Regularized logistic models for probabilistic forecasting and diagnostics

Logistic models are studied as a tool to convert dynamical forecast information (deterministic and ensemble) into probability forecasts. A logistic model is obtained by setting the logarithmic odds ratio equal to a linear combination of the inputs. As with any statistical model, logistic models will suffer from overfitting if the number of inputs is comparable to the number of forecast instances. Computational approaches to avoid overfitting by regularization are discussed, and efficient techniques for model assessment and selection are presented. A logit version of the lasso (originally a linear regression technique), is discussed. In lasso models, less important inputs are identified and the corresponding coefficient is set to zero, providing an efficient and automatic model reduction procedure. For the same reason, lasso models are particularly appealing for diagnostic purposes.

Probabilistic evaluation of time series models : a comparison of several approaches

Several methods are examined which allow to produce forecasts for time series in the form of probability assignments. The necessary concepts are presented, addressing questions such as how to assess the performance of a probabilistic forecast. A particular class of models, cluster weighted models (CWMs), is given particular attention. CWMs, originally proposed for deterministic forecasts, can be employed for probabilistic forecasting with little modification. Two examples are presented. The first involves estimating the state of (numerically simulated) dynamical systems from noise corrupted measurements, a problem also known as filtering. There is an optimal solution to this problem, called the optimal filter, to which the considered time series models are compared. (The optimal filter requires the dynamical equations to be known.) In the second example, we aim at forecasting the chaotic oscillations of an experimental bronze spring system. Both examples demonstrate that the considered time series models, and especially the CWMs, provide useful probabilistic information about the underlying dynamical relations. In particular, they provide more than just an approximation to the conditional mean.

Calibrating flood inundation models: identifying and addressing uncertainty in satellite observed flood extents

Improvements in the resolution of satellite imagery have enabled extraction of water surface elevations at the margins of the flood. Comparison between modelled and observed water surface elevations provides a new means for calibrating and validating flood inundation models, however the uncertainty in this observed data has yet to be addressed. Here a flood inundation model is calibrated using a probabilistic treatment of the observed data. A LiDAR guided snake algorithm is used to determine an outline of a flood event in 2006 on the River Dee, North Wales, UK, using a 12.5m ERS-1 image. Points at approximately 100m intervals along this outline are selected, and the water surface elevation recorded as the LiDAR DEM elevation at each point. With a planar water surface from the gauged upstream to downstream water elevations as an approximation, the water surface elevations at points along this flooded extent are compared to their ‘expected’ value. The pattern of errors between the two show a roughly normal distribution, however when plotted against coordinates there is obvious spatial autocorrelation. The source of this spatial dependency is investigated by comparing errors to the slope gradient and aspect of the LiDAR DEM. A LISFLOOD-FP model of the flood event is set-up to investigate the effect of observed data uncertainty on the calibration of flood inundation models. Multiple simulations are run using different combinations of friction parameters, from which the optimum parameter set will be selected. For each simulation a T-test is used to quantify the fit between modelled and observed water surface elevations. The points chosen for use in this T-test are selected based on their error. The criteria for selection enables evaluation of the sensitivity of the choice of optimum parameter set to uncertainty in the observed data. This work explores the observed data in detail and highlights possible causes of error. The identification of significant error (RMSE = 0.8m) between approximate expected and actual observed elevations from the remotely sensed data emphasises the limitations of using this data in a deterministic manner within the calibration process. These limitations are addressed by developing a new probabilistic approach to using the observed data.

Probabilistic simulations of crop yield over western India using the DEMETER seasonal hindcast ensembles

Process-based integrated modelling of weather and crop yield over large areas is becoming an important research topic. The production of the DEMETER ensemble hindcasts of weather allows this work to be carried out in a probabilistic framework. In this study, ensembles of crop yield (groundnut, Arachis hypogaea L.) were produced for 10 2.5 degrees x 2.5 degrees grid cells in western India using the DEMETER ensembles and the general large-area model (GLAM) for annual crops. Four key issues are addressed by this study. First, crop model calibration methods for use with weather ensemble data are assessed. Calibration using yield ensembles was more successful than calibration using reanalysis data (the European Centre for Medium-Range Weather Forecasts 40-yr reanalysis, ERA40). Secondly, the potential for probabilistic forecasting of crop failure is examined. The hindcasts show skill in the prediction of crop failure, with more severe failures being more predictable. Thirdly, the use of yield ensemble means to predict interannual variability in crop yield is examined and their skill assessed relative to baseline simulations using ERA40. The accuracy of multi-model yield ensemble means is equal to or greater than the accuracy using ERA40. Fourthly, the impact of two key uncertainties, sowing window and spatial scale, is briefly examined. The impact of uncertainty in the sowing window is greater with ERA40 than with the multi-model yield ensemble mean. Subgrid heterogeneity affects model accuracy: where correlations are low on the grid scale, they may be significantly positive on the subgrid scale. The implications of the results of this study for yield forecasting on seasonal time-scales are as follows. (i) There is the potential for probabilistic forecasting of crop failure (defined by a threshold yield value); forecasting of yield terciles shows less potential. (ii) Any improvement in the skill of climate models has the potential to translate into improved deterministic yield prediction. (iii) Whilst model input uncertainties are important, uncertainty in the sowing window may not require specific modelling. The implications of the results of this study for yield forecasting on multidecadal (climate change) time-scales are as follows. (i) The skill in the ensemble mean suggests that the perturbation, within uncertainty bounds, of crop and climate parameters, could potentially average out some of the errors associated with mean yield prediction. (ii) For a given technology trend, decadal fluctuations in the yield-gap parameter used by GLAM may be relatively small, implying some predictability on those time-scales.

Detecting recombination in 4-taxa DNA sequence alignments with Bayesian hidden Markov models and Markov chain Monte Carlo

This article presents a statistical method for detecting recombination in DNA sequence alignments, which is based on combining two probabilistic graphical models: (1) a taxon graph (phylogenetic tree) representing the relationship between the taxa, and (2) a site graph (hidden Markov model) representing interactions between different sites in the DNA sequence alignments. We adopt a Bayesian approach and sample the parameters of the model from the posterior distribution with Markov chain Monte Carlo, using a Metropolis-Hastings and Gibbs-within-Gibbs scheme. The proposed method is tested on various synthetic and real-world DNA sequence alignments, and we compare its performance with the established detection methods RECPARS, PLATO, and TOPAL, as well as with two alternative parameter estimation schemes.

Case study part 2: probabilistic modelling of long-term effects of pesticides on individual breeding success in birds and mammals

Abstract: Long-term exposure of skylarks to a fictitious insecticide and of wood mice to a fictitious fungicide were modelled probabilistically in a Monte Carlo simulation. Within the same simulation the consequences of exposure to pesticides on reproductive success were modelled using the toxicity-exposure-linking rules developed by R.S. Bennet et al. (2005) and the interspecies extrapolation factors suggested by R. Luttik et al.(2005). We built models to reflect a range of scenarios and as a result were able to show how exposure to pesticide might alter the number of individuals engaged in any given phase of the breeding cycle at any given time and predict the numbers of new adults at the season’s end.

Potential Predictability and AMIP Implications of Midlatitude Climate Variability in Two General Circulation Models

Ensembles of extended Atmospheric Model Intercomparison Project (AMIP) runs from the general circulation models of the National Centers for Environmental Prediction (formerly the National Meteorological Center) and the Max-Planck Institute (Hamburg, Germany) are used to estimate the potential predictability (PP) of an index of the Pacific–North America (PNA) mode of climate change. The PP of this pattern in “perfect” prediction experiments is 20%–25% of the index’s variance. The models, particularly that from MPI, capture virtually all of this variance in their hindcasts of the winter PNA for the period 1970–93. The high levels of internally generated model noise in the PNA simulations reconfirm the need for an ensemble averaging approach to climate prediction. This means that the forecasts ought to be expressed in a probabilistic manner. It is shown that the models’ skills are higher by about 50% during strong SST events in the tropical Pacific, so the probabilistic forecasts need to be conditional on the tropical SST. Taken together with earlier studies, the present results suggest that the original set of AMIP integrations (single 10-yr runs) is not adequate to reliably test the participating models’ simulations of interannual climate variability in the midlatitudes.

Contrasting probabilistic scoring rules

There are several scoring rules that one can choose from in order to score probabilistic forecasting models or estimate model parameters. Whilst it is generally agreed that proper scoring rules are preferable, there is no clear criterion for preferring one proper scoring rule above another. This manuscript compares and contrasts some commonly used proper scoring rules and provides guidance on scoring rule selection. In particular, it is shown that the logarithmic scoring rule prefers erring with more uncertainty, the spherical scoring rule prefers erring with lower uncertainty, whereas the other scoring rules are indifferent to either option.

Wind gust estimation for Mid-European winter storms: towards a probabilistic view

Three wind gust estimation (WGE) methods implemented in the numerical weather prediction (NWP) model COSMO-CLM are evaluated with respect to their forecast quality using skill scores. Two methods estimate gusts locally from mean wind speed and the turbulence state of the atmosphere, while the third one considers the mixing-down of high momentum within the planetary boundary layer (WGE Brasseur). One hundred and fifty-eight windstorms from the last four decades are simulated and results are compared with gust observations at 37 stations in Germany. Skill scores reveal that the local WGE methods show an overall better behaviour, whilst WGE Brasseur performs less well except for mountain regions. The here introduced WGE turbulent kinetic energy (TKE) permits a probabilistic interpretation using statistical characteristics of gusts at observational sites for an assessment of uncertainty. The WGE TKE formulation has the advantage of a ‘native’ interpretation of wind gusts as result of local appearance of TKE. The inclusion of a probabilistic WGE TKE approach in NWP models has, thus, several advantages over other methods, as it has the potential for an estimation of uncertainties of gusts at observational sites.

Can dynamically downscaled windstorm footprints be improved by observations through a probabilistic approach?

Windstorms are a main feature of the European climate and exert strong socioeconomic impacts. Large effort has been made in developing and enhancing models to simulate the intensification of windstorms, resulting footprints, and associated impacts. Simulated wind or gust speeds usually differ from observations, as regional climate models have biases and cannot capture all local effects. An approach to adjust regional climate model (RCM) simulations of wind and wind gust toward observations is introduced. For this purpose, 100 windstorms are selected and observations of 173 (111) test sites of the German Weather Service are considered for wind (gust) speed. Theoretical Weibull distributions are fitted to observed and simulated wind and gust speeds, and the distribution parameters of the observations are interpolated onto the RCM computational grid. A probability mapping approach is applied to relate the distributions and to correct the modeled footprints. The results are not only achieved for single test sites but for an area-wide regular grid. The approach is validated using root-mean-square errors on event and site basis, documenting that the method is generally able to adjust the RCM output toward observations. For gust speeds, an improvement on 88 of 100 events and at about 64% of the test sites is reached. For wind, 99 of 100 improved events and ~84% improved sites can be obtained. This gives confidence on the potential of the introduced approach for many applications, in particular those considering wind data.

Probabilistic parameterisation of the surface mass balance--elevation feedback in regional climate model simulations of the Greenland ice sheet

We present a new parameterisation that relates surface mass balance (SMB: the sum of surface accumulation and surface ablation) to changes in surface elevation of the Greenland ice sheet (GrIS) for the MAR (Modèle Atmosphérique Régional: Fettweis, 2007) regional climate model. The motivation is to dynamically adjust SMB as the GrIS evolves, allowing us to force ice sheet models with SMB simulated by MAR while incorporating the SMB–elevation feedback, without the substantial technical challenges of coupling ice sheet and climate models. This also allows us to assess the effect of elevation feedback uncertainty on the GrIS contribution to sea level, using multiple global climate and ice sheet models, without the need for additional, expensive MAR simulations. We estimate this relationship separately below and above the equilibrium line altitude (ELA, separating negative and positive SMB) and for regions north and south of 77� N, from a set of MAR simulations in which we alter the ice sheet surface elevation. These give four “SMB lapse rates”, gradients that relate SMB changes to elevation changes. We assess uncertainties within a Bayesian framework, estimating probability distributions for each gradient from which we present best estimates and credibility intervals (CI) that bound 95% of the probability. Below the ELA our gradient estimates are mostly positive, because SMB usually increases with elevation: 0.56 (95% CI: −0.22 to 1.33) kgm−3 a−1 for the north, and 1.91 (1.03 to 2.61) kgm−3 a−1 for the south. Above the ELA, the gradients are much smaller in magnitude: 0.09 (−0.03 to 0.23) kgm−3 a−1 in the north, and 0.07 (−0.07 to 0.59) kgm−3 a−1 in the south, because SMB can either increase or decrease in response to increased elevation. Our statistically founded approach allows us to make probabilistic assessments for the effect of elevation feedback uncertainty on sea level projections (Edwards et al., 2014).

An evaluation of decadal probability forecasts from state-of-the-art climate models

While state-of-the-art models of Earth's climate system have improved tremendously over the last 20 years, nontrivial structural flaws still hinder their ability to forecast the decadal dynamics of the Earth system realistically. Contrasting the skill of these models not only with each other but also with empirical models can reveal the space and time scales on which simulation models exploit their physical basis effectively and quantify their ability to add information to operational forecasts. The skill of decadal probabilistic hindcasts for annual global-mean and regional-mean temperatures from the EU Ensemble-Based Predictions of Climate Changes and Their Impacts (ENSEMBLES) project is contrasted with several empirical models. Both the ENSEMBLES models and a “dynamic climatology” empirical model show probabilistic skill above that of a static climatology for global-mean temperature. The dynamic climatology model, however, often outperforms the ENSEMBLES models. The fact that empirical models display skill similar to that of today's state-of-the-art simulation models suggests that empirical forecasts can improve decadal forecasts for climate services, just as in weather, medium-range, and seasonal forecasting. It is suggested that the direct comparison of simulation models with empirical models becomes a regular component of large model forecast evaluations. Doing so would clarify the extent to which state-of-the-art simulation models provide information beyond that available from simpler empirical models and clarify current limitations in using simulation forecasting for decision support. Ultimately, the skill of simulation models based on physical principles is expected to surpass that of empirical models in a changing climate; their direct comparison provides information on progress toward that goal, which is not available in model–model intercomparisons.

Probabilistic skill in ensemble seasonal forecasts

Simulation models are widely employed to make probability forecasts of future conditions on seasonal to annual lead times. Added value in such forecasts is reflected in the information they add, either to purely empirical statistical models or to simpler simulation models. An evaluation of seasonal probability forecasts from the Development of a European Multimodel Ensemble system for seasonal to inTERannual prediction (DEMETER) and ENSEMBLES multi-model ensemble experiments is presented. Two particular regions are considered: Nino3.4 in the Pacific and the Main Development Region in the Atlantic; these regions were chosen before any spatial distribution of skill was examined. The ENSEMBLES models are found to have skill against the climatological distribution on seasonal time-scales. For models in ENSEMBLES that have a clearly defined predecessor model in DEMETER, the improvement from DEMETER to ENSEMBLES is discussed. Due to the long lead times of the forecasts and the evolution of observation technology, the forecast-outcome archive for seasonal forecast evaluation is small; arguably, evaluation data for seasonal forecasting will always be precious. Issues of information contamination from in-sample evaluation are discussed and impacts (both positive and negative) of variations in cross-validation protocol are demonstrated. Other difficulties due to the small forecast-outcome archive are identified. The claim that the multi-model ensemble provides a ‘better’ probability forecast than the best single model is examined and challenged. Significant forecast information beyond the climatological distribution is also demonstrated in a persistence probability forecast. The ENSEMBLES probability forecasts add significantly more information to empirical probability forecasts on seasonal time-scales than on decadal scales. Current operational forecasts might be enhanced by melding information from both simulation models and empirical models. Simulation models based on physical principles are sometimes expected, in principle, to outperform empirical models; direct comparison of their forecast skill provides information on progress toward that goal.

Towards improving the framework for probabilistic forecast evaluation

The evaluation of forecast performance plays a central role both in the interpretation and use of forecast systems and in their development. Different evaluation measures (scores) are available, often quantifying different characteristics of forecast performance. The properties of several proper scores for probabilistic forecast evaluation are contrasted and then used to interpret decadal probability hindcasts of global mean temperature. The Continuous Ranked Probability Score (CRPS), Proper Linear (PL) score, and IJ Good’s logarithmic score (also referred to as Ignorance) are compared; although information from all three may be useful, the logarithmic score has an immediate interpretation and is not insensitive to forecast busts. Neither CRPS nor PL is local; this is shown to produce counter intuitive evaluations by CRPS. Benchmark forecasts from empirical models like Dynamic Climatology place the scores in context. Comparing scores for forecast systems based on physical models (in this case HadCM3, from the CMIP5 decadal archive) against such benchmarks is more informative than internal comparison systems based on similar physical simulation models with each other. It is shown that a forecast system based on HadCM3 out performs Dynamic Climatology in decadal global mean temperature hindcasts; Dynamic Climatology previously outperformed a forecast system based upon HadGEM2 and reasons for these results are suggested. Forecasts of aggregate data (5-year means of global mean temperature) are, of course, narrower than forecasts of annual averages due to the suppression of variance; while the average “distance” between the forecasts and a target may be expected to decrease, little if any discernible improvement in probabilistic skill is achieved.

Probabilistic wind power forecasts with an inverse power curve transformation and censored regression

Forecasting wind power is an important part of a successful integration of wind power into the power grid. Forecasts with lead times longer than 6 h are generally made by using statistical methods to post-process forecasts from numerical weather prediction systems. Two major problems that complicate this approach are the non-linear relationship between wind speed and power production and the limited range of power production between zero and nominal power of the turbine. In practice, these problems are often tackled by using non-linear non-parametric regression models. However, such an approach ignores valuable and readily available information: the power curve of the turbine's manufacturer. Much of the non-linearity can be directly accounted for by transforming the observed power production into wind speed via the inverse power curve so that simpler linear regression models can be used. Furthermore, the fact that the transformed power production has a limited range can be taken care of by employing censored regression models. In this study, we evaluate quantile forecasts from a range of methods: (i) using parametric and non-parametric models, (ii) with and without the proposed inverse power curve transformation and (iii) with and without censoring. The results show that with our inverse (power-to-wind) transformation, simpler linear regression models with censoring perform equally or better than non-linear models with or without the frequently used wind-to-power transformation.