=Swets |title=Signal detection theory and psychophysics |publisher=John Wiley and Sons Inc. |year=1966 |location=New York, NY |isbn=978-0-471-32420-1 }}</ref> Following the [[attack on Pearl Harbor]] in 1941, the United States military began new research to increase the prediction of correctly detected Japanese aircraft from their radar signals. For these purposes they measured the ability of a radar receiver operator to make these important distinctions, which was called the Receiver Operating Characteristic.<ref name="roc etymology">{{cite web|title=Using the Receiver Operating Characteristic (ROC) curve to analyze a classification model: A final note of historical interest|url=http://www.math.utah.edu/~gamez/files/ROC-Curves.pdf|url-status=live|archive-url=https://web.archive.org/web/20200822005346/http://www.math.utah.edu/~gamez/files/ROC-Curves.pdf |archive-date=2020-08-22 |access-date=May 25, 2017|website=Department of Mathematics, University of Utah}}</ref>

In the 1950s, ROC curves were employed in [[psychophysics]] to assess human (and occasionally non-human animal) detection of weak signals.<ref name="green66" /> In [[medicine]], ROC analysis has been extensively used in the evaluation of [[diagnostic test]]s.<ref>{{cite journal |first1=Mark H. |last1=Zweig |first2=Gregory |last2=Campbell |journal=Clinical Chemistry |pmid=8472349 |title=Receiver-operating characteristic (ROC) plots: a fundamental evaluation tool in clinical medicine |volume=39 |issue=8 |year=1993 |pages=561–577 |doi=10.1093/clinchem/39.4.561 |url=http://www.clinchem.org/content/39/4/561.full.pdf |doi-access=free }}</ref><ref>{{cite book |first=Margaret S. |last=Pepe |title=The statistical evaluation of medical tests for classification and prediction |location=New York, NY |publisher=Oxford |year=2003 |isbn=978-0-19-856582-6 }}</ref> ROC curves are also used extensively in [[epidemiology]] and [[medical research]] and are frequently mentioned in conjunction with [[evidence-based medicine]]. In [[radiology]], ROC analysis is a common technique to evaluate new radiology techniques.<ref>{{cite journal |first=Nancy A. |last=Obuchowski|author-link=Nancy Obuchowski |title=Receiver operating characteristic curves and their use in radiology |pmid=14519861 |journal=Radiology |volume=229 |issue=1 |year=2003 |pages=3–8 |doi=10.1148/radiol.2291010898 }}</ref> In the social sciences, ROC analysis is often called the ROC Accuracy Ratio, a common technique for judging the accuracy of default probability models.
ROC curves are widely used in laboratory medicine to assess the diagnostic accuracy of a test, to choose the optimal cut-off of a test and to compare diagnostic accuracy of several tests.

ROC curves also proved useful for the evaluation of [[machine learning]] techniques. The first application of ROC in machine learning was by Spackman who demonstrated the value of ROC curves in comparing and evaluating different classification [[algorithm]]s.<ref>{{cite conference |last=Spackman |first=Kent A. |year=1989 |title=Signal detection theory: Valuable tools for evaluating inductive learning |book-title=Proceedings of the Sixth International Workshop on Machine Learning |location=San Mateo, CA |pages=160–163 |publisher=[[Morgan Kaufmann]] }}</ref>

ROC curves are also used in verification of forecasts in meteorology.<ref>{{ cite journal |first=Viatcheslav |last=Kharin| title=On the ROC score of probability forecasts| journal=Journal of Climate| volume=16|number=24|  pages=4145–4150|year=2003|doi=10.1175/1520-0442(2003)016<4145:OTRSOP>2.0.CO;2
|bibcode=2003JCli...16.4145K|doi-access=free}}</ref>

=== Radar in detail ===
As mentioned ROC curves are critical to [[radar]] operation and theory. The signals received at a receiver station, as reflected by a target, are often of very low energy, in comparison to the [[noise floor]]. The ratio of [[Signal-to-noise ratio|signal to noise]] is an important metric when determining if a target will be detected. This signal to noise ratio is directly correlated to the receiver operating characteristics of the whole radar system, which is used to quantify the ability of a radar system.

Consider the development of a radar system. A specification for the abilities of the system may be provided in terms of probability of detect, <math>P_{D}</math>, with a certain tolerance for false alarms, <math>P_{FA}</math>. A simplified approximation of the required signal to noise ratio at the receiver station can be calculated by solving<ref>{{Citation |title=Fundamentals of Radar |date=2008-01-29 |url=http://dx.doi.org/10.1002/9780470377765.ch4 |work=Digital Signal Processing Techniques and Applications in Radar Image Processing |pages=93–115 |access-date=2023-05-20 |place=Hoboken, NJ, USA |publisher=John Wiley & Sons, Inc.|doi=10.1002/9780470377765.ch4 |isbn=9780470377765 }}</ref>

: <math>P_{D}=\frac{1}{2}\operatorname{erfc}\left(\operatorname{erfc}^{-1}\left(2P_{FA}\right)-\sqrt{\mathcal{X}}\right)</math>

for the signal to noise ratio <math>\mathcal{X}</math>. Here, <math>\mathcal{X}</math> is not in [[decibel]]s, as is common in many radar applications. Conversion to decibels is through <math>\mathcal{X}_{dB}=10\log_{10}\mathcal{X}</math>. From this figure, the common entries in the radar range equation (with noise factors) may be solved, to estimate the required [[effective radiated power]].