Estimation of Extreme Values by the Average Conditional Exceedance Rate Method

Abstract

This paper details a method for extreme value prediction on the basis of a sampled time series. The method is specifically designed to account for statistical dependence between the sampled data points in a precise manner. In fact, if properly used, the new method will provide statistical estimates of the exact extreme value distribution provided by the data in most cases of practical interest. It avoids the problem of having to decluster the data to ensure independence, which is a requisite component in the application of, for example, the standard peaks-over-threshold method. The proposed method also targets the use of subasymptotic data to improve prediction accuracy. The method will be demonstrated by application to both synthetic and real data. From a practical point of view, it seems to perform better than the POT and block extremes methods, and, with an appropriate modification, it is directly applicable to nonstationary time series.

1. Introduction

Extreme value statistics, even in applications, are generally based on asymptotic results. This is done either by assuming that the epochal extremes, for example, yearly extreme wind speeds at a given location, are distributed according to the generalized (asymptotic) extreme value distribution with unknown parameters to be estimated on the basis of the observed data [1, 2]. Or it is assumed that the exceedances above high thresholds follow a generalized (asymptotic) Pareto distribution with parameters that are estimated from the data [1–4]. The major problem with both of these approaches is that the asymptotic extreme value theory itself cannot be used in practice to decide to what extent it is applicable for the observed data. And since the statistical tests to decide this issue are rarely precise enough to completely settle this problem, the assumption that a specific asymptotic extreme value distribution is the appropriate distribution for the observed data is based more or less on faith or convenience.

On the other hand, one can reasonably assume that in most cases long time series obtained from practical measurements do contain values that are large enough to provide useful information about extreme events that are truly asymptotic. This cannot be strictly proved in general, of course, but the accumulated experience indicates that asymptotic extreme value distributions do provide reasonable, if not always very accurate, predictions when based on measured data. This is amply documented in the vast literature on the subject, and good references to this literature are [2, 5, 6]. In an effort to improve on the current situation, we have tried to develop an approach to the extreme value prediction problem that is less restrictive and more flexible than the ones based on asymptotic theory. The approach is based on two separate components which are designed to improve on two important aspects of extreme value prediction based on observed data. The first component has the capability to accurately capture and display the effect of statistical dependence in the data, which opens for the opportunity of using all the available data in the analysis. The second component is then constructed so as to make it possible to incorporate to a certain extent also the subasymptotic part of the data into the estimation of extreme values, which is of some importance for accurate estimation. We have used the proposed method on a wide variety of estimation problems, and our experience is that it represents a very powerful addition to the toolbox of methods for extreme value estimation. Needless to say, what is presented in this paper is by no means considered a closed chapter. It is a novel method, and it is to be expected that several aspects of the proposed approach will see significant improvements.

2. Cascade of Conditioning Approximations

In this section, a sequence of nonparametric distribution functions will be constructed that converges to the exact extreme value distribution for the time series considered. This constitutes the core of the proposed approach.

Consider a stochastic process 𝑍(𝑡), which has been observed over a time interval, (0,𝑇) say. Assume that values 𝑋₁,…,𝑋_𝑁, which have been derived from the observed process, are allocated to the discrete times 𝑡₁,…,𝑡_𝑁 in (0,𝑇). This could be simply the observed values of 𝑍(𝑡) at each 𝑡_𝑗, 𝑗=1,…,𝑁, or it could be average values or peak values over smaller time intervals centered at the 𝑡_𝑗’s. Our goal in this paper is to accurately determine the distribution function of the extreme value 𝑀_𝑁=max{𝑋_𝑗;𝑗=1,…,𝑁}. Specifically, we want to estimate 𝑃(𝜂)=Prob(𝑀_𝑁≤𝜂) accurately for large values of 𝜂. An underlying premise for the development in this paper is that a rational approach to the study of the extreme values of the sampled time series is to consider exceedances of the individual random variables 𝑋_𝑗 above given thresholds, as in classical extreme value theory. The alternative approach of considering the exceedances by upcrossing of given thresholds by a continuous stochastic process has been developed in [7, 8] along lines similar to that adopted here. The approach taken in the present paper seems to be the appropriate way to deal with the recorded data time series of, for example, the hourly or daily largest wind speeds observed at a given location.

From the definition of 𝑃(𝜂) it follows that 𝑃𝜂=Prob𝑀_𝑁≤𝜂=Prob𝑋_𝑁≤𝜂,…,𝑋₁𝑋≤𝜂=Prob_𝑁≤𝜂∣𝑋_𝑁−1≤𝜂,…,𝑋₁𝑋≤𝜂⋅Prob_𝑁−1≤𝜂,…,𝑋₁=≤𝜂_𝑁𝑗=2Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,…,𝑋₁𝑋≤𝜂⋅Prob₁≤𝜂. (1)

In general, the variables 𝑋_𝑗 are statistically dependent. Hence, instead of assuming that all the 𝑋_𝑗 are statistically independent, which leads to the classical approximation 𝑃𝜂≈𝑃₁𝜂∶=^𝑁_𝑗=1Prob𝑋_𝑗≤𝜂, (2) where ∶= means “by definition”, the following one-step memory approximation will, to a certain extent, account for the dependence between the 𝑋_𝑗’s, Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,…,𝑋₁𝑋≤𝜂≈Prob_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂, (3) for 2≤𝑗≤𝑁. With this approximation, it is obtained that 𝑃𝜂≈𝑃₂𝜂∶=^𝑁_𝑗=2Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂Prob𝑋₁≤𝜂. (4) By conditioning on one more data point, the one-step memory approximation is extended to Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,…,𝑋₁𝑋≤𝜂≈Prob_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,𝑋_𝑗−2≤𝜂, (5) where 3≤𝑗≤𝑁, which leads to the approximation 𝑃𝜂≈𝑃₃𝜂∶=^𝑁_𝑗=3Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,𝑋_𝑗−2𝑋≤𝜂⋅Prob₂≤𝜂∣𝑋₁≤𝜂Prob𝑋₁≤𝜂. (6) For a general 𝑘, 2≤𝑘≤𝑁, it is obtained that 𝑃𝜂≈𝑃_𝑘𝜂∶=^𝑁_𝑗=𝑘Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂,…,𝑋_{𝑗−𝑘+1}⋅≤𝜂_{𝑘−1𝑗=2}Prob𝑋_𝑗≤𝜂∣𝑋_𝑗−1≤𝜂…,𝑋₁𝑋≤𝜂⋅Prob₁≤𝜂, (7) where 𝑃(𝜂)=𝑃_𝑁(𝜂).

It should be noted that the one-step memory approximation adopted above is not a Markov chain approximation [9–11], nor do the 𝑘-step memory approximations lead to 𝑘th-order Markov chains [12, 13]. An effort to relinquish the Markov chain assumption to obtain an approximate distribution of clusters of extremes is reported in [14].

It is of interest to have a closer look at the values for 𝑃(𝜂) obtained by using (7) as compared to (2). Now, (2) can be rewritten in the form 𝑃𝜂≈𝑃₁𝜂=^𝑁_𝑗=11−𝛼_1𝑗𝜂, (8) where 𝛼_1𝑗(𝜂)=Prob{𝑋_𝑗>𝜂}, 𝑗=1,…,𝑁. Then the approximation based on assuming independent data can be written as 𝑃𝜂≈𝐹₁𝜂∶=exp−^𝑁_𝑗=1𝛼_1𝑗𝜂,𝜂→∞. (9)

Alternatively, (7) can be expressed as, 𝑃𝜂≈𝑃_𝑘𝜂=^𝑁_𝑗=𝑘1−𝛼_𝑘𝑗𝜂^𝑘−1_𝑗=11−𝛼_𝑗𝑗𝜂, (10) where 𝛼_𝑘𝑗(𝜂)=Prob{𝑋_𝑗>𝜂|𝑋_𝑗−1≤𝜂,…,𝑋_{𝑗−𝑘+1}≤𝜂}, for 𝑗≥𝑘≥2, denotes the exceedance probability conditional on 𝑘−1 previous nonexceedances. From (10) it is now obtained that 𝑃𝜂≈𝐹_𝑘𝜂∶=exp−^𝑁_𝑗=𝑘𝛼_𝑘𝑗𝜂−^𝑘−1_𝑗=1𝛼_𝑗𝑗𝜂,𝜂→∞, (11) and 𝐹_𝑘(𝜂)→𝑃(𝜂) as 𝑘→𝑁 with 𝐹_𝑁(𝜂)=𝑃(𝜂) for 𝜂→∞.

For the cascade of approximations 𝐹_𝑘(𝜂) to have practical significance, it is implicitly assumed that there is a cut-off value 𝑘_𝑐 satisfying 𝑘_𝑐≪𝑁 such that effectively 𝐹_𝑘𝑐(𝜂)=𝐹_𝑁(𝜂). It may be noted that for 𝑘-dependent stationary data sequences, that is, for data where 𝑋_𝑖 and 𝑋_𝑗 are independent whenever |𝑗−𝑖|>𝑘, then 𝑃(𝜂)=𝑃_𝑘+1(𝜂) exactly, and, under rather mild conditions on the joint distributions of the data, lim_𝑁→∞𝑃₁(𝜂)=lim_𝑁→∞𝑃(𝜂) [15]. In fact, it can be shown that lim_𝑁→∞𝑃₁(𝜂)=lim_𝑁→∞𝑃(𝜂) is true for weaker conditions than 𝑘-dependence [16]. However, for finite values of 𝑁 the picture is much more complex, and purely asymptotic results should be used with some caution. Cartwright [17] used the notion of 𝑘-dependence to investigate the effect on extremes of correlation in sea wave data time series.

Returning to (11), extreme value prediction by the conditioning approach described above reduces to estimation of (combinations) of the 𝛼_𝑘𝑗(𝜂) functions. In accordance with the previous assumption about a cut-off value 𝑘_𝑐, for all 𝑘-values of interest, 𝑘≪𝑁, so that ∑^𝑘−1_𝑗=1𝛼_𝑗𝑗(𝜂) is effectively negligible compared to ∑^𝑁_𝑗=𝑘𝛼_𝑘𝑗(𝜂). Hence, for simplicity, the following approximation is adopted, which is applicable to both stationary and nonstationary data, 𝐹_𝑘𝜂=exp−^𝑁_𝑗=𝑘𝛼_𝑘𝑗𝜂,𝑘≥1. (12)

Going back to the definition of 𝛼_1𝑗(𝜂), it follows that ∑^𝑁_𝑗=1𝛼_1𝑗(𝜂) is equal to the expected number of exceedances of the threshold 𝜂 during the time interval (0,𝑇). Equation (9) therefore expresses the approximation that the stream of exceedance events constitute a (nonstationary) Poisson process. This opens for an understanding of (12) by interpreting the expressions ∑^𝑁_𝑗=𝑘𝛼_𝑘𝑗(𝜂) as the expected effective number of independent exceedance events provided by conditioning on 𝑘−1 previous observations.

3. Empirical Estimation of the Average Conditional Exceedance Rates

The concept of average conditional exceedance rate (ACER) of order 𝑘 is now introduced as follows: 𝜀_𝑘𝜂=1𝑁−𝑘+1^𝑁_𝑗=𝑘𝛼_𝑘𝑗𝜂,𝑘=1,2,…. (13) In general, this ACER function also depends on the number of data points 𝑁.

In practice, there are typically two scenarios for the underlying process 𝑍(𝑡). Either we may consider to be a stationary process, or, in fact, even an ergodic process. The alternative is to view 𝑍(𝑡) as a process that depends on certain parameters whose variation in time may be modelled as an ergodic process in its own right. For each set of values of the parameters, the premise is that 𝑍(𝑡) can then be modelled as an ergodic process. This would be the scenario that can be used to model long-term statistics [18, 19].

For both these scenarios, the empirical estimation of the ACER function 𝜀_𝑘(𝜂) proceeds in a completely analogous way by counting the total number of favourable incidents, that is, exceedances combined with the requisite number of preceding nonexceedances, for the total data time series and then finally dividing by 𝑁−𝑘+1≈𝑁. This can be shown to apply for the long-term situation.

A few more details on the numerical estimation of 𝜀_𝑘(𝜂) for 𝑘≥2 may be appropriate. We start by introducing the following random functions: 𝐴_𝑘𝑗𝜂=𝟏𝑋_𝑗>𝜂,𝑋_𝑗−1≤𝜂,…,𝑋_{𝑗−𝑘+1},𝐵≤𝜂𝑗=𝑘,…,𝑁,𝑘=2,3,…,_𝑘𝑗𝜂=𝟏𝑋_𝑗−1≤𝜂,…,𝑋_{𝑗−𝑘+1}≤𝜂,𝑗=𝑘,…,𝑁,𝑘=2,…, (14) where 𝟏{𝒜} denotes the indicator function of some event 𝒜. Then, 𝛼_𝑘𝑗𝜂=𝔼𝐴_𝑘𝑗𝜂𝔼𝐵_𝑘𝑗𝜂,𝑗=𝑘,…,𝑁,𝑘=2,…, (15) where 𝔼[⋅] denotes the expectation operator. Assuming an ergodic process, then obviously 𝜀_𝑘(𝜂)=𝛼_𝑘𝑘(𝜂)=⋯=𝛼_𝑘𝑁(𝜂), and by replacing ensemble means with corresponding time averages, it may be assumed that for the time series at hand 𝜀_𝑘𝜂=lim_𝑁→∞∑^𝑁_𝑗=𝑘𝑎_𝑘𝑗𝜂∑^𝑁_𝑗=𝑘𝑏_𝑘𝑗𝜂, (16) where 𝑎_𝑘𝑗(𝜂) and 𝑏_𝑘𝑗(𝜂) are the realized values of 𝐴_𝑘𝑗(𝜂) and 𝐵_𝑘𝑗(𝜂), respectively, for the observed time series.

Clearly, lim_𝜂→∞𝔼[𝐵_𝑘𝑗(𝜂)]=1. Hence, lim_𝜂→∞𝜀_𝑘̃(𝜂)/𝜀_𝑘(𝜂)=1, where 𝜀_𝑘̃𝜂=∑^𝑁_𝑗=𝑘𝔼𝐴_𝑘𝑗𝜂.𝑁−𝑘+1 (17) The advantage of using the modified ACER function 𝜀_𝑘̃(𝜂) for 𝑘≥2 is that it is easier to use for nonstationary or long-term statistics than 𝜀_𝑘(𝜂). Since our focus is on the values of the ACER functions at the extreme levels, we may use any function that provides correct predictions of the appropriate ACER function at these extreme levels.

To see why (17) may be applicable for nonstationary time series, it is recognized that 𝑃𝜂≈exp−^𝑁_𝑗=𝑘𝛼_𝑘𝑗𝜂=exp−^𝑁_𝑗=𝑘𝔼𝐴_𝑘𝑗𝜂𝔼𝐵_𝑘𝑗𝜂_𝜂→≃_∞exp−^𝑁_𝑗=𝑘𝔼𝐴_𝑘𝑗𝜂. (18) If the time series can be segmented into 𝐾 blocks, such that 𝔼[𝐴_𝑘𝑗(𝜂)] remains approximately constant within each block and such that ∑_{𝑗∈𝐶𝑖}𝔼[𝐴_𝑘𝑗(𝜂)]≈∑_{𝑗∈𝐶𝑖}𝑎_𝑘𝑗(𝜂) for a sufficient range of 𝜂-values, where 𝐶_𝑖 denotes the set of indices for block no. 𝑖,  𝑖=1,…,𝐾, then ∑^𝑁_𝑗=𝑘𝔼[𝐴_𝑘𝑗(𝜂)]≈∑^𝑁_𝑗=𝑘𝑎_𝑘𝑗(𝜂). Hence, 𝑃𝜂≈exp−(𝑁−𝑘+1)𝜀_𝑘̂𝜂, (19) where 𝜀_𝑘̂𝜂=1𝑁−𝑘+1^𝑁_𝑗=𝑘𝑎_𝑘𝑗𝜂. (20)

It is of interest to note what events are actually counted for the estimation of the various 𝜀_𝑘(𝜂), 𝑘≥2. Let us start with 𝜀₂(𝜂). It follows from the definition of 𝜀₂(𝜂) that 𝜀₂(𝜂)(𝑁−1) can be interpreted as the expected number of exceedances above the level 𝜂, satisfying the condition that an exceedance is counted only if it is immediately preceded by a non-exceedance. A reinterpretation of this is that 𝜀₂(𝜂)(𝑁−1) equals the average number of clumps of exceedances above 𝜂, for the realizations considered, where a clump of exceedances is defined as a maximum number of consecutive exceedances above 𝜂. In general, 𝜀_𝑘(𝜂)(𝑁−𝑘+1) then equals the average number of clumps of exceedances above 𝜂 separated by at least 𝑘−1 nonexceedances. If the time series analysed is obtained by extracting local peak values from a narrow band response process, it is interesting to note the similarity between the ACER approximations and the envelope approximations for extreme value prediction [7, 20]. For alternative statistical approaches to account for the effect of clustering on the extreme value distribution, the reader may consult [21–26]. In these works, the emphasis is on the notion of an extremal index, which characterizes the clumping or clustering tendency of the data and its effect on the extreme value distribution. In the ACER functions, these effects are automatically accounted for.

Now, let us look at the problem of estimating a confidence interval for 𝜀_𝑘(𝜂), assuming a stationary time series. If 𝑅 realizations of the requisite length of the time series is available, or, if one long realization can be segmented into 𝑅 subseries, then the sample standard deviation 𝑠_𝑘̂(𝜂) can be estimated by the standard formula 𝑠_𝑘̂𝜂²=1𝑅−1^𝑅_𝑟=1𝜀̂_𝑘^(𝑟)𝜂−𝜀_𝑘̂𝜂², (21) where 𝜀̂_𝑘^(𝑟)(𝜂) denotes the ACER function estimate from realization no. 𝑟, and 𝜀_𝑘̂(𝜂)=∑^𝑅_𝑟=1𝜀̂_𝑘^(𝑟)(𝜂)/𝑅.

Assuming that realizations are independent, for a suitable number 𝑅, for example, 𝑅≥20, (21) leads to a good approximation of the 95% confidence interval CI=(𝐶⁻(𝜂),𝐶⁺(𝜂)) for the value 𝜀_𝑘(𝜂), where 𝐶^±𝜂=𝜀_𝑘̂𝜂±1.96𝑠_𝑘̂𝜂√𝑅. (22)

Alternatively, and which also applies to the non-stationary case, it is consistent with the adopted approach to assume that the stream of conditional exceedances over a threshold 𝜂 constitute a Poisson process, possibly non-homogeneous. Hence, the variance of the estimator 𝐸_𝑘(𝜂) of 𝜀_𝑘̃(𝜂), where 𝐸_𝑘𝜂=∑^𝑁_𝑗=𝑘𝐴_𝑘𝑗𝜂𝑁−𝑘+1 (23) is Var[𝐸_𝑘(𝜂)]=𝜀_𝑘̃(𝜂). Therefore, for high levels 𝜂, the approximate limits of a 95% confidence interval of 𝜀_𝑘̃(𝜂), and also 𝜀_𝑘(𝜂), can be written as 𝐶^±𝜂=𝜀_𝑘̂𝜂1±1.96(𝑁−𝑘+1)𝜀_𝑘̂𝜂. (24)

4. Estimation of Extremes for the Asymptotic Gumbel Case

The second component of the approach to extreme value estimation presented in this paper was originally derived for a time series with an asymptotic extreme value distribution of the Gumbel type, compared with [27]. We have therefore chosen to highlight this case first, also because the extension of the asymptotic distribution to a parametric class of extreme value distribution tails that are capable of capturing to some extent subasymptotic behaviour is more transparent, and perhaps more obvious, for the Gumbel case. The reason behind the efforts to extend the extreme value distributions to the subasymptotic range is the fact that the ACER functions allow us to use not only asymptotic data, which is clearly an advantage since proving that observed extremes are truly asymptotic is really a nontrivial task.

The implication of the asymptotic distribution being of the Gumbel type on the possible subasymptotic functional forms of 𝜀_𝑘(𝜂) cannot easily be decided in any detail. However, using the asymptotic form as a guide, it is assumed that the behaviour of the mean exceedance rate in the tail is dominated by a function of the form exp{−𝑎(𝜂−𝑏)^𝑐} (𝜂≥𝜂₁≥𝑏), where 𝑎, 𝑏, and 𝑐 are suitable constants, and 𝜂₁ is an appropriately chosen tail marker. Hence, it will be assumed that, 𝜀_𝑘𝜂=𝑞_𝑘𝜂exp−𝑎_𝑘𝜂−𝑏_𝑘^𝑐𝑘,𝜂≥𝜂₁, (25) where the function 𝑞_𝑘(𝜂) is slowly varying, compared with the exponential function exp{−𝑎_𝑘(𝜂−𝑏_𝑘)^𝑐𝑘} and 𝑎_𝑘, 𝑏_𝑘, and 𝑐_𝑘 are suitable constants, that in general will be dependent on 𝑘. Note that the value 𝑐_𝑘=𝑞_𝑘(𝜂)=1 corresponds to the asymptotic Gumbel distribution, which is then a special case of the assumed tail behaviour.

From (25) it follows that 𝜀−loglog^𝑘𝜂𝑞_𝑘𝜂=−𝑐_𝑘log𝜂−𝑏_𝑘−log𝑎_𝑘. (26) Therefore, under the assumptions made, a plot of −log|log(𝜀_𝑘(𝜂)/𝑞_𝑘(𝜂))| versus log(𝜂−𝑏_𝑘) will exhibit a perfectly linear tail behaviour.

It is realized that if the function 𝑞_𝑘(𝜂) could be replaced by a constant value, say 𝑞_𝑘, one would immediately be in a position to apply a linear extrapolation strategy for deep tail prediction problems. In general, 𝑞_𝑘(𝜂) is not constant, but its variation in the tail region is often sufficiently slow to allow for its replacement by a constant, possibly by adjusting the tail marker 𝜂₁. The proposed statistical approach to the prediction of extreme values is therefore based on the assumption that we can write, 𝜀_𝑘𝜂=𝑞_𝑘exp−𝑎_𝑘𝜂−𝑏_𝑘^𝑐𝑘,𝜂≥𝜂₁, (27) where 𝑎_𝑘, 𝑏_𝑘, 𝑐_𝑘, and 𝑞_𝑘 are appropriately chosen constants. In a certain sense, this is a minimal class of parametric functions that can be used for this purpose which makes it possible to achieve three important goals. Firstly, the parametric class contains the asymptotic form given by 𝑐_𝑘=𝑞_𝑘=1 as a special case. Secondly, the class is flexible enough to capture, to a certain extent, subasymptotic behaviour of any extreme value distribution, that is, asymptotically Gumbel. Thirdly, the parametric functions agree with a wide range of known special cases, of which a very important example is the extreme value distribution for a regular stationary Gaussian process, which has 𝑐_𝑘=2.

The viability of this approach has been successfully demonstrated by the authors for mean up-crossing rate estimation for extreme value statistics of the response processes related to a wide range of different dynamical systems, compared with [7, 8].

As to the question of finding the parameters 𝑎, 𝑏, 𝑐, 𝑞 (the subscript 𝑘, if it applies, is suppressed), the adopted approach is to determine these parameters by minimizing the following mean square error function, with respect to all four arguments, 𝐹𝑎,𝑏,𝑐,𝑞=^𝐽_𝑗=1𝑤_𝑗log𝜀̂𝜂_𝑗−log𝑞+𝑎𝜂_𝑗−𝑏^𝑐2, (28) where 𝜂₁<⋯<𝜂_𝐽 denotes the levels where the ACER function has been estimated, 𝑤_𝑗 denotes a weight factor that puts more emphasis on the more reliably estimated 𝜀(̂𝜂_𝑗). The choice of weight factor is to some extent arbitrary. We have previously used 𝑤_𝑗=(log𝐶⁺(𝜂_𝑗)−log𝐶⁻(𝜂_𝑗))^−𝜃 with 𝜃=1 and 2, combined with a Levenberg-Marquardt least squares optimization method [28]. This has usually worked well provided reasonable and initial values for the parameters were chosen. Note that the form of 𝑤_𝑗 puts some restriction on the use of the data. Usually, there is a level 𝜂_𝑗 beyond which 𝑤_𝑗 is no longer defined, that is, 𝐶⁻(𝜂_𝑗) becomes negative. Hence, the summation in (28) has to stop before that happens. Also, the data should be preconditioned by establishing the tail marker 𝜂₁ based on inspection of the empirical ACER functions.

In general, to improve robustness of results, it is recommended to apply a nonlinearly constrained optimization [29]. The set of constraints is written as log𝑞−𝑎𝜂_𝑖−𝑏^𝑐≤0,0<𝑞<+∞,min_𝑗𝑋_𝑗<𝑏≤𝜂₁,0<𝑎<+∞,0<𝑐<5. (29) Here, the first nonlinear inequality constraint is evident, since under our assumption we have 𝜀_𝑘̂(𝜂_𝑖)=𝑞exp{−𝑎(𝜂_𝑖−𝑏)^𝑐}, and 𝜀_𝑘̂(𝜂_𝑖)<1 by definition.

A Note of Caution. When the parameter 𝑐 is equal to 1.0 or close to it, that is, the distribution is close to the Gumbel distribution, the optimization problem becomes ill-defined or close to ill-defined. It is seen that when 𝑐=1.0, there is an infinity of (𝑏,𝑞) values that gives exactly the same value of 𝐹(𝑎,𝑏,𝑐,𝑞). Hence, there is no well-defined optimum in parameter space. There are simply too many parameters. This problem is alleviated by fixing the 𝑞-value, and the obvious choice is 𝑞=1.

Although the Levenberg-Marquardt method generally works well with four or, when appropriate, three parameters, we have also developed a more direct and transparent optimization method for the problem at hand. It is realized by scrutinizing (28) that if 𝑏 and 𝑐 are fixed, the optimization problem reduces to a standard weighted linear regression problem. That is, with both 𝑏 and 𝑐 fixed, the optimal values of 𝑎 and log𝑞 are found using closed form weighted linear regression formulas in terms of 𝑤_𝑗, 𝑦_𝑗=log𝜀(𝜂_𝑗) and 𝑥_𝑗=(𝜂_𝑗−𝑏)^𝑐. In that light, it can also be concluded that the best linear unbiased estimators (BLUE) are obtained for 𝑤_𝑗=𝜎⁻²_𝑦𝑗, where 𝜎²_𝑦𝑗=Var[𝑦_𝑗] (empirical) [30, 31]. Unfortunately, this is not a very practical weight factor for the kind of problem we have here because the summation in (28) then typically would have to stop at undesirably small values of 𝜂_𝑗.

It is obtained that the optimal values of 𝑎 and 𝑞 are given by the relations 𝑎^∗()∑𝑏,𝑐=−^𝑁𝑗=1𝑤_𝑗𝑥_𝑗−𝑥𝑦_𝑗−𝑦∑^𝑁_𝑗=1𝑤_𝑗𝑥_𝑗−𝑥²,log𝑞^∗(𝑏,𝑐)=𝑦+𝑎^∗(𝑏,𝑐)𝑥, (30) where 𝑥=∑^𝑁_𝑗=1𝑤_𝑗𝑥_𝑗/∑^𝑁_𝑗=1𝑤_𝑗, with a similar definition of 𝑦.

To calculate the final optimal set of parameters, one may use the Levenberg-Marquardt method on the function 𝐹(𝑏,𝑐)=𝐹(𝑎^∗(𝑏,𝑐),𝑏,𝑐,𝑞^∗(𝑏,𝑐)) to find the optimal values 𝑏^∗ and 𝑐^∗, and then use (30) to calculate the corresponding 𝑎^∗ and 𝑞^∗.

For a simple construction of a confidence interval for the predicted, deep tail extreme value given by a particular ACER function as provided by the fitted parametric curve, the empirical confidence band is reanchored to the fitted curve by centering the individual confidence intervals CI_0.95 for the point estimates of the ACER function on the fitted curve. Under the premise that the specified class of parametric curves fully describes the behaviour of the ACER functions in the tail, parametric curves are fitted as described above to the boundaries of the reanchored confidence band. These curves are used to determine a first estimate of a 95% confidence interval of the predicted extreme value. To obtain a more precise estimate of the confidence interval, a bootstrapping method would be recommended. A comparison of estimated confidence intervals by both these methods will be presented in the section on extreme value prediction for synthetic data. As a final point, it has been observed that the predicted value is not very sensitive to the choice of 𝜂₁, provided it is chosen with some care. This property is easily recognized by looking at the way the optimized fitting is done. If the tail marker is in the appropriate domain of the ACER function, the optimal fitted curve does not change appreciably by moving the tail marker.

5. Estimation of Extremes for the General Case

For independent data in the general case, the ACER function 𝜀₁(𝜂) can be expressed asymptotically as 𝜀₁𝜂≃_𝜂→∞1+𝜉𝑎𝜂−𝑏^−1/𝜉, (31) where 𝑎>0, 𝑏, 𝜉 are constants. This follows from the explicit form of the so-called generalized extreme value (GEV) distribution Coles [1].

Again, the implication of this assumption on the possible subasymptotic functional forms of 𝜀_𝑘(𝜂) in the general case is not a trivial matter. The approach we have chosen is to assume that the class of parametric functions needed for the prediction of extreme values for the general case can be modelled on the relation between the Gumbel distribution and the general extreme value distribution. While the extension of the asymptotic Gumbel case to the proposed class of subasymptotic distributions was fairly transparent, this is not equally so for the general case. However, using a similar kind of approximation, the behaviour of the mean exceedance rate in the subasymptotic part of the tail is assumed to follow a function largely of the form [1+𝜉(𝑎(𝜂−𝑏)^𝑐)]^−1/𝜉 (𝜂≥𝜂₁≥𝑏), where 𝑎>0, 𝑏, 𝑐>0, and 𝜉>0 are suitable constants, and 𝜂₁ is an appropriately chosen tail level. Hence, it will be assumed that [32] 𝜀_𝑘𝜂=𝑞_𝑘𝜂1+𝜉_𝑘𝑎_𝑘𝜂−𝑏_𝑘^{𝑐𝑘−1/𝜉𝑘},𝜂≥𝜂₁, (32) where the function 𝑞_𝑘(𝜂) is weakly varying, compared with the function [1+𝜉_𝑘(𝑎_𝑘(𝜂−𝑏_𝑘)^𝑐𝑘)]^{−1/𝜉𝑘} and 𝑎_𝑘>0,  𝑏_𝑘, 𝑐_𝑘>0 and 𝜉_𝑘>0 are suitable constants, that in general will be dependent on 𝑘. Note that the values 𝑐_𝑘=1 and 𝑞_𝑘(𝜂)=1 corresponds to the asymptotic limit, which is then a special case of the general expression given in (25). Another method to account for subasymptotic effects has recently been proposed by Eastoe and Tawn [33], building on ideas developed by Tawn [34], Ledford and Tawn [35] and Heffernan and Tawn [36]. In this approach, the asymptotic form of the marginal distribution of exceedances is kept, but it is modified by a multiplicative factor accounting for the dependence structure of exceedances within a cluster.

An alternative form to (32) would be to assume that 𝜀_𝑘𝜂=1+𝜉_𝑘𝑎_𝑘𝜂−𝑏_𝑘^𝑐𝑘+𝑑_𝑘𝜂^{−1/𝜉𝑘},𝜂≥𝜂₁, (33) where the function 𝑑_𝑘(𝜂) is weakly varying compared with the function 𝑎_𝑘(𝜂−𝑏_𝑘)^𝑐𝑘. However, for estimation purposes, it turns out that the form given by (25) is preferable as it leads to simpler estimation procedures. This aspect will be discussed later in the paper.

For practical identification of the ACER functions given by (32), it is expedient to assume that the unknown function 𝑞_𝑘(𝜂) varies sufficiently slowly to be replaced by a constant. In general, 𝑞_𝑘(𝜂) is not constant, but its variation in the tail region is assumed to be sufficiently slow to allow for its replacement by a constant. Hence, as in the Gumbel case, it is in effect assumed that 𝑞_𝑘(𝜂) can be replaced by a constant for 𝜂≥𝜂₁, for an appropriate choice of tail marker 𝜂₁. For simplicity of notation, in the following we will suppress the index 𝑘 on the ACER functions, which will then be written as 𝜀𝜂=𝑞1+𝑎𝜂−𝑏^𝑐−𝛾,𝜂≥𝜂₁, (34) where 𝛾=1/𝜉, 𝑎=𝑎𝜉.

For the analysis of data, first the tail marker 𝜂₁ is provisionally identified from visual inspection of the log plot (𝜂,ln𝜀_𝑘̂(𝜂)). The value chosen for 𝜂₁ corresponds to the beginning of regular tail behaviour in a sense to be discussed below.

The optimization process to estimate the parameters is done relative to the log plot, as for the Gumbel case. The mean square error function to be minimized is in the general case written as 𝐹𝑎,𝑏,𝑐,𝑞,𝛾=^𝑁_𝑗=1𝑤_𝑗log𝜀̂𝜂_𝑗𝑎𝜂−log𝑞+𝛾log1+_𝑗−𝑏^𝑐2, (35) where 𝑤_𝑗 is a weight factor as previously defined.

An option for estimating the five parameters 𝑎, 𝑏, 𝑐, 𝑞, 𝛾 is again to use the Levenberg-Marquardt least squares optimization method, which can be simplified also in this case by observing that if 𝑎, 𝑏, and 𝑐 are fixed in (28), the optimization problem reduces to a standard weighted linear regression problem. That is, with 𝑎, 𝑏, and 𝑐 fixed, the optimal values of 𝛾 and log𝑞 are found using closed form weighted linear regression formulas in terms of 𝑤_𝑗, 𝑦_𝑗=log𝜀(̂𝜂_𝑗) and 𝑥_𝑗=1+𝑎(𝜂_𝑗−𝑏)^𝑐.

It is obtained that the optimal values of 𝛾 and log𝑞 are given by relations similar to (30). To calculate the final optimal set of parameters, the Levenberg-Marquardt method may then be used on the function 𝐹(𝑎,𝑏,𝑐)=𝐹(𝑎,𝑏,𝑐,𝑞^∗(𝑎,𝑏,𝑐),𝛾^∗(𝑎,𝑏,𝑐)) to find the optimal values 𝑎^∗, 𝑏^∗, and 𝑐^∗, and then the corresponding 𝛾^∗ and 𝑞^∗ can be calculated. The optimal values of the parameters may, for example, also be found by a sequential quadratic programming (SQP) method [37].

6. The Gumbel Method

To offer a comparison of the predictions obtained by the method proposed in this paper with those obtained by other methods, we will use the predictions given by the two methods that seem to be most favored by practitioners, the Gumbel method and the peaks-over-threshold (POT) method, provided, of course, that the correct asymptotic extreme value distribution is of the Gumbel type.

The Gumbel method is based on recording epochal extreme values and fitting these values to a corresponding Gumbel distribution [38]. By assuming that the recorded extreme value data are Gumbel distributed, then representing the obtained data set of extreme values as a Gumbel probability plot should ideally result in a straight line. In practice, one cannot expect this to happen, but on the premise that the data follow a Gumbel distribution, a straight line can be fitted to the data. Due to its simplicity, a popular method for fitting this straight line is the method of moments, which is also reasonably stable for limited sets of data. That is, writing the Gumbel distribution of the extreme value 𝑀_𝑁 as Prob𝑀_𝑁≤𝜂=exp−exp−𝑎𝜂−𝑏, (36) it is known that the parameters 𝑎 and 𝑏 are related to the mean value 𝑚_𝑀 and standard deviation 𝜎_𝑀 of 𝑀(𝑇) as follows: 𝑏=𝑚_𝑀−0.57722𝑎⁻¹ and 𝑎=1.28255/𝜎_𝑀 [39]. The estimates of 𝑚_𝑀 and 𝜎_𝑀 obtained from the available sample therefore provides estimates of 𝑎 and 𝑏, which leads to the fitted Gumbel distribution by the moment method.

Typically, a specified quantile value of the fitted Gumbel distribution is then extracted and used in a design consideration. To be specific, let us assume that the requested quantile value is the 100(1−𝛼)% fractile, where 𝛼 is usually a small number, for example, 𝛼=0.1. To quantify the uncertainty associated with the obtained 100(1−𝛼)% fractile value based on a sample of size 𝑁, the 95% confidence interval of this value is often used. A good estimate of this confidence interval can be obtained by using a parametric bootstrapping method [40, 41]. Note that the assumption that the initial 𝑁 extreme values are actually generated with good approximation from a Gumbel distribution cannot easily be verified with any accuracy in general, which is a drawback of this method. Compared with the POT method, the Gumbel method would also seem to use much less of the information available in the data. This may explain why the POT method has become increasingly popular over the past years, but the Gumbel method is still widely used in practice.

7. The Peaks-over-Threshold Method

7.1. The Generalized Pareto Distribution

The POT method for independent data is based on what is called the generalized Pareto (GP) distribution (defined below) in the following manner: it has been shown in [42] that asymptotically the excess values above a high level will follow a GP distribution if and only if the parent distribution belongs to the domain of attraction of one of the extreme value distributions. The assumption of a Poisson process model for the exceedance times combined with GP distributed excesses can be shown to lead to the generalized extreme value (GEV) distribution for corresponding extremes, see below. The expression for the GP distribution is 𝐺𝑦𝑦=𝐺𝑦;𝑎,𝑐=Prob𝑌≤𝑦=1−1+𝑐𝑎₊^−1/𝑐. (37) Here 𝑎>0 is a scale parameter and 𝑐(−∞<𝑐<∞) determines the shape of the distribution. (𝑧)₊=max(0,𝑧).

The asymptotic result referred to above implies that (37) can be used to represent the conditional cumulative distribution function of the excess 𝑌=𝑋−𝑢 of the observed variates 𝑋 over the threshold 𝑢, given that 𝑋>𝑢 for 𝑢 is sufficiently large [42]. The cases 𝑐>0, 𝑐=0, and 𝑐<0 correspond to Fréchet (Type II), Gumbel (Type I), and reverse Weibull (Type III) domains of attraction, respectively, compared with section below.

For 𝑐=0, which corresponds to the Gumbel extreme value distribution, the expression between the parentheses in (37) is understood in a limiting sense as the exponential distribution 𝐺𝑦−𝑦=𝐺𝑦;𝑎,0=exp𝑎. (38)

Since the recorded data in practice are rarely independent, a declustering technique is commonly used to filter the data to achieve approximate independence [1, 2].

7.2. Return Periods

The return period 𝑅 of a given value 𝑥_𝑅 of 𝑋 in terms of a specified length of time 𝜏, for example, a year, is defined as the inverse of the probability that the specified value will be exceeded in any time interval of length 𝜏. If 𝜆 denotes the mean exceedance rate of the threshold 𝑢 per length of time 𝜏 (i.e., the average number of data points above the threshold 𝑢 per 𝜏), then the return period 𝑅 of the value of 𝑋 corresponding to the level 𝑥_𝑅=𝑢+𝑦 is given by the relation 1𝑅=𝜆Prob𝑋>𝑥_𝑅=1.𝜆Prob𝑌>𝑦 (39) Hence, it follows that 1Prob𝑌≤𝑦=1−().𝜆𝑅 (40) Invoking (1) for 𝑐≠0 leads to the result 𝑥_𝑅𝑎()=𝑢−1−𝜆𝑅^𝑐𝑐. (41) Similarly, for 𝑐=0, it is found that, 𝑥_𝑅=𝑢+𝑎ln(𝜆𝑅), (42) where 𝑢 is the threshold used in the estimation of 𝑐 and 𝑎.

8. Extreme Value Prediction for Synthetic Data

In this section, we illustrate the performance of the ACER method and also the 95% CI estimation. We consider 20 years of synthetic wind speed data, amounting to 2000 data points, which is not much for detailed statistics. However, this case may represent a real situation when nothing but a limited data sample is available. In this case, it is crucial to provide extreme value estimates utilizing all data available. As we will see, the tail extrapolation technique proposed in this paper performs better than asymptotic methods such as POT or Gumbel.

The extreme value statistics will first be analyzed by application to synthetic data for which the exact extreme values can be calculated [43]. In particular, it is assumed that the underlying (normalized) stochastic process 𝑍(𝑡) is stationary and Gaussian with mean value zero and standard deviation equal to one. It is also assumed that the mean zero up-crossing rate 𝜈⁺(0) is such that the product 𝜈⁺(0)𝑇=10³, where 𝑇=1 year, which seems to be typical for the wind speed process. Using the Poisson assumption, the distribution of the yearly extreme value of 𝑍(𝑡) is then calculated by the formula 𝐹^1yr𝜂=exp−𝜈⁺𝜂𝑇=exp−10³−𝜂exp²2, (43) where 𝑇=1 year and 𝜈⁺(𝜂) is the mean up-crossing rate per year, 𝜂 is the scaled wind speed. The 100-year return period value 𝜂^100yr is then calculated from the relation 𝐹^1yr(𝜂^100yr)=1−1/100, which gives 𝜂^100yr=4.80.

The Monte Carlo simulated data to be used for the synthetic example are generated based on the observation that the peak events extracted from measurements of the wind speed process, are usually separated by 3-4 days. This is done to obtain approximately independent data, as required by the POT method. In accordance with this, peak event data are generated from the extreme value distribution 𝐹^3d𝜂−𝜂=exp−𝑞exp²2, (44) where 𝑞=𝜈⁺(0)𝑇=10, which corresponds to 𝑇=3.65 days, and 𝐹^1yr(𝜂)=(𝐹^3d(𝜂))¹⁰⁰.

Since the data points (i.e., 𝑇=3.65 days maxima) are independent, 𝜀_𝑘(𝜂) is independent of 𝑘. Therefore, we put 𝑘=1. Since we have 100 data from one year, the data amounts to 2000 data points. For estimation of a 95% confidence interval for each estimated value of the ACER function 𝜀₁(𝜂) for the chosen range of 𝜂-values, the required standard deviation in (22) was based on 20 estimates of the ACER function using the yearly data. This provided a 95% confidence band on the optimally fitted curve based on 2000 data. From these data, the predicted 100-year return level is obtained from 𝜀₁̂(𝜂^100yr)=10⁻⁴. A nonparametric bootstrapping method was also used to estimate a 95% confidence interval based on 1000 resamples of size 2000.

The POT prediction of the 100-year return level was based on using maximum likelihood estimates (MLE) of the parameters in (37) for a specific choice of threshold. The 95% confidence interval was obtained from the parametrically bootstrapped PDF of the POT estimate for the given threshold. A sample of 1000 data sets was used. One of the unfortunate features of the POT method is that the estimated 100 year value may vary significantly with the choice of threshold. So also for the synthetic data. We have followed the standard recommended procedures for identifying a suitable threshold [1].

Note that in spite of the fact that the true asymptotic distribution of exceedances is the exponential distribution in (38), the POT method used here is based on adopting (37). The reason is simply that this is the recommended procedure [1], which is somewhat unfortunate but understandable. The reason being that the GP distribution provides greater flexibility in terms of curve fitting. If the correct asymptotic distribution of exceedances had been used on this example, poor results for the estimated return period values would be obtained. The price to pay for using the GP distribution is that the estimated parameters may easily lead to an asymptotically inconsistent extreme value distribution.

The 100-year return level predicted by the Gumbel method was based on using the method of moments for parameter estimation on the sample of 20 yearly extremes. This choice of estimation method is due to the small sample of extreme values. The 95% confidence interval was obtained from the parametrically bootstrapped PDF of the Gumbel prediction. This was based on a sample of size 10,000 data sets of 20 yearly extremes. The results obtained by the method of moments were compared with the corresponding results obtained by using the maximum likelihood method. While there were individual differences, the overall picture was one of very good agreement.

In order to get an idea about the performance of the ACER, POT, and Gumbel methods, 100 independent 20-year MC simulations as discussed above were done. Table 1 compares predicted values and confidence intervals for a selection of 10 cases together with average values over the 100 simulated cases. It is seen that the average of the 100 predicted 100-year return levels is slightly better for the ACER method than for both the POT and the Gumbel methods. But more significantly, the range of predicted 100-year return levels by the ACER method is 4.34–5.36, while the same for the POT method is 4.19–5.87 and for the Gumbel method is 4.41–5.71. Hence, in this case the ACER method performs consistently better than both these methods. It is also observed from the estimated 95% confidence intervals that the ACER method, as implemented in this paper, provides slightly higher accuracy than the other two methods. Lastly, it is pointed out that the confidence intervals of the 100-year return level estimated by the ACER method obtained by either the simplified extrapolated confidence band approach or by nonparametric bootstrapping are very similar, except for a slight mean shift. As a final comparison, the 100 bootstrapped confidence intervals obtained for the ACER and Gumbel methods missed the target value three times, while for the POT method this number was 18.

Table 1 

100-year return level estimates and 95% CI (BCI = CI by bootstrap) for A = ACER, G = Gumbel, and P = POT. Exact value = 4.80.

An example of the ACER plot and results obtained for one set of data is presented in Figure 1. The predicted 100-year value is 4.85 with a predicted 95% confidence interval (4.45, 5.09). Figure 2 presents POT predictions based on MLE for different thresholds in terms of the number 𝑛 of data points above the threshold. The predicted value is 4.7 at 𝑛=204, while the 95% confidence interval is (4.25, 5.28). The same data set as in Figure 1 was used. This was also used for the Gumbel plot shown in Figure 3. In this case the predicted value based on the method of moments (MM) is 𝜂^100yr_MM=4.75 with a parametric bootstrapped 95% confidence interval of (4.34, 5.27). Prediction based on the Gumbel-Lieblein BLUE method (GL), compared with for example, Cook [44], is 𝜂^100yr_GL=4.73 with a parametric bootstrapped 95% confidence interval equal to (4.35, 5.14).

Figure 1 

Synthetic data ACER 𝜀1̂, Monte Carlo simulation (*); optimized curve fit (—); empirical 95% confidence band (- -); optimized confidence band (⋯). Tail marker 𝜂1=2.3.

Figure 2 

The point estimate 𝜂100yr of the 100-year return period value based on 20 years synthetic data as a function of the number 𝑛 of data points above threshold. The return level estimate = 4.7 at 𝑛=204.

Figure 3 

The point estimate 𝜂100yr of the 100-year return period value based on 20 years synthetic data. Lines are fitted by the method of moments—solid line (—) and the Gumbel-Lieblein BLUE method—dash-dotted lite (- · -). The return level estimate by the method of moments is 4.75, by the Gumbel-Lieblein BLUE method is 4.73.

9. Measured Wind Speed Data

In this section, we analyze real wind speed data, measured at two weather stations off the coast of Norway: at Nordøyan and at Hekkingen, see Figure 4. Extreme wind speed prediction is an important issue for design of structures exposed to the weather variations. Significant efforts have been devoted to the problem of predicting extreme wind speeds on the basis of measured data by various authors over several decades, see, for example, [45–48] for extensive references to previous work.

Figure 4 

Wind speed measurement stations.

Hourly maximum gust wind was recorded during the 13 years 1999–2012 at Nordøyan and the 14 years 1998–2012 at Hekkingen. The objective is to estimate a 100-year wind speed. Variation in the wind speed caused by seasonal variations in the wind climate during the year makes the wind speed a non-stationary process on the scale of months. Moreover, due to global climate change, yearly statistics may vary on the scale of years. The latter is, however, a slow process, and for the purpose of long-term prediction we assume here that within a time span of 100 years a quasi-stationary model of the wind speeds applies. This may not be entirely true, of course.

9.1. Nordøyan

Figure 5 highlights the cascade of ACER estimates 𝜀₁̂,…,𝜀̂₉₆, for the case of 13 years of hourly data recorded at the Nordøyan weather station. Here, 𝜀̂₉₆ is considered to represent the final converged results. By “converged,” we mean that 𝜀̂₉₆≈𝜀_𝑘̂ for 𝑘>96 in the tail, so that there is no need to consider conditioning of an even higher order than 96. Figure 5 reveals a rather strong statistical dependence between consecutive data, which is clearly reflected in the effect of conditioning on previous data values. It is also interesting to observe that this effect is to some extent captured already by 𝜀₂̂, that is, by conditioning only on the value of the previous data point. Subsequent conditioning on more than one previous data point does not lead to substantial changes in ACER values, especially for tail values. On the other hand, to bring out fully the dependence structure of these data, it was necessary to carry the conditioning process to (at least) the 96th ACER function, as discussed above.

Figure 5 

Nordøyan wind speed statistics, 13 years hourly data. Comparison between ACER estimates for different degrees of conditioning. 𝜎=6.01m/s.

However, from a practical point of view, the most important information provided by the ACER plot of Figure 5 is that for the prediction of a 100-year value, one may use the first ACER function. The reason for this is that Figure 5 shows that all the ACER functions coalesce in the far tail. Hence, we may use any of the ACER functions for the prediction. Then, the obvious choice is to use the first ACER function, which allows us to use all the data in its estimation and thereby increase accuracy.

In Figure 6 is shown the results of parametric estimation of the return value and its 95% CI for 13 years of hourly maxima. The predicted 100-year return speed is 𝜂^100yr=51.85 m/s with 95% confidence interval (48.4,53.1). 𝑅=13 years of data may not be enough to guarantee (22), since we required 𝑅≥20. Nevertheless, for simplicity, we use it here even with 𝑅=13, accepting that it may not be very accurate.

Figure 6 

Nordøyan wind speed statistics, 13 years hourly data. 𝜀1̂(𝜂) (*); optimized curve fit (—); empirical 95% confidence band (- -); optimized confidence band (⋯). Tail marker 𝜂1=12.5m/s=2.08𝜎 (𝜎=6.01m/s).

Figure 7 presents POT predictions for different threshold numbers based on MLE. The POT prediction is 𝜂^100yr=47.8 m/s at threshold 𝑛=161, while the bootstrapped 95% confidence interval is found to be (44.8, 52.7) m/s based on 10,000 generated samples. It is interesting to observe the unstable characteristics of the predictions over a range of threshold values, while they are quite stable on either side of this range giving predictions that are more in line with the results from the other two methods.

Figure 7 

The point estimate 𝜂100yr of the 100-year return level based on 13 years hourly data as a function of the number 𝑛 of data points above threshold. 𝜎=6.01m/s.

Figure 8 presents a Gumbel plot based on the 13 yearly extremes extracted from the 13 years of hourly data. The Gumbel prediction based on the method of moments (MM) is 𝜂^100yr_MM=51.5 m/s, with a parametric bootstrapped 95% confidence interval equal to (45.2, 59.3) m/s, while prediction based on the Gumbel-Lieblein BLUE method (GL) is 𝜂^100yr_GL=55.5 m/s, with a parametric bootstrapped 95% confidence interval equal to (48.0, 64.9) m/s.

Figure 8 

Nordøyan wind speed statistics, 13 years of hourly data. Gumbel plot of yearly extremes. Lines are fitted by the method of moments—solid line (—) and the Gumbel-Lieblein BLUE method—dash-dotted lite (- · -). 𝜎=6.01m/s.

In Table 2 the 100-year return period values for the Nordøyan station are listed together with the predicted 95% confidence intervals for all methods.

Table 2 

Predicted 100-year return period levels for Nordøyan Fyr weather station by the ACER method for different degrees of conditioning, annual maxima, and POT methods, respectively.

9.2. Hekkingen

Figure 9 shows the cascade of estimated ACER functions 𝜀₁̂,…,𝜀̂₉₆ for the case of 14 years of hourly data. As for Nordøyan, 𝜀̂₉₆ is used to represent the final converged results. Figure 9 also reveals a rather strong statistical dependence between consecutive data at moderate wind speed levels. This effect is again to some extent captured already by 𝜀₂̂, so that subsequent conditioning on more than one previous data point does not lead to substantial changes in ACER values, especially for tail values.

Figure 9 

Hekkingen wind speed statistics, 14 years hourly data. Comparison between ACER estimates for different degrees of conditioning. 𝜎=5.72m/s.

Also, for the Hekkingen data, the ACER plot of Figure 9 indicates that the ACER functions coalesce in the far tail. Hence, for the practical prediction of a 100-year value, one may use the first ACER function.

In Figure 10 is shown the results of parametric estimation of the return value and its 95% CI for 14 years of hourly maxima. The predicted 100-year return speed is 𝜂^100yr=60.47 m/s with 95% confidence interval (53.1,64.9). Equation (22) has been used also for this example with 𝑅=14.

Figure 10 

Hekkingen wind speed statistics, 14 years hourly data. 𝜀1̂(𝜂) (*); optimized curve fit (—); empirical 95% confidence band (- -); optimized confidence band (⋯). Tail marker 𝜂1=23m/s=4.02𝜎 (𝜎=5.72m/s).

Figure 11 presents POT predictions for different threshold numbers based on MLE. The POT prediction is 𝜂^100yr=53.48 m/s at threshold 𝑛=183, while the bootstrapped 95% confidence interval is found to be (48.9, 57.0) m/s based on 10,000 generated samples. It is interesting to observe the unstable characteristics of the predictions over a range of threshold values, while they are quite stable on either side of this range giving predictions that are more in line with the results from the other two methods.

Figure 11 

The point estimate 𝜂100yr of the 100-year return level based on 14 years hourly data as a function of the number 𝑛 of data points above threshold. 𝜎=5.72m/s.

Figure 12 presents a Gumbel plot based on the 14 yearly extremes extracted from the 14 years of hourly data. The Gumbel prediction based on the method of moments (MM) is 𝜂^100yr_MM=58.10 m/s, with a parametric bootstrapped 95% confidence interval equal to (50.8, 67.3) m/s. Prediction based on the Gumbel-Lieblein BLUE method (GL) is 𝜂^100yr_GL=60.63 m/s, with a parametric bootstrapped 95% confidence interval equal to (53.0, 70.1) m/s.

Figure 12 

Hekkingen wind speed statistics, 14 years of hourly data. Gumbel plot of yearly extremes. Lines are fitted by the method of moments—solid line (—) and the Gumbel-Lieblein BLUE method—dash-dotted lite (- · -). 𝜎=5.72m/s.

In Table 3, the 100-year return period values for the Hekkingen station are listed together with the predicted 95% confidence intervals for all methods.

Table 3 

Predicted 100-year return period levels for Nordøyan Fyr weather station by the ACER method for different degrees of conditioning, annual maxima, and POT methods, respectively.

10. Extreme Value Prediction for a Narrow Band Process

In engineering mechanics, a classical extreme response prediction problem is the case of a lightly damped mechanical oscillator subjected to random forces. To illustrate this prediction problem, we will investigate the response process of a linear mechanical oscillator driven by a Gaussian white noise. Let 𝑋(𝑡) denote the displacement response; the dynamic model can then be expressed as, 𝑋̈(𝑡)+2𝜁𝜔_𝑒𝑋̇(𝑡)+𝜔²_𝑒𝑋(𝑡)=𝑊(𝑡), where 𝜁 = relative damping, 𝜔_𝑒 = undamped eigenfrequency, and 𝑊(𝑡) = a stationary Gaussian white noise (of suitable intensity). By choosing a small value for 𝜁, the response time series will exhibit narrow band characteristics, that is, the spectral density of the response process 𝑋(𝑡) will assume significant values only over a narrow range of frequencies. This manifests itself by producing a strong beating of the response time series, which means that the size of the response peaks will change slowly in time, see Figure 13. A consequence of this is that neighbouring peaks are strongly correlated, and there is a conspicuous clumping of the peak values. Hence the problem with accurate prediction, since the usual assumption of independent peak values is then violated.

Figure 13 

Part of the narrow-band response time series of the linear oscillator with fully sampled and peak values indicated.

Many approximations have been proposed to deal with this correlation problem, but no completely satisfactory solution has been presented. In this section, we will show that the ACER method solves this problem efficiently and elegantly in a statistical sense. In Figure 14 are shown some of the ACER functions for the example time series. It may be verified from Figure 13 that there are approximately 32 sample points between two neighbouring peaks in the time series. To illustrate a point, we have chosen to analyze the time series consisting of all sample points. Usually, in practice, only the time series obtained by extracting the peak values would be used for the ACER analysis. In the present case, the first ACER function is then based on assuming that all the sampled data points are independent, which is obviously completely wrong. The second ACER function, which is based on counting each exceedance with an immediately preceding nonexceedance, is nothing but an upcrossing rate. Using this ACER function is largely equivalent to assuming independent peak values. It is now interesting to observe that the 25th ACER function can hardly be distinguished from the second ACER function. In fact, the ACER functions after the second do not change appreciably until one starts to approach the 32nd, which corresponds to hitting the previous peak value in the conditioning process. So, the important information concerning the dependence structure in the present time series seems to reside in the peak values, which may not be very surprising. It is seen that the ACER functions show a significant change in value as a result of accounting for the correlation effects in the time series. To verify the full dependence structure in the time series, it is necessary to continue the conditioning process down to at least the 64th ACER function. In the present case, there is virtually no difference between the 32nd and the 64th, which shows that the dependence structure in this particular time series is captured almost completely by conditioning on the previous peak value. It is interesting to contrast the method of dealing with the effect of sampling frequency discussed here with that of [49].

Figure 14 

Comparison between ACER estimates for different degrees of conditioning for the narrow-band time series.

To illustrate the results obtained by extracting only the peak values from the time series, which would be the approach typically chosen in an engineering analysis, the ACER plots for this case is shown in Figure 15. By comparing results from Figures 14 and 15, it can be verified that they are in very close agreement by recognizing that the second ACER function in Figure 14 corresponds to the first ACER function in Figure 15, and by noting that there is a factor of approximately 32 between corresponding ACER functions in the two figures. This is due to the fact that the time series of peak values contains about 32 times less data than the original time series.

Figure 15 

Comparison between ACER estimates for different degrees of conditioning based on the time series of the peak values, compared with Figure 13.

11. Concluding Remarks

This paper studies a new method for extreme value prediction for sampled time series. The method is based on the introduction of a conditional average exceedance rate (ACER), which allows dependence in the time series to be properly and easily accounted for. Declustering of the data is therefore avoided, and all the data are used in the analysis. Significantly, the proposed method also aims at capturing to some extent the subasymptotic form of the extreme value distribution.

Results for wind speeds, both synthetic and measured, are used to illustrate the method. An estimation problem related to applications in mechanics is also presented. The validation of the method is done by comparison with exact results (when available), or other widely used methods for extreme value statistics, such as the Gumbel and the peaks-over-threshold (POT) methods. Comparison of the various estimates indicate that the proposed method provides more accurate results than the Gumbel and POT methods.

Subject to certain restrictions, the proposed method also applies to nonstationary time series, but it cannot directly predict for example, the effect of climate change in the form of long-term trends in the average exceedance rates extending beyond the data. This must be incorporated into the analysis by explicit modelling techniques.

As a final remark, it may be noted that the ACER method as described in this paper has a natural extension to higher dimensional distributions. The implication is that, it is then possible to provide estimates of for example, the exact bivariate extreme value distribution for a suitable set of data [50]. However, as is easily recognized, the extrapolation problem is not as simply dealt with as for the univariate case studied in this paper.

Acknowledgment

This work was supported by the Research Council of Norway (NFR) through the Centre for Ships and Ocean Structures (CeSOS) at the Norwegian University of Science and Technology.