page_title: Dragon king theory
section_title: Detecting DKs as statistical outliers
hierarchical_section_title: Dragon king theory / Detecting DKs as statistical outliers
caption_attribution_description: Probability density function schematic where the dragon king regime is represented by a lump of mass deep in the tail.
context_page_description: Dragon king is a double metaphor for an event that is both extremely large in size or impact and born of unique origins relative to its peers. DK events are generated by or correspond to mechanisms such as positive feedback, tipping points, bifurcations, and phase transitions, that tend to occur in nonlinear and complex systems, and serve to amplify DK events to extreme levels. By understanding and monitoring these dynamics, some predictability of such events may be obtained.
  The dragon king theory was developed by Didier Sornette, who hypothesizes that many crises are in fact DKs rather than black swans—i.e., they may be predictable to some degree. Given the importance of crises to the long-term organization of a variety of systems, the DK theory urges that special attention be given to the study and monitoring of extremes, and that a dynamic view be taken. From a scientific viewpoint, such extremes are interesting because they may reveal underlying, often hidden, organizing principles.
context_section_description: DKs are outliers by definition. However, when calling DKs outliers there is an important proviso: In standard statistics outliers are typically erroneous values and are discarded, or statistical methods are chosen that are somehow insensitive to outliers. Contrarily, DKs are outliers that are highly informative, and should be the focus of much statistical attention. Thus a first step is identifying DKs in historical data. Existing tests are either based on the asymptotic properties of the empirical distribution function (EDF) or on an assumption about the underlying cumulative distribution function (CDF) of the data.
  It turns out that testing for outliers relative to an exponential distribution is very general. The latter follows from the Pickands–Balkema–de Haan theorem of extreme value theory which states that a wide range of distributions asymptotically (above high thresholds) have exponential or power law tails. As an aside, this is one explanation why power law tails are so common when studying extremes. To finish the point, since the natural logarithm of a power law tail is exponential, one can take the logarithm of power law data and then test for outliers relative to an exponential tail. There are many test statistics and techniques for testing for outliers in an exponential sample. An inward test sequentially tests the largest point, then the second largest, and so on, until the first test that is not rejected (i.e., the null hypothesis that the point is not an outlier is not rejected). The number of rejected tests identifies the number of outliers. For instance, where  is the sorted sample, the inward robust test uses the test statistic  where r is the point being tested , and  where m is the pre-specified maximum number of outliers. At each step the p-value for the test statistic must be computed and, if lower than some level, the test rejected. This test has many desirable properties: It does not require that the number of outliers be specified, it is not prone to under (masking) and over (swamping) estimation of the number outliers, it is easy to implement, and the test is independent of the value of the parameter of the exponential tail.
caption: Probability density function schematic where the dragon king regime is represented by a lump of mass deep in the tail
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