Statistical Extremes and Applications

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Extreme value theory provides the statistical framework to make inferences about the probability of very rare or extreme events. The GEV distribution is parameterized with a shape paramter, location parameter and scale parameter. Based on the extreme value theorem the GEV distribution is the limit distribution of properly normalized maxima of a sequence of independent and identically distributed random variables. Thus, the GEV distribution is used as an approximation to model the maxima of long finite sequences of random variables.

The shape and location parameter can take on any real value. The resulting probability distribution function PDF for two category of shape parameter i. Application of GEV distribution Return value calculation : Based on the extreme value theory that derives the GEV distribution, we can fit a sample of extremes to the GEV distribution to obtain the parameters that best explains the probability distribution of the extremes. The modeling approach we introduce here is an important step in this direction, although such a complete analysis is beyond the scope of the present work.

Inferring extreme behavior for random variables in a spatial setting is still an active area of research see, e. The present study differs from these previous works in that, here, we characterize the spatial nature of a variable WmSh conditioned on its exhibiting extreme conditions somewhere in the field. Despite the fact that large regions may not experience extreme conditions at all, overall, a summary measure across the field is used to recognize that at least one or more regions experience particularly high values.

Because these are large-scale variables available on a grid everywhere in the region, we are not concerned with spatial interpolation. To this end, we employ the conditional extreme value model introduced by Heffernan and Tawn , which allows for modeling the distribution function of one or more variables, conditional on another variable's being extreme. They showed that for a wide class of extreme dependence structures, the joint df, conditional on one variable's exceeding a high threshold, results in a dependence model with a specific structure.

When modeling several variables conditional on the same variable, simultaneously, any dependence among these variables is also established. To identify extreme conditions, we consider the overall field energy, rather than focusing on extreme values at individual locations. The approach is to condition on a univariate variable that measures the overall spatial energy of the processes at each point in time. There are many choices for such a measure, such as the field mean, sum, or a specific quantile. Several such measures were investigated, and here, we report our findings for the upper quartile, q75, as our measure of choice, which consistently yields physically meaningful results.

In this way, it is possible to model the spatial distribution of the large-scale indicators given an extreme amount of spatial energy in the field regardless of whether every grid point in space is extreme or not. The objective is focused on the description of the field overall, its typical spatial coherence, relation to large-scale processes, and ultimately how variability and eventually future change in the large scale influences the spatio-temporal variability in severe weather indicators at finer scales.


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A variation of the previous approach is also explored. In particular, WmSh is conditioned on extreme river flow from three rivers in northwestern Tennessee in order to qualitatively study the spatial patterns of WmSh when river flows become dangerously high. A further analysis not described here investigated river flow conditioned on extreme values of WmSh at nearby grid cells. The association was found to be positive but weak. Univariate EVA is concerned with the distribution of intense events that have a very low probability of occurrence.

Theoretical results provide justification for using the generalized extreme value df for modeling maxima of a process taken over long blocks e. A Poisson process characterization for extreme events ties both of these approaches together and provides a mechanism for studying both the frequency and intensity of extreme events. It is possible to determine sources of variability for extremes of a random variable through covariate modeling within the parameters of the extreme value distributions e.

Incorporating covariates in this manner addresses how the distribution of extremes of one variable varies according to the covariates, be they extreme or not, and statistical tests, such as the likelihood ratio test, are available to test the significance of inclusion of the covariate s of interest in a similar manner as is performed in linear regression see, e. Heuristically, the previous probability is approximately equivalent to the GP df for a sufficiently large threshold, u assuming it is non-degenerate in the limit , which is given by.

For the present work, we will assume that Y follows a standard exponential df i. Before discussing the conditional model, it is helpful to review the copula dependence model formulation, which models the dependence of random variables on a common marginal df.


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That is,. The conditional model framework employed here is that introduced by Heffernan and Tawn hereafter, HT HT found that, for a wide class of copula dependence models, using Gumbel margins for X and Y , that the forms for a Y and b Y fell into the simple class. For negatively associated X and Y , another more complicated form was found; however, here, we follow Keef et al. Because these two parameters are not independent of each other, interpretation of their values is not straightforward.

No simple closed-form expression exists for G. From Equations 2 and 3 , we have that. Heffernan and Resnick, The proposed estimation method from HT is semi-parametric and involves several steps:. Transform each variable in order that they each follow a marginal standard Gumbel or, e. Estimate the parameters of the parametric model conditional on large values of the conditioning variable. Information about G e.

Back transformation can be used to put these estimates onto the original scale. HT suggest using a hybrid, semi-parametric, model for step 1 of the following form that accounts for both the extreme and non-extreme values cf. Coles and Tawn, , In this study, we transform the variables using the Laplace transformation in step 2 following Keef et al. To counteract the inherent estimation bias from this approach, Keef et al. From these estimates in step 3, HT obtain new estimates from which simulations from are obtained. In this step, it is important to keep the dependence structure inherent between the variates by keeping them together when performing the resampling of the residual vectors Z.

Uncertainty inference is carried out presently through bootstrap sampling in order to incorporate the uncertainty at each stage of the estimation procedure. We follow the strategy suggested by HT, the key component of which is the data generation step, which must be implemented carefully in order to maintain both the marginal and dependence features of the multivariate data. In particular,. Next, a nonparametric bootstrap is obtained by sampling with replacement from the transformed data. The marginal values of this bootstrap sample are rearranged in order to preserve the associations between the ranked values in each component by replacing the ordered samples with ordered samples of the same size from the standardized Laplace df.

The resulting sample is back transformed to the original marginal scales.

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Statistics of Extremes: Theory and Applications

In order to maintain relationships between the X variates, the nonparametric bootstrap samples are kept together. That is, for each variate X i , if the fourth entry for variate i is selected, then the fourth entry for all other X j is also selected. Model fit can be diagnosed by plotting the residual terms as well as from the fitted model against the quantiles of the conditioning variable. A loess curve is also graphed. Any trend in these graphs suggests a violation in the assumption of independence between Z and the conditioning variable cf. Equations 2 and 4. Further diagnostics include plotting the original data on the original scale against the fitted quantiles of the conditional df.

A good model fit should have good agreement in their quantiles conditioned on high values of the conditioning variable.

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Note that this approach differs from that of incorporating covariates into the parameters of a univariate extreme value df in that a distribution for values of one variate is conditioned on only the extreme values of another variable. Therefore, the dependence is on the processes themselves rather than indirectly through distributional parameters. The conditional EVA carried out in this study uses the R R Development Core Team, package texmex Southworth and Heffernan, , which allows for the constrained estimation of the dependence parameters with the Laplace transformation on the marginal variables.

For additional helpful references on the HT conditional modeling approach, see Heffernan and Resnick , Keef , Keef et al. The original output employed here contained six hourly values every day for 42 years — Because the interest is in extremes, the daily maximum for each variable at each grid point is first taken to reduce the number of data points, which tend to occur at the same time every day, yielding a total of 15, non-missing data points at each grid location.

Top left: map of the three River measuring locations in Tennessee at Port Royal and near Kingston Springs and Lobelville, respectively. Dotted rectangle shows the areal coverage of a reanalysis grid point 1. Gray lines indicate the full available record, and black lines indicate the record available coincidentally with the WmSh reanalysis data used in the analyses in Section 4.

Dashed horizontal lines are the 90th percentile for each river for data points shown in black. Following previous studies cf. Brooks et al. Note that while this product appears to discriminate severe weather environments fairly well cf. Figure 1 , it is possible for one of the variables to dominate the product so that both are not concurrently extreme. One might add complexity see, e.

However, we note that even with such added complexity, an exact correspondence between high values and severe weather does not necessarily exist, and yet, in certain places, the occurrence of high values of just one of these variables can be important. Therefore, in the present work, we simply use WmSh without any additional requirements. In order to investigate the full behavior of WmSh and how different processes can potentially contribute to the spatial extent of extreme weather, we stratify our dataset by four seasons defined here as: winter DJF , spring MAM , summer JJA , and fall SON.

This approach of segmenting the data can also be used to compare results for different periods. Here, we segment the data into three periods —, —, and — , reflecting important changes in the quality of input data used in the reanalysis. The final short segment covers a period over which changes in large-scale climate have become quite noticeable. However, any other segmentation is possible, and future work will investigate how the spatio-temporal structures have changed and will change in the future based on climate model projections.

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To explore connections on the impact level, we also investigate daily mean stream flow discharge cfs from three rivers in Tennessee Figure 2 : The Red River at Port Royal top right , the Harpeth River near Kingston Springs, bottom left , and the Buffalo River near Lobelville bottom right. The full record for the Red River is from 1 August through 28 September All data were retrieved on 29 September They largely overlap with the reanalyis data with the Red River series exhibiting some missing values in the common years.

Extreme statistics: theory and applications Part I

The EVA approach of HT is well suited to handle the data properties considered here: in particular, WmSh shares some properties in common with precipitation fields in that there are often many grid points and days with zero WmSh. Further, some grid points tend to have considerably larger values of WmSh than others e. The conditional EVA approach allows for these discrepancies among the different spatial locations because their entire distributions are modeled not just the extremes.

To confirm appropriateness of the assumptions of independence between Z and q75 when q75 is large and the appropriateness of the underlying model, different diagnostic plots for several locations were consulted not shown. Evaluations of these diagnostics suggest the appropriateness of the model fit e. Conditioning requires a variable, which one could consider as varying over space e. However, this is not necessary here because our interest is in the overall evolution of severe weather spatially over time.

For the present study, therefore, we consider a more general conditioning variable that summarizes the amount of WmSh energy over space for the entire field at each time point. Different options for such a measure of the energy could include, for example, an overall sum, a field average, or a high quantile of WmSh. The result is a univariate time series that captures a balanced summary of the intensity of the larger values of WmSh over space at each time point.

We also investigated other quantiles and overall integrated field measures sum, average as measures of field energy. Briefly, the field sum naturally yields higher values, but otherwise, spatial patterns and properties tend to mimic those for q For the present study, we focus on the method rather than the specific choice of field-energy summaries. Note that by conditioning on extreme values of a measure of field energy in this case q75 using the HT model, we are able to ascertain the spatial cohesiveness, locations, and patterns of the WmSh field at time points when WmSh is intense typically over one or more sub-regions.

Such an approach differs considerably from previous extreme value analyses for such indicators. For example, Heaton et al. The present analysis allows for a study of the spatial structure and its physical properties of WmSh when severe storm environments exist. It is important to understand that conditioning on extreme values of q75 can ignore important high values of WmSh if they are restricted to limited areas.

One way to account for such bias is to repeat the study for smaller regions or different time intervals, such as different decades, or as in this study, seasons. Such stratifications enable a more thorough analysis of underlying changes in intensity, spatial cohesiveness, spread, and the relationships with other large-scale processes. However, the focus, here, is on the most extreme activity over time, and other analyses may be necessary to fully capture all of the pertinent behavior of WmSh as it relates to severe weather environments.

Spatial extreme value analysis to project extremes of large-scale indicators for severe weather

Figure 3 shows summaries of the annual distribution of q Although variability is evident top , no obvious long term trend exists in these data. A marked seasonality middle and bottom is apparent, however, where the peak field energy clearly occurs in the spring and summer months. This observation agrees with the availability of convective energy that is maximized when cool air from mid-latitudes flows across increasingly hot and humid air from the south: a maximization of vertical instability W max and significant Shear toward summer.

The first step of the conditional EVA method, after deciding on a variable on which to condition the analyses, is to fit marginal dfs to each variable in order to apply the Laplace transformation. To do so, a hybrid of a GP df for values above a high threshold and the empirical df for values below the threshold is fit for q75 and WmSh cf. Reiss and Thomas ; MacDonald et al. It is found that the 90th percentile results in good fit diagnostics not shown for q75 and arbitrarily selected locations. Our key interest lies in the spatial distribution of WmSh given high field energy i.

That is, we do not obtain a closed-form parametric df, but simulated realizations from it. To investigate its properties, such as the mean and low and high quantiles, we graph the associated values spatially. Given that our interest is in extremes, it may at first appear strange to focus on low quantiles or even the mean of this df. However, because the df is conditional on high values of q75, such maps are informative of, in the case of low quantiles, a best-case scenario, or in the case of the mean, the expected spatial patterns and locations of high WmSh when it is extremely high over a large subregion of the domain.

Clearly, the model characterizes the observed behavior well. The advantage to having a valid model for WmSh conditioned on the extremes of q75 or other field-energy measure is that one can now make conjectures about field behavior under extreme return levels of q75; values that may not have been observed, but are reasonable to expect, as well as having a framework from which hypothesis testing can be conducted.

Note that the values in the bottom row of Figure 4 are generally considerably more extreme than those in the top row, and that they are maxima taken separately so that dependence is modeled over variables occurring at different time points. This illustrates a fundamental difference in approaches between the conditional approach adopted here and other spatial EVA models that utilize multivariate extreme value dfs or univariate extreme value dfs with spatially varying parameters.

The conditional EVA model employed here demonstrates how the spatial WmSh process behaves when an extreme amount of energy exists over a relatively large portion of the spatial domain, even if many of the grid points are not at all extreme. The mean simulated WmSh for this region may be associated with increased sea surface temperatures in the most recent 5 years of the data record for the winter season.

High values are defined here as being above the 90th percentile taken over the season for — left column , — middle column , and — last column. Not surprisingly, the Spring season Figure 5 , second row is characterized by the emergence of thunderstorm activity in a clear band of extreme simulated WmSh stretching from about Nebraska and Iowa down to eastern Mexico and the central Gulf of Mexico, with the most intense values hovering over the eastern coast of Mexico.

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The upper branch of this overall region is sometimes referred to as tornado alley because it is a distinct region with frequent heavy thunderstorms and tornados in the spring and summer months. The large values on the east coast of Mexico are attributed to heating over dry land in this season that is associated with high CAPE not shown. Further north, the values also reflect the still strong shear from westerlies aloft, which represent the cold-season conditions that taper out or retract northward as the summer progresses.

Similar activity is found in Figure 5 for the summer third row , but noticeably lower WmSh is simulated south of the USA than in the spring months. The slim band of high simulated values in the central USA in the spring is considerably wider in summer, covering most of the plains and midwestern states in period 1 and stretching through to the southeastern states as well in the later periods. The most intense predictions, however, occur during period 2.

Results for fall are shown in Figure 5 bottom row , where a strong increase in simulated WmSh in period 3 over the first two periods is evident, particularly around the eastern border of Texas and the western border of Louisiana.

Sebastian Engelke

WmSh predictions also increase in the Gulf of Mexico and especially stretching south from this region out into the Atlantic Ocean for period 3. Of course, the most important question to be addressed from these analyses concerns how WmSh is changing over time. Is it becoming more intense in critical areas? Are high intensity values migrating in space? For the present study, these questions are addressed by investigating differences between the periods of interest.

Figure 6 shows results for the differences in the mean simulated WmSh based on this conditional EVA model for the winter season. The top two panels are the same as the top row and first two columns in Figure 5 but with a scale particular to only the first two winter periods. The bottom two panels show the difference between the two means. The significance test is made using a normal approximation interval with the variance obtained from the bootstrap procedure.

What can be gleaned from the differences is that the majority of the region is not showing statistically significant changes in WmSh for winter from the first to second periods. Although spatial correlation is taken into account in the significance testing, no attempt has been made to account for multiple testing issues, which could be handled, for example, using a false discovery method e. The top two panels are the same as the top row first two columns in Figure 5 but with a scale representing the range of data only for winter rather than for all seasons.

Bottom right panel is the same but only differences found to be statistically significant based on bootstrapping are displayed. A few areas have grid points with lower WmSh activity, such as the Oregon coast and the North Atlantic just east of the Canadian and New England coast, as well as one grid point in the Gulf of Mexico due south of the eastern most part of Louisiana.

These large-scale structures could be associated with variations in regional modes of variability typical to the climate system e. When accounting for statistically significant changes, this band is still apparent, only thinner. Close inspection of the upper two panels in the figure suggests that these changes may mostly be the result of a change in the location of severe WmSh activity, rather than an intensification. One could check this by first applying a technique such as image warping e.

Together, they indicate a westward shift of the severe weather activity. Same as Figure 6 but for fall instead of winter. An observation that can be made about the use of q75 as a conditioning variable within the conditional EVA approach is that lower WmSh predictions occur outside the regions where the bulk of the energy is apparent. That is, for the average simulated values from the conditional model over the entire record Figure 4 , bottom row , WmSh is very high in the tornado alley and hurricane prone regions, as well as slightly higher in the southern North Pacific Ocean. However, in other areas, the predictions are much lower e.

Similarly, in the winter, large regions of zero WmSh are simulated on average for most of the northern USA, as well as for pockets of regions in other areas depending on the period. In spring, the coastal areas of the northern Atlantic and again Baja California and extending southward, show average WmSh simulated to be essentially zero. These same areas exhibit values considerably larger for the summer, except for the most recent period.

Fall generally has lower WmSh, but large areas of zero WmSh are simulated in the most recent period. A more physically meaningful picture of WmSh is achieved from this perspective than from simply investigating the marginal GP df fits, or from the spatial EVA approaches that implicitly assume simultaneously extreme events. At the same time, the method provides a mechanism for making projections of probabilities for extreme events taking into account the spatial structure of the field.

Because WmSh is a large-scale indicator for severe weather, we investigate how extreme events may be related to WmSh at an impact-level variable; that is, a variable that occurs at a fine spatial scale where their extremes can have a major impact on society e. To that end, WmSh is modeled conditional on large values of stream flow from three rivers in northeast Tennessee. While it may seem strange to condition on high stream flow, rather than the other way around, we note that we are after the spatial distribution of WmSh over the region associated with such high stream flow episodes.

It is these patterns that are of interest. We choose Tennessee because of a heavy precipitation event that occurred in that area in and resulted in a devastating flood in Nashville, Tennessee see, e. Although our data record does not cover this exceptional event in , it is of interest to investigate the types of spatial patterns that relate WmSh to extreme river flows in this region, and if these analyses would yield similar results as the conditions that actually occurred in spring In conjunction with heavy precipitation, the events, which occurred from 1—2 May , included a series of strong thunderstorms with reports of 41 tornados, 57 severe winds, and 43 severe hail episodes, conditions generally closely associated with high WmSh see previous texts.

For each river, a threshold equal to the 99th percentile of river flow data is found to yield the best fit diagnostics not shown for fitting the marginal GP df the 90th percentile resulted in very poor diagnostics. The 90th percentile of WmSh is used as a selection of fit diagnostics not shown revealed that this value is adequate for the threshold for this variable. For the subsequent conditional EVA fits step 2 of the HT estimation method , the 99th percentile is used. Selected plots of Z from Equation 4 against high quantiles of the conditioning variables do not show any obvious trends indicating that the threshold choice is appropriate.

For the Red River at Port Royal, Tennessee, we analyze a subset of the data that are available for the same days as the WmSh reanalysis product black lines in Figure 2. Scatter plots not shown of WmSh against daily mean discharge flow for the Red River at Port Royal, Tennessee for nearby reanalysis grid points do not reveal any obvious association between the variables. Nevertheless, because of the rarity of extremes, by definition, it is of interest to determine if there is any dependence between the two when at least one of the two variables is extreme.

To gain an understanding of possible associations in their extremes, we concentrate on a couple of grid points from the WmSh reanalysis that are nearby the river locations. The conditional value of WmSh is greatest for the spring. Previous results from the conditional modeling approach highlight an advantage of the method over other spatial EV methods in that more complicated relationships between the extremes of variables can be identified and then estimated. In particular, for WmSh conditioned on high values of river flow in the selected Tennessee Rivers, physical characteristics of high WmSh advection from the south are evident coincidentally with high river flow in this region, whereas WmSh is considerably less extreme nearly everywhere else cf.

Figure 8. Top left panel is the simulated mean WmSh, top right is the simulated 5th percentile WmSh, lower left is the median simulated WmSh, and lower right is the 95th percentile of simulated WmSh. Figure 8 displays results of simulated WmSh given that the stream flow discharge is larger than its 99th percentile for the Red River at Port Royal, Tennessee. High simulated values of WmSh are evident near the measuring location of the river Figure 2 top left and to the south.

The distribution of WmSh conditional on high values of river flow is characteristic of advection from the south but also demonstrates that the dependence between the two variables is relatively weak. Results at the other two rivers not shown are analogous to the Red River. These analyses were also carried out by season, and similar results not shown are obtained, though it is clear that the strongest associations are evident in the spring and summer seasons.

Of course, as pointed out by a reviewer, it would be more interesting to study how good of a predictor the spatial patterns of WmSh are for extreme river flow. A full analyses of such predictability is beyond the scope of the present treatment. However, an important consideration concerns the prevalent values of WmSh.

Figure 9 shows empirical averages for WmSh during the record for the Red River in Tennessee, as well as under lower river flows. Further, considerable WmSh prevails elsewhere, where it typically does not when river flow is extreme. Top left is for all values of river flow, top right is the average conditional on river flow less than its 99th percentile, and bottom left is conditioned on river flow less than cfs. The conditional EVA approach introduced by Heffernan and Tawn , HT is applied to a large-scale indicator for severe weather, namely, the product of the vertical instability indicator W max and 0—6 km wind shear WmSh reanalysis data over North America by conditioning on large values of its 75th percentile across the grid at each time point.

It is found that physically meaningful patterns are discerned from the approach making it a useful new tool for analyzing extremes under a changing climate. We demonstrate how this type of analysis can be used to make inferences about future extremes when the character of the spatial processes is complex. Most previous spatial extreme analyses require simultaneous extremes to be valid; otherwise, the dependence is over extremes that occur at possibly different times of the year, etc. Some recent exceptions include Wadsworth and Tawn and Davison et al. The conditional approach taken here allows for studying the entire spatial distribution of WmSh under severe storm environments, and spatial correlation is taken into account in the significance testing; although not multiple testing issues, which could be handled, for example, using a false discovery method e.

The analysis here does not account for temporal dependence or structure, which may be important when making inferences for WmSh as a severe weather indicator. Such dependence could be accounted for by imposing covariates on the parameters in Equation 3 ; Keef et al.

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