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Gamma, lognormal and Weibull distributions are most commonly used in modeling asymmetric data coming from the areas of life testing and reliability engineering. In this study, we deal with the prob...

Portugal is recurrently affected by large wildfire events that have serious impacts at the socio-economic and environmental levels and dramatic consequences associated with the loss of lives and the destruction of the landscape. Accordingly, seasonal forecasts are required to assist fire managers, thus contributing to alter the historically based purely reactive response. In this context, we present and discuss a statistical model to estimate the probability that the total burned area during summer will exceed a given threshold. The statistical model uses meteorological information that rates the accumulation of thermal and vegetation stress. Outlooks for the 39-year study period (1980–2018) show that, when the statistical model is applied from 26 May to 30 June, out of the six severe years, only one year is not anticipated as potentially severe and, out of the six weak years, only one is not anticipated as potentially weak. The availability of outlooks of wildfire potential with an anticipation of up to 1 month before the starting of the fire season, such as the one proposed here, may serve to provide clear directions for the fire community when planning prevention and combating fire events.

Assessing the exposure of an individual or population is a challenging statistical task. Borrowing from the concept of the genome in genetics, recently the concept of an exposome has emerged to help explain the complexities of exposure assessment and guide research in this area. According to the Centers for Disease Control and Prevention, “The exposome can be defined as the measure of all the exposures of an individual in a lifetime and how those exposures relate to health.” The simplest algorithm to estimate concentration at an unmonitored locations is to use the measured value for the nearest monitoring location. The estimated nearest-neighbor concentration surface is then piecewice constant over Voronoi polygons around the monitors. This is a special case of K-nearest neighbor (KNN) interpolation. Inverse distance weighting (also known as kernel smoothing) is an extension to KNN that weights locations according to their distance from the prediction location.

Across the globe, human activities have been gaining importance relatively to climate and ecology as the main controls on fire regimes and consequently human activity became an important driver of the frequency, extent and intensity of vegetation burning worldwide. Our objective in the present study is to look for weekly cycles in vegetation fire activity at global scale as evidence of human agency, relying on the original MODIS active fire detections at 1 km spatial resolution (MCD14ML) and using novel statistical methodologies to detect significant periodicities in time series data. We tested the hypotheses that global fire activity displays weekly cycles and that the weekday with the fewest fires is Sunday. We also assessed the effect of land use and land cover on weekly fire cycle significance by testing those hypotheses separately for the Villages, Settlements, Croplands, Rangelands, Seminatural, and Wildlands anthromes. Based on a preliminary data analysis of the daily global active fire counts periodogram, we developed an harmonic regression model for the mean function of daily fire activity and assumed a linear model for the de-seasonalized time series. For inference purposes, we used a Bayesian methodology and constructed a simultaneous 95% credible band for the mean function. The hypothesis of a Sunday weekly minimum was directly investigated by computing the probabilities that the mean functions of every weekday (Monday to Saturday) are inside the credible band corresponding to mean Sunday fire activity. Since these probabilities are small, there is statistical evidence of significantly fewer fires on Sunday than on the other days of the week. Cropland, rangeland, and seminatural anthromes, which cover 70% of the global land area and account for 94% of the active fires analysed, display weekly cycles in fire activity. Due to lower land management intensity and less strict control over fire size and duration, weekly cycles in Rangelands and Seminatural anthromes, which jointly account for 53.46% of all fires, although statistically significant are weaker than those detected in Croplands.

The Portuguese National Statistical Institute is responsible for estimating and publishing quarterly labour market figures at national level for both NUTS I and NUTS II regions. Over recent years it has become increasinglyimportant to identify these figures at more disaggregated levels. However, based on the established direct estimation method, it is not possible to produce satisfactorily precise estimates at higher spatial resolutions. From the 4th quarter of 2014 onwards, all the sampling units, namely the residential buildings, of the Portuguese Labour Force Survey (PLFS) were georeferenced. To take full advantage of this information, Pereira et al. (2019) proposed applying a spatial marked point process approach to unemployment estimation, in which the estimation of the unemployment intensity becomes the focal point. There, the sampling units were assumed to be a realization of a spatial point process, specifically a log Gaussian Cox process, with the number of unemployed in each unit being their respective marks. Recently, further information on the geo-referenced locations of all units of the population, namely all residential buildings in the national territory, became available. Consequently, it is no longer necessary to model the spatial configuration of the units of the population. Thus, we propose a new point referenced model for the marks based on the sampled units and extrapolate this to all units of the population. As expected, this extra information, and as a consequence the new model itself, produce estimates with higher precision.

In this work we reassess the impacts of blocking patterns on precipitation regimes in the Iberian Peninsula, distinguishing between north Atlantic and European blocking. For this we take full advantage of the recently developed high-resolution datasets for the Iberian countries. Precipitation anomalies during blocking events obtained with this dataset allow a much finer regional characterization of the impacts in both average and extreme daily precipitation, particularly when compared to widely used low-resolution reanalysis datasets. Blocked patterns induce a negative-positive dipole of precipitation anomalies from northwest to southeast Iberia. Increases are widespread during Atlantic blocks and pronounced in southern and eastern areas of Iberia, while during European blocks they are more spatially restricted, with increases above 50 % in coastal Mediterranean areas, which represents a considerable fraction of the annual precipitation. Blocking impacts in precipitation are nearly opposite to those found during strong zonal flow situations, but there are also some asymmetries in the precipitation responses. A significant increase in cyclones and cut-off lows frequency southwards of blocking structures is related to precipitation excesses over southern and eastern areas, where dynamical factors and local processes play a crucial role. On the contrary, precipitation deficits in northwest Iberia during blocking episodes are better explained by a reduction in north Atlantic frontal activity and simultaneous decreases in large-scale moisture advection towards northern Iberia. We show that these anomalies during blocking result from changes in precipitation amount rather than from increases in rainy days, pointing to more extreme rainfall regimes, particularly in southeastern Iberia. Finally, an Extreme Value Analysis was performed, fitting Generalized Pareto Distributions to precipitation extremes. Results show that the different extreme precipitation regimes of northwest and Mediterranean regions are partially determined by opposite anomalies of the zonal flow. Thus, heavy precipitation events in Mediterranean areas are usually short-lived and frequently associated with blocking conditions, while in northwest Iberia the total accumulations during rainfall episodes are more important for triggering extreme events and they are mainly related to strong westerly flows.

From the fourth quarter of 2014, the Portuguese Labour Force Survey (PLFS) started geo-referencing the sampling units, namely the dwellings, in which the surveys are carried out. This opens new possibilities in analysing and estimating unemployment and its spatial distribution across any region by employing point referenced methods and models. According to a preestablished sampling criteria, the labour force survey selects a certain number of dwellings from across the nation to study, and establishes the number of unemployed people in each dwelling. Based on this survey, the National Statistical Institute of Portugal presently uses direct estimation methods to estimate the national unemployment figures. Recently however, there has been increased interest in estimating these figures in smaller areas. Due to reduced sampling sizes in small areas, direct estimation methods tend to produce fairly large sampling variations. Therefore, model based methods should be favoured as these tend to borrow strength from area to area by making use of the areal dependence. These model based methods tend to use areal counting processes as models and typically introduce spatial dependence through the model parameters using a latent random effect. In this paper, we suggest using point referenced models as an alternative to the traditional small area estimation methods for unemployment estimation. Specifically, we model the spatial distribution of residential buildings across Portugal using a log Gaussian Cox process, and the number of unemployed people per residential unit as a mark attached to these random points. Thus, the main focus of the study is to model the spatial intensity function of this marked point process. The number of unemployed people in any region can then be estimated using a proper functional of this marked point process. The principal objective of this point referenced method for unemployment estimation is to produce reliable estimates at higher spatial resolutions, and at the same time to incorporate into the model any available auxiliary information of the residential units, such as mean age or education level as compared to areal unit averages used in small area estimation.

Vegetation fires are an important environmental and socioeconomic problem, and large budgets are spent in fire prevention and fire fighting. Detailed knowledge of spatiotemporal patterns of fire occurrence is required for effective and efficient fire management, and annual fire risk maps can be an important tool to support strategic decisions relating to location–allocation of equipment and human resources. Here, we define risk of fire in the narrow sense as the probability of its occurrence, without addressing the loss component. We propose and evaluate two alternative approaches to the development of annual fire risk maps, using an atlas of annual burned area maps of Portugal (1975–2009), derived from the classification of satellite imagery, and a set of environmental maps representing vegetation, climatic, and topographic covariates. We look at current approaches for producing annual fire risk maps, and suggest improvements by incorporating the strong spatial and temporal dependence that exists in the ...

Periodic autoregressive (PAR) models extend the classical autoregressive models by allowing the parameters tovary with seasons. Selecting PAR time-series models can be computationally expensive, and the results are notalways satisfactory. In this article, we propose a new automatic procedure to the model selection problem by usingthe genetic algorithm. The Bayesian information criterion is used as a tool to identify the order of the PAR model.The success of the proposed procedure is illustrated in a small simulation study, and an application with monthlydata is presented.Keywords: Periodic time series; identification; genetic algorithms; parameter constraints; BIC.

Policy responses for local and global fire management as well as international green-gas inventories depend heavily on the proper understanding of the annual fire extend as well as its spatial variation across any given study area. Proper statistical models are important tools in quantifying these fire risks. We propose Bayesian methods to model jointly the probability of ignition and fire sizes in Australia and New Zeland. The data set on which we base our model and results consists of annual observations of several meteorological and topographical explanatory variables, together with the percentage of land burned over a grid with resolution of 1° across Austalia and New Zealand. Our model and conclusions bring improvements on the results reported by Russell-Smith et al. in Int J Wildland Fire, 16:361–377 (2007) based on a similar data set.

Recently there has been a lot of effort to model extremes of spatially dependent data. These efforts seem to be divided into two distinct groups: the study of max-stable processes, together with the development of statistical models within this framework; the use of more pragmatic, flexible models using Bayesian hierarchical models (BHM) and simulation based inference techniques. Each modeling strategy has its strong and weak points. While max-stable models capture the local behavior of spatial extremes correctly, hierarchical models based on the conditional independence assumption, lack the asymptotic arguments the max-stable models enjoy. On the other hand, they are very flexible in allowing the introduction of physical plausibility into the model. When the objective of the data analysis is to estimate return levels or kriging of extreme values in space, capturing the correct dependence structure between the extremes is crucial and max-stable processes are better suited for these purposes. However when the primary interest is to explain the sources of variation in extreme events Bayesian hierarchical modeling is a very flexible tool due to the ease with which random effects are incorporated in the model. In this paper we model a data set on Portuguese wildfires to show the flexibility of BHM in incorporating spatial dependencies acting at different resolutions.

In Portugal, due to the combination of climatological and ecological factors, large wildfires are a constant threat and due to their economic impact, a big policy issue. In order to organize efficient fire fighting capacity and resource management, correct quantification of the risk of large wildfires are needed. In this paper, we quantify the regional risk of large wildfire sizes, by fitting a Generalized Pareto distribution to excesses over a suitably chosen high threshold. Spatio-temporal variations are introduced into the model through model parameters with suitably chosen link functions. The inference on these models are carried using Bayesian Hierarchical Models and Markov chain Monte Carlo methods.

Let {X(s, t), s = (s1, s2) ∈ ℝ2, t ∈ ℝ} be a stationary random field defined over a discrete lattice. In this paper, we consider a set of domain of attraction criteria giving the notion of extremal index for random fields. Together with the extremal-types theorem given by Leadbetter and Rootzen (1997), this will give a characterization of the limiting distribution of the maximum of such random fields.

In this paper, we introduce a Bayesian Hierarchical model to estimate the abundance of hake recruits, as well as to study their spatial distributional patterns in the Portuguese territorial waters. Main objective of the paper is to improve on traditional empirical methods based on sampling averages and variances by using probabilistic models that capture spatial dependence structures. Copyright © 2006 John Wiley & Sons, Ltd.

Precipitation over the Western part of Iberian Peninsula is known to be related to the large-scale sea level pressure field and thus to advection of humidity into this area. The major problem is to downscale this synoptic atmospheric information to local daily precipitation patterns. One way to handle this problem is by weather-state models, where, based on the pressure field, each day is classified into a weather state and precipitation is then modeled within each weather state via multivariate distributions. In this paper, we propose a spatiotemporal Bayesian hierarchical model for precipitation. Basic objective and novelty of the paper is to capture and model the essential spatiotemporal relationships that exist between large-scale sea level pressure field and local daily precipitation. A specific local spatial ordering that mimics the essential large-scale patterns is used in the likelihood. The model is then applied to a network of rain gauge stations in the river Tagus valley. The inference is then carried out using appropriate MCMC methods. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper, we look at the extremal behavior of Volterra series expansions generated by heavy-tailed innovations, via a point process formulation. Volterra series expansions are known to be the most general nonlinear representation for any stationary sequence. The so called complete convergence theorem on point processes we prove enable us to give in detail, the weak limiting behavior of various functionals of the underlying process including the asymptotic distribution of upper and lower order statistics. In particular, we investigate the limiting distribution of the sample maxima and the corresponding extremal index. The study of the extremal properties of finite order Volterra series expansions would be highly valuable in understanding the extremal behavior of nonlinear processes as well as understanding of order identification and adequacy of Volterra series when used as models in signal processing. In fact, such extremal properties may suggest a way of finding the order of a finite Volterra expansions which is consistent with the nonlinearities of the observed process.

In a series of papers, Lindgren (1975a, 1985) and de Mare (1980) set the principles of optimal alarm systems and obtained the basic results. Application of these ideas to linear discrete time-series models was carried out by Svensson et al. (1996). In this paper, we suggest a Bayesian predictive approach to event prediction and optimal alarm systems for discrete time series. There are two novelties in this approach: first, the variation in the model parameters is incorporated in the analysis; second, this method allows ‘on-line prediction’ in the sense that, as we observe the process, our posterior probabilities and predictions are updated at each time point.

Let Xk be a stationary time series and yk=XkM be the sub-sampled series corresponding to a fixed systematic sampling interval M > 1. In this paper, we use a point process approach to study the effect of the sub sampling on the extremal properties of Yk when Xk is a linear process with heavy-tailed innovations. We prove complete point process convergence theorems which enable us to give in detail the weak limiting behaviour of maxima of the sub-sampled process and to compare it with that of the original process. The results both exemplify the findings of a study by Robinson and Tawn (2000) and offer more precise details for the class of linear models. Motivation comes from the comparison of schemes for monitoring financial and environmental processes.

In this paper we study the spatio-temporal behaviour of air pollutants measured daily over the city of Lisbon, Portugal. Our specific aim is to predict air pollutant levels in time and space over a fine grid of locations based on observations from a small number of monitoring sites. Our suggested prediction procedure is based on the simple and intuitive idea of first making predictions in time at the monitoring sites and then extending these predictions in space to locations other than the monitoring sites using kriging methods. Copyright © 2002 John Wiley & Sons, Ltd.

We have seen in Sect. 2.1.4 that nonlinear processes, due to their dependence on initial conditions, often magnify error causing unstable behavior. Even when stationary solutions exist, this noise magnification and dependence on initial conditions reflects on the tails of the stationary distribution, as well as on how large values cluster.

Suppose we have an observed time series \(x_{1},x_{2},\ldots,x_{n}\) and want to know if a linear time series model is adequate for the data, or an alternative nonlinear model should be considered. Linear models are often taken as the null hypotheses against a nonlinear alternative due to the simplicity of inference. Often we know much about the underlying process which generate the data set. Therefore it is possible to decide if a linear model will be adequate and if not, what aspects of nonlinearity should be modeled as alternative.

Assume that for \(t \in \mathbb{Z}\), (Z t ) and \((Z_{t}^{{\ast}})\) are respectively uncorrelated and independent sequences of r.v’s having identical marginal distribution F(⋅ ), with zero mean and variance \(\sigma _{Z}^{2}

One of the issues contributing to the success of any extreme value modeling is the choice of the number of upper order statistics used for inference, or equivalently, the selection of an appropriate threshold. In this paper we propose a Bayesian predictive approach to the peaks over threshold method with the purpose of estimating extreme quantiles beyond the range of the data. In the peaks over threshold (POT) method, we assume that the threshold identifies a model with a specified prior probability, from a set of possible models. For each model, the predictive distribution of a future excess over the corresponding threshold is computed, as well as a conditional estimate for the corresponding tail probability. The unconditional tail probability for a given future extreme observation from the unknown distribution is then obtained as an average of the conditional tail estimates with weights given by the posterior probability of each model.

The class of bilinear time series models is an obvious generalization of linear ARMA models and has found many applications in time series modeling. It is known that the sample paths of even the simplest bilinear process may have sudden bursts of large negative and positive values that vary in form and amplitude depending on the model parameters. Yet, little is known about the extremal properties of this class. In this paper, we look at the extremal properties of bilinear processes and explain how model parameters affect the extremal behavior.

Summary Leading economic indicators are often used to anticipate changes in key economic variables. Understanding the dynamics of these indicators is of primary interest for policy-making objectives and for sustainable economic welfare. We are concerned with the problem of setting a dynamic threshold above which the value of leading indicators would be considered as extreme. We propose a dynamic threshold modelling approach based on fractionally integrated processes where a semiparametric method is used to determine the amount of differencing that is required to obtain a weakly stationary process—to which standard methods of statistics of extremes apply. Given that our approach is linked to the Box–Jenkins method, we refer to the procedure proposed and applied herein as the Box–Jenkins–Pareto procedure. We use our approach to analyse the weekly number of unemployment insurance claims in the USA and explore the connection between its threshold exceedances and the US business cycle.

In many applications, the primary interest is the supremum of some continuous-time process over a specified period. However, data are usually available over a discrete set of times and the inference can only be made for the maximum of the process over this discrete set of times. If we want to estimate the extremes of the continuous-time process based on discrete time data, we need to understand the relationship between the continuous and discrete extremes. Thus, we look at asymptotic joint distributions of the maxima of stationary processes and their discrete versions.

We extend the results on the extremal properties of chain-dependent sequences considered in Turkman and Walker (1983) by assuming conditions similar to those given by Leadbetter and Nandagopalan (1987) which permit clustering of high values.

Let { X n } be a stationary sequence. The optimal alarm policy for detecting future upcrossings of the sequence is studied in a Bayesian predictive context and particular calculations are carried for an autoregressive process of order 1.

Here we present a procedure that allows the operational generation of daily maps of fire danger over Mediterranean Europe. These are based on integrated use of vegetation cover maps, weather data and fire activity as detected by remote sensing from space. The study covers the period of July–August 2007 to 2009. It is demonstrated that statistical models based on two-parameter generalised Pareto (GP) distributions adequately fit the observed samples of fire duration and that these models are significantly improved when the Fire Weather Index (FWI), which rates fire danger, is integrated as a covariate of scale parameters of GP distributions. Probabilities of fire duration exceeding specified thresholds are then used to calibrate FWI leading to the definition of five classes of fire danger. Fire duration is estimated on the basis of 15-min data provided by Meteosat Second Generation (MSG) satellites and corresponds to the total number of hours in which fire activity is detected in a single MSG pixel during one day. Considering all observed fire events with duration above 1h, the relative number of events steeply increases with classes of increasing fire danger and no fire activity was recorded in the class of low danger. Defined classes of fire danger provide useful information for wildfire management and are based on the Fire Risk Mapping product that is being disseminated on a daily basis by the EUMETSAT Satellite Application Facility on Land Surface Analysis.

Let $\{X_t\}^\infty_{t=1}$ be a stationary Gaussian time series with zero mean, unit variance, absolutely summable autocorrelation function and at least once differentiable spectral density function which is strictly positive in $\lbrack 0, \pi \rbrack$. In this paper it is shown that, if $M_n$ denotes the maximum of the normalized periodogram of $\{X_1,\ldots, X_n\}$ over the interval $\lbrack 0, \pi \rbrack$, then, almost surely, \begin{equation*}\tag{1} \lim \inf_{n\rightarrow\infty} \lbrack M_n - 2 \log n + \log \log n \rbrack \geq 0\end{equation*} and \begin{equation*}\tag{2} \lim \sup_{n\rightarrow\infty} \lbrack M_n - 2 \log n - 2(\log n)^\delta \rbrack = -\infty\end{equation*} for any $\delta > 0$.

The relationship lictwecn the statistics of environmcmal meHsurcmcntii averaged over different time scales is related to extreme levels of the Vdriabtc;. Results on the asymptotic joint distributions of extreme averages over different time periods arc treated. Key tvords: averages; extremes; extremal index; Puisson random measures.

Let {ε t, t = 1, 2, ···, n} be a sequence of mutually independent standard normal random variables. Let Xn(λ) and Yn(λ) be respectively the real and imaginary parts of exp iλ t, and let . It is shown that as n tends to∞, the distribution functions of the normalized maxima of the processes {Xn(λ)}, (Yn(λ)}, {In(λ)} over the interval λ∈ [0,π] each converge to the extremal distribution function exp [–e–x], —∞ < x It is also shown that these results can be extended to the case where {ε t} is a stationary Gaussian sequence with a moving-average representation.

We propose Akaike's information criterion for the choice of extremal models and by simulation we analyse its effectiveness in choosing the most likely among the Gumbel, Frechet and Weibull models.

A class of quasi-stationary sequences of random variables is introduced. After giving the definition, it is shown that the limiting distributions of the maxima of such sequences, when suitably normalized, converge to one of the three extreme-value distributions. EXTREME-VALUE DISTRIBUTIONS 1. Definition and some examples of the class We consider a class 16 of sequences of random variables {X., n = 1, 2, . } having the following properties. (la) There exists a bivariate sequence of random variables {(S", X,),n = 1,2, " } such that, for each n, (1.1) P[S.=k]=6 ,,k k=1,2,..,m, XZ n k=1. k-•l (1.2) P[X. -x I Sx = s,, . . ,S. = s.] = P[X. -5x IS. = s.] = F,.(x), say, s. = 1, -... m, (thus depending only on the value of S.). (lb) Conditional on S =s..-* , S., = s., the set of random variables (X1, X2, ..., X) is independent of (S+,, S~+2,. *"), for each n; thus for any integer k > 0 we can write P[X X1, . , X. < s=x. S , , S, = s.] (1.3) = P[X, x, -, X. = x. 5 IS, = s,,i..,Snk = sk. Received 28 August 1981; revision received 1 February 1983. * Present address: Faculdade de Ciencias da Universidade de Lisboa, Departamento de Estatistica, InvestigaCAo Operacional e Computaqbo, 58 Rua da Escola Polit6cnica, 1294 Lisboa Codex, Portugal. ** Postal address: Department of Probability and Statistics, The University, Sheffield S3 7RH, U.K.