Extreme Value Distributions Theory And Applications Pdf
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We further explored the daily number of emergency department visits in a network of 37 hospitals over —
- Extreme Value Theory and Applications
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- Generalized extreme value distribution
- Generalized extreme value distribution
By the extreme value theorem the GEV distribution is the only possible limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Despite this, the GEV distribution is often used as an approximation to model the maxima of long finite sequences of random variables.
Extreme Value Theory and Applications
We further explored the daily number of emergency department visits in a network of 37 hospitals over — Maxima of grouped consecutive observations were fitted to a generalized extreme value distribution. The distribution was used to estimate the probability of extreme values in specified time periods.
Over the past 10 years, the observed maximum increase in the daily number of visits from the same weekday between two consecutive weeks was We estimated at 0. The EVT method can be applied to various topics in epidemiology thus contributing to public health planning for extreme events. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Data Availability: All relevant data are within the paper and its Supporting Information files. A central question for resource planning in public health is to predict the likelihood that exceptional or extreme events will occur in the not too distant future [ 1 ]. Such events might be, for example, an unusual community epidemic, a major heat wave, or an accidental toxic exposure.
Statistical approaches focused on extreme values have shown promising results in forecasting unusual events in earth sciences, genetics and finance. For instance, Extreme Value Theory EVT was developed in the s [ 2 ] and has been used to predict the occurrence of events as varied as droughts and flooding [ 3 ] or financial crashes [ 4 ].
To our knowledge, applications of EVT in public health are scarce. In a first unpublished work we applied the method to predict extreme influenza mortality in the US [ 5 ].
EVT was also used to detect outliers here seen as extreme events in time series surveillance data, rather than estimate the risk of future extreme events [ 6 ]. More recently, a study applied EVT to predict monthly incidence of avian influenza cases [ 7 ].
The main goal of EVT is to assess, from a series of observations, the probability of events that are more extreme than those previously recorded. The height of dikes can be calculated from storm data collected for around years using EVT, so that the risk of flooding would be less than one every 10, years [ 8 ]. Similarly, the likelihood of epidemics of unusual sizes could be determined by applying EVT to past epidemic observations, which could then help planning resources for mitigating the burden of these epidemics.
The aim of this paper is to present how EVT can be applied in public health. We illustrate its use on two different applications—to predict extremes of annual seasonal influenza mortality or variations by weekday in daily number of emergency department visits. Data used in this paper were counts of deaths per age group and per month or counts of emergency visits per day or counts of population per age group. All data were received by the authors in de-identified form.
These data were strictly anonymous and did not require approval from an ethics committee. A classical method for modelling the extremes of a stationary time series is the method of block maxima, in which consecutive observations are grouped into non-overlapping blocks of length n , generating a series of m block maxima, M n,1 ,…, M n,m , say, to which the GEV distribution can be fitted for some large value of n.
The usual approach is to consider blocks of a given time length e. Once a GEV distribution is fitted to n empirical observations, it becomes possible 1 to estimate the probability of an event that has not been observed yet, e. This is an extreme quantile because only n observations are included in the block. The first estimate is simply given by the distribution function of the GEV. For blocks of one year, the level z p is expected to be exceeded on average once every years t p.
More precisely, z p is exceeded by the annual maximum in any particular year with probability p. The level z p can be expressed in terms of the GEV parameters:. We used the evd and extRemes packages in R v 3. We then calculated weekly age-standardized death rates using the French population structure as reference. The eight-week period coincides with the length of a typical influenza epidemic, which is estimated to last approximately 2 months on average. In our application, M n,1 ,…, M n,m stand for the maxima of n cPI observations within a respiratory year that is, from July to June to encompass annual influenza epidemics.
We obtained 32 annual maxima, denoted cPIM Fig 1. The highest maximum 12 deaths per , was observed during the — respiratory year.
Return level plots were then calculated for return periods up to 50 years Fig 2C. Note that the linear aspect of the plot was the consequence of the close-to-zero shape parameter. Finally, we computed the probabilities to exceed some cPIM values greater than the largest maxima ever observed. As an example, there is a 1. Thirty-seven hospitals participated to the network and reported, on a daily basis, the total number of visits in the emergency departments. Because of expected variations of the number of visits according to the day of the week, we considered weekly increments of emergency visits iEV , that is, the difference between the number of visits on the same weekday between two consecutive weeks S1 Fig.
The empirical distribution of iEVM had a mean of minimum 82; maximum visits. The return level plot showed here again a linear aspect due to the zero scale parameter S2C Fig. The monthly risk of an increase greater than in the number of emergency department visits between the same weekday from two consecutive weeks is estimated to be 0.
Using simple illustrative examples, we showed the applicability of EVT to epidemiologic data. A GEV distribution was fitted to block maxima and was used to calculate estimates of return levels and of risks of exceeding a defined threshold value over given time periods.
In this work, we assumed the stationarity of the underlying working time series. This was likely the case for the two applications presented: means and standard deviations calculated over moving windows of different lengths did not vary over the study periods and the autocorrelations coefficients for both time series decreased rapidly towards the null results not shown. Moreover, the model fits were good except for one outlier value of weekly increment of emergency department visits in summer Methods for dealing with non-stationary distributions of maxima have been suggested in EVT.
For other applications, it might be useful to consider a cyclical GEV model, that is a GEV model with time-varying location and scale parameters [ 2 , 3 , 5 ]. Yet, this method requires the estimation of at least two more parameters than the model presented in this paper: this might produce large confidence intervals due to the small numbers of observations available.
To improve forecasting with relatively few annual observations, one might leverage other available information that could represent covariates associated with the outcome [ 3 ] e.
While such refinements might improve the accuracy of extreme value estimates, they are beyond the scope of this study as the choice of the specific approach would depend on the intended use of the forecasts.
Return level estimates should be helpful in planning resource needs, much like the statistical rationale for building dikes in the Netherlands. In our illustrative application on emergency department visits, EVT can be useful to estimate the surge capacity of the institution. For example, one could recommend sizing complementary health care resources beds, staff on a value that might be exceeded once in the next ten years—in our case an increase of visits in the emergency rooms.
Taking the example of seasonal influenza epidemics, if one assumes that antivirals, vaccines or face-masks stockpiles should be amassed, they can easily be dimensioned using estimates of EVT analysis based on an annual risk of exceeding an a priori defined threshold of cumulative influenza incidence.
Other examples of potential applications include anticipating the impact of extreme environmental exposures such as heat waves, pollutants, radiations…. For these types of extreme events, other methods such as risk analysis or modeling should be used.
However, when data are available, we believe that extreme value theory offers a statistical rationale for public health planning of extreme events, and could be applied to a various range of topics in epidemiology. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Conclusion The EVT method can be applied to various topics in epidemiology thus contributing to public health planning for extreme events. Funding: The authors received no specific funding for this work.
Competing interests: The authors have declared that no competing interests. Introduction A central question for resource planning in public health is to predict the likelihood that exceptional or extreme events will occur in the not too distant future [ 1 ]. Materials and Methods Ethics statement Data used in this paper were counts of deaths per age group and per month or counts of emergency visits per day or counts of population per age group. Download: PPT.
Fig 1. Fig 2. Discussion Using simple illustrative examples, we showed the applicability of EVT to epidemiologic data. Supporting Information.
S1 Fig. Weekly increments of emergency visits — in Paris. S2 Fig. S1 File. Pneumonia and Influenza data. S2 File. Emergency visits data. Contains date and number of emergency visits. References 1. Khan AS, Lurie N. Health security in building on preparedness knowledge for emerging health threats.
Coles S. An introduction to statistical modeling of extreme values. Springer-Verlag; Statistics of extremes in hydrology. Adv Water Resour. View Article Google Scholar 4.
Modelling extremal events for insurance and finance. Lee HC, Wackernagel H. An extreme value theory approach for the early detection of time clusters. A simulation-based assessment and an illustration to the surveillance of Salmonella. Stat Med. Using extreme value theory approaches to forecast the probability of outbreak of highly pathogenic influenza in Zhejiang, China.
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The Gumbel distribution, named after one of the pioneer scientists in practical applications of the Extreme Value Theory (EVT), the German mathematician Emil.
Generalized extreme value distribution
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Generalized extreme value distribution
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