Extreme value theory

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, economics, 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.

Data analysis

Two main approaches exist for practical extreme value analysis.

The first method relies on deriving block maxima (minima) series as a preliminary step. In many situations it is customary and convenient to extract the annual maxima (minima), generating an annual maxima series (AMS).

The second method relies on extracting, from a continuous record, the peak values reached for any period during which values exceed a certain threshold (falls below a certain threshold). This method is generally referred to as the peak over threshold method (POT).[1]

For AMS data, the analysis may partly rely on the results of the Fisher–Tippett–Gnedenko theorem, leading to the generalized extreme value distribution being selected for fitting.[2][3] However, in practice, various procedures are applied to select between a wider range of distributions. The theorem here relates to the limiting distributions for the minimum or the maximum of a very large collection of independent random variables from the same distribution. Given that the number of relevant random events within a year may be rather limited, it is unsurprising that analyses of observed AMS data often lead to distributions other than the generalized extreme value distribution (GEVD) being selected.[4]

For POT data, the analysis may involve fitting two distributions: One for the number of events in a time period considered and a second for the size of the exceedances.

A common assumption for the first is the Poisson distribution, with the generalized Pareto distribution being used for the exceedances. A tail-fitting can be based on the Pickands–Balkema–de Haan theorem.[5][6]

Novak (2011) reserves the term "POT method" to the case where the threshold is non-random, and distinguishes it from the case where one deals with exceedances of a random threshold.[7]

Applications

Applications of extreme value theory include predicting the probability distribution of:

History

The field of extreme value theory was pioneered by L. Tippett (1902–1985). Tippett was employed by the British Cotton Industry Research Association, where he worked to make cotton thread stronger. In his studies, he realized that the strength of a thread was controlled by the strength of its weakest fibres. With the help of R.A. Fisher, Tippet obtained three asymptotic limits describing the distributions of extremes assuming independent variables. E.J. Gumbel (1958)[22] codified this theory. These results can be extended to allow for slight correlations between variables, but the classical theory does not extend to strong correlations of the order of the variance. One universality class of particular interest is that of log-correlated fields, where the correlations decay logarithmically with the distance.

Univariate theory

The theory for extreme values of a single variable is governed by the extreme value theorem, also called the Fisher–Tippett–Gnedenko theorem, which describes which of the three possible distributions for extreme values applies for a particular statistical variable ${\displaystyle \ X\ ,}$ which is summarized in this section.

Let ${\displaystyle \ X_{1},\ \dots \ X_{n}\ }$ be a sample of independent and identically distributed random variables with cumulative distribution function ${\displaystyle \ F\ }$ and let ${\displaystyle \ M_{n}=\max \ \{\ X_{1},\ \dots \ X_{n}\ \}\ }$ denote the sample maximum.

In theory, the exact distribution of the maximum can be derived:

{\displaystyle {\begin{aligned}{\boldsymbol {\operatorname {\mathcal {P}} }}\left\{\ M_{n}\leq z\ \right\}&={\boldsymbol {\operatorname {\mathcal {P}} }}\left\{\ X_{1}\leq z,\ \dots \ ,\ X_{n}\leq z\ \right\}\\&={\boldsymbol {\operatorname {\mathcal {P}} }}\left\{\ X_{1}\leq z\ \right\}\times \cdots \times {\boldsymbol {\operatorname {\mathcal {P}} }}\left\{\ X_{n}\leq z\ \right\}={\bigl (}\ F(z)\ {\bigr )}^{n}~.\end{aligned}}}

The value of the associated indicator function ${\displaystyle \ I_{n}={\boldsymbol {\operatorname {\mathcal {I}} }}\left[\ M_{n}>z\ \right]\ }$ is a Bernoulli process with a success probability ${\displaystyle \ p(z)=1-{\bigl (}\ F(z)\ {\bigr )}^{n}\ }$ that depends on the magnitude ${\displaystyle \ z\ }$ of the extreme event. The number of extreme events within ${\displaystyle \ n\ }$ trials thus follows a binomial distribution and the number of trials until an event occurs follows a geometric distribution with expected value and standard deviation of the same order ${\displaystyle \ {\boldsymbol {\operatorname {\mathcal {O}} }}\left({\tfrac {\ 1\ }{p(z)}}\right)~.}$

In practice, we might not have the distribution function ${\displaystyle \ F\ }$ but the Fisher–Tippett–Gnedenko theorem provides an asymptotic result. If there exist sequences of paired constants ${\displaystyle \ (a_{n},b_{n})\ ,}$ with ${\displaystyle \ a_{n}>0\ }$ and ${\displaystyle \ b_{n}\in \mathbb {R} \ ,}$ such that

${\displaystyle {\boldsymbol {\operatorname {\mathcal {P}} }}\left\{{\frac {\ M_{n}-b_{n}\ }{\ a_{n}\ }}\leq z\right\}\rightarrow G(z)}$

as ${\displaystyle \ n\rightarrow \infty \ }$ then

${\displaystyle G(z)\ \propto \ \exp \left[-{\bigl (}1+\gamma \cdot z{\bigr )}^{-{\frac {\ 1\ }{\gamma }}}\right]}$

where the parameter ${\displaystyle \ \gamma \ }$ depends on how steeply of the distribution's tail(s) diminish (called "ordinary" tail(s), "thin" tail(s), and "fat" tail(s), with the normal distribution put in the "thin" tailed group instead of "ordinary" for this context, at least). When normalized, ${\displaystyle \ G\ }$ belongs to one of the following non-degenerate distribution families:

Type 1: Gumbel distribution, for ${\displaystyle \ \gamma =0\ }$
${\displaystyle G(z)=\exp \left[-\exp \left(-{\tfrac {\ z\ -\ b\ }{a}}\right)\right]}$
when the distribution of ${\displaystyle \ M_{n}\ }$ has an "ordinary" exponentially diminishing tail.

Type 2: Fréchet distribution, for ${\displaystyle \ \gamma <0\ }$
${\displaystyle G(z)={\begin{cases}0\quad &z\leq b\\\exp \left[-\left({\tfrac {\ z\ -\ b\ }{a}}\right)^{-\left|\gamma \right|}\right]&z>b\end{cases}}}$
when the distribution of ${\displaystyle \ M_{n}\ }$ has a heavy tail (including polynomial decay).

Type 3: Weibull distribution, ${\displaystyle \ \gamma >0\ }$
${\displaystyle G(z)={\begin{cases}\exp \left[-\left(-{\tfrac {\ z\ -\ b\ }{a}}\right)^{\gamma }\right]&z for ${\displaystyle \ z\in \mathbb {R} \ }$
when the distribution of ${\displaystyle \ M_{n}\ }$ has a thin tail with finite upper bound.

Multivariate theory

Extreme value theory in more than one variable introduces additional issues that have to be addressed. One problem that arises is that one must specify what constitutes an extreme event.[23] Although this is straightforward in the univariate case, there is no unambiguous way to do this in the multivariate case. The fundamental problem is that although it is possible to order a set of real-valued numbers, there is no natural way to order a set of vectors.

As an example, in the univariate case, given a set of observations ${\displaystyle \ x_{i}\ }$ it is straightforward to find the most extreme event simply by taking the maximum (or minimum) of the observations. However, in the bivariate case, given a set of observations ${\displaystyle \ (x_{i},y_{i})\ }$, it is not immediately clear how to find the most extreme event. Suppose that one has measured the values ${\displaystyle \ (3,4)\ }$ at a specific time and the values ${\displaystyle \ (5,2)\ }$ at a later time. Which of these events would be considered more extreme? There is no universal answer to this question.

Another issue in the multivariate case is that the limiting model is not as fully prescribed as in the univariate case. In the univariate case, the model (GEV distribution) contains three parameters whose values are not predicted by the theory and must be obtained by fitting the distribution to the data. In the multivariate case, the model not only contains unknown parameters, but also a function whose exact form is not prescribed by the theory. However, this function must obey certain constraints.[24][25] It is not straightforward to devise estimators that obey such constraints though some have been recently constructed.[26][27][28]

As an example of an application, bivariate extreme value theory has been applied to ocean research.[23][29]

Non-stationary extremes

Statistical modeling for nonstationary time series was developed in the 1990s.[30] Methods for nonstationary multivariate extremes have been introduced more recently.[31] The latter can be used for tracking how the dependence between extreme values changes over time, or over another covariate.[32][33][34]

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