Bayesian parameter estimation for latent Markov random fields and social networks

Undirected graphical models are widely used in statistics, physics and machine vision. However Bayesian parameter estimation for undirected models is extremely challenging, since evaluation of the posterior typically involves the calculation of an intractable normalising constant. This problem has received much attention, but very little of this has focussed on the important practical case where the data consists of noisy or incomplete observations of the underlying hidden structure. This paper specifically addresses this problem, comparing two alternative methodologies. In the first of these approaches particle Markov chain Monte Carlo (Andrieu et al., 2010) is used to efficiently explore the parameter space, combined with the exchange algorithm (Murray et al., 2006) for avoiding the calculation of the intractable normalising constant (a proof showing that this combination targets the correct distribution in found in a supplementary appendix online). This approach is compared with approximate Bayesian computation (Pritchard et al., 1999). Applications to estimating the parameters of Ising models and exponential random graphs from noisy data are presented. Each algorithm used in the paper targets an approximation to the true posterior due to the use of MCMC to simulate from the latent graphical model, in lieu of being able to do this exactly in general. The supplementary appendix also describes the nature of the resulting approximation.

A linear model of gravity wave drag for hydrostatic sheared flow over elliptical mountains

An analytical model of orographic gravity wave drag due to sheared flow past elliptical mountains is developed. The model extends the domain of applicability of the well-known Phillips model to wind profiles that vary relatively slowly in the vertical, so that they may be treated using a WKB approximation. The model illustrates how linear processes associated with wind profile shear and curvature affect the drag force exerted by the airflow on mountains, and how it is crucial to extend the WKB approximation to second order in the small perturbation parameter for these effects to be taken into account. For the simplest wind profiles, the normalized drag depends only on the Richardson number, Ri, of the flow at the surface and on the aspect ratio, γ, of the mountain. For a linear wind profile, the drag decreases as Ri decreases, and this variation is faster when the wind is across the mountain than when it is along the mountain. For a wind that rotates with height maintaining its magnitude, the drag generally increases as Ri decreases, by an amount depending on γ and on the incidence angle. The results from WKB theory are compared with exact linear results and also with results from a non-hydrostatic nonlinear numerical model, showing in general encouraging agreement, down to values of Ri of order one.

Endogenous business cycles with frictional labour markets

We present a neoclassical model of capital accumulation with frictional labour markets. Under standard parameter values the equilibrium of the model is indeterminate and consequently displays expectations-driven business cycles – so-called endogenous business cycles. We study the properties of such cycles, and find that the model predicts the high autocorrelation in output growth and the hump-shaped impulse response of output found in US data – important features that existing endogenous real business cycle models fail to explain. The indeterminacy of the equilibrium stems from job search externalities and does not rely on increasing returns to scale as in most models.

Predictability and uncertainty in a regional climate model

The evaluation of the quality and usefulness of climate modeling systems is dependent upon an assessment of both the limited predictability of the climate system and the uncertainties stemming from model formulation. In this study a methodology is presented that is suited to assess the performance of a regional climate model (RCM), based on its ability to represent the natural interannual variability on monthly and seasonal timescales. The methodology involves carrying out multiyear ensemble simulations (to assess the predictability bounds within which the model can be evaluated against observations) and multiyear sensitivity experiments using different model formulations (to assess the model uncertainty). As an example application, experiments driven by assimilated lateral boundary conditions and sea surface temperatures from the ECMWF Reanalysis Project (ERA-15, 1979–1993) were conducted. While the ensemble experiment demonstrates that the predictability of the regional climate varies strongly between different seasons and regions, being weakest during the summer and over continental regions, important sensitivities of the modeling system to parameterization choices are uncovered. In particular, compensating mechanisms related to the long-term representation of the water cycle are revealed, in which summer dry and hot conditions at the surface, resulting from insufficient evaporation, can persist despite insufficient net solar radiation (a result of unrealistic cloud-radiative feedbacks).

An evaluation of the spatiotemporal structure of large-scale European drought within the HiGEM climate model

We compare the characteristics of synthetic European droughts generated by the HiGEM1 coupled climate model run with present day atmospheric composition with observed drought events extracted from the CRU TS3 data set. The results demonstrate consistency in both the rate of drought occurrence and the spatiotemporal structure of the events. Estimates of the probability density functions for event area, duration and severity are shown to be similar with confidence > 90%. Encouragingly, HiGEM is shown to replicate the extreme tails of the observed distributions and thus the most damaging European drought events. The soil moisture state is shown to play an important role in drought development. Once a large-scale drought has been initiated it is found to be 50% more likely to continue if the local soil moisture is below the 40th percentile. In response to increased concentrations of atmospheric CO2, the modelled droughts are found to increase in duration, area and severity. The drought response can be largely attributed to temperature driven changes in relative humidity. 1 HiGEM is based on the latest climate configuration of the Met Office Hadley Centre Unified Model (HadGEM1) with the horizontal resolution increased to 1.25 x 0.83 degrees in longitude and latitude in the atmosphere and 1/3 x 1/3 degrees in the ocean.

Climatology and Forcing of the Quasi-Biennial Oscillation in the MAECHAM5 Model

The quasi-biennial oscillation (QBO) in the equatorial zonal wind is an outstanding phenomenon of the atmosphere. The QBO is driven by a broad spectrum of waves excited in the tropical troposphere and modulates transport and mixing of chemical compounds in the whole middle atmosphere. Therefore, the simulation of the QBO in general circulation models and chemistry climate models is an important issue. Here, aspects of the climatology and forcing of a spontaneously occurring QBO in a middle-atmosphere model are evaluated, and its influence on the climate and variability of the tropical middle atmosphere is investigated. Westerly and easterly phases are considered separately, and 40-yr ECMWF Re-Analysis (ERA-40) data are used as a reference where appropriate. It is found that the simulated QBO is realistic in many details. Resolved large-scale waves are particularly important for the westerly phase, while parameterized gravity wave drag is more important for the easterly phase. Advective zonal wind tendencies are important for asymmetries between westerly and easterly phases, as found for the suppression of the easterly phase downward propagation. The simulation of the QBO improves the tropical upwelling and the atmospheric tape recorder compared to a model without a QBO. The semiannual oscillation is simulated realistically only if the QBO is represented. In sensitivity tests, it is found that the simulated QBO is strongly sensitive to changes in the gravity wave sources. The sensitivity to the tested range of horizontal resolutions is small. The stratospheric vertical resolution must be better than 1 km to simulate a realistic QBO.

Past and future conditions for polar stratospheric cloud formation simulated by the Canadian Middle Atmosphere Model

We analyze here the polar stratospheric temperatures in an ensemble of three 150-year integrations of the Canadian Middle Atmosphere Model (CMAM), an interactive chemistry-climate model which simulates ozone depletion and recovery, as well as climate change. A key motivation is to understand possible mechanisms for the observed trend in the extent of conditions favourable for polar stratospheric cloud (PSC) formation in the Arctic winter lower stratosphere. We find that in the Antarctic winter lower stratosphere, the low temperature extremes required for PSC formation increase in the model as ozone is depleted, but remain steady through the twenty-first century as the warming from ozone recovery roughly balances the cooling from climate change. Thus, ozone depletion itself plays a major role in the Antarctic trends in low temperature extremes. The model trend in low temperature extremes in the Arctic through the latter half of the twentieth century is weaker and less statistically robust than the observed trend. It is not projected to continue into the future. Ozone depletion in the Arctic is weaker in the CMAM than in observations, which may account for the weak past trend in low temperature extremes. In the future, radiative cooling in the Arctic winter due to climate change is more than compensated by an increase in dynamically driven downwelling over the pole.

Conditioning model output statistics of regional climate model precipitation on circulation patterns

Dynamical downscaling of Global Climate Models (GCMs) through regional climate models (RCMs) potentially improves the usability of the output for hydrological impact studies. However, a further downscaling or interpolation of precipitation from RCMs is often needed to match the precipitation characteristics at the local scale. This study analysed three Model Output Statistics (MOS) techniques to adjust RCM precipitation; (1) a simple direct method (DM), (2) quantile-quantile mapping (QM) and (3) a distribution-based scaling (DBS) approach. The modelled precipitation was daily means from 16 RCMs driven by ERA40 reanalysis data over the 1961–2000 provided by the ENSEMBLES (ENSEMBLE-based Predictions of Climate Changes and their Impacts) project over a small catchment located in the Midlands, UK. All methods were conditioned on the entire time series, separate months and using an objective classification of Lamb's weather types. The performance of the MOS techniques were assessed regarding temporal and spatial characteristics of the precipitation fields, as well as modelled runoff using the HBV rainfall-runoff model. The results indicate that the DBS conditioned on classification patterns performed better than the other methods, however an ensemble approach in terms of both climate models and downscaling methods is recommended to account for uncertainties in the MOS methods.

Extended Canadian middle atmosphere model: zonal-mean climatology and physical parameterizations

This paper describes the energetics and zonal-mean state of the upward extension of the Canadian Middle Atmosphere Model, which extends from the ground to ~210 km. The model includes realistic parameterizations of the major physical processes from the ground up to the lower thermosphere and exhibits a broad spectrum of geophysical variability. The rationale for the extended model is to examine the nature of the physical and dynamical processes in the mesosphere/lower thermosphere (MLT) region without the artificial effects of an imposed sponge layer which can modify the circulation in an unrealistic manner. The zonal-mean distributions of temperature and zonal wind are found to be in reasonable agreement with observations in most parts of the model domain below ~150 km. Analysis of the global-average energy and momentum budgets reveals a balance between solar extreme ultraviolet heating and molecular diffusion and a thermally direct viscous meridional circulation above 130 km, with the viscosity coming from molecular diffusion and ion drag. Below 70 km, radiative equilibrium prevails in the global mean. In the MLT region between ~70 and 120 km, many processes contribute to the global energy budget. At solstice, there is a thermally indirect meridional circulation driven mainly by parameterized nonorographic gravity-wave drag. This circulation provides a net global cooling of up to 25 K d^-1.

Empirical test of an agricultural landscape model: the importance of farmer preference for risk aversion and crop complexity

Developing models to predict the effects of social and economic change on agricultural landscapes is an important challenge. Model development often involves making decisions about which aspects of the system require detailed description and which are reasonably insensitive to the assumptions. However, important components of the system are often left out because parameter estimates are unavailable. In particular, measurements of the relative influence of different objectives, such as risk, environmental management, on farmer decision making, have proven difficult to quantify. We describe a model that can make predictions of land use on the basis of profit alone or with the inclusion of explicit additional objectives. Importantly, our model is specifically designed to use parameter estimates for additional objectives obtained via farmer interviews. By statistically comparing the outputs of this model with a large farm-level land-use data set, we show that cropping patterns in the United Kingdom contain a significant contribution from farmer’s preference for objectives other than profit. In particular, we found that risk aversion had an effect on the accuracy of model predictions, whereas preference for a particular number of crops grown was less important. While nonprofit objectives have frequently been identified as factors in farmers’ decision making, our results take this analysis further by demonstrating the relationship between these preferences and actual cropping patterns.

Metabifurcation analysis of a mean field model of the cortex

Mean field models (MFMs) of cortical tissue incorporate salient, average features of neural masses in order to model activity at the population level, thereby linking microscopic physiology to macroscopic observations, e.g., with the electroencephalogram (EEG). One of the common aspects of MFM descriptions is the presence of a high-dimensional parameter space capturing neurobiological attributes deemed relevant to the brain dynamics of interest. We study the physiological parameter space of a MFM of electrocortical activity and discover robust correlations between physiological attributes of the model cortex and its dynamical features. These correlations are revealed by the study of bifurcation plots, which show that the model responses to changes in inhibition belong to two archetypal categories or “families”. After investigating and characterizing them in depth, we discuss their essential differences in terms of four important aspects: power responses with respect to the modeled action of anesthetics, reaction to exogenous stimuli such as thalamic input, and distributions of model parameters and oscillatory repertoires when inhibition is enhanced. Furthermore, while the complexity of sustained periodic orbits differs significantly between families, we are able to show how metamorphoses between the families can be brought about by exogenous stimuli. We here unveil links between measurable physiological attributes of the brain and dynamical patterns that are not accessible by linear methods. They instead emerge when the nonlinear structure of parameter space is partitioned according to bifurcation responses. We call this general method “metabifurcation analysis”. The partitioning cannot be achieved by the investigation of only a small number of parameter sets and is instead the result of an automated bifurcation analysis of a representative sample of 73,454 physiologically admissible parameter sets. Our approach generalizes straightforwardly and is well suited to probing the dynamics of other models with large and complex parameter spaces.

Dynamical complexity in a mean-field model of human EEG

A recently proposed mean-field theory of mammalian cortex rhythmogenesis describes the salient features of electrical activity in the cerebral macrocolumn, with the use of inhibitory and excitatory neuronal populations (Liley et al 2002). This model is capable of producing a range of important human EEG (electroencephalogram) features such as the alpha rhythm, the 40 Hz activity thought to be associated with conscious awareness (Bojak & Liley 2007) and the changes in EEG spectral power associated with general anesthetic effect (Bojak & Liley 2005). From the point of view of nonlinear dynamics, the model entails a vast parameter space within which multistability, pseudoperiodic regimes, various routes to chaos, fat fractals and rich bifurcation scenarios occur for physiologically relevant parameter values (van Veen & Liley 2006). The origin and the character of this complex behaviour, and its relevance for EEG activity will be illustrated. The existence of short-lived unstable brain states will also be discussed in terms of the available theoretical and experimental results. A perspective on future analysis will conclude the presentation.

Regional changes in wind energy potential over Europe using regional climate model ensemble projections

The impact of climate change on wind power generation potentials over Europe is investigated by considering ensemble projections from two regional climate models (RCMs) driven by a global climate model (GCM). Wind energy density and its interannual variability are estimated based on hourly near-surface wind speeds. Additionally, the possible impact of climatic changes on the energy output of a sample 2.5-MW turbine is discussed. GCM-driven RCM simulations capture the behavior and variability of current wind energy indices, even though some differences exist when compared with reanalysis-driven RCM simulations. Toward the end of the twenty-first century, projections show significant changes of energy density on annual average across Europe that are substantially stronger in seasonal terms. The emergence time of these changes varies from region to region and season to season, but some long-term trends are already statistically significant in the middle of the twenty-first century. Over northern and central Europe, the wind energy potential is projected to increase, particularly in winter and autumn. In contrast, energy potential over southern Europe may experience a decrease in all seasons except for the Aegean Sea. Changes for wind energy output follow the same patterns but are of smaller magnitude. The GCM/RCM model chains project a significant intensification of both interannual and intra-annual variability of energy density over parts of western and central Europe, thus imposing new challenges to a reliable pan-European energy supply in future decades.

Analysis of frequency and intensity of European winter storm events from a multi-model perspective, at synoptic and regional scales

This study focuses on the analysis of winter (October-November-December-January-February-March; ONDJFM) storm events and their changes due to increased anthropogenic greenhouse gas concentrations over Europe. In order to assess uncertainties that are due to model formulation, 4 regional climate models (RCMs) with 5 high resolution experiments, and 4 global general circulation models (GCMs) are considered. Firstly, cyclone systems as synoptic scale processes in winter are investigated, as they are a principal cause of the occurrence of extreme, damage-causing wind speeds. This is achieved by use of an objective cyclone identification and tracking algorithm applied to GCMs. Secondly, changes in extreme near-surface wind speeds are analysed. Based on percentile thresholds, the studied extreme wind speed indices allow a consistent analysis over Europe that takes systematic deviations of the models into account. Relative changes in both intensity and frequency of extreme winds and their related uncertainties are assessed and related to changing patterns of extreme cyclones. A common feature of all investigated GCMs is a reduced track density over central Europe under climate change conditions, if all systems are considered. If only extreme (i.e. the strongest 5%) cyclones are taken into account, an increasing cyclone activity for western parts of central Europe is apparent; however, the climate change signal reveals a reduced spatial coherency when compared to all systems, which exposes partially contrary results. With respect to extreme wind speeds, significant positive changes in intensity and frequency are obtained over at least 3 and 20% of the European domain under study (35–72°N and 15°W–43°E), respectively. Location and extension of the affected areas (up to 60 and 50% of the domain for intensity and frequency, respectively), as well as levels of changes (up to +15 and +200% for intensity and frequency, respectively) are shown to be highly dependent on the driving GCM, whereas differences between RCMs when driven by the same GCM are relatively small.

Nonlinear stability of Eady's model

A nonlinear stability theorem is established for Eady's model of baroclinic flow. In particular, the Eady basic state is shown to be nonlinearly stable (for arbitrary shear) provided (Δz)/(Δy) > 2(5)^1/2f/(πN),where Δz is the height of the domain, Δy the channel width, f the Coriolis parameter, and N the buoyancy frequency. When this criterion is satisfied, explicit bounds can be derived on the disturbance potential enstrophy, the disturbance energy, and the disturbance available potential energy on the rigid lids, which are expressed in terms of the initial disturbance fields. The disturbances are completely general (with nonzero potential vorticity) and are not assumed to be of small amplitude. The results may be regarded as an extension of Arnol'd's second nonlinear stability theorem to continuously stratified quasigeostrophic baroclinic flow.