Distribution Of Extreme Events Research Articles

Estimating the distribution of extreme events

Modeling the distribution of extreme events such as major floods, severe rainfall, and high wind speeds can be challenging. Researchers in many fields commonly use the generalized extreme value (GEV) distribution to characterize how these events are distributed over time. However, for many types of extreme events, there are limited data available, making it difficult to fit the GEV distribution. To help deal with this problem, Ailliot et al. generalized two common methods (maximum likelihood and method of moments) for estimating the parameters and high quantiles of the GEV distribution. Using simple parameter constraints, the scientists developed efficient and numerically robust estimation procedures that are suitable for automatic batch processing of many small‐to medium‐sized data sets. They illustrate their methods by applying them to annual maximum rainfall data in New Zealand. (Water Resources Research, doi:10.1029/2010WR009417, 2011)

A note of caution when interpreting parameters of the distribution of excesses

In climatology and hydrology, univariate Extreme Value Theory has become a powerful tool to model the distribution of extreme events. The Generalized Pareto Distribution (GPD) is routinely applied to model excesses in space or time by letting the two GPD parameters depend on appropriate covariates. Two possible pitfalls of this strategy are the modeling and the interpretation of the scale and shape GPD parameters estimates which are often and incorrectly viewed as independent variables. In this note we first recall a statistical technique that makes the GPD estimates less correlated within a Maximum Likelihood (ML) estimation approach. In a second step we propose novel reparametrizations for two method-of-moments particularly popular in hydrology: the Probability Weighted Moment (PWM) method and its generalized version (GPWM). Finally these three inference methods (ML, PWM and GPWM) are compared and discussed with respect to the issue of correlations.

Estimation of the Degree of Extremality of High-Water Flows in Selected Rivers in Poland in 1971–2006

Abstract The paper presents the results of estimation and comparison of risk of extreme floods on rivers of various hydrological regime. The hypothesis that extreme events occur with the same frequency in all rivers was rejected. The limit between extreme and common floods on 30 rivers from different geographical regions of Poland was defined on the basis of standardized flow-duration curve in 1971-2006. These analyses resulted in designing five curve groups. Four measures of flood magnitude have been proposed. The time distribution of extreme events during the last decades was estimated for the most dynamic rivers.

Deducing the Implications of Jump Models for the Structure of Stock Market Crashes, Rallies, Jump Arrival Rates, and Extremes

This article studies the structure of stock market crashes, rallies, their jump arrival rates, and extremes. Large market moves are characterized in a pure-jump modeling framework. Based on both raw and devolatized returns, it is shown empirically that crashes are more severe in intensity than rallies, and have higher arrival rates. At the same time, both left-tail and right-tail extreme events conform with Fréchet limit laws. Pure-jump models which describe well the tail properties of market returns are identified via their Lévy measures. The distribution of extreme events implied by our model’s Lévy measure is closer to the actual realization of extremes than those of competing models. Finally, there is information content in the Lévy measure of pure-jump models for forward arrival rate of jumps.

Vulnerability assessment of Central-East Sardinia (Italy) to extreme rainfall events

Abstract. In Sardinia (Italy), the highest frequency of extreme events is recorded in the Central-East area (3–4 events per year). The presence of high and steep mountains near the sea on the central and south-eastern coast, causes an East-West precipitation gradient in autumn especially, due to hot and moist currents coming from Africa. Soil structure and utilization make this area highly vulnerable to flash flooding and landslides. The specific purpose of this work is to provide a description of the heavy rainfall phenomenon on a statistical basis. The analysis mainly focuses on i) the existence of trends in heavy rainfall and ii) the characterization of the distribution of extreme events. First, to study possible trends in extreme events a few indices have been analyzed by the linear regression test. The analysis has been carried out at annual and seasonal scales. Then, extreme values analysis has been carried out by fitting a Generalized Pareto Distribution (GPD) to the data. As far as trends are concerned, different results are obtained at the two temporal scales: significant trends are obtained at the seasonal scale which are masked at the annual scale. By combining trend analysis and GPD analysis, the vulnerability of the study area to the occurrence of heavy rainfall has been characterized. Therefore, this work might support the improvement of land use planning and the application of suitable prevention systems. Future work will consider the extension of the analysis to all Sardinia and the application of statistical methods taking into account the spatial correlation of extreme events.

Deducing the Implications of Jump Models for the Structure of Stock Market Crashes, Rallies, Jump Arrival Rates, and Extremes

This paper studies the structure of stock market crashes, rallies, their jump arrival rates, and extremes. Large market moves are characterized in a pure-jump modeling framework. Based on both raw and devolatized returns, it is shown empirically that crashes are more severe in intensity than rallies, and have higher arrival rates. At the same time, both left-tail and right-tail extreme events conform with Frechet limit laws. Pure-jump models which describe well the tail properties of market returns are identified via their Levy measures. The distribution of extreme events implied by our model's Levy measure is closer to the actual realization of extremes than those of competing models. Finally, there is information content in the Levy measure of pure-jump models for forward arrival rate of jumps.

Return interval distribution of extreme events and long-term memory

The distribution of recurrence times or return intervals between extreme events is important to characterize and understand the behavior of physical systems and phenomena in many disciplines. It is well known that many physical processes in nature and society display long-range correlations. Hence, in the last few years, considerable research effort has been directed towards studying the distribution of return intervals for long-range correlated time series. Based on numerical simulations, it was shown that the return interval distributions are of stretched exponential type. In this paper, we obtain an analytical expression for the distribution of return intervals in long-range correlated time series which holds good when the average return intervals are large. We show that the distribution is actually a product of power law and a stretched exponential form. We also discuss the regimes of validity and perform detailed studies on how the return interval distribution depends on the threshold used to define extreme events.

On the tails of extreme event distributions in hydrology

The study of the tail behaviour of extreme event distributions is important in several fields such as hydrology, finance, and telecommunications. Based on two classifications and five graphical criteria, this paper presents a practical procedure to select the class of distributions that provides the best fit to a dataset, especially for the right tail (large extreme events). Some numerical illustrations show that, almost all graphical tools allow discriminating between the regularly varying class and the sub-exponential class of distributions and lead to coherent conclusions.

Return times for stochastic processes with power-law scaling

An analytical study of the return time distribution of extreme events for stochastic processes with power-law correlation has been carried out. The calculation is based on an epsilon expansion in the correlation exponent: C(t)=/t/-1+epsilon. The fixed point of the theory is associated with stretched exponential scaling of the distribution; analytical expressions have been provided in the preasymptotic regime. Also, the permanence time distribution appears to be characterized by stretched exponential scaling. The conditions for application of the theory to non-Gaussian processes have been analyzed and the relations with the issue of return times in the case of multifractal measures have been discussed.

MULTIFRACTAL SCALING OF DAILY RUNOFF TIME SERIES IN AGRICULTURAL WATERSHEDS1

Abstract: Multifractal scaling behavior of long‐term records of daily runoff time series in 32 subwatersheds covering a wide range of sizes was examined. These subwatersheds were associated with four agricultural watersheds with different climates and topography. The empirical moment scaling curves obtained using the trace moment method showed that the runoff time series exhibited a multifractal behavior, which was valid over a time scale range from one day to about three years. The multi‐fractal scaling of the runoff time series was well described by the Universal Multifractal Model. The spectral analysis (β < 1) and the order of fractional integration (H ⋍; 0) indicated that the runoff time series were conservative. The multifractal parameters, α (multifractal index) and C1 (co‐dimension), were reasonably close to each other for subwatersheds within each of the watersheds and were generally similar among the four watersheds. The α values of the four watersheds were 1.10 ± 0.13, 1.61 ± 0.06,1.61 ± 0.24, and 1.63 ± 0.19. The C1 values of four watersheds were 0.19 ± 0.01, 0.17 ± 0.01, 0.17 ± 0.04, and 0.11 ± 0.02. The multifractal analyses provided useful insight into the runoff time series, especially the occurrence and distribution of extreme events.

Characterization and interpretation of strongly nonlinear phenomena in fusion, space and astrophysical plasmas

Much of plasma behaviour is governed by multiple distinct nonlinear processes, operating on a wide range of lengthscales and timescales, that are coupled together in innumerable feedback loops. Capturing and quantifying this nonlinear behaviour is crucial at all levels of description, ranging from individual events to global phenomenology. In recent years, a range of techniques derived from complex systems science has been applied successfully to nonlinear plasma datasets. The present paper reviews several of these techniques in the context of applications spanning fusion, space, solar and astrophysical plasmas. Topics include non-Gaussian probability density functions, notably extreme event distributions in fusion and astrophysics and power law distributions in the solar context; differencing and rescaling of fluctuation data, which has yielded information on the number of dominant plasma turbulent processes, and the spatiotemporal ranges over which they operate, in plasmas ranging from microquasar accretion discs to L-mode and dithering H-mode fusion plasmas in the MAST tokamak; quantitative measures of mutual information content and pattern repetition between causally linked but spatiotemporally separated nonlinear events in solar wind and magnetospheric plasmas; global statistics of full-disc solar irradiance; and ELMing, considered as a sequence of pulsed events, in H-mode fusion plasmas in the JET tokamak. These developments in nonlinear plasma data characterization provide fresh additional insights into the underlying plasma physics. They also provide new opportunities for comparing models with data, and with each other, and open avenues for the development of a more rigorous predictive capability in this field.

SCALING PROPERTIES OF EPIDEMIOLOGICAL TIME SERIES

The statistical analysis monthly Hungarian (1967-2001) and Austrian (1989-2003) rabies and Hungarian bovine tuberculosis (1950-1979) cases were statistically analyzed. There were eradication campaigns against both diseases, which gave us the unique opportunity to observe how the statistical properties of the time series changed due to the eradication program. The fluctuations around the trend were got by removing the 12 month moving averages from the time series. In order to characterize the structure of the contacts among the individuals exposed to the diseases and the underlying processes governing the behavior of the epidemics, the fluctuations were analyzed before and after eradication separately. It turned out that the tails of the complementary cumulative distribution functions differ in the rabies cases, and do not differ in the case of tuberculosis. In each case the tail of the distribution follows an Inverse Power Law (IPL) function and describes the distribution of extreme events. It is possible to make conclusions about the dimensions of the processes from the fitted exponents. These dimensions are not related necessarily to the spatial dimensions, but to the possible connections. The knowledge about the distributions give us the opportunity to asses the risk of epidemic outbreaks.

Analysis of the effect of immunization in rabies time series

The authors analysed time series of rabies cases diagnosed in Hungary between 1967 and 2001. In Transdanubia (West Hungary), an oral immunization program started in 1992 and in East Hungary in 2001. Both long term and seasonal trends were identified in the time series of rabies cases. In order to characterize the underlying processes governing the behaviour of the epidemic, the fluctuations around the trend were analysed before and after immunization separately. It turned out that the tail of the complementary cumulative distribution functions differ. The tail of the distribution follows an inverse power law (IPL) function and describes the distribution of extreme events. The significant difference in the IPL exponents before and after immunization can be explained by the theory of Highly Optimized Tolerance (HOT).

Accounting for Extreme Events in the Economic Assessment of Climate Change

Extreme events are one of the main channels through which climate and socio-economic systems interact. It is likely that climate change will modify their probability distributions and their consequences. The long-term growth models used in climate change assessments, however, cannot capture the effects of short-term shocks; they thus model extreme events in a very crude manner. To assess the importance of this limitation, a non-equilibrium dynamic model (NEDyM) is used to model the macroeconomic consequences of extreme events. Its conclusions are the following: (i) Dynamic processes multiply the extreme event direct costs by a factor 20; half of this increase comes from short-term processes; (ii) A possible modication of the extreme event distribution due to climate change can be responsible for significant GDP losses; (iii) The production losses caused by extreme events depend, with strong non-linearity, both on the changes in the extreme distribution and on the ability to fund the rehabilitation after each disaster. These conclusions illustrate that the economic assessment of climate change does not only depend on beliefs on climate change but also on beliefs on the economy. Moreover, they suggest that averaging short-term processes like extreme events over the five- or ten-year time step of a classical long-term growth model can lead to inaccurately low assessments of the climate change damages.

Observations and Regional Climate Model Simulations of Heavy Precipitation Events and Seasonal Anomalies: A Comparison

A regional climate model simulation of the period of 1979–88 over the contiguous United States, driven by lateral boundary conditions from the National Centers for Environmental Prediction–National Center for Atmospheric Research reanalysis, was analyzed to assess the ability of the model to simulate heavy precipitation events and seasonal precipitation anomalies. Heavy events were defined by precipitation totals that exceed the threshold value for a specified return period and duration. The model magnitudes of the thresholds for 1-day heavy precipitation events were in good agreement with observed thresholds for much of the central United States. Model thresholds were greater than observed for the eastern and intermountain western portions of the region and were smaller than observed for the lower Mississippi River basin. For 7-day events, model thresholds were in good agreement with observed thresholds for the eastern United States and Great Plains, were less than observed for the most of the Mississippi River valley, and were greater than observed for the intermountain western region. The interannual variability in frequency of heavy events in the model simulation exhibited similar behavior to that of the observed variability in the South, Southwest, West, and North-Central study regions. The agreement was poorer for the Midwest and Northeast, although the magnitude of variability was similar for both model and observations. There was good agreement between the model and observational data in the seasonal distribution of extreme events for the West and North-Central study regions; in the Southwest, Midwest, and Northeast, there were general similarities but some differences in the details of the distributions. The most notable differences occurred for the southern Gulf Coast region, for which the model produced a summer peak that is not present in the observational data. There was not a very high correlation in the timing of individual heavy events between the model and observations, reflecting differences between model and observations in the speed and path of many of the synoptic-scale events triggering the precipitation.

Regional frequency analysis methods for evaluating changes in hydrologic extremes

A common assumption in frequency analysis is that hydrologic extremes (floods or heavy precipitation) are generated by a random process. This implies that natural climatic variability does not change the distribution of extreme events. A regional frequency analysis approach is proposed to test the hypothesis of randomness over secular timescales. Observed regional occurrences of extreme events are compared to those from a random process. Significant departures may indicate nonrandomness due to climatic variability. Application of the approach to a region in the Southern Plains indicates nonrandomness in annual maximum precipitation.

In search of a gust definition

We propose a simple gust definition based on the theory of excursions by Rice (1944 and 1945). We discuss the relation to the distribution of extreme events and demonstrate theoretically and experimentally that the most probable extreme event is very close to being identical to the gust according to our definition. We demonstrate how it is possible to predict the gust on the basis of the measured mean wind and variance rather than rely on actually measured extreme excursions. Our gust definition also allows us to predict the average duration of a gust.

Climatological analysis of rainfall for the wet Pampa and northwest of Buenos Aires province

A climatological analysis of rainfall, particularly for the NW of the province of Buenos Aires, was carried out using data series from 40 locations extending over a period of more than 80 years. The annual rainfall distribution has shown a high degree of variability over this century and the series present general positive trends. Climatic change has produced a net shift towards the SW of the border between the dry and the humid Pampa and has had a very important impact on agricultural production throughout the region. The spatial distribution of main spectral peaks for different frequency bands was analysed and a yearly areal distribution of extreme events (wet and dry) was determined for the present century.