Technical note: Preliminary estimation of rockfall runout zones

Rockfall propagation areas can be determined using a simple geometric rule known as shadow angle or energy line method based on a simple Coulomb frictional model implemented in the CONEFALL computer program. Runout zones are estimated from a digital terrain model (DTM) and a grid file containing the cells representing rockfall potential source areas. The cells of the DTM that are lowest in altitude and located within a cone centered on a rockfall source cell belong to the potential propagation area associated with that grid cell. In addition, the CONEFALL method allows estimation of mean and maximum velocities and energies of blocks in the rockfall propagation areas. Previous studies indicate that the slope angle cone ranges from 27° to 37° depending on the assumptions made, i.e. slope morphology, probability of reaching a point, maximum run-out, field observations. Different solutions based on previous work and an example of an actual rockfall event are presented here.

Improved estimation of the covariance matrix of stock returns with an application to portofolio selection

This paper proposes to estimate the covariance matrix of stock returnsby an optimally weighted average of two existing estimators: the samplecovariance matrix and single-index covariance matrix. This method isgenerally known as shrinkage, and it is standard in decision theory andin empirical Bayesian statistics. Our shrinkage estimator can be seenas a way to account for extra-market covariance without having to specifyan arbitrary multi-factor structure. For NYSE and AMEX stock returns from1972 to 1995, it can be used to select portfolios with significantly lowerout-of-sample variance than a set of existing estimators, includingmulti-factor models.

Biomass estimation of Triplectides egleri Sattler (Trichoptera, Leptoceridae) in a stream at Ducke Reserve, Central Amazonia

ABSTRACT Biomass is a fundamental measure for understanding the structure and functioning (e.g. fluxes of energy and nutrients in the food chain) of aquatic ecosystems. We aim to provide predictive models to estimate the biomass of Triplectides egleri Sattler, 1963, in a stream in Central Amazonia, based on body and case dimensions. We used body length, head-capsule width, interocular distance and case length and width to derive biomass estimations. Linear, exponential and power regression models were used to assess the relationship between biomass and body or case dimensions. All regression models used in the biomass estimation of T. egleri were significant. The best fit between biomass and body or case dimensions was obtained using the power model, followed by the exponential and linear models. Body length provided the best estimate of biomass. However, the dimensions of sclerotized structures (interocular distance and head-capsule width) also provided good biomass predictions, and may be useful in estimating biomass of preserved and/or damaged material. Case width was the dimension of the case that provided the best estimate of biomass. Despite the low relation, case width may be useful in studies that require low stress on individuals.

Scaling of semivariograms and the kriging estimation of field-measured properties

Two methods were evaluated for scaling a set of semivariograms into a unified function for kriging estimation of field-measured properties. Scaling is performed using sample variances and sills of individual semivariograms as scale factors. Theoretical developments show that kriging weights are independent of the scaling factor which appears simply as a constant multiplying both sides of the kriging equations. The scaling techniques were applied to four sets of semivariograms representing spatial scales of 30 x 30 m to 600 x 900 km. Experimental semivariograms in each set successfully coalesced into a single curve by variances and sills of individual semivariograms. To evaluate the scaling techniques, kriged estimates derived from scaled semivariogram models were compared with those derived from unscaled models. Differences in kriged estimates of the order of 5% were found for the cases in which the scaling technique was not successful in coalescing the individual semivariograms, which also means that the spatial variability of these properties is different. The proposed scaling techniques enhance interpretation of semivariograms when a variety of measurements are made at the same location. They also reduce computational times for kriging estimations because kriging weights only need to be calculated for one variable. Weights remain unchanged for all other variables in the data set whose semivariograms are scaled.

Unfolding nature "in silico" : overcoming practical and metodological limitations of species distribution models

Abstract : The existence of a causal relationship between the spatial distribution of living organisms and their environment, in particular climate, has been long recognized and is the central principle of biogeography. In turn, this recognition has led scientists to the idea of using the climatic, topographic, edaphic and biotic characteristics of the environment to predict its potential suitability for a given species or biological community. In this thesis, my objective is to contribute to the development of methodological improvements in the field of species distribution modeling. More precisely, the objectives are to propose solutions to overcome limitations of species distribution models when applied to conservation biology issues, or when .used as an assessment tool of the potential impacts of global change. The first objective of my thesis is to contribute to evidence the potential of species distribution models for conservation-related applications. I present a methodology to generate pseudo-absences in order to overcome the frequent lack of reliable absence data. I also demonstrate, both theoretically (simulation-based) and practically (field-based), how species distribution models can be successfully used to model and sample rare species. Overall, the results of this first part of the thesis demonstrate the strong potential of species distribution models as a tool for practical applications in conservation biology. The second objective this thesis is to contribute to improve .projections of potential climate change impacts on species distributions, and in particular for mountain flora. I develop and a dynamic model, MIGCLIM, that allows the implementation of dispersal limitations into classic species distribution models and present an application of this model to two virtual species. Given that accounting for dispersal limitations requires information on seed dispersal, distances, a general methodology to classify species into broad dispersal types is also developed. Finally, the M~GCLIM model is applied to a large number of species in a study area of the western Swiss Alps. Overall, the results indicate that while dispersal limitations can have an important impact on the outcome of future projections of species distributions under climate change scenarios, estimating species threat levels (e.g. species extinction rates) for a mountainous areas of limited size (i.e. regional scale) can also be successfully achieved when considering dispersal as unlimited (i.e. ignoring dispersal limitations, which is easier from a practical point of view). Finally, I present the largest fine scale assessment of potential climate change impacts on mountain vegetation that has been carried-out to date. This assessment involves vegetation from 12 study areas distributed across all major western and central European mountain ranges. The results highlight that some mountain ranges (the Pyrenees and the Austrian Alps) are expected to be more affected by climate change than others (Norway and the Scottish Highlands). The results I obtain in this study also indicate that the threat levels projected by fine scale models are less severe than those derived from coarse scale models. This result suggests that some species could persist in small refugias that are not detected by coarse scale models. Résumé : L'existence d'une relation causale entre la répartition des espèces animales et végétales et leur environnement, en particulier le climat, a été mis en évidence depuis longtemps et est un des principes centraux en biogéographie. Ce lien a naturellement conduit à l'idée d'utiliser les caractéristiques climatiques, topographiques, édaphiques et biotiques de l'environnement afin d'en prédire la qualité pour une espèce ou une communauté. Dans ce travail de thèse, mon objectif est de contribuer au développement d'améliorations méthodologiques dans le domaine de la modélisation de la distribution d'espèces dans le paysage. Plus précisément, les objectifs sont de proposer des solutions afin de surmonter certaines limitations des modèles de distribution d'espèces dans des applications pratiques de biologie de la conservation ou dans leur utilisation pour évaluer l'impact potentiel des changements climatiques sur l'environnement. Le premier objectif majeur de mon travail est de contribuer à démontrer le potentiel des modèles de distribution d'espèces pour des applications pratiques en biologie de la conservation. Je propose une méthode pour générer des pseudo-absences qui permet de surmonter le problème récurent du manque de données d'absences fiables. Je démontre aussi, de manière théorique (par simulation) et pratique (par échantillonnage de terrain), comment les modèles de distribution d'espèces peuvent être utilisés pour modéliser et améliorer l'échantillonnage des espèces rares. Ces résultats démontrent le potentiel des modèles de distribution d'espèces comme outils pour des applications de biologie de la conservation. Le deuxième objectif majeur de ce travail est de contribuer à améliorer les projections d'impacts potentiels des changements climatiques sur la flore, en particulier dans les zones de montagnes. Je développe un modèle dynamique de distribution appelé MigClim qui permet de tenir compte des limitations de dispersion dans les projections futures de distribution potentielle d'espèces, et teste son application sur deux espèces virtuelles. Vu que le fait de prendre en compte les limitations dues à la dispersion demande des données supplémentaires importantes (p.ex. la distance de dispersion des graines), ce travail propose aussi une méthode de classification simplifiée des espèces végétales dans de grands "types de disperseurs", ce qui permet ainsi de d'obtenir de bonnes approximations de distances de dispersions pour un grand nombre d'espèces. Finalement, j'applique aussi le modèle MIGCLIM à un grand nombre d'espèces de plantes dans une zone d'études des pré-Alpes vaudoises. Les résultats montrent que les limitations de dispersion peuvent avoir un impact considérable sur la distribution potentielle d'espèces prédites sous des scénarios de changements climatiques. Cependant, quand les modèles sont utilisés pour évaluer les taux d'extinction d'espèces dans des zones de montages de taille limitée (évaluation régionale), il est aussi possible d'obtenir de bonnes approximations en considérant la dispersion des espèces comme illimitée, ce qui est nettement plus simple d'un point dé vue pratique. Pour terminer je présente la plus grande évaluation à fine échelle d'impact potentiel des changements climatiques sur la flore des montagnes conduite à ce jour. Cette évaluation englobe 12 zones d'études réparties sur toutes les chaines de montages principales d'Europe occidentale et centrale. Les résultats montrent que certaines chaines de montagnes (les Pyrénées et les Alpes Autrichiennes) sont projetées comme plus sensibles aux changements climatiques que d'autres (les Alpes Scandinaves et les Highlands d'Ecosse). Les résultats obtenus montrent aussi que les modèles à échelle fine projettent des impacts de changement climatiques (p. ex. taux d'extinction d'espèces) moins sévères que les modèles à échelle large. Cela laisse supposer que les modèles a échelle fine sont capables de modéliser des micro-niches climatiques non-détectées par les modèles à échelle large.

Estimation of percentage body fat in 6- to 13-year-old children by skinfold thickness, body mass index and waist circumference.

We evaluated the accuracy of skinfold thicknesses, BMI and waist circumference for the prediction of percentage body fat (PBF) in a representative sample of 372 Swiss children aged 6-13 years. PBF was measured using dual-energy X-ray absorptiometry. On the basis of a preliminary bootstrap selection of predictors, seven regression models were evaluated. All models included sex, age and pubertal stage plus one of the following predictors: (1) log-transformed triceps skinfold (logTSF); (2) logTSF and waist circumference; (3) log-transformed sum of triceps and subscapular skinfolds (logSF2); (4) log-transformed sum of triceps, biceps, subscapular and supra-iliac skinfolds (logSF4); (5) BMI; (6) waist circumference; (7) BMI and waist circumference. The adjusted determination coefficient (R² adj) and the root mean squared error (RMSE; kg) were calculated for each model. LogSF4 (R² adj 0.85; RMSE 2.35) and logSF2 (R² adj 0.82; RMSE 2.54) were similarly accurate at predicting PBF and superior to logTSF (R² adj 0.75; RMSE 3.02), logTSF combined with waist circumference (R² adj 0.78; RMSE 2.85), BMI (R² adj 0.62; RMSE 3.73), waist circumference (R² adj 0.58; RMSE 3.89), and BMI combined with waist circumference (R² adj 0.63; RMSE 3.66) (P < 0.001 for all values of R² adj). The finding that logSF4 was only modestly superior to logSF2 and that logTSF was better than BMI and waist circumference at predicting PBF has important implications for paediatric epidemiological studies aimed at disentangling the effect of body fat on health outcomes.

Estimation of jump-diffusion processes via empirical characteristic functions

Preface The starting point for this work and eventually the subject of the whole thesis was the question: how to estimate parameters of the affine stochastic volatility jump-diffusion models. These models are very important for contingent claim pricing. Their major advantage, availability T of analytical solutions for characteristic functions, made them the models of choice for many theoretical constructions and practical applications. At the same time, estimation of parameters of stochastic volatility jump-diffusion models is not a straightforward task. The problem is coming from the variance process, which is non-observable. There are several estimation methodologies that deal with estimation problems of latent variables. One appeared to be particularly interesting. It proposes the estimator that in contrast to the other methods requires neither discretization nor simulation of the process: the Continuous Empirical Characteristic function estimator (EGF) based on the unconditional characteristic function. However, the procedure was derived only for the stochastic volatility models without jumps. Thus, it has become the subject of my research. This thesis consists of three parts. Each one is written as independent and self contained article. At the same time, questions that are answered by the second and third parts of this Work arise naturally from the issues investigated and results obtained in the first one. The first chapter is the theoretical foundation of the thesis. It proposes an estimation procedure for the stochastic volatility models with jumps both in the asset price and variance processes. The estimation procedure is based on the joint unconditional characteristic function for the stochastic process. The major analytical result of this part as well as of the whole thesis is the closed form expression for the joint unconditional characteristic function for the stochastic volatility jump-diffusion models. The empirical part of the chapter suggests that besides a stochastic volatility, jumps both in the mean and the volatility equation are relevant for modelling returns of the S&P500 index, which has been chosen as a general representative of the stock asset class. Hence, the next question is: what jump process to use to model returns of the S&P500. The decision about the jump process in the framework of the affine jump- diffusion models boils down to defining the intensity of the compound Poisson process, a constant or some function of state variables, and to choosing the distribution of the jump size. While the jump in the variance process is usually assumed to be exponential, there are at least three distributions of the jump size which are currently used for the asset log-prices: normal, exponential and double exponential. The second part of this thesis shows that normal jumps in the asset log-returns should be used if we are to model S&P500 index by a stochastic volatility jump-diffusion model. This is a surprising result. Exponential distribution has fatter tails and for this reason either exponential or double exponential jump size was expected to provide the best it of the stochastic volatility jump-diffusion models to the data. The idea of testing the efficiency of the Continuous ECF estimator on the simulated data has already appeared when the first estimation results of the first chapter were obtained. In the absence of a benchmark or any ground for comparison it is unreasonable to be sure that our parameter estimates and the true parameters of the models coincide. The conclusion of the second chapter provides one more reason to do that kind of test. Thus, the third part of this thesis concentrates on the estimation of parameters of stochastic volatility jump- diffusion models on the basis of the asset price time-series simulated from various "true" parameter sets. The goal is to show that the Continuous ECF estimator based on the joint unconditional characteristic function is capable of finding the true parameters. And, the third chapter proves that our estimator indeed has the ability to do so. Once it is clear that the Continuous ECF estimator based on the unconditional characteristic function is working, the next question does not wait to appear. The question is whether the computation effort can be reduced without affecting the efficiency of the estimator, or whether the efficiency of the estimator can be improved without dramatically increasing the computational burden. The efficiency of the Continuous ECF estimator depends on the number of dimensions of the joint unconditional characteristic function which is used for its construction. Theoretically, the more dimensions there are, the more efficient is the estimation procedure. In practice, however, this relationship is not so straightforward due to the increasing computational difficulties. The second chapter, for example, in addition to the choice of the jump process, discusses the possibility of using the marginal, i.e. one-dimensional, unconditional characteristic function in the estimation instead of the joint, bi-dimensional, unconditional characteristic function. As result, the preference for one or the other depends on the model to be estimated. Thus, the computational effort can be reduced in some cases without affecting the efficiency of the estimator. The improvement of the estimator s efficiency by increasing its dimensionality faces more difficulties. The third chapter of this thesis, in addition to what was discussed above, compares the performance of the estimators with bi- and three-dimensional unconditional characteristic functions on the simulated data. It shows that the theoretical efficiency of the Continuous ECF estimator based on the three-dimensional unconditional characteristic function is not attainable in practice, at least for the moment, due to the limitations on the computer power and optimization toolboxes available to the general public. Thus, the Continuous ECF estimator based on the joint, bi-dimensional, unconditional characteristic function has all the reasons to exist and to be used for the estimation of parameters of the stochastic volatility jump-diffusion models.

Bayesian estimation of speciation and extinction from incomplete fossil occurrence data.

The temporal dynamics of species diversity are shaped by variations in the rates of speciation and extinction, and there is a long history of inferring these rates using first and last appearances of taxa in the fossil record. Understanding diversity dynamics critically depends on unbiased estimates of the unobserved times of speciation and extinction for all lineages, but the inference of these parameters is challenging due to the complex nature of the available data. Here, we present a new probabilistic framework to jointly estimate species-specific times of speciation and extinction and the rates of the underlying birth-death process based on the fossil record. The rates are allowed to vary through time independently of each other, and the probability of preservation and sampling is explicitly incorporated in the model to estimate the true lifespan of each lineage. We implement a Bayesian algorithm to assess the presence of rate shifts by exploring alternative diversification models. Tests on a range of simulated data sets reveal the accuracy and robustness of our approach against violations of the underlying assumptions and various degrees of data incompleteness. Finally, we demonstrate the application of our method with the diversification of the mammal family Rhinocerotidae and reveal a complex history of repeated and independent temporal shifts of both speciation and extinction rates, leading to the expansion and subsequent decline of the group. The estimated parameters of the birth-death process implemented here are directly comparable with those obtained from dated molecular phylogenies. Thus, our model represents a step towards integrating phylogenetic and fossil information to infer macroevolutionary processes.

Estimation of the noisy component of anatomical backgrounds.

The knowledge of the relationship that links radiation dose and image quality is a prerequisite to any optimization of medical diagnostic radiology. Image quality depends, on the one hand, on the physical parameters such as contrast, resolution, and noise, and on the other hand, on characteristics of the observer that assesses the image. While the role of contrast and resolution is precisely defined and recognized, the influence of image noise is not yet fully understood. Its measurement is often based on imaging uniform test objects, even though real images contain anatomical backgrounds whose statistical nature is much different from test objects used to assess system noise. The goal of this study was to demonstrate the importance of variations in background anatomy by quantifying its effect on a series of detection tasks. Several types of mammographic backgrounds and signals were examined by psychophysical experiments in a two-alternative forced-choice detection task. According to hypotheses concerning the strategy used by the human observers, their signal to noise ratio was determined. This variable was also computed for a mathematical model based on the statistical decision theory. By comparing theoretical model and experimental results, the way that anatomical structure is perceived has been analyzed. Experiments showed that the observer's behavior was highly dependent upon both system noise and the anatomical background. The anatomy partly acts as a signal recognizable as such and partly as a pure noise that disturbs the detection process. This dual nature of the anatomy is quantified. It is shown that its effect varies according to its amplitude and the profile of the object being detected. The importance of the noisy part of the anatomy is, in some situations, much greater than the system noise. Hence, reducing the system noise by increasing the dose will not improve task performance. This observation indicates that the tradeoff between dose and image quality might be optimized by accepting a higher system noise. This could lead to a better resolution, more contrast, or less dose.

Accuracy of indirect estimation of power output from uphill performance in cycling.

PURPOSE: To use measurement by cycling power meters (Pmes) to evaluate the accuracy of commonly used models for estimating uphill cycling power (Pest). Experiments were designed to explore the influence of wind speed and steepness of climb on accuracy of Pest. The authors hypothesized that the random error in Pest would be largely influenced by the windy conditions, the bias would be diminished in steeper climbs, and windy conditions would induce larger bias in Pest. METHODS: Sixteen well-trained cyclists performed 15 uphill-cycling trials (range: length 1.3-6.3 km, slope 4.4-10.7%) in a random order. Trials included different riding position in a group (lead or follow) and different wind speeds. Pmes was quantified using a power meter, and Pest was calculated with a methodology used by journalists reporting on the Tour de France. RESULTS: Overall, the difference between Pmes and Pest was -0.95% (95%CI: -10.4%, +8.5%) for all trials and 0.24% (-6.1%, +6.6%) in conditions without wind (<2 m/s). The relationship between percent slope and the error between Pest and Pmes were considered trivial. CONCLUSIONS: Aerodynamic drag (affected by wind velocity and orientation, frontal area, drafting, and speed) is the most confounding factor. The mean estimated values are close to the power-output values measured by power meters, but the random error is between ±6% and ±10%. Moreover, at the power outputs (>400 W) produced by professional riders, this error is likely to be higher. This observation calls into question the validity of releasing individual values without reporting the range of random errors.

Gas-liquid mass transfer in bubbly flow: Estimation of mass transfer, bubble size and reactor performance in various applications

Gas-liquid mass transfer is an important issue in the design and operation of many chemical unit operations. Despite its importance, the evaluation of gas-liquid mass transfer is not straightforward due to the complex nature of the phenomena involved. In this thesis gas-liquid mass transfer was evaluated in three different gas-liquid reactors in a traditional way by measuring the volumetric mass transfer coefficient (kLa). The studied reactors were a bubble column with a T-junction two-phase nozzle for gas dispersion, an industrial scale bubble column reactor for the oxidation of tetrahydroanthrahydroquinone and a concurrent downflow structured bed.The main drawback of this approach is that the obtained correlations give only the average volumetric mass transfer coefficient, which is dependent on average conditions. Moreover, the obtained correlations are valid only for the studied geometry and for the chemical system used in the measurements. In principle, a more fundamental approach is to estimate the interfacial area available for mass transfer from bubble size distributions obtained by solution of population balance equations. This approach has been used in this thesis by developing a population balance model for a bubble column together with phenomenological models for bubble breakage and coalescence. The parameters of the bubble breakage rate and coalescence rate models were estimated by comparing the measured and calculated bubble sizes. The coalescence models always have at least one experimental parameter. This is because the bubble coalescence depends on liquid composition in a way which is difficult to evaluate using known physical properties. The coalescence properties of some model solutions were evaluated by measuring the time that a bubble rests at the free liquid-gas interface before coalescing (the so-calledpersistence time or rest time). The measured persistence times range from 10 msup to 15 s depending on the solution. The coalescence was never found to be instantaneous. The bubble oscillates up and down at the interface at least a coupleof times before coalescence takes place. The measured persistence times were compared to coalescence times obtained by parameter fitting using measured bubble size distributions in a bubble column and a bubble column population balance model. For short persistence times, the persistence and coalescence times are in good agreement. For longer persistence times, however, the persistence times are at least an order of magnitude longer than the corresponding coalescence times from parameter fitting. This discrepancy may be attributed to the uncertainties concerning the estimation of energy dissipation rates, collision rates and mechanisms and contact times of the bubbles.

Estimation of age- and stage-specific Catalan breast cancer survival functions using US and Catalan survival data

Background: During the last part of the 1990s the chance of surviving breast cancer increased. Changes in survival functions reflect a mixture of effects. Both, the introduction of adjuvant treatments and early screening with mammography played a role in the decline in mortality. Evaluating the contribution of these interventions using mathematical models requires survival functions before and after their introduction. Furthermore, required survival functions may be different by age groups and are related to disease stage at diagnosis. Sometimes detailed information is not available, as was the case for the region of Catalonia (Spain). Then one may derive the functions using information from other geographical areas. This work presents the methodology used to estimate age- and stage-specific Catalan breast cancer survival functions from scarce Catalan survival data by adapting the age- and stage-specific US functions. Methods: Cubic splines were used to smooth data and obtain continuous hazard rate functions. After, we fitted a Poisson model to derive hazard ratios. The model included time as a covariate. Then the hazard ratios were applied to US survival functions detailed by age and stage to obtain Catalan estimations. Results: We started estimating the hazard ratios for Catalonia versus the USA before and after the introduction of screening. The hazard ratios were then multiplied by the age- and stage-specific breast cancer hazard rates from the USA to obtain the Catalan hazard rates. We also compared breast cancer survival in Catalonia and the USA in two time periods, before cancer control interventions (USA 1975–79, Catalonia 1980–89) and after (USA and Catalonia 1990–2001). Survival in Catalonia in the 1980–89 period was worse than in the USA during 1975–79, but the differences disappeared in 1990–2001. Conclusion: Our results suggest that access to better treatments and quality of care contributed to large improvements in survival in Catalonia. On the other hand, we obtained detailed breast cancer survival functions that will be used for modeling the effect of screening and adjuvant treatments in Catalonia.

Falsification and corroboration of conceptual hydrological models using geophysical data

Geophysical data may provide crucial information about hydrological properties, states, and processes that are difficult to obtain by other means. Large data sets can be acquired over widely different scales in a minimally invasive manner and at comparatively low costs, but their effective use in hydrology makes it necessary to understand the fidelity of geophysical models, the assumptions made in their construction, and the links between geophysical and hydrological properties. Geophysics has been applied for groundwater prospecting for almost a century, but it is only in the last 20 years that it is regularly used together with classical hydrological data to build predictive hydrological models. A largely unexplored venue for future work is to use geophysical data to falsify or rank competing conceptual hydrological models. A promising cornerstone for such a model selection strategy is the Bayes factor, but it can only be calculated reliably when considering the main sources of uncertainty throughout the hydrogeophysical parameter estimation process. Most classical geophysical imaging tools tend to favor models with smoothly varying property fields that are at odds with most conceptual hydrological models of interest. It is thus necessary to account for this bias or use alternative approaches in which proposed conceptual models are honored at all steps in the model building process.

Bayesian Estimation of Noise

In mathematical modeling the estimation of the model parameters is one of the most common problems. The goal is to seek parameters that fit to the measurements as well as possible. There is always error in the measurements which implies uncertainty to the model estimates. In Bayesian statistics all the unknown quantities are presented as probability distributions. If there is knowledge about parameters beforehand, it can be formulated as a prior distribution. The Bays’ rule combines the prior and the measurements to posterior distribution. Mathematical models are typically nonlinear, to produce statistics for them requires efficient sampling algorithms. In this thesis both Metropolis-Hastings (MH), Adaptive Metropolis (AM) algorithms and Gibbs sampling are introduced. In the thesis different ways to present prior distributions are introduced. The main issue is in the measurement error estimation and how to obtain prior knowledge for variance or covariance. Variance and covariance sampling is combined with the algorithms above. The examples of the hyperprior models are applied to estimation of model parameters and error in an outlier case.

Estimation of age- and stage-specific Catalan breast cancer survival functions using US and Catalan survival data

During the last part of the 1990s the chance of surviving breast cancer increased. Changes in survival functions reflect a mixture of effects. Both, the introduction of adjuvant treatments and early screening with mammography played a role in the decline in mortality. Evaluating the contribution of these interventions using mathematical models requires survival functions before and after their introduction. Furthermore, required survival functions may be different by age groups and are related to disease stage at diagnosis. Sometimes detailed information is not available, as was the case for the region of Catalonia (Spain). Then one may derive the functions using information from other geographical areas. This work presents the methodology used to estimate age- and stage-specific Catalan breast cancer survival functions from scarce Catalan survival data by adapting the age- and stage-specific US functions. Methods: Cubic splines were used to smooth data and obtain continuous hazard rate functions. After, we fitted a Poisson model to derive hazard ratios. The model included time as a covariate. Then the hazard ratios were applied to US survival functions detailed by age and stage to obtain Catalan estimations. Results: We started estimating the hazard ratios for Catalonia versus the USA before and after the introduction of screening. The hazard ratios were then multiplied by the age- and stage-specific breast cancer hazard rates from the USA to obtain the Catalan hazard rates. We also compared breast cancer survival in Catalonia and the USA in two time periods, before cancer control interventions (USA 1975–79, Catalonia 1980–89) and after (USA and Catalonia 1990–2001). Survival in Catalonia in the 1980–89 period was worse than in the USA during 1975–79, but the differences disappeared in 1990–2001. Conclusion: Our results suggest that access to better treatments and quality of care contributed to large improvements in survival in Catalonia. On the other hand, we obtained detailed breast cancer survival functions that will be used for modeling the effect of screening and adjuvant treatments in Catalonia