Peter Guttorp

The Pacific Northwest (PNW) of North America has high seismic hazard due to numerous earthquake sources under populated areas. It hosts several tectonic regimes and subregional seismic zones that are hypothesized to have different patterns of earthquake and aftershock occurrence. It is also predisposed to earthquake swarms, which can complicate the statistical modeling of these patterns. We present the first statistical seismicity model of the PNW catalog using the epidemic-type aftershock sequence (ETAS) framework. We develop a Bayesian inference procedure that provides a stable estimation of both ETAS parameters and their uncertainties for different sets of PNW earthquakes, even those with very sparse catalogs. The Bayesian approach allows us to investigate how parameter estimates change between the intraslab and crustal tectonic regimes, the northern and southern PNW, and when swarms are included and excluded from the catalog. We also utilize our Bayesian framework to calculate parameter estimates under different prior beliefs about PNW seismicity, as well as to propagate catalog measurement errors into ETAS parameter estimates. We discuss the implications of parameter differences across the region for aftershock forecasting for the PNW.

In the study of the global climate, ocean temperature estimates use sea surface temperature (SST) anomalies instead of marine air temperature (MAT) anomalies. A key question to ask is whether biases result from this choice. In this article we employ hierarchical statistical models to investigate spatiotemporal differences between SST and MAT and their anomalies in the tropical Pacific. The analysis uses observations from the Tropical Atmosphere Ocean (TAO) buoy network and the ERA5 data product. Our spatiotemporal modeling approach accounts for missing data in the observation network and allows for full uncertainty quantification. Our findings indicate evidence that SST and MAT are interchangeable in the tropical Pacific when we calculate seasonally adjusted monthly anomalies.

Fitting parametric seismological models to earthquake catalogs often comes with numerical challenges, especially when catalogs are small. An alternative way to quantify parameter values for a seismic region is by eliciting expert opinions on the seismological characteristics that each parameter corresponds to. For instance, expert beliefs on aftershock patterns can be formulated into prior distributions for aftershock parameters, for example, for the epidemic‐type aftershock sequence (ETAS) model. We illustrate such a method by not only eliciting priors for ETAS parameters for the Pacific Northwest (PNW), a subduction zone with a complex tectonic environment, but also a relatively small catalog. We compare these priors with those suggested by the ETAS literature for global subduction zones, discussing implications for aftershock forecasting for the PNW.

We introduce a class of proper scoring rules for evaluating spatial point process forecasts based on summary statistics. These scoring rules rely on Monte Carlo approximations of expectations and can therefore easily be evaluated for any point process model that can be simulated. In this regard, they are more flexible than the commonly used logarithmic score and other existing proper scores for point process predictions. The scoring rules allow for evaluating the calibration of a model to specific aspects of a point process, such as its spatial distribution or tendency toward clustering. Using simulations, we analyze the sensitivity of our scoring rules to different aspects of the forecasts and compare it to the logarithmic score. Applications to earthquake occurrences in northern California, United States and the spatial distribution of Pacific silver firs in Findley Lake Reserve in Washington highlight the usefulness of our scores for scientific model selection.

A new statistical approach to validating climate models is introduced. First, five observational estimates of global mean surface temperature with estimated standard errors are combined into one data product, latent observed annual global temperature anomalies for the years 1880–2014, using a Bayesian hierarchical statistical approach. Summarizing these observed anomalies, estimates of smooth trend, levels of warming, and residual dependence as summarized by the spectral density function, are provided with simultaneous 95% credible bands. Then, corresponding estimates of smooth trend, levels of warming, and residual dependence are produced for sixth Climate Model Intercomparison Project (CMIP6) historical simulations analyzed at the annual global temperature anomaly scale, and compared to these bands. Among our results, we find that 93 out of the 318 CMIP6 historical model runs contain trends fitting inside the simultaneous bands for the smooth trend constructed from the data products, and for residual temporal dependence 69 out of 318 model runs contain spectral density functions that are within the corresponding data-product-based-bands. We estimate the mean global temperature increase from 1995–2014 relative to 1880–1899, from the data product, to be 0.896°C with a 95% credible interval of between 0.877 and 0.915. We find that 14 CMIP6 model runs agree with this interval, 197 model runs lead to a smaller temperature increase globally, and 107 model runs lead to a larger temperature increase.

The Pacific Northwest (PNW) of North America encompasses diverse tectonic settings that can produce damaging earthquakes near population centers. Seismicity in this region is often clustered into aftershock sequences and swarms, and their patterns and frequencies differ across subregions or tectonic regimes. Characterizing the seismicity of the PNW requires a catalog of observed earthquakes. Furthermore, applications with the catalog may require earthquake clusters to be identified and regarded separately. Unlike previous studies, we explicate how to overcome challenges when combining catalogs from different countries, particularly in accounting for duplicate events and other discrepancies. We apply this to merge authoritative catalogs for the United States and Canadian portions of the PNW, along with a third dataset with data quality measures. We also perform a window‐based search for earthquake clusters, which then get labeled as possible or definite swarms or aftershock sequences. We further split the catalog into its two primary tectonic regimes. We then study the PNW catalog’s completeness, and the extent to which this varies between the northern and southern parts of the region. We provide a harmonized international PNW catalog with derived variables describing earthquake clustering and tectonic regimes. This entire processing pipeline has also been fully documented and is supported with software, enabling its use in other seismic regions.

Earthquake models can produce aftershock forecasts, which have recently been released to lay audiences. While visualization literature suggests that displaying forecast uncertainty can improve how forecast maps are used, research on uncertainty visualization is missing from earthquake science. We designed a pre-registered online experiment to test the effectiveness of three visualization techniques for displaying aftershock forecast maps and their uncertainty. These maps showed the forecasted number of aftershocks at each location for a week following a hypothetical mainshock, along with the uncertainty around each location's forecast. Three different uncertainty visualizations were produced: (1) forecast and uncertainty maps adjacent to one another; (2) the forecast map depicted in a color scheme, with the uncertainty shown by the transparency of the color; and (3) two maps that showed the lower and upper bounds of the forecast distribution at each location. We compared the three uncertainty visualizations using tasks that were specifically designed to address broadly applicable and user-generated communication goals. We compared task responses between participants using uncertainty visualizations and using the forecast map shown without its uncertainty (the current practice). Participants completed two map-reading tasks that targeted several dimensions of the readability of uncertainty visualizations. Participants then performed a Comparative Judgment task, which demonstrated whether a visualization was successful in reaching two key communication goals: indicating where many aftershocks and no aftershocks are likely (sure bets) and where the forecast is low but the uncertainty is high enough to imply potential risk (surprises). All visualizations performed equally well in the goal of communicating sure bet situations. But the visualization with lower and upper bounds was substantially better than the other designs at communicating surprises. These results have implications for the visual communication of forecast uncertainty both within and beyond earthquake science.

Recently, several global temperature series have been updated using new data sets, new methods, and importantly, assessments of their uncertainties. This enables us to produce a timely estimate of the annual global mean temperature with a smaller combined estimate of uncertainty. We describe the hierarchical model we propose, and a Bayesian scheme for fitting the model, allowing for dependence between the data sets, which all use some of the same observations. The discrepancy between individual data series and the combined estimate illustrates potential sources of deviation between them. In addition, we test the sensitivity of the results to each of the series, using a leave-one-out approach. This is a way of combining all the data sets in a way that improves on the straight or precision weighted ensemble mean, thus providing a more authoritative global temperature series with corresponding standard errors, which are smaller than that of individual products. Using the combined estimate of the global temperature series, we estimate that the global temperature has increased 1.2°C with a standard error of 0.03°C over the 1880–1900 average. By taking into account the uncertainties of the estimates rather than just comparing the estimates, we find that the probability that 2020 was the warmest year on record is 0.44, while the years 2015–2020 are virtually certain to have been the six warmest years in recorded history. We show that our estimate performs similarly to the reanalysis product ERA5, and that the satellite record from University of Alabama does not agree very well neither with ERA5 nor with our product.

Recently several global temperature series have been updated using new data sets, new methods, and for a statistician most importantly, assessments of their uncertainties. This enables us to produce a timely estimate of the annual global mean temperature with a combined estimate of uncertainty. We describe the hierarchical model we propose, and a Bayesian scheme for fitting the model. In addition, we test the sensitivity to the results to each of the series, identifying groups of data products that act similar to one another. Using the combined estimate of the global temperature series, we estimate that the probability that 2020 was the warmest year on record is 0.41.

We focus on the discussion of modeling processes that are observed at fixed locations of a region (geostatistics). A standard approach is to assume that the process of interest follows a Gaussian Process with some mean and (valid) covariance functions. It is common to model the covariance function as the product between a variance parameter, and a correlation function which is a function of the Euclidean distance between locations. This implies that the distribution of the process is unchanged when the origin of the index set is translated, and the process is invariant under rotation about the origin; that is the process is stationary and isotropic or homogeneous. However, the assumption of stationarity and isotropy (homogeneity) rarely holds in practice. Commonly, the correlation structures of such processes are influenced by local characteristics resulting in different behaviors in neighborhoods of different spatial locations. We review models that allow for heterogeneous covariance structures and point to some avenues of future research.

We introduce a class of proper scoring rules for evaluating spatial point process forecastsbased on summary statistics. These scoring rules rely on Monte-Carlo approximation ofan expectation and can therefore easily be evaluated for any point process model that canbe simulated. In this regard, they are more flexible than the commonly used logarithmicscore; they are also fruitful for evaluating the calibration of a model to specific aspectsof a point process, such as its spatial distribution or tendency towards clustering. Weshow using simulations that our scoring rules are able to discern between competingmodels better than the logarithmic score. An application on growth in Pacific silver firtrees demonstrates the promise of our scores for scientific model selection.

We propose an interacting cluster model for the spatial distribution of epidermal nerve fibers (ENF). The model consists of a spatial process of parent points modeling the base points of nerve fiber bundles. To each base point there is an offspring point process of fibers end points associated. The parent process is, possibly, inhibited by fiber endings that belong to different base bundles. The fibers themselves have random length and spatial orientation. We consider a non-orphan process where we can connect each offspring to a parent. We examine the processes that can be described with our model, how coefficient estimation can be performed under the Bayesian paradigm via Markov chain Monte Carlo methods, and detail approaches for the implementation of efficient MCMC sampling schemes. We study the performance of the estimation procedure through a simulation study. An application to data from skin blister biopsy images of ENFs is presented.

Statisticians are in general agreement that there are flaws in how science is currently practiced; there is less agreement in how to make repairs. Our prescription for a Post-p < 0.05 Era is to develop and teach courses that expand our view of what constitutes the domain of statistics and thereby bridge undergraduate statistics coursework and the graduate student experience of applying statistics in research. Such courses can speed up the process of gaining statistical wisdom by giving students insight into the human propensity to make statistical errors, the meaning of a single test within a research project, ways in which p-values work and don't work as expected, the role of statistics in the lifecycle of science, and best practices for statistical communication. The course we have developed follows the story of how we use data to understand the world, leveraging simulation-based approaches to perform customized analyses and evaluate the behavior of statistical procedures. We provide ideas for expanding beyond the traditional classroom, two example activities, and a course syllabus as well as the set of statistical best practices for creating and consuming scientific information that we develop during the course.

We introduce a class of proper scoring rules for evaluating spatial point process forecastsbased on summary statistics. These scoring rules rely on Monte-Carlo approximation ofan expectation and can therefore easily be evaluated for any point process model thatcan be simulated. In this regard they are more flexible than the commonly used logar-ithmic score which cannot be evaluated for many point process models, as their densityis only known up to an untractable constant. In simulation studies we demonstrate theusefulness of our scores. Furthermore we consider a scoring rule, the quantile score, thatis commonly used to validate earthquake rate predictions, and show that it lacks propri-ety. As a consequence, several tests that are commonly applied in this context are biasedand systematically favour predictive distributions that are too uniform. We suggest toremedy this issue by replacing the commonly used one-sided by two-sided tests.

Sea level rise poses a threat to the Norwegian coastal city of Bergen and its historic harbour. The threat could be reduced, but greater flood protection comes at greater cost. And, of course, no one knows for certain how far sea level will rise in future. Decision‐makers must therefore decide what to do, and how much to spend, without knowing exactly how bad things could get. Peter Guttorp and Thordis L. Thorarinsdottir explain the problem, and how to deal with the uncertainty

Wildfire behaviors are complex and are of interest to fire managers and scientists for a variety of reasons. Many of these important behaviors are directly measured from the cumulative burn area time series of individual wildfires; however, estimating cumulative burn area time series is challenging due to the magnitude of measurement errors and missing entries. To resolve this, we introduce two state space models for reconstructing wildfire burn area using repeated observations from multiple data sources that include different levels of measurement error and temporal gaps. The constant growth parameter model uses a few parameters and assumes a burn area time series that follows a logistic growth curve. The non-constant growth parameter model uses a time-varying logistic growth curve to produce detailed estimates of the burn area time series that permit sudden pauses and pulses of growth. We apply both reconstruction models to burn area data from 13 large wildfire incidents to compare the quality of the burn area time series reconstructions and computational requirements. The constant growth parameter model reconstructs burn area time series with minimal computational requirements, but inadequately fits observed data in most cases. The non-constant growth parameter model better describes burn area time series, but can also be highly computationally demanding. Sensitivity analyses suggest that in a typical application, the reconstructed cumulative burn area time series is fairly robust to minor changes in the prior distributions.

A point process data set on epidermal nerve fiber bundles is used as the basis for a series of experiments in identifying clusters. In this data set we know which secondary points are connected to which primary points.We will pretend that we do not have this information, and using Bayesian tools estimate the information from data. For comparison we also use k-means clustering. We do this both for known cluster centers, and when the cluster centers must be estimated from data.

A separation between the academic subjects statistics and mathematical statistics has existed in Sweden almost as long as there have been statistics professors. The same distinction has not been maintained in other countries. Why has it been kept for so long in Sweden, and what consequences may it have had? In May 2015, it was 100 years since Mathematical Statistics was formally established as an academic discipline at a Swedish university where Statistics had existed since the turn of the century. We give an account of the debate in Lund and elsewhere about this division during the first decades after 1900 and present two of its leading personalities. The Lund University astronomer (and mathematical statistician) C. V. L. Charlier was a leading proponent for a position in mathematical statistics at the university. Charlier’s adversary in the debate was Pontus Fahlbeck, professor in political science and statistics, who reserved the word statistics for ‘statistics as a social science’. Charlier not only secured the first academic position in Sweden in mathematical statistics for his former PhD student Sven Wicksell but also demonstrated that a mathematical statistician can be influential in matters of state, finance as well as in different natural sciences. Fahlbeck saw mathematical statistics as a set of tools that sometimes could be useful in his brand of statistics. After a summary of the organisational, educational and scientific growth of the statistical sciences in Sweden that has taken place during the last 50 years, we discuss what effects the Charlier–Fahlbeck divergence might have had on this development.

Sea level rise has serious consequences for harbor infrastructure, storm drains and sewer systems, and many other issues. Adapting to sea level rise requires comparing different possible adaptation strategies, comparing the cost of different actions (including no action), and assessing where and at what point in timethe chosenstrategy should be implemented. All these decisions must be madeunderconsiderableuncertainty–in the amount of sea level rise, in the cost and prioritization of adaptation actions, and in the implications of no action. Here we develop two illustrative examples: for Bergen on Norway’s west coast and for Esbjerg on the west coast of Denmark, to highlight how technical efforts to understand and quantify uncertainties in hydrologic projections can be coupled with concrete decision-problems framed by the needs of the end-users using statistical formulations. Different components of uncertainty are visualized. We demonstrate the value of uncertainties and show for example that failing to take uncertainty into account can result in the median projected damage costs being an order of magnitude smaller.

Sea level rise has serious consequences for harbor infrastructure, storm drains and sewer systems, and many other issues. Adapting to sea level rise requires comparing different possible adaptation strategies, comparing the cost of different actions (including no action), and assessing where and at what point in time the chosen strategy should be implemented. All these decisions must be made under considerable uncertainty—in the amount of sea level rise, in the cost and prioritization of adaptation actions, and in the implications of no action. Here we develop two illustrative examples: for Bergen on Norway's west coast and for Esbjerg on the west coast of Denmark, to highlight how technical efforts to understand and quantify uncertainties in hydrologic projections can be coupled with concrete decision-problems framed by the needs of the end-users using statistical formulations. Different components of uncertainty are visualized. We demonstrate the value of uncertainties and show for example that failing to take uncertainty into account can result in the median-projected damage costs being an order of magnitude smaller

Abstract• References• Author information Climate simulation data comprise a range of different phenomena with complex and interacting processes. Yet our understanding of the climate is incomplete despite the huge volumes of data, of which only a small fraction has been explored, and many questions remain, particularly those on the character and origin of uncertainties associated with model simulations and how further modelling efforts can improve understanding. Here, we question whether climate model information could be used more effectively and how so-called 'ensembles of opportunity' should be interpreted. Statisticians can contribute substantially to designing 'smarter' ensemble experiments, improving the distillation of information from ensembles, and helping interpret the relative merits of additional simulations. Future progress may be enhanced by increasing collaborations with statisticians.

Climate studies often fit linear trends to data. In many cases simplifying assumptions such as independent errors and constant variance are used. We review a variety of approaches to estimating linear trends, and illustrate with US temperature data how oversimplified assumptions may lead to false significance. We outline a variety of methods to fit nonlinear trend models. Using the Berkeley Earth global data set we show that a bent cable fit is better than a linear fit for this series. We also review spatial and spatiotemporal trend models in mean, variance and extremes, as well as models with long term memory structure.

Epidermal nerve fibre (ENF) density and morphology are used to study small fibre involvement in diabetic, HIV, chemotherapy induced and other neuropathies. ENF density and summed length of ENFs per epidermal surface area are reduced, and ENFs may appear more clustered within the epidermis in subjects with small fibre neuropathy than in healthy subjects. Therefore, it is important to understand the spatial structure of ENFs. In this paper, we compare the ENF patterns between healthy subjects and subjects suffering from mild diabetic neuropathy. The study is based on suction skin blister specimens from the right foot of 32 healthy subjects and eight subjects with mild diabetic neuropathy. We regard the ENF entry point (location where the trunks of a nerve enters the epidermis) and ENF end point (termination of the nerve fibres) patterns as realizations of spatial point processes, and develop tools that can be used in the analysis and modelling of ENF patterns. We use spatial summary statistics and shift plots and define a new tool, reactive territory, to study the spatial patterns and to compare the patterns of the two groups. We will also introduce a simple model for these data in order to understand the growth process of the nerve fibres.

Dynamical downscaling of earth system models is intended to produce high-resolution climate information at regional to local scales. Current models, while adequate for describing temperature distributions at relatively small scales, struggle when it comes to describing precipitation distributions. In order to better match the distribution of observed precipitation over Norway, we consider approaches to statistical adjustment of the output from a regional climate model when forced with ERA-40 reanalysis boundary conditions. As a second step, we try to correct downscalings of historical climate model runs using these transformations built from downscaled ERA-40 data. Unless such calibrations are successful, it is difficult to argue that scenario-based downscaled climate projections are realistic and useful for decision makers. We study both full quantile calibrations and several different methods that correct individual quantiles separately using random field models. Results based on cross-validation show that while a full quantile calibration is not very effective in this case, one can correct individual quantiles satisfactorily if the spatial structure in the data are accounted for. Interestingly, different methods are favoured depending on whether ERA-40 data or historical climate model runs are adjusted.

How do people look at art? Are there any differences between how experienced and inexperienced art viewers look at a painting? We approach these questions by analyzing and modeling eye movement data from a cognitive art research experiment, where the eye movements of twenty test subjects, ten experienced and ten inexperienced art viewers, were recorded while they were looking at paintings. Eye movements consist of stops of the gaze as well as jumps between the stops. Hence, the observed gaze stop locations can be thought of as a spatial point pattern, which can be modeled by a spatio-temporal point process. We introduce some statistical tools to analyze the spatio-temporal eye movement data, and compare the eye movements of experienced and inexperienced art viewers. In addition, we develop a stochastic model, which is rather simple but fits quite well to the eye movement data, to further investigate the differences between the two groups through functional summary statistics.

Sea level rise is a threat to many coastal communities, and projection of future sea level for different climate change scenarios is an important societal task. In this paper, we first construct a time series regression model to predict global sea level from global temperature. The model is fitted to two sea level data sets (with and without corrections for reservoir storage of water) and three temperature data sets. The effect of smoothing before regression is also studied. Finally, we apply a novel methodology to develop confidence bands for the projected sea level, simultaneously for 2000-2100, under different scenarios, using temperature projections from the latest climate modeling experiment. The main finding is that different methods for sea level projection, which appear to disagree, have confidence intervals that overlap, when taking into account the different sources of variability in the analyses.

For a statistician, climate is the distribution of weather and other variables that are part of the climate system. This distribution changes over time. This review considers some aspects of climate data, climate model assessment, and uncertainty estimation pertinent to climate issues, focusing mainly on temperatures. Some interesting methodological needs that arise from these issues are also considered.

The process of moving from an ensemble of global climate model temperature projections to local sea level projections requires several steps. Sea level was estimated in Olympia, Washington (a city that is very concerned with sea level rise because parts of downtown are barely above mean highest high tide), by relating global mean temperature to global sea level; relating global sea level to sea levels at Seattle, Washington; and finally relating Seattle to Olympia. There has long been a realization that accurate assessment of the precision of projections is needed for science-based policy decisions. When a string of statistical and/or deterministic models is connected, the uncertainty of each individual model needs to be accounted for. Here the uncertainty is quantified for each model in the described system and the total uncertainty is assessed in a cascading effect throughout the system. The projected sea level rise over time and its total estimated uncertainty are visualized simultaneously for the years 2000–2100, the increased uncertainty due to each of the component models at a particular projection year is identified, and estimates of the time at which a certain sea level rise will first be reached are made.

A recent observation in NOAA's National Climatic Data Center's monthly assessment of the state of the climate was that contiguous U.S. average monthly temperatures were in the top third of monthly ranked historical temperatures for 13 straight months from June 2011 to June 2012. The chance of such a streak occurring randomly was quoted as (1/3)13, or about one in 1.6 million. The streak continued for three more months before the October 2012 value dropped below the upper tercile. The climate system displays a degree of persistence that increases this probability relative to the assumption of independence. This paper puts forth different statistical techniques that more accurately quantify the probability of this and other such streaks. We consider how much more likely streaks are when an underlying warming trend is accounted for in the record, the chance of streaks occurring anywhere in the record, and the distribution of the record's longest streak.

Wind direction plays an important role in the spread of air pollutants over a geographical region. We discuss how to include wind directional information in the covariance function of spatial models. Our models are based on a constructive convolution approach, wherein a spatial process is described as a convolution between a spatially varying smoothing kernel and a white noise process. We describe two different ways of accounting for wind direction: one makes use of a non-stationary version of the Matérn covariance function and the other a kernel function that resembles the exponential correlation function. We fit the models proposed to ground level ozone observed at a monitoring network in north-eastern USA. We compare our wind-based covariance models with three other models: two that make use of standard covariance functions, and one whose kernel function varies across space according to latent spatial processes. The inference procedure is performed under the Bayesian paradigm, and uncertainty about parameter estimation is naturally accounted for when performing spatial interpolation. Samples from the posterior distribution under our proposed models are obtained much faster when compared with the model based on latent spatial processes. Although fitted values that are obtained under our proposed models and those obtained based on latent processes are quite similar, our models provided smaller ranges of the predictive posterior credible intervals.

Ranking years based on statistical estimates of regional and temporal averages is subject to uncertainty. This uncertainty can in fact be quite substantial and can be described by the rank distribution of an ensemble of such averages. The authors develop a method for estimating it using simulation. The effect of temporal correlation is quite limited in the case studied in this paper: the contiguous United States' annual-mean temperature. The method also allows assessment of derived quantities such as the probability of a given year being one of the 10 warmest in the historical record.

We compare some different approaches to estimating daily mean temperature (DMT). In many countries, the routine approach is to calculate the average of the directly measured minimum and maximum daily temperature. In some, the maximum and minimum are obtained from hourly measurements. In other countries, temperature readings at specific times throughout the day are taken into account. For example, the Swedish approach uses a linear combination of five temperature readings, including the minimum and the maximum, with coefficients that depend on longitude and month. We first look at data with very high temporal resolution, and compare some different approaches to estimating DMT. Then, we compare the Swedish formula to various averages of the daily minimum and maximum, finding the latter method being substantially less precise. We finally compare the Swedish formula to hourly averages, and find that a recalibrated linear combination improves estimation accuracy.

Climate models are mathematical models that describe the temporal evolution of climate, oceans, atmosphere, ice, and land-use processes, across a spatial domain via systems of partial differential equations. Because these models cannot be solved analytically, the model output is generated numerically over grid boxes. Regional climate models (RCMs), or the dynamic downscaling of global climate models to regional scales, are often used for planning purposes, and it is important to assess carefully the uncertainty of such models. We evaluate the Swedish Meteorological and Hydrological Institute (SMHI) RCM by comparing its model output at the grid box level, with the predictions obtained from two observation-driven spatio-temporal statistical models. The “downscaling model” combines the spatially and temporally smoothed climate model output with temperature observations at synoptic stations in a spatio-temporal linear statistical model. The “upscaling model” describes the observational temperature alone at the daily scale, via a spatio-temporal model that includes a wavelet-based trend, spatially varying seasonality, along with volatility and long-range dependence terms. Both statistical models have the ability to make predictions at a seasonal scale, both at point and grid box level. In the years 1962–2007 in South Central Sweden, we show that the climate model performs well in predicting the annual and seasonal average temperature at three reserved stations, but there are interesting differences among the model output and the statistical model-based predictions at the grid box level.

Predictions of climate change are uncertain mainly because of uncertainties in the emissions of greenhouse gases and how sensitive the climate is to changes in the abundance of the atmospheric constituents. The equilibrium climate sensitivity is defined as the temperature increase because of a doubling of the CO2 concentration in the atmosphere when the climate reaches a new steady state. CO2 is only one out of the several external factors that affect the global temperature, called radiative forcing mechanisms as a collective term. In this paper, we present a model framework for estimating the climate sensitivity. The core of the model is a simple, deterministic climate model based on elementary physical laws such as energy balance. It models yearly hemispheric surface temperature and global ocean heat content as a function of historical radiative forcing. This deterministic model is combined with an empirical, stochastic model and fitted to observations on global temperature and ocean heat content, conditioned on estimates of historical radiative forcing. We use a Bayesian framework, with informative priors on a subset of the parameters and flat priors on the climate sensitivity and the remaining parameters. The model is estimated by Markov Chain Monte Carlo techniques

The use of properties of a Poisson process to study the randomness of stars is traced back to a 1767 paper. The process was used and rediscovered many times, and we mention some of the early scientific areas. The name Poisson process was first used in print in 1940, and we believe the term was coined in the corridors of Stockholm University some time between 1936 and 1939.We follow the early developments of doubly stochastic processes and cluster processes, and describe different efforts to apply the Markov property to point processes.

The Bayesian approach to statistical inference has in recent years become very popular, especially in the analysis of complex data sets. This is largely due to the development of Markov chain Monte Carlo methods, which expand the scope of application of Bayesian methods considerably. In this paper, we review the Bayesian contributions to inference for point processes. We focus on non-Markovian processes, specifically Poisson and related models, doubly stochastic models, and cluster models. We also discuss Bayesian model selection for these models and give examples in which Bayes factors are applied both directly and indirectly through a reversible jump algorithm.

Many problems in current climate research deal with extreme events. Since by definition there are few observations of really extreme events, it is a statistical challenge to assess whether observed trends are significant. In this paper we illustrate one method to look for climate signals in extreme temperature data, and how to compare the data to a climate simulation based on a regional climate model.

Classical assessments of temperature trends are based on the analysis of a small number of time series. Considering trend to be only smooth changes of the mean value of a stochastic process through time is limiting, because it does not provide a mechanism to study changes of the mean that could also occur over space. Thus, in studies of climate there is a substantial interest in being able to jointly characterize temperature trends over time and space. In this article we build wavelet-based space-time hierarchical Bayesian models that can be used to simultaneously model trend, seasonality, and error, allowing for the possibility that the error process may exhibit space-time long-range dependence. We demonstrate how these statistical models can be used to assess the significance of trend over time and space. We motivate and apply our methods to the analysis of space-time temperature trends, based on data collected in the last five decades from central Sweden.