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The theory of DK is related to concepts such as black swan theory , outliers , complex systems , nonlinear dynamics , power laws , extreme value theory , prediction , extreme risks , and risk management.

A black swan can be considered a metaphor for an event that is surprising to the observer , has a major effect, and, after being observed, is rationalized in hindsight.

The theory of black swans is epistemological , relating to the limited knowledge and understanding of the observer. The black swan concept is important and poses a valid criticism of people, firms, and societies that are irresponsible in the sense that they are overly confident in their ability to anticipate and manage risk.

However, claiming that extreme events are — in general — unpredictable may also lead to a lack of accountability in risk management roles.

In fact, it is known that in a wide range of physical systems that extreme events are predictable to some degree.

This is the domain of the dragon kings. Such events have been referred to as Grey Swans by Taleb.

A more rigorous distinction between black swans, grey swans, and dragon kings is difficult as black swans are not precisely defined in physical and mathematical terms.

However technical elaboration of concepts in the Black Swan book are elaborated in the Silent Risk document. An analysis of the precise definition of a black swan in a risk management context was written by Prof.

It is well known that many phenomena in both the natural and social sciences have power law statistics Pareto distribution [8] [9] [10].

Furthermore, from extreme value theory , it is known that a broad range of distributions the Frechet class have tails that are asymptotically power law.

This property implies that all events — both large and small — are generated by the same mechanism, and thus there will be no distinct precursors by which the largest events may be predicted.

A well known conceptual framework for events of this type is self-organized criticality. Such concepts are compatible with the theory of the black swan.

However Taleb has also stated that considering the power law as a model instead of a model with lighter tails e.

In a variety of studies it has been found that, despite the fact that a power law models the tail of the empirical distribution well, the largest events are significantly outlying i.

Such events are interpreted as dragon kings as they indicate a departure from the generic process underlying the power law. Examples of this include the largest radiation release events occurring in nuclear power plant accidents, the largest city agglomeration within the sample of cities in a country, the largest crashes in financial markets, and intraday wholesale electricity prices.

Physically speaking, dragon kings may be associated with the regime changes, bifurcations , and tipping points of complex out-of-equilibrium systems.

Many observers would be surprised by such a dramatic change of state. However, it is well known that in dynamic systems, there are many precursors as the system approaches the catastrophe.

Positive feedback is also a mechanism that can spawn dragon kings. For instance, in a stampede the number of cattle running increases the level of panic which causes more cattle to run, and so on.

In human dynamics such herding and mob behavior has also been observed in crowds, stock markets, and so on see herd behavior.

Dragon kings are also caused by attractor bubbling in coupled oscillator systems. These excursions form the dragon kings, as illustrated in the figure.

It is claimed that such models can describe many real phenomena such as earthquakes, brain activity, etc.

It could also be the case that dragon kings are created as a result of system control or intervention.

That is, trying to suppress the release of stress or death in dynamic complex systems may lead to an accumulation of stress or a maturation towards instability.

Such fires are inconvenient and thus we may wish that they are diligently extinguished. This leads to long periods without inconvenient fires, however, in the absence of fires, dead wood accumulates.

Once this accumulation reaches a critical point, and a fire starts, the fire becomes so large that it cannot be controlled — a singular event that could be considered to be a dragon king.

Other policies, such as doing nothing allowing for small fires to occur naturally , or performing strategic controlled burning , would avoid enormous fires by allowing for frequent small ones.

Another example is monetary policy. Quantitative easing programs and low interest rate policies are common, with the intention of avoiding recessions, promoting growth, etc.

However, such programs build instability by increasing income inequality, keeping weak firms alive, and inflating asset bubbles. DK are outliers by definition.

However, when calling DK 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, DK are outliers that are highly informative, and should be the focus of much statistical attention.

Thus a first step is identifying DK in historical data. Existing tests are either based on the asymptotic properties of the empirical distribution function EDF [13] 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.

The number of rejected tests identifies the 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.

Some examples of where dragon kings have been detected as outliers include: I financial crashes as measured by drawdowns , where the outliers correspond to terrorist attacks e.

III the largest city measured by the population in its agglomeration in the population of cities within a country, where the largest city plays a disproportionately important role in the dynamics of the country, and benefits from unique growth; and,.

How one models and predicts dragon kings depends on the underlying mechanism. However, the common approach will require continuous monitoring of the focal system and comparing measurements with a non-linear or complex dynamic model.

It has been proposed that the more homogeneous the system, and the stronger its interactions, the more predictable it will be.

For instance, in non-linear systems with phase transitions at a critical point, it is well known that a window of predictability occurs in the neighborhood of the critical point due to precursory signs: For the phenomena of unsustainable growth e.

In systems that are discrete scale invariant such a model is power law growth, decorated with a log-periodic function. This has been applied to many problems, [3] for instance: An interesting dynamic to consider, that may reveal the development of a block-buster success, is Epidemic phenomena: Given a model and data, one can obtain a statistical model estimate.

This model estimate can then be used to compute interesting quantities such as the conditional probability of the occurrence of a dragon king event in a future time interval, and the most probable occurrence time.

As a 'challenger', Jerome Miles had the option of choosing several celebrity teams, including those of Wu Chun and Zheng Shuang.

However, a quick turn by the latter meant that Dragon King didn't have any time to use either of its weapons. Getting back out to the middle of arena, the two machines spent several seconds driving in circles around each other, neither initially getting an advantage.

This stalemate lasted until ORBY Blade, now at full speed, managed to get in a massive hit onto the side of Dragon King that dislodged its entire track module.

Capable of driving without its left track, Dragon King made another assault at ORBY Blade, taking another hit that sent both robots flying in opposite directions.

By now the lost track was obviously causing Dragon King control issues, as it drove around in circles for a short while before bumping into a wall.

Managing to stabilize itself enough for a final attack, it took yet another hit from ORBY Blade head-on, but this seemed to have been too much for the machine as it didn't move afterwards, and was subsequently declared immobile, costing its chance to enter the competition.

They first did this with Knome at the first ever live event in , armed with a side-mounted axe to make the robot a thwackbot, as well as an active axe at the rear.

However, it lost to Stinger and W. The team competed in Series 1. Team Duct Tape then competed in three different seasons with Sublime.

Sublime's experience on television allowed it to skip the preliminary rounds in Season 3. Sublime was much more successful in Season 5.

Team Duct Tape also competed in both seasons of Robot Wars: For Season 1, the team competed with Unibite , an invertible robot armed with a horizontal saw blade and a rear lifter, but Unibite did not win its opening melee in the US Championship, and fell in the second round of the Annihilator.

Unibite returned for Season 2 as Unibite 2. The team also entered the Annihilator and the Tag Team Terror with a vertical spinner named Hyperactive , yet in spite of immobilizing Thor's Hammer in the first round of the Annihilator to collect a win, it lost its other two battles.

Miles also converted his Hyperactive design into a featherweight, completing two separate builds of the machine in and Red Devil defeated Wrecks in the first round, although this victory required little input from Red Devil, so it was given the third-lowest seed in the Round of 32, drawing it against the highly favored Witch Doctor.

In one of the biggest upsets of the series, Red Devil used its saw to cut through the batteries of Witch Doctor and flip its opponent over, allowing it to progress to the Round of

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As a 'challenger', it fought for its position in the competition during Episode 7, but lost to ORBY Blade , and did not join Wu Chun's team as was intended.

Much like Red Devil, it runs on tracks, attached to the robot be separate detachable pods, which could articulate up and down For weaponry, the robot is equipped with two yellow circular saws on moving arms, allowing them to be brought down onto the surface of other robots at an angle.

Additionally, the front of the robot is armed with a 'dragon head', serving as a wedge to hold other robots in place for attacks, in combination with the actuated crushing jaw atop it, giving the robot a pincer weapon.

However, the robot is weak to horizontal spinners and full-body spinners. As a 'challenger', Jerome Miles had the option of choosing several celebrity teams, including those of Wu Chun and Zheng Shuang.

However, a quick turn by the latter meant that Dragon King didn't have any time to use either of its weapons. Getting back out to the middle of arena, the two machines spent several seconds driving in circles around each other, neither initially getting an advantage.

This stalemate lasted until ORBY Blade, now at full speed, managed to get in a massive hit onto the side of Dragon King that dislodged its entire track module.

Capable of driving without its left track, Dragon King made another assault at ORBY Blade, taking another hit that sent both robots flying in opposite directions.

By now the lost track was obviously causing Dragon King control issues, as it drove around in circles for a short while before bumping into a wall.

Managing to stabilize itself enough for a final attack, it took yet another hit from ORBY Blade head-on, but this seemed to have been too much for the machine as it didn't move afterwards, and was subsequently declared immobile, costing its chance to enter the competition.

They first did this with Knome at the first ever live event in , armed with a side-mounted axe to make the robot a thwackbot, as well as an active axe at the rear.

However, it lost to Stinger and W. The team competed in Series 1. Team Duct Tape then competed in three different seasons with Sublime.

Sublime's experience on television allowed it to skip the preliminary rounds in Season 3. Sublime was much more successful in Season 5.

Team Duct Tape also competed in both seasons of Robot Wars: For Season 1, the team competed with Unibite , an invertible robot armed with a horizontal saw blade and a rear lifter, but Unibite did not win its opening melee in the US Championship, and fell in the second round of the Annihilator.

Unibite returned for Season 2 as Unibite 2. Dragon king DK is a double metaphor for an event that is both extremely large in size or impact a "king" and born of unique origins a "dragon" relative to its peers other events from the same system.

By understanding and monitoring these dynamics, some predictability of such events may be obtained. The theory has been developed by Didier Sornette , who hypothesizes that many of the crises that we face are in fact DK rather than black swans —i.

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. Practically speaking, one should ambitiously study extreme risks, but not forget that significant uncertainty will almost always be present, and should be rigorously considered in decisions regarding risk management and design.

The theory of DK is related to concepts such as black swan theory , outliers , complex systems , nonlinear dynamics , power laws , extreme value theory , prediction , extreme risks , and risk management.

A black swan can be considered a metaphor for an event that is surprising to the observer , has a major effect, and, after being observed, is rationalized in hindsight.

The theory of black swans is epistemological , relating to the limited knowledge and understanding of the observer. The black swan concept is important and poses a valid criticism of people, firms, and societies that are irresponsible in the sense that they are overly confident in their ability to anticipate and manage risk.

However, claiming that extreme events are — in general — unpredictable may also lead to a lack of accountability in risk management roles.

In fact, it is known that in a wide range of physical systems that extreme events are predictable to some degree. This is the domain of the dragon kings.

Such events have been referred to as Grey Swans by Taleb. A more rigorous distinction between black swans, grey swans, and dragon kings is difficult as black swans are not precisely defined in physical and mathematical terms.

However technical elaboration of concepts in the Black Swan book are elaborated in the Silent Risk document. An analysis of the precise definition of a black swan in a risk management context was written by Prof.

It is well known that many phenomena in both the natural and social sciences have power law statistics Pareto distribution [8] [9] [10]. Furthermore, from extreme value theory , it is known that a broad range of distributions the Frechet class have tails that are asymptotically power law.

This property implies that all events — both large and small — are generated by the same mechanism, and thus there will be no distinct precursors by which the largest events may be predicted.

A well known conceptual framework for events of this type is self-organized criticality. Such concepts are compatible with the theory of the black swan.

However Taleb has also stated that considering the power law as a model instead of a model with lighter tails e. In a variety of studies it has been found that, despite the fact that a power law models the tail of the empirical distribution well, the largest events are significantly outlying i.

Such events are interpreted as dragon kings as they indicate a departure from the generic process underlying the power law. Examples of this include the largest radiation release events occurring in nuclear power plant accidents, the largest city agglomeration within the sample of cities in a country, the largest crashes in financial markets, and intraday wholesale electricity prices.

Physically speaking, dragon kings may be associated with the regime changes, bifurcations , and tipping points of complex out-of-equilibrium systems.

Many observers would be surprised by such a dramatic change of state. However, it is well known that in dynamic systems, there are many precursors as the system approaches the catastrophe.

Positive feedback is also a mechanism that can spawn dragon kings. For instance, in a stampede the number of cattle running increases the level of panic which causes more cattle to run, and so on.

In human dynamics such herding and mob behavior has also been observed in crowds, stock markets, and so on see herd behavior.

Dragon kings are also caused by attractor bubbling in coupled oscillator systems. These excursions form the dragon kings, as illustrated in the figure.

It is claimed that such models can describe many real phenomena such as earthquakes, brain activity, etc. It could also be the case that dragon kings are created as a result of system control or intervention.

That is, trying to suppress the release of stress or death in dynamic complex systems may lead to an accumulation of stress or a maturation towards instability.

Such fires are inconvenient and thus we may wish that they are diligently extinguished. This leads to long periods without inconvenient fires, however, in the absence of fires, dead wood accumulates.

Once this accumulation reaches a critical point, and a fire starts, the fire becomes so large that it cannot be controlled — a singular event that could be considered to be a dragon king.

Other policies, such as doing nothing allowing for small fires to occur naturally , or performing strategic controlled burning , would avoid enormous fires by allowing for frequent small ones.

Another example is monetary policy. Quantitative easing programs and low interest rate policies are common, with the intention of avoiding recessions, promoting growth, etc.

However, such programs build instability by increasing income inequality, keeping weak firms alive, and inflating asset bubbles. DK are outliers by definition.

However, when calling DK 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, DK are outliers that are highly informative, and should be the focus of much statistical attention. Thus a first step is identifying DK in historical data.

Existing tests are either based on the asymptotic properties of the empirical distribution function EDF [13] 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.

The number of rejected tests identifies the 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.

Some examples of where dragon kings have been detected as outliers include: I financial crashes as measured by drawdowns , where the outliers correspond to terrorist attacks e.

III the largest city measured by the population in its agglomeration in the population of cities within a country, where the largest city plays a disproportionately important role in the dynamics of the country, and benefits from unique growth; and,.

How one models and predicts dragon kings depends on the underlying mechanism. However, the common approach will require continuous monitoring of the focal system and comparing measurements with a non-linear or complex dynamic model.

It has been proposed that the more homogeneous the system, and the stronger its interactions, the more predictable it will be.

For instance, in non-linear systems with phase transitions at a critical point, it is well known that a window of predictability occurs in the neighborhood of the critical point due to precursory signs:

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