Probabilistic structure of an annual extreme rainfall series of a coastal area of the State of São Paulo, Brazil

This study aimed to describe the probabilistic structure of the annual series of extreme daily rainfall (Preabs), available from the weather station of Ubatuba, State of São Paulo, Brazil (1935-2009), by using the general distribution of extreme value (GEV). The autocorrelation function, the Mann-Kendall test, and the wavelet analysis were used in order to evaluate the presence of serial correlations, trends, and periodical components. Considering the results obtained using these three statistical methods, it was possible to assume the hypothesis that this temporal series is free from persistence, trends, and periodicals components. Based on quantitative and qualitative adhesion tests, it was found that the GEV may be used in order to quantify the probabilities of the Preabs data. The best results of GEV were obtained when the parameters of this function were estimated using the method of maximum likelihood. The method of L-moments has also shown satisfactory results.

The best are never normal : exploring the distribution of firm performance

Competitive Strategy literature predicts three different mechanisms of performance generation, thus distinguishing between firms that have competitive advantage, firms that have competitive disadvantage or firms that have neither. Nonetheless, previous works in the field have fitted a single normal distribution to model firm performance. Here, we develop a new approach that distinguishes among performance generating mechanisms and allows the identification of firms with competitive advantage or disadvantage. Theorizing on the positive feedback loops by which firms with competitive advantage have facilitated access to acquire new resources, we proposed a distribution we believe data on firm performance should follow. We illustrate our model by assessing its fit to data on firm performance, addressing its theoretical implications and comparing it to previous works.

Análise clássica e bayesiana do modelo Weibull modificado generalizado

Pós-graduação em Matematica Aplicada e Computacional - FCT

Robust modeling using the generalized epsilon-skew-t distribution

In this paper, an alternative skew Student-t family of distributions is studied. It is obtained as an extension of the generalized Student-t (GS-t) family introduced by McDonald and Newey [10]. The extension that is obtained can be seen as a reparametrization of the skewed GS-t distribution considered by Theodossiou [14]. A key element in the construction of such an extension is that it can be stochastically represented as a mixture of an epsilon-skew-power-exponential distribution [1] and a generalized-gamma distribution. From this representation, we can readily derive theoretical properties and easy-to-implement simulation schemes. Furthermore, we study some of its main properties including stochastic representation, moments and asymmetry and kurtosis coefficients. We also derive the Fisher information matrix, which is shown to be nonsingular for some special cases such as when the asymmetry parameter is null, that is, at the vicinity of symmetry, and discuss maximum-likelihood estimation. Simulation studies for some particular cases and real data analysis are also reported, illustrating the usefulness of the extension considered.

SEMIPARAMETRIC METHODS IN THE ESTIMATION OF TAIL PROBABILITIES AND EXTREME QUANTILES

In quantitative risk analysis, the problem of estimating small threshold exceedance probabilities and extreme quantiles arise ubiquitously in bio-surveillance, economics, natural disaster insurance actuary, quality control schemes, etc. A useful way to make an assessment of extreme events is to estimate the probabilities of exceeding large threshold values and extreme quantiles judged by interested authorities. Such information regarding extremes serves as essential guidance to interested authorities in decision making processes. However, in such a context, data are usually skewed in nature, and the rarity of exceedance of large threshold implies large fluctuations in the distribution's upper tail, precisely where the accuracy is desired mostly. Extreme Value Theory (EVT) is a branch of statistics that characterizes the behavior of upper or lower tails of probability distributions. However, existing methods in EVT for the estimation of small threshold exceedance probabilities and extreme quantiles often lead to poor predictive performance in cases where the underlying sample is not large enough or does not contain values in the distribution's tail. In this dissertation, we shall be concerned with an out of sample semiparametric (SP) method for the estimation of small threshold probabilities and extreme quantiles. The proposed SP method for interval estimation calls for the fusion or integration of a given data sample with external computer generated independent samples. Since more data are used, real as well as artificial, under certain conditions the method produces relatively short yet reliable confidence intervals for small exceedance probabilities and extreme quantiles.

Influence of memory in deterministic walks in random media: Analytical calculation within a mean-field approximation

Consider a random medium consisting of N points randomly distributed so that there is no correlation among the distances separating them. This is the random link model, which is the high dimensionality limit (mean-field approximation) for the Euclidean random point structure. In the random link model, at discrete time steps, a walker moves to the nearest point, which has not been visited in the last mu steps (memory), producing a deterministic partially self-avoiding walk (the tourist walk). We have analytically obtained the distribution of the number n of points explored by the walker with memory mu=2, as well as the transient and period joint distribution. This result enables us to explain the abrupt change in the exploratory behavior between the cases mu=1 (memoryless walker, driven by extreme value statistics) and mu=2 (walker with memory, driven by combinatorial statistics). In the mu=1 case, the mean newly visited points in the thermodynamic limit (N >> 1) is just < n >=e=2.72... while in the mu=2 case, the mean number < n > of visited points grows proportionally to N(1/2). Also, this result allows us to establish an equivalence between the random link model with mu=2 and random map (uncorrelated back and forth distances) with mu=0 and the abrupt change between the probabilities for null transient time and subsequent ones.

An Extension of Gompertzian Growth Dynamics Weibull and Frechet Models

In this work a new probabilistic and dynamical approach to an extension of the Gompertz law is proposed. A generalized family of probability density functions, designated by Beta* (p, q), which is proportional to the right hand side of the Tsoularis-Wallace model, is studied. In particular, for p = 2, the investigation is extended to the extreme value models of Weibull and Frechet type. These models, described by differential equations, are proportional to the hyper-Gompertz growth model. It is proved that the Beta* (2, q) densities are a power of betas mixture, and that its dynamics are determined by a non-linear coupling of probabilities. The dynamical analysis is performed using techniques of symbolic dynamics and the system complexity is measured using topological entropy. Generally, the natural history of a malignant tumour is reflected through bifurcation diagrams, in which are identified regions of regression, stability, bifurcation, chaos and terminus.

Occupational exposure to particulate matter in 2 Portuguese waste-sorting units

Objectives - This study intended to characterize work environment contamination by particles in 2 waste-sorting plants. Material and Methods - Particles were measured by portable direct-reading equipment. Besides mass concentration in different sizes, data related with the number of particles concentration were also obtained. Results - Both sorting units showed the same distribution concerning the 2 exposure metrics: particulate matter 5 (PM5) and particulate matter 10 (PM10) reached the highest levels and 0.3 μm was the fraction with a higher number of particles. Unit B showed higher (p < 0.05) levels for both exposure metrics. For instance, in unit B the PM10 size is 9-fold higher than in unit A. In unit A, particulate matter values obtained in pre-sorting and in the sequential sorting cabinet were higher without ventilation working. Conclusions - Workers from both waste-sorting plants are exposed to particles. Particle counting provided additional information that is of extreme value for analyzing the health effects of particles since higher values of particles concentration were obtained in the smallest fraction.

Modeling and mitigation of supply-demand mismatches

In this thesis, I develop analytical models to price the value of supply chain investments under demand uncer¬tainty. This thesis includes three self-contained papers. In the first paper, we investigate the value of lead-time reduction under the risk of sudden and abnormal changes in demand forecasts. We first consider the risk of a complete and permanent loss of demand. We then provide a more general jump-diffusion model, where we add a compound Poisson process to a constant-volatility demand process to explore the impact of sudden changes in demand forecasts on the value of lead-time reduction. We use an Edgeworth series expansion to divide the lead-time cost into that arising from constant instantaneous volatility, and that arising from the risk of jumps. We show that the value of lead-time reduction increases substantially in the intensity and/or the magnitude of jumps. In the second paper, we analyze the value of quantity flexibility in the presence of supply-chain dis- intermediation problems. We use the multiplicative martingale model and the "contracts as reference points" theory to capture both positive and negative effects of quantity flexibility for the downstream level in a supply chain. We show that lead-time reduction reduces both supply-chain disintermediation problems and supply- demand mismatches. We furthermore analyze the impact of the supplier's cost structure on the profitability of quantity-flexibility contracts. When the supplier's initial investment cost is relatively low, supply-chain disin¬termediation risk becomes less important, and hence the contract becomes more profitable for the retailer. We also find that the supply-chain efficiency increases substantially with the supplier's ability to disintermediate the chain when the initial investment cost is relatively high. In the third paper, we investigate the value of dual sourcing for the products with heavy-tailed demand distributions. We apply extreme-value theory and analyze the effects of tail heaviness of demand distribution on the optimal dual-sourcing strategy. We find that the effects of tail heaviness depend on the characteristics of demand and profit parameters. When both the profit margin of the product and the cost differential between the suppliers are relatively high, it is optimal to buffer the mismatch risk by increasing both the inventory level and the responsive capacity as demand uncertainty increases. In that case, however, both the optimal inventory level and the optimal responsive capacity decrease as the tail of demand becomes heavier. When the profit margin of the product is relatively high, and the cost differential between the suppliers is relatively low, it is optimal to buffer the mismatch risk by increasing the responsive capacity and reducing the inventory level as the demand uncertainty increases. In that case, how¬ever, it is optimal to buffer with more inventory and less capacity as the tail of demand becomes heavier. We also show that the optimal responsive capacity is higher for the products with heavier tails when the fill rate is extremely high.

A Study on the Reversed Lack of Memory Property and its Generalizations

In this thesis, the concept of reversed lack of memory property and its generalizations is studied.We we generalize this property which involves operations different than the ”addition”. In particular an associative, binary operator ” * ” is considered. The univariate reversed lack of memory property is generalized using the binary operator and a class of probability distributions which include Type 3 extreme value, power function, reflected Weibull and negative Pareto distributions are characterized (Asha and Rejeesh (2009)). We also define the almost reversed lack of memory property and considered the distributions with reversed periodic hazard rate under the binary operation. Further, we give a bivariate extension of the generalized reversed lack of memory property and characterize a class of bivariate distributions which include the characterized extension (CE) model of Roy (2002a) apart from the bivariate reflected Weibull and power function distributions. We proved the equality of local proportionality of the reversed hazard rate and generalized reversed lack of memory property. Study of uncertainty is a subject of interest common to reliability, survival analysis, actuary, economics, business and many other fields. However, in many realistic situations, uncertainty is not necessarily related to the future but can also refer to the past. Recently, Di Crescenzo and Longobardi (2009) introduced a new measure of information called dynamic cumulative entropy. Dynamic cumulative entropy is suitable to measure information when uncertainty is related to the past, a dual concept of the cumulative residual entropy which relates to uncertainty of the future lifetime of a system. We redefine this measure in the whole real line and study its properties. We also discuss the implications of generalized reversed lack of memory property on dynamic cumulative entropy and past entropy.In this study, we extend the idea of reversed lack of memory property to the discrete set up. Here we investigate the discrete class of distributions characterized by the discrete reversed lack of memory property. The concept is extended to the bivariate case and bivariate distributions characterized by this property are also presented. The implication of this property on discrete reversed hazard rate, mean past life, and discrete past entropy are also investigated.

Some bivariate life time models in discrete time"

The term reliability of an equipment or device is often meant to indicate the probability that it carries out the functions expected of it adequately or without failure and within specified performance limits at a given age for a desired mission time when put to use under the designated application and operating environmental stress. A broad classification of the approaches employed in relation to reliability studies can be made as probabilistic and deterministic, where the main interest in the former is to device tools and methods to identify the random mechanism governing the failure process through a proper statistical frame work, while the latter addresses the question of finding the causes of failure and steps to reduce individual failures thereby enhancing reliability. In the probabilistic attitude to which the present study subscribes to, the concept of life distribution, a mathematical idealisation that describes the failure times, is fundamental and a basic question a reliability analyst has to settle is the form of the life distribution. It is for no other reason that a major share of the literature on the mathematical theory of reliability is focussed on methods of arriving at reasonable models of failure times and in showing the failure patterns that induce such models. The application of the methodology of life time distributions is not confined to the assesment of endurance of equipments and systems only, but ranges over a wide variety of scientific investigations where the word life time may not refer to the length of life in the literal sense, but can be concieved in its most general form as a non-negative random variable. Thus the tools developed in connection with modelling life time data have found applications in other areas of research such as actuarial science, engineering, biomedical sciences, economics, extreme value theory etc.

Das Einzugsgebiet der Fulda - Untersuchung der Methoden der Hochwasserstatistik im Zeichen des Klimawandels

Numerous studies have proven an effect of a probable climate change on the hydrosphere’s different subsystems. In the 21st century global and regional redistribution of water has to be expected and it is very likely that extreme weather phenomenon will occur more frequently. From a global view the flood situation will exacerbate. In contrast to these discoveries the classical approach of flood frequency analysis provides terms like “mean flood recurrence interval”. But for this analysis to be valid there is a need for the precondition of stationary distribution parameters which implies that the flood frequencies are constant in time. Newer approaches take into account extreme value distributions with time-dependent parameters. But the latter implies a discard of the mentioned old terminology that has been used up-to-date in engineering hydrology. On the regional scale climate change affects the hydrosphere in various ways. So, the question appears to be whether in central Europe the classical approach of flood frequency analysis is not usable anymore and whether the traditional terminology should be renewed. In the present case study hydro-meteorological time series of the Fulda catchment area (6930 km²), upstream of the gauging station Bonaforth, are analyzed for the time period 1960 to 2100. At first a distributed catchment area model (SWAT2005) is build up, calibrated and finally validated. The Edertal reservoir is regulated as well by a feedback control of the catchments output in case of low water. Due to this intricacy a special modeling strategy has been necessary: The study area is divided into three SWAT basin models and an additional physically-based reservoir model is developed. To further improve the streamflow predictions of the SWAT model, a correction by an artificial neural network (ANN) has been tested successfully which opens a new way to improve hydrological models. With this extension the calibration and validation of the SWAT model for the Fulda catchment area is improved significantly. After calibration of the model for the past 20th century observed streamflow, the SWAT model is driven by high resolution climate data of the regional model REMO using the IPCC scenarios A1B, A2, and B1, to generate future runoff time series for the 21th century for the various sub-basins in the study area. In a second step flood time series HQ(a) are derived from the 21st century runoff time series (scenarios A1B, A2, and B1). Then these flood projections are extensively tested with regard to stationarity, homogeneity and statistical independence. All these tests indicate that the SWAT-predicted 21st-century trends in the flood regime are not significant. Within the projected time the members of the flood time series are proven to be stationary and independent events. Hence, the classical stationary approach of flood frequency analysis can still be used within the Fulda catchment area, notwithstanding the fact that some regional climate change has been predicted using the IPCC scenarios. It should be noted, however, that the present results are not transferable to other catchment areas. Finally a new method is presented that enables the calculation of extreme flood statistics, even if the flood time series is non-stationary and also if the latter exhibits short- and longterm persistence. This method, which is called Flood Series Maximum Analysis here, enables the calculation of maximum design floods for a given risk- or safety level and time period.

Modelling European winter wind storm losses in current and future climate

Severe wind storms are one of the major natural hazards in the extratropics and inflict substantial economic damages and even casualties. Insured storm-related losses depend on (i) the frequency, nature and dynamics of storms, (ii) the vulnerability of the values at risk, (iii) the geographical distribution of these values, and (iv) the particular conditions of the risk transfer. It is thus of great importance to assess the impact of climate change on future storm losses. To this end, the current study employs—to our knowledge for the first time—a coupled approach, using output from high-resolution regional climate model scenarios for the European sector to drive an operational insurance loss model. An ensemble of coupled climate-damage scenarios is used to provide an estimate of the inherent uncertainties. Output of two state-of-the-art global climate models (HadAM3, ECHAM5) is used for present (1961–1990) and future climates (2071–2100, SRES A2 scenario). These serve as boundary data for two nested regional climate models with a sophisticated gust parametrizations (CLM, CHRM). For validation and calibration purposes, an additional simulation is undertaken with the CHRM driven by the ERA40 reanalysis. The operational insurance model (Swiss Re) uses a European-wide damage function, an average vulnerability curve for all risk types, and contains the actual value distribution of a complete European market portfolio. The coupling between climate and damage models is based on daily maxima of 10 m gust winds, and the strategy adopted consists of three main steps: (i) development and application of a pragmatic selection criterion to retrieve significant storm events, (ii) generation of a probabilistic event set using a Monte-Carlo approach in the hazard module of the insurance model, and (iii) calibration of the simulated annual expected losses with a historic loss data base. The climate models considered agree regarding an increase in the intensity of extreme storms in a band across central Europe (stretching from southern UK and northern France to Denmark, northern Germany into eastern Europe). This effect increases with event strength, and rare storms show the largest climate change sensitivity, but are also beset with the largest uncertainties. Wind gusts decrease over northern Scandinavia and Southern Europe. Highest intra-ensemble variability is simulated for Ireland, the UK, the Mediterranean, and parts of Eastern Europe. The resulting changes on European-wide losses over the 110-year period are positive for all layers and all model runs considered and amount to 44% (annual expected loss), 23% (10 years loss), 50% (30 years loss), and 104% (100 years loss). There is a disproportionate increase in losses for rare high-impact events. The changes result from increases in both severity and frequency of wind gusts. Considerable geographical variability of the expected losses exists, with Denmark and Germany experiencing the largest loss increases (116% and 114%, respectively). All countries considered except for Ireland (−22%) experience some loss increases. Some ramifications of these results for the socio-economic sector are discussed, and future avenues for research are highlighted. The technique introduced in this study and its application to realistic market portfolios offer exciting prospects for future research on the impact of climate change that is relevant for policy makers, scientists and economists.

Characterising the variability and extremes of the stratospheric polar vortices using 2D moment analysis

The mean state, variability and extreme variability of the stratospheric polar vortices, with an emphasis on the Northern Hemisphere vortex, are examined using 2-dimensional moment analysis and Extreme Value Theory (EVT). The use of moments as an analysis to ol gives rise to information about the vortex area, centroid latitude, aspect ratio and kurtosis. The application of EVT to these moment derived quantaties allows the extreme variability of the vortex to be assessed. The data used for this study is ECMWF ERA-40 potential vorticity fields on interpolated isentropic surfaces that range from 450K-1450K. Analyses show that the most extreme vortex variability occurs most commonly in late January and early February, consistent with when most planetary wave driving from the troposphere is observed. Composites around sudden stratospheric warming (SSW) events reveal that the moment diagnostics evolve in statistically different ways between vortex splitting events and vortex displacement events, in contrast to the traditional diagnostics. Histograms of the vortex diagnostics on the 850K (∼10hPa) surface over the 1958-2001 period are fitted with parametric distributions, and show that SSW events comprise the majority of data in the tails of the distributions. The distribution of each diagnostic is computed on various surfaces throughout the depth of the stratosphere, and shows that in general the vortex becomes more circular with higher filamentation at the upper levels. The Northern Hemisphere (NH) and Southern Hemisphere (SH) vortices are also compared through the analysis of their respective vortex diagnostics, and confirm that the SH vortex is less variable and lacks extreme events compared to the NH vortex. Finally extreme value theory is used to statistically mo del the vortex diagnostics and make inferences about the underlying dynamics of the polar vortices.

Symmetry breaking, mixing, instability, and low frequency variability in a minimal Lorenz-like system

Starting from the classical Saltzman two-dimensional convection equations, we derive via a severe spectral truncation a minimal 10 ODE system which includes the thermal effect of viscous dissipation. Neglecting this process leads to a dynamical system which includes a decoupled generalized Lorenz system. The consideration of this process breaks an important symmetry and couples the dynamics of fast and slow variables, with the ensuing modifications to the structural properties of the attractor and of the spectral features. When the relevant nondimensional number (Eckert number Ec) is different from zero, an additional time scale of O(Ec−1) is introduced in the system, as shown with standard multiscale analysis and made clear by several numerical evidences. Moreover, the system is ergodic and hyperbolic, the slow variables feature long-term memory with 1/f3/2 power spectra, and the fast variables feature amplitude modulation. Increasing the strength of the thermal-viscous feedback has a stabilizing effect, as both the metric entropy and the Kaplan-Yorke attractor dimension decrease monotonically with Ec. The analyzed system features very rich dynamics: it overcomes some of the limitations of the Lorenz system and might have prototypical value in relevant processes in complex systems dynamics, such as the interaction between slow and fast variables, the presence of long-term memory, and the associated extreme value statistics. This analysis shows how neglecting the coupling of slow and fast variables only on the basis of scale analysis can be catastrophic. In fact, this leads to spurious invariances that affect essential dynamical properties (ergodicity, hyperbolicity) and that cause the model losing ability in describing intrinsically multiscale processes.