Deformed SPDE models with an application to spatial modeling of significant wave height

A non-stationary Gaussian random field model is developed based on a combination of the stochastic partial differential equation (SPDE) approach and the classical deformation method. With the deformation method, a stationary field is defined on a domain which is deformed so that the field becomes non-stationary. We show that if the stationary field is a Mat'ern field defined as a solution to a fractional SPDE, the resulting non-stationary model can be represented as the solution to another fractional SPDE on the deformed domain. By defining the model in this way, the computational advantages of the SPDE approach can be combined with the deformation method's more intuitive parameterisation of non-stationarity. In particular it allows for independent control over the non-stationary practical correlation range and the variance, which has not been possible with previously proposed non-stationary SPDE models. The model is tested on spatial data of significant wave height, a characteristic of ocean surface conditions which is important when estimating the wear and risks associated with a planned journey of a ship. The model parameters are estimated to data from the north Atlantic using a maximum likelihood approach. The fitted model is used to compute wave height exceedance probabilities and the distribution of accumulated fatigue damage for ships traveling a popular shipping route. The model results agree well with the data, indicating that the model could be used for route optimization in naval logistics.Comment: 22 pages, 12 figure

On flexible random field models for spatial statistics: Spatial mixture models and deformed SPDE models

Spatial random fields are one of the key concepts in statistical analysis of spatial data. The random field explains the spatial dependency and serves the purpose ofregularizing interpolation of measured values or to act as an explanatory model. In this thesis, models for applications in medical imaging, spatial point pattern analysis, and maritime engineering are developed. They are constructed to be flexible yet interpretable. Since spatial data in several dimensions tend to be large, the methods considered for estimation, prediction, and approximation are focused on reducing computational complexity. The novelty of this work is based on two main ideas. First, the idea of a spatial mixture model, i.e., a stochastic partitioning of the spatial domain using a latent categorically valued random field. This makes it possible to explain discontinuities in otherwise smoothly varying random fields. It also introduces a different perspective that of a spatial classification problem. This idea is used to model the spatial distribution of tissue types in the human head; an application important in reducing cell damage due to ionizing radiation in medical imaging. The idea is also used to introduce an extension of the popular log-Gaussian Cox process. This extension adds an extra layer of a latent random partitioning of the spatial domain. Using this model,it is possible to classify spatial domains based on observed point patterns. The second main idea of this thesis is that of spatially deforming a solution to a stochastic partial differential equation. In this way, a random field with a needed degree of non-stationarity and anisotropy can be acquired. A coupled system of two such stochastic partial differential equations is used to model the joint distribution of significant wave heights and wave periods in the north Atlantic. The model is used to assess risks in naval logistics

Joint spatial modeling of significant wave height and wave period using the SPDE approach

The ocean wave distribution in a specific region of space and time is described by its sea state. Knowledge about the sea states a ship encounters on a journey can be used to assess various parameters of risk and wear associated with this journey. Two important characteristics of the sea state are significant wave height and mean wave period. We propose a joint spatial model of these two quantities on the north Atlantic ocean. The model describes the distribution of the logarithm of the two quantities as a bivariate Gaussian random field, modeled as a solution to a system of coupled fractional stochastic partial differential equations. The bivariate random field is non-stationary and allows for arbitrary, and different, smoothness for the two marginal fields. The parameters of the model are estimated from data using a stepwise maximum likelihood method. The fitted model is used to derive the distribution of accumulated fatigue damage for a ship sailing a transatlantic route. Also, a method for estimating the risk of capsizing due to broaching-to based on the joint distribution of the two sea state characteristics is investigated. The risks are calculated for a transatlantic route between America and Europe using both data and the fitted model. The results show that the model compares well with observed data. It further shows that the bivariate model is needed and cannot simply be approximated by a model of significant wave height alone

A nationwide survey of the influence of month of birth on the risk of developing multiple sclerosis in Sweden and Iceland

Previous studies have shown that the risk of multiple sclerosis (MS) is associated with season of birth with a higher proportion of MS patients being born in spring. However, this relationship has recently been questioned and may be due to confounding factors. Our aim was to assess the influence from season or month of birth on the risk of developing MS in Sweden and Iceland. Information about month of birth, gender, and phenotype of MS for patients born 1940–1996 was retrieved from the Swedish MS registry (SMSR), and their place of birth was retrieved from the Swedish Total Population Registry (TPR). The corresponding information was retrieved from medical journals of Icelandic MS patients born 1981–1996. The control groups consisted of every person born in Sweden 1940–1996, their gender and county of birth (TPR), and in Iceland all persons born between 1981 and 1996 and their gender (Statistics Iceland). We calculated the expected number of MS patients born during each season and in every month and compared it with the observed number. Adjustments were made for gender, birth year, and county of birth. We included 12,020 Swedish and 108 Icelandic MS patients in the analyses. There was no significant difference between expected and observed MS births related to season or month of birth in Sweden or Iceland. This was even the results before adjustments were made for birth year and birth place. No significant differences were found in subgroup analyses including data of latitude of birth, gender, clinical phenotype, and MS onset of 30 years or less. Our results do not support the previously reported association between season or month of birth and MS risk. Analysis of birth place and birth year as possible confounding factors showed no major influence of them on the seasonal MS risk in Sweden and Iceland

Bürgerbildung in Slowenien und Diskurs der Permissivität

Tekst ima dva dijela koji predstavljaju dva različita koncepta odnosa između građanskog obrazovanja i prihvaćenog vrijednosnog sustava u državnim školama. U prvom dijelu analizirat ćemo javne rasprave koje se odnose na građansko obrazovanje: želimo provjeriti koliko su izložene utjecaju javnih i političkih očekivanja i stajalištima koja se formiraju kroz mehanizme političke moći. Koncepte građanskog obrazovanja analiziramo kao diskurzivni fenomen. Želimo pokazati kako diskurs utječe na koncepte građanskog obrazovanja. Pritom diskurs interpretiramo kao društvenu vezu ili kao točku identifikacije u kojoj se možemo prepoznati. U drugom dijelu rada nastojimo dati odgovor na pitanje o uspostavi građanskog obrazovanja kao posljedice izgubljenih iluzija na kraju Velikih priča i poslije ‘’sloma’’ u bivšim socijalističkim ili komunističkim zemljama, koji se dogodio nakon uspostavljanja parlamentarne demokracije i tržišne ekonomije. Pitamo se: Kako nastavnici promatraju građansko obrazovanje kao nastavni predmet? Što bi trebali biti sadržaji građanskog obrazovanja ako određeni predmet promatraju učenici, i to kao predmet koji nema vrijednost ili koji im je nepoznat? Trebaju li sadržaji građanskog obrazovanja reflektirati želje učenika?The text is divided in two opposing parts, presenting two different concepts of the relation between civil education and the accepted system of values in state schools. The first part of the text is an analysis of public debates about civil education: the authors wanted to determine to what extent they are influenced by public and political expectations or convictions formed through political power mechanisms. The concepts of civil education are analysed as discursive phenomena. The authors’ aim is to show how a discourse influences the concepts of civil education. A discourse is interpreted as a social connection or a point of identification for an individual. The second part of the paper tries to answer the questions raised by the introduction of civil education as a reaction to the disillusionment at the end of the big stories and after the breakdown in former socialist or communist countries, made necessary by the establishment of parliamentary democracy and market economy. The questions raised in that part are: How do teachers perceive Civil Education as a subject? What should Civil Education contain, from students’ point of view, especially if such a subject is thought to be worthless or totally alienated from their reality? Should the content of Civil Education reflect students’ wishes?Der Text hat zwei Teile, deren Einschränkungen mit Hilfe von zwei verschiedenen Beziehungskonzepten zwischen der Bürgerbildung und dem geltenden Wertsystem an den Staatsschulen veranschaulicht werden. Im ersten Teil wird die öffentliche Diskussion über die Bürgerbildung analysiert: Wir möchten überprüfen, in welchem Ausmass sie von den öffentlichen und politischen Erwartungen und Überzeugungen beeinflusst wird, die durch Mechanismen der politischen Macht gebildet werden. Konzepte der Bürgerbildung werden als diskursives Phänomen analysiert. Wir möchten aufzeigen, wie der Diskurs die Konzepte der Bürgerbildung beeinflusst. Dabei wird der Diskurs als gesellschaftliche Beziehung oder als Identifikationspunkt gedeutet, in dem wir alle uns wiedererkennen können. Im zweiten Teil unserer Arbeit versuchen wir, die Frage zu beantworten, ob die Einführung der Bürgerbildung als eine Folge von verlorenen Illusionen am Ende der grossen Geschichten nach dem «Zusammenbruch» in den ex-sozialistischen oder kommunistischen Ländern anzusehen sei, der nach der Einführung der parlamentarischen Demokratie und Marktwirtschaft erfolgte. In diesem Teil stellen wir die Fragen: Wie betrachten die Lehrer die Bürgerbildung als Unterrichtsfach? Was sollten die Inhalte der Bürgerbildung sein, wenn wir sie vom Gesichtspunkt der Schüler aus betrachten als ein Schulfach, das für sie keinen Wert besitzt oder ihnen völlig fremd ist? Sollten die Inhalte der Bürgerbildung die Wünsche der Schüler wiederspiegeln

Spatial Mixture Models with Applications in Medical Imaging and Spatial Point Processes

Finite mixture models have proven to be a great tool for both modeling non-standard probability distributions and for classification problems (using the latent variable interpretation). In this thesis we are building spatial models by incorporating spatially dependent categorical latent random fields in a hierarchical manner similar to that of finite mixture models. This allows for non-linear prediction, better interpretation of estimated model parameters, and the added possibility of addressing questions related to classification. This thesis consists of two papers. The first paper concerns a problem in medical imaging where substitutes of computed tomography (CT) images are demanded due to the risks associated with X-radiation. This problem is addressed by modeling the dependency between CT images and magnetic resonance (MR) images. The model proposed incorporates multidimensional normal inverse Gaussian distributions and a spatially dependent Potts model for the latent classification. Parameter estimation is suggested using a maximum pseudo-likelihood approach implemented using the EM gradient method. The model is evaluated using cross-validation on three dimensional data of human brains.The second paper concerns modeling of spatial point patterns. A novel hierarchical Bayesian model is constructed by using Gaussian random fields and level sets in a Cox process. The model is an extension to the popular log-Gaussian Cox process and incorporates a latent classification field in order to handle sudden jumps in the intensity surface and to address classification problems.For inference, a Markov chain Monte Carlo method based on the preconditioned Crank-Nicholson MALA method is suggested. Finally, the model is applied to a popular data set of tree locations in a rainforest and the results show the advantage of the proposed model compared to the log-Gaussian Cox process that has been applied to the very same data set in several earlier publications

An evaluation of conditional spatial predictions of significant wave height based on the nonstationary spde model

The sea state plays an important role in offshore-and marine operations. It affects both direct costs as well as risks for human and/or material loss. A better understanding of the present-, near-future-, and far-future sea states will increase efficiency and safety in shipping since it allow a ship to reroute to a safer and/or more cost effective route. In the offshore industry it allows for minimizing downtime and aids in planning the construction of new offshore sites. Due to the complex nature of the sea state, its spatial distribution over a large region of ocean should be modeled using a probabilistic model. In this way, uncertainties due to lack of information and/or computing power can be quantified and decisions can be taken based on both what is known and what is not known. We analyze such a spatial probabilistic model in order to assess its ability to predict the significant wave height in the whole north Atlantic based only on measurements on a small line path, i.e., conditional prediction. This work is relevant for several applications, for instance data assimilation, oceanographic forecasting, and routing of ships