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How Are Rare Events Formed?
Published in Yair Neuman, How to Find a Needle in a Haystack, 2023
By definition, a rare event (or object) is an event (or object) whose frequency is relatively low by some standard. Winning a lottery is a rare event and so is dying in a shark attack (Sepulveda, 2021) or being struck by lightning (National Weather Service, n.d.). We can think about a rare object or a rare event by examining a probability distribution, which is just a fancy name for “the probabilities of occurrence of different possible outcomes” (Wikipedia, 2022b). If you are counting the two possible outcomes of a coin toss, then the probabilities of the outcomes (heads or tails) make up the probability distribution of the coin toss. Figure 3.1 shows a possible probability distribution describing the distribution of height among adult men. As you can see, height is normally distributed in the general population. This means that the distribution of the different outcomes (i.e. heights) is concentrated around the mean and the distribution is symmetric. The right side of the distribution is a mirror image of its left side. You may also notice that being taller or shorter than a certain height is a rare event. You will hardly ever meet adult men taller than 2 m or shorter than 1.5 m.
Newly enhanced computing algorithm to quantify unavailability of maintained multi-component systems
Published in Stein Haugen, Anne Barros, Coen van Gulijk, Trond Kongsvik, Jan Erik Vinnem, Safety and Reliability – Safe Societies in a Changing World, 2018
Estimating (un)availability of a highly-reliable multi-component and maintained system is a problem of great interest in different areas such as computer systems, telecommunications, mechanics, aircraft design, power utilities, and many other engineering fields. Increasing demand for system reliability cannot depend on the increasing reliability of components due to technological restrictions. Safety systems of nuclear power stations represent other example of highly reliable complex systems. They have to be reliable enough to comply with still increasing internationally agreed safety criteria and moreover they are mostly so called sleeping systems which start and operate only in the case of big accidents. Their hypothetical failures are not apparent (hidden or latent failures) and thus repairable only at optimally selected inspection times. Wide class of highly reliable fault-tolerant systems, which, through the use of redundancy, have the ability to operate properly in the presence of faults are investigated in (VillénAltamirano 2014). Any system failure should have a small probability of occurring; that is, it should be a rare event. It is important to estimate such probabilities because when a rare event does occur, its consequences may be catastrophic. For example, network servers play an increasingly important role due to the rapid growth in demand for internet services, and a server breakdown event may cause significant financial losses. As a result, redundancy is usually built in to prevent services from breaking down.
Extreme Event Theory
Published in C. Ariel Pinto, Paul R. Garvey, Advanced Risk Analysis in Engineering Enterprise Systems, 2016
C. Ariel Pinto, Paul R. Garvey
Extreme and rare events have been defined earlier as those having relatively very low frequency of occurrence and at the same time have relatively extreme high or extreme low degree of magnitude. Through several examples, extreme and rare events have been exemplified in both natural and engineering systems. In data measurement, some extreme data may not be recorded for various reasons such as limitation of measuring equipment or too few samples to capture the truly rare events. All these result in what analysts may refer to as data censoring. For the few possible instances that rare events do get recorded, classical data analysis may be performed resulting in extreme and rare measurements being labeled as outliers. This is often the result of hasty data filtering and forcing the assumption of stationarity. What may not be very apparent is the relationship among these notions: extreme and rare events and the challenges in their analysis, system complexity, probabilistic causation (Chapter 9), and evidence-based analysis (i.e., Bayes’ rule, Chapter 3). These notions can converge and pose themselves as two familiar problems of causalities and correlations.
Some Structural Properties Related to the Borel-Taner Distribution and its’ Application
Published in American Journal of Mathematical and Management Sciences, 2023
US Earthquakes data is available in the Wolfram Mathematica Data Repository. It gives earthquake magnitudes recorded by the US Geological Survey ranging from 1.0 to 9.9, in the US, and for illustrative purposes we select a sample data set between 1935 and 1990. To exhibit the flexibility of the p.m.f. in (1), we tallied the magnitudes and used those frequencies as our dataset, labeled earthquakes. In this case the unit of time is year. For this data, we provide some useful descriptive summary given as follows: Mean = 17.586; Median = 14.486; Variance = 7.289. Clearly, the data is discrete and positively skewed. Therefore, we conjecture at this point that a univariate discrete distribution might be a good fit that exhibits positive skewness. Next, earthquakes are rare events (except for certain geographical regions, such as Japan, etc.) in many parts of the world, including the US. Therefore, we can think of discrete distributions that are useful for modeling rare events might be utilized to model this data. Consequently, we select BT distribution along with Poisson, Poisson-Consul and the Negative Binomial to examine which one among them provides a reasonably better fit.
Machine learning in drying
Published in Drying Technology, 2020
Besides the challenge with obtaining high-quality data and managing such data, there are other challenges that need to be addressed for practical application of ML in drying. The algorithms of machine learning based on the probability functions have been developed mostly to detect trends and frequent patterns rather than anomalies or rare events. In case of unexpected failure of sensor or actuator, such as heating element or vacuum pump, a ML based model could miserably fail. Therefore, when building ML models for drying process control, such low probability situations need to be accounted for in some form of computer rules,[100] suggesting compensation of failure or activation of alarm. Accounting for these rarely predictable dynamic situations will increase robustness of drying systems and also open new area of applications ML for fault control.
Improved estimation of ultra-deep water pipe collapse pressures by using extreme value theory
Published in Quality Engineering, 2019
Janet E. Heffernan, Alastair Walker, Ping Liu
Design criteria for installation require the probability of collapse to be around 1e − 07. The collapse event is sufficiently rare as to be effectively unobservable from any data sample of a practical and economically feasible size. Statistical models motivated by EVT have been developed expressly for the estimation of occurrence probabilities of extremely rare events. Many applications in engineering, environmental science, finance and others require estimation of the risk of rare events (such events include flooding, financial crashes, extreme loadings on structures caused by large weather events, etc.). These catastrophic events often lie well beyond levels observed in available data. Statisticians are naturally wary of extrapolating their fitted models beyond levels at which data can be used to validate model fit. Many practical applications therefore rely on statistical models motivated by EVT which provides additional theoretical justification for model extrapolation. EVT has been established for many years and offers a practical solution to the challenge of estimating the risk of rare events: see Finkenstadt and Rootzèn (2003), Coles (2001), and references therein. Statistical models motivated by EVT are widely used in coastal and offshore engineering applications where methods specifically suited to this setting have been developed by Haigh, Nicholls, and Wells (2010), Tawn (1992), and Dhanak and Xiros (2016). There is some history of these models being used in manufacturing engineering but the methods are still considered somewhat esoteric, Anderson et al. (2003) and Fougères, Holm, and Rootzèn (2006). Here we hope to illustrate the value of the proposed methods in this setting.