Malcolm Rivera
Extreme value theory, or extreme value analysis (EVA), is a valuable tool for predicting the probability of rare and extreme events in various fields, such as finance, environmental science, and engineering. The challenge lies in defining what constitutes an extreme event, particularly when dealing with multiple variables. To address this, two primary methods are employed in EVA: the first involves identifying annual or block maxima (or minima), while the second focuses on extracting peak values that surpass a predetermined threshold. After selecting an extreme value distribution, such as Gumbel, Fréchet, or Weibull, the distribution is fitted to the data by estimating parameters that align with the observed values. This enables the prediction of the likelihood of extreme events beyond the scope of the original data. The accuracy of EVA can be enhanced through methods like VWLS, which minimize the variance of ranked observations, yielding improved estimates for small data sets.
| Characteristics | Values |
|---|---|
| Definition | Extreme value theory or extreme value analysis (EVA) is the study of extremes in statistical distributions. |
| Applications | Used in hydrology to estimate the probability of an unusually large flooding event, such as the 100-year flood. |
| Used by | Coastal engineers to estimate the 50-year wave and design structures accordingly. |
| Two main approaches | Deriving block maxima (minima) series as a preliminary step and extracting peak values from a continuous record. |
| Data Preprocessing | Identify and extract extreme events by selecting a threshold above which events are considered extreme. |
| Model Fitting | Model the extreme events using extreme value distributions (EVDs): Gumbel, Fréchet, and Weibull distributions. |
| GEV Distribution | The generalized extreme value (GEV) distribution is a better fit than individual Gumbel, Fréchet, and Weibull models, especially in hydrology. |
| POT Method | Used when dealing with exceedances of a non-random threshold; applicable in structural engineering and finance. |
| Improved Methods | VWLS method improves accuracy by minimizing variance of order-ranked observations and applying least squares fitting. |
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What You'll Learn
Extreme value distributions
Extreme value theory, also known as extreme value analysis (EVA), is a statistical theory that deals with extreme deviations from the median of probability distributions. It is widely applied in various fields, including structural engineering, finance, economics, and earth sciences. The primary goal of EVA is to predict the probability distribution of extreme events.
There are two main approaches to practical extreme value analysis. The first approach involves deriving block maxima or minima series. This often includes generating an annual maxima series (AMS) by extracting the annual extreme values. The second approach is based on identifying peak values from a continuous record that exceed a certain threshold. This threshold can be determined using methods like the peak-over-threshold (POT) method.
Once the extreme events are identified, the next step is to model them using extreme value distributions. Extreme value theory provides three main types of distributions: the Gumbel distribution, the Fréchet distribution, and the Weibull distribution. These distributions are suitable for modelling maximum or minimum values, depending on whether upper or lower extremes are of interest. The Gumbel distribution, for example, is commonly used in hydrology to analyse monthly and annual maximum values of daily rainfall and river discharge volumes.
The generalized extreme value (GEV) distribution is a family of continuous probability distributions that includes the Gumbel, Fréchet, and Weibull cases. The GEV distribution is often preferred as it provides a better fit than the individual Gumbel, Fréchet, and Weibull models. This is because it avoids assuming a lower bound on the distribution, which is a requirement of the Fréchet distribution. However, the GEV distribution has been found to lead to undefined means and variances in some cases, making reliable data analysis challenging.
In conclusion, extreme value distributions are a crucial component of extreme value analysis. By selecting an appropriate distribution and fitting it to the data, EVA enables the estimation of the probability of extreme events beyond the observed data range. This makes EVA a valuable tool for gaining insights into rare and significant events across various domains.
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Univariate vs multivariate cases
When considering what constitutes a good extreme value analysis (EVA) fit, it is important to distinguish between univariate and multivariate cases.
In the univariate case, the model is well-defined, and the three parameters of the GEV distribution can be obtained by fitting the distribution to the data. This makes it relatively straightforward to identify the most extreme event by taking the maximum or minimum of the observations. For example, in hydrology, the Gumbel distribution is commonly used to analyse monthly and annual maximum values of daily rainfall and river discharge volumes. The Gumbel distribution is also applied to drought analysis.
However, in the multivariate case, additional issues arise. One key problem is that there is no unambiguous way to specify what constitutes an extreme event. While it is possible to order a set of real-valued numbers, there is no natural ordering for a set of vectors. This makes it more challenging to identify the most extreme event in a multivariate setting. Furthermore, the multivariate model contains not only unknown parameters but also a function whose exact form is not prescribed by the theory, adding complexity to the analysis.
Another distinction between univariate and multivariate EVA is the range of interest for the extreme value distributions. In the univariate case, the three main distributions are Gumbel, Fréchet, and Weibull, each with distinct characteristics. Gumbel is unlimited, Fréchet has a lower limit, and reversed Weibull has an upper limit. In the multivariate case, the appropriate distribution may depend on the specific application and the nature of the data being analysed.
Despite the challenges in the multivariate case, recent advancements have been made in constructing estimators that obey the constraints imposed by the theory. For example, bivariate extreme value theory has been successfully applied to ocean research, demonstrating the potential for meaningful analysis in the multivariate domain.
In conclusion, while the univariate EVA is more straightforward due to the well-defined model and clear identification of extreme events, the multivariate EVA offers valuable insights into complex phenomena. The choice between univariate and multivariate EVA depends on the specific problem at hand, the nature of the data, and the availability of appropriate tools and methodologies.
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Threshold determination
Understanding Extreme Value Theory
Extreme value theory, also known as extreme value analysis (EVA), focuses on the extremes in statistical distributions. It aims to identify and analyse rare and significant events that deviate from the median of probability distributions. The theory provides a framework for understanding and modelling these extremes, which can have a substantial impact on various domains.
Selecting a Threshold Method
- The POT method, as described by Novak (2011), involves defining a non-random threshold. Events that exceed this threshold are considered extreme.
- The block maxima approach involves extracting the annual maxima or minima series from the data. This method is particularly useful when dealing with time series data, where identifying the highest or lowest values within specific periods is of interest.
Application of Threshold in EVA
Once the threshold is determined, it serves as a critical filter for identifying extreme events. Any data points that surpass the predefined threshold are labelled as extreme events and become the focus of further analysis. This step is crucial for distinguishing between regular occurrences and events that warrant special attention due to their rarity or magnitude.
Modelling Extreme Events
After identifying the extreme events, the next step is to model them using appropriate distributions. Extreme value theory provides three main types of distributions: the Gumbel distribution, the Fréchet distribution, and the Weibull distribution. These distributions are suitable for modelling maximum or minimum values, depending on whether the focus is on upper or lower extremes. The Gumbel distribution, for example, is commonly used in hydrology to analyse monthly and annual maximum values of daily rainfall and river discharge volumes.
Evaluating Model Fit
The selected extreme value distribution is then fitted to the extracted extreme events. This involves estimating the parameters of the distribution that best fit the data. The goodness of fit of the model can be evaluated using statistical techniques. A well-fitted model allows for reliable estimations of the probability of extreme events beyond the observed data range. This, in turn, enhances our understanding of rare events and supports informed decision-making in various fields.
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Model fitting
Extreme value theory (EVT) is a branch of statistics that deals with extreme deviations from the median of probability distributions. It involves two main steps: data preprocessing and model fitting. This answer will focus on the model-fitting step.
The Gumbel distribution is unlimited, the Fréchet distribution has a lower limit, and the reversed Weibull distribution has an upper limit. The generalized extreme value (GEV) distribution is a combination of these three distributions and is often used as an approximation to model the minima or maxima of long sequences of random variables. The GEV distribution is particularly useful in hydrology, where it is applied to extreme events such as annual maximum one-day rainfalls and river discharges.
Once the extreme value distribution is selected, it can be fitted to the extracted extreme events from the time series data. This involves estimating the parameters of the distribution that best fit the data. The fitted distribution can then be used to estimate the probability of extreme events beyond the observed data range.
An improved method for extreme value analysis is the VWLS method, which is based on minimizing the variance of order-ranked observations plotted according to their true probability and applying the least squares fitting. This method provides better estimates for the extremes, especially for small data sets, and does not require any subjective methodological decisions from the user.
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Applications
Extreme value analysis (EVA) is a powerful tool with a wide range of applications across various domains. One of its key applications is in hydrology, where it is used to estimate the probability of rare and extreme events such as flooding. For instance, in the Netherlands, EVT has been crucial in determining the required height of dikes to protect the country from flooding, given the low probability of such events occurring. Similarly, coastal engineers use EVA to estimate the "50-year wave" and design breakwaters accordingly.
In finance, EVA is used to study the impact of large financial losses and their occurrence probabilities. It provides insights into price shocks and helps in risk management by modelling extreme events separately, which is challenging with normal distribution methods. Actuarial science is another field that benefits from EVA, as it helps in understanding extreme claim sizes.
Environmental science is a domain where EVA is applied to address concerns related to the environment. For example, it can be used to analyse air pollutant concentrations, contributing to improved risk management. Additionally, in ocean research, bivariate extreme value theory has been utilised.
Extreme value analysis also has applications in engineering, specifically in structural engineering and geological engineering. It can aid in designing structures that can withstand extreme events, such as estimating the height and number of floors in skyscrapers. Furthermore, EVA is useful in traffic prediction, helping to optimise transportation systems and manage traffic flow, especially during peak or unusual conditions.
In addition to the above, EVA has been applied in various other contexts, such as modelling athletic records in sports and determining the limits of human lifespan using data on ages at death.
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Frequently asked questions
Extreme value analysis (EVA) is a tool used to gain insights into rare and extreme events that may have a significant impact on various domains, such as finance, environmental science, and engineering. It involves identifying and extracting extreme events from time series data, selecting an appropriate extreme value distribution, and then fitting that distribution to the extracted data.
Good fits for extreme value analysis include the use of the generalized extreme value (GEV) distribution, which is often used to model the minima or maxima of long sequences of random variables. The GEV distribution includes the Gumbel, Fréchet, and Weibull distributions as special cases, and is suitable for situations where the lower bound is unknown. Another example is the VWLS method, which improves the accuracy of flood frequency analysis by minimizing the variance of order-ranked observations.
Extreme value analysis has a wide range of applications, including hydrology, where it can be used to estimate the probability of extreme flooding events or to analyze monthly and annual maximum values of daily rainfall and river discharge. It is also used in structural engineering, finance, economics, earth sciences, traffic prediction, and coastal engineering for designing structures that can withstand extreme events such as large waves.
