Nonparametric estimation of mean and dispersion functions in extended generalized linear models.

In this paper the interest is in regression analysis for data that show possibly overdispersion or underdispersion. The starting point for modeling are generalized linear models in which we no longer admit a linear form for the mean regression function, but allow it to be any smooth function of the covariate(s). In view of analyzing overdispersed or underdispersed data, we additionally bring in an unknown dispersion function. The mean regression function and the dispersion function are then estimated using P-splines with difference type of penalty to prevent from overfitting. We discuss two approaches: one based on an extended quasi-likelihood idea and one based on a pseudo-likelihood approach. The choices of smoothing parameters and implementation issues are discussed. The performance of the estimation method is investigated via simulations and its use is illustrated on several data, including continuous data, counts and proportions.Double exponential family; Extended quasi-likelihood; Modeling; Overdispersion; Pseudo likelihood; P-splines; Regression; Variance estimation; Underdispersion;

Robust estimation of mean and dispersion functions in extended generalized additive models.

Generalized Linear Models are a widely used method to obtain parametric estimates for the mean function. They have been further extended to allow the relationship between the mean function and the covariates to be more flexible via Generalized Additive Models. However the fixed variance structure can in many cases be too restrictive. The Extended Quasi-Likelihood (EQL) framework allows for estimation of both the mean and the dispersion/variance as functions of covariates. As for other maximum likelihood methods though, EQL estimates are not resistant to outliers: we need methods to obtain robust estimates for both the mean and the dispersion function. In this paper we obtain functional estimates for the mean and the dispersion that are both robust and smooth. The performance of the proposed method is illustrated via a simulation study and some real data examples.Dispersion; Generalized additive modelling; Mean regression function; M-estimation; P-splines; Robust estimation;

Assessment of trends in hydrological extremes using regional magnification factors

Detection and attribution of trends in individual at-site series of hydrological extremes is routinely undertaken using simple linear regression-based models. However, the available records are often too short to allow a consistent assessment of trends across different stations in a region. The theoretical developments presented in this paper propose a new method for estimating a regional regression slope parameter across a region, or pooling group, of catchment considered hydrologically similar, and where annual maximum events at different sites are cross-correlated. Assuming annual maximum events to follow a two-parameter log-normal distribution, a series of Monte Carlo simulations demonstrate the ability of the new framework to accurately identify the regional slope, and provide estimates with a reduced sampling variability as compared to the equivalent at-site estimates, thereby enhancing the statistical power of the trend test. This regionally-based trend estimates would allow for a clear characterization of changes across several stations in a region. Finally, the new method is applied to national dataset of annual maximum series of peak flow from 662 gauging sites located across the United Kingdom. The results show that the regional slope estimates are significantly positive (p &lt; 0.05) consistently in the west and north of the country, while mostly not significant in the east and south. This translate into a corresponding increase in design flood (as measured by regional magnification factors) of up-to 50% for time horizon of 50-years into the future.</p

Parametrisation of change-permitting extreme value models and its impact on the description of change

The potential for changes in environmental extremes is routinely investigated by fitting change-permitting extreme value models to long-term observations, allowing one or more distribution parameters to change as a function of time or some other covariate. In most extreme value analyses, the main quantity of interest is typically the upper quantiles of the distribution, which are often needed for practical applications such as engineering design. This study focuses on the changes in quantile estimates under different change-permitting models. First, metrics which measure the impact of changes in parameters on changes in quantiles are introduced. The mathematical structure of these change metrics is investigated for several change-permitting models based on the Generalised Extreme Value (GEV) distribution. It is shown that for the most commonly used models, the predicted changes in the quantiles are a non-intuitive function of the distribution parameters, leading to results which are difficult to interpret. Next, it is posited that commonly used change-permitting GEV models do not preserve a constant coefficient of variation, a property that is typically assumed to hold for environmental extremes. To address these shortcomings a new (parsimonious) model is proposed: the model assumes a constant coefficient of variation, allowing the location and scale parameters to change simultaneously. The proposed model results in changes in the quantile function that are easier to interpret. Finally, the consequences of the different modelling choices on quantile estimates are exemplified using a dataset of extreme peak river flow measurements in Massachusetts, USA. It is argued that the decision on which model structure to adopt to describe change in extremes should also take into consideration any requirements on the behaviour of the quantiles of interest

Assessment of trends in hydrological extremes using regional magnification factors

Detection and attribution of trends in individual at-site series of hydrological extremes is routinely undertaken using simple linear regression-based models. However, the available records are often too short to allow a consistent assessment of trends across different stations in a region. The theoretical developments presented in this paper propose a new method for estimating a regional regression slope parameter across a region, or pooling group, of catchment considered hydrologically similar, and where annual maximum events at different sites are cross-correlated. Assuming annual maximum events to follow a two-parameter log-normal distribution, a series of Monte Carlo simulations demonstrate the ability of the new framework to accurately identify the regional slope, and provide estimates with a reduced sampling variability as compared to the equivalent at-site estimates, thereby enhancing the statistical power of the trend test. This regionally-based trend estimates would allow for a clear characterization of changes across several stations in a region. Finally, the new method is applied to national dataset of annual maximum series of peak flow from 662 gauging sites located across the United Kingdom. The results show that the regional slope estimates are significantly positive (p &lt; 0.05) consistently in the west and north of the country, while mostly not significant in the east and south. This translate into a corresponding increase in design flood (as measured by regional magnification factors) of up-to 50% for time horizon of 50-years into the future.</p

Attribution of long-term changes in peak river flows in Great Britain

We investigate the evidence for changes in the magnitude of peak river flows in Great Britain. We focus on a set of 117 near-natural "benchmark" catchments to detect trends not driven by land use and other human impacts, and aim to attribute trends in peak river flows to some climate indices such as the North Atlantic Oscillation (NAO) and the East Atlantic (EA) index. We propose modelling all stations together in a Bayesian multilevel framework to be better able to detect any signal that is present in the data by pooling information across several stations. This approach leads to the detection of a clear countrywide time trend. Additionally, in a univariate approach, both the EA and NAO indices appear to have a considerable association with peak river flows. When a multivariate approach is taken to unmask the collinearity between climate indices and time, the association between NAO and peak flows disappears, while the association with EA remains clear. This demonstrates the usefulness of a multivariate and multilevel approach when it comes to accurately attributing trends in peak river flows

Statistical distributions for monthly aggregations of precipitation and streamflow in drought indicator applications

Drought indicators are used as triggers for action and so are the foundation of drought monitoring and early warning. The computation of drought indicators like the standardized precipitation index (SPI) and standardized streamflow index (SSI) require a statistical probability distribution to be fitted to the observed data. Both precipitation and streamflow have a lower bound at zero, and their empirical distributions tend to have positive skewness. For deriving the SPI, the Gamma distribution has therefore often been a natural choice. The concept of the SSI is newer and there is no consensus regarding distribution. In the present study, twelve different probability distributions are fitted to streamflow and catchment average precipitation for four durations (1, 3, 6, and 12 months), for 121 catchments throughout the United Kingdom. The more flexible three- and four-parameter distributions generally do not have a lower bound at zero, and hence may attach some probability to values below zero. As a result, there is a censoring of the possible values of the calculated SPIs and SSIs. This can be avoided by using one of the bounded distributions, such as the reasonably flexible three-parameter Tweedie distribution, which has a lower bound (and potentially mass) at zero. The Tweedie distribution has only recently been applied to precipitation data, and only for a few sites. We find it fits both precipitation and streamflow data nearly as well as the best of the traditionally used three-parameter distributions, and should improve the accuracy of drought indices used for monitoring and early warning

Going Beyond the Ensemble Mean: Assessment of Future Floods From Global Multi‐Models

Future changes in the occurrence of flood events can be estimated using multi-model ensembles to inform adaption and mitigation strategies. In the near future, these estimates could be used to guide the updating of exceedance probabilities for flood control design and water resources management. However, the estimate of return levels from ensemble experiments represents a challenge: model runs are affected by biases and uncertainties and by inconsistencies in simulated peak flows when compared with observed data. Moreover, extreme value distributions are generally fit to ensemble members individually and then averaged to obtain the ensemble fit with loss of information. To overcome these limitations, we propose a Bayesian hierarchical model for assessing changes in future peak flows, and the uncertainty coming from global climate, global impact models and their interaction. The model we propose allows use of all members of the ensemble at once for estimating changes in the parameters of an extreme value distribution from historical to future peak flows. The approach is applied to a set of grid-cells in the eastern United States to the full and to a constrained version of the ensemble. We find that, while the dominant source of uncertainty in the changes varies across the domain, there is a consensus on a decrease in flood magnitudes toward the south. We conclude that projecting future flood magnitude under climate change remains elusive due to large uncertainty mostly coming from global models and from the intrinsic uncertain nature of extreme values

Assessing the element of surprise of record-breaking flood events

The occurrence of record-breaking flood events continues to cause damage and disruption despite significant investments in flood defences, suggesting that these events are in some sense surprising. This study develops a new statistical test to help assess if a flood event can be considered surprising or not. The test statistic is derived from annual maximum series (AMS) of extreme events, and Monte Carlo simulations were used to derive critical values for a range of significance levels based on a Generalised Logistic distribution. The method is tested on a national data set of AMS of peak flow from the United Kingdom, and is found to correctly identify recent large events that have been identified elsewhere as causing a significant change in UK flood management policy. No temporal trend in the frequency or magnitude of surprising events was identified, and no link could be established between the occurrences of surprising events and large-scale drivers. Finally, the implications of the findings for future research examining the most extreme flood events are discussed.</p