Extreme Value Analysis Task

AKA: Extreme Value Theory, EVA.
Context:
- It can be used to describe an Anomaly Detection Algorithm.
- …
Example(s):
- EVA(???prob. family???, {1, 2, 2, 50, 98, 98, 99}) ⇒ [{1,2,2},{98,98,99}].
- …
Counter-Example(s):
- Distance-based Outlier Detection.
See: 100-Year Flood, Breakwater (Structure), Anomaly Prediction.

References

(Wikipedia, 2016) ⇒ http://wikipedia.org/wiki/Extreme_value_theory Retrieved:2016-4-12.
- Extreme value theory or extreme value analysis (EVA) is a branch of statistics dealing with the extreme deviations from the median of probability distributions. It seeks to assess, from a given ordered sample of a given random variable, the probability of events that are more extreme than any previously observed. Extreme value analysis is widely used in many disciplines, such as structural engineering, finance, earth sciences, traffic prediction, and geological engineering. For example, EVA might be used in the field of hydrology to estimate the probability of an unusually large flooding event, such as the 100-year flood. Similarly, for the design of a breakwater, a coastal engineer would seek to estimate the 50-year wave and design the structure accordingly.

(Aggarwal, 2013) ⇒ Charu C. Aggarwal. (2013). “Outlier Analysis." Springer Publishing Company, Incorporated. ISBN:1461463955, 9781461463955 doi:10.1007/978-1-4614-6396-2
- QUOTE: Extreme value statistics (Pickands, 1975) is distinct from the traditional definition of outliers. The traditional definition of outliers, as provided by Hawkins, defines such objects by their generative probabilities rather than the extremity in their values. For example, in the data set {1, 2, 2, 50, 98, 98, 99} of 1-dimensional values, the values 1 and 99, could very mildly, be considered extreme values. On the other hand, the value 50 is the average of the data set, and is most definitely not an extreme value. However, most probabilistic and density-based models would classify the value 50 as the strongest outlier in the data …

(Roberts, 1999) ⇒ S. J. Roberts. (1999) "Novelty Detection Using Extreme Value Statistics.” In: IEEE Proceedings Vision, Image & Signal Processing. (146:3) doi:10.1049/ip-vis:19990428
- ABSTRACT: Extreme value theory is a branch of statistics that concerns the distribution of data of unusually low or high value, i.e. in the tails of some distribution. These extremal points are important in many applications as they represent the outlying regions of normal events against which we may wish to define abnormal events. In the context of density modelling, novelty detection or radial-basis function systems, points that lie outside of the range of expected extreme values may be flagged as outliers. There has been interest in the area of novelty detection, but decisions as to whether a point is an outlier or not tend to be made on the basis of exceeding some (heuristic) threshold. It is shown that a more principled approach may be taken using extreme value statistics.

(Pickands, 1975) ⇒ J. Pickands. (1975). “Statistical Inference using Extreme Order Statistics.” In: The Annals of Statistics, 3(1).
- ABSTRACT: A method is presented for making statistical inferences about the upper tail of a distribution function. It is useful for estimating the probabilities of future extremely large observations. The method is applicable if the underlying distribution function satisfies a condition which holds for all common continuous distribution functions.