Discrete and continuous compositions

This paper examines a dataset which is modeled well by the Poisson-Log Normal process and by this process mixed with Log Normal data, which are both turned into compositions. This generates compositional data that has zeros without any need for conditional models or assuming that there is missing or censored data that needs adjustment. It also enables us to model dependence on covariates and within the composition

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

Improvements in the ray tracing of implicit surfaces based on interval arithmetic

Las superfícies implícitas son útiles en muchas áreasde los gráficos por ordenador. Una de sus principales ventajas es que pueden ser fácilmente usadas como primitivas para modelado. Aun asi, no son muy usadas porque su visualización toma bastante tiempo. Cuando se necesita una visualización precisa, la mejor opción es usar trazado de rayos. Sin embargo, pequeñas partes de las superficies desaparecen durante la visualización. Esto ocurre por la truncación que se presenta en la representación en punto flotante de los ordenadores; algunos bits se puerden durante las operaciones matemáticas en los algoritmos de intersección. En este tesis se presentan algoritmos para solucionar esos problemas. La investigación se basa en el uso del Análisis Intervalar Modal el cual incluye herramientas para resolver problemas con incertidumbe cuantificada. En esta tesis se proporcionan los fundamentos matemáticos necesarios para el desarrollo de estos algoritmos.

SBR technology for wastewater treatment: suitable operational conditions for a nutrient removal

Actualment, la legislació ambiental ha esdevingut més restrictiva pel que fa a la descàrrega d'aigües residuals amb nutrients, especialment en les anomenades àrees sensibles o zones vulnerables. Arran d'aquest fet, s'ha estimulat el coneixement, desenvolupament i millora dels processos d'eliminació de nutrients. El Reactor Discontinu Seqüencial (RDS) o Sequencing Batch Reactor (SBR) en anglès, és un sistema de tractament de fangs actius que opera mitjançant un procediment d'omplerta-buidat. En aquest tipus de reactors, l'aigua residual és addicionada en un sol reactor que treballa per càrregues repetint un cicle (seqüència) al llarg del temps. Una de les característiques dels SBR és que totes les diferents operacions (omplerta, reacció, sedimentació i buidat) es donen en un mateix reactor. La tecnologia SBR no és nova d'ara. El fet, és que va aparèixer abans que els sistema de tractament continu de fangs actius. El precursor dels SBR va ser un sistema d'omplerta-buidat que operava en discontinu. Entre els anys 1914 i 1920, varen sorgir certes dificultats moltes d'elles a nivell d'operació (vàlvules, canvis el cabal d'un reactor a un altre, elevat temps d'atenció per l'operari...) per aquests reactors. Però no va ser fins a finals de la dècada dels '50 principis del '60, amb el desenvolupament de nous equipaments i noves tecnologies, quan va tornar a ressorgir l'interès pels SBRs. Importants millores en el camp del subministrament d'aire (vàlvules motoritzades o d'acció pneumàtica) i en el de control (sondes de nivell, mesuradors de cabal, temporitzadors automàtics, microprocessadors) han permès que avui en dia els SBRs competeixin amb els sistemes convencional de fangs actius. L'objectiu de la present tesi és la identificació de les condicions d'operació adequades per un cicle segons el tipus d'aigua residual a l'entrada, les necessitats del tractament i la qualitat desitjada de la sortida utilitzant la tecnologia SBR. Aquestes tres característiques, l'aigua a tractar, les necessitats del tractament i la qualitat final desitjada determinen en gran mesura el tractament a realitzar. Així doncs, per tal d'adequar el tractament a cada tipus d'aigua residual i les seves necessitats, han estat estudiats diferents estratègies d'alimentació. El seguiment del procés es realitza mitjançant mesures on-line de pH, OD i RedOx, els canvis de les quals donen informació sobre l'estat del procés. Alhora un altre paràmetre que es pot calcular a partir de l'oxigen dissolt és la OUR que és una dada complementària als paràmetres esmentats. S'han avaluat les condicions d'operació per eliminar nitrogen d'una aigua residual sintètica utilitzant una estratègia d'alimentació esglaonada, a través de l'estudi de l'efecte del nombre d'alimentacions, la definició de la llargada i el número de fases per cicle, i la identificació dels punts crítics seguint les sondes de pH, OD i RedOx. S'ha aplicat l'estratègia d'alimentació esglaonada a dues aigües residuals diferents: una procedent d'una indústria tèxtil i l'altra, dels lixiviats d'un abocador. En ambdues aigües residuals es va estudiar l'eficiència del procés a partir de les condicions d'operació i de la velocitat del consum d'oxigen. Mentre que en l'aigua residual tèxtil el principal objectiu era eliminar matèria orgànica, en l'aigua procedent dels lixiviats d'abocador era eliminar matèria orgànica i nitrogen. S'han avaluat les condicions d'operació per eliminar nitrogen i fòsfor d'una aigua residual urbana utilitzant una estratègia d'alimentació esglaonada, a través de la definició del número i la llargada de les fases per cicle, i la identificació dels punts crítics seguint les sondes de pH, OD i RedOx. S'ha analitzat la influència del pH i la font de carboni per tal d'eliminar fòsfor d'una aigua sintètica a partir de l'estudi de l'increment de pH a dos reactors amb diferents fonts de carboni i l'estudi de l'efecte de canviar la font de carboni. Tal i com es pot veure al llarg de la tesi, on s'han tractat diferents aigües residuals per a diferents necessitats, un dels avantatges més importants d'un SBR és la seva flexibilitat.

Modelling, control and supervision for a class of hybrid systems

The aim of this thesis is to narrow the gap between two different control techniques: the continuous control and the discrete event control techniques DES. This gap can be reduced by the study of Hybrid systems, and by interpreting as Hybrid systems the majority of large-scale systems. In particular, when looking deeply into a process, it is often possible to identify interaction between discrete and continuous signals. Hybrid systems are systems that have both continuous, and discrete signals. Continuous signals are generally supposed continuous and differentiable in time, since discrete signals are neither continuous nor differentiable in time due to their abrupt changes in time. Continuous signals often represent the measure of natural physical magnitudes such as temperature, pressure etc. The discrete signals are normally artificial signals, operated by human artefacts as current, voltage, light etc. Typical processes modelled as Hybrid systems are production systems, chemical process, or continuos production when time and continuous measures interacts with the transport, and stock inventory system. Complex systems as manufacturing lines are hybrid in a global sense. They can be decomposed into several subsystems, and their links. Another motivation for the study of Hybrid systems is the tools developed by other research domains. These tools benefit from the use of temporal logic for the analysis of several properties of Hybrid systems model, and use it to design systems and controllers, which satisfies physical or imposed restrictions. This thesis is focused in particular types of systems with discrete and continuous signals in interaction. That can be modelled hard non-linealities, such as hysteresis, jumps in the state, limit cycles, etc. and their possible non-deterministic future behaviour expressed by an interpretable model description. The Hybrid systems treated in this work are systems with several discrete states, always less than thirty states (it can arrive to NP hard problem), and continuous dynamics evolving with expression: with Ki ¡ Rn constant vectors or matrices for X components vector. In several states the continuous evolution can be several of them Ki = 0. In this formulation, the mathematics can express Time invariant linear system. By the use of this expression for a local part, the combination of several local linear models is possible to represent non-linear systems. And with the interaction with discrete events of the system the model can compose non-linear Hybrid systems. Especially multistage processes with high continuous dynamics are well represented by the proposed methodology. Sate vectors with more than two components, as third order models or higher is well approximated by the proposed approximation. Flexible belt transmission, chemical reactions with initial start-up and mobile robots with important friction are several physical systems, which profits from the benefits of proposed methodology (accuracy). The motivation of this thesis is to obtain a solution that can control and drive the Hybrid systems from the origin or starting point to the goal. How to obtain this solution, and which is the best solution in terms of one cost function subject to the physical restrictions and control actions is analysed. Hybrid systems that have several possible states, different ways to drive the system to the goal and different continuous control signals are problems that motivate this research. The requirements of the system on which we work is: a model that can represent the behaviour of the non-linear systems, and that possibilities the prediction of possible future behaviour for the model, in order to apply an supervisor which decides the optimal and secure action to drive the system toward the goal. Specific problems can be determined by the use of this kind of hybrid models are: - The unity of order. - Control the system along a reachable path. - Control the system in a safe path. - Optimise the cost function. - Modularity of control The proposed model solves the specified problems in the switching models problem, the initial condition calculus and the unity of the order models. Continuous and discrete phenomena are represented in Linear hybrid models, defined with defined eighth-tuple parameters to model different types of hybrid phenomena. Applying a transformation over the state vector : for LTI system we obtain from a two-dimensional SS a single parameter, alpha, which still maintains the dynamical information. Combining this parameter with the system output, a complete description of the system is obtained in a form of a graph in polar representation. Using Tagaki-Sugeno type III is a fuzzy model which include linear time invariant LTI models for each local model, the fuzzyfication of different LTI local model gives as a result a non-linear time invariant model. In our case the output and the alpha measure govern the membership function. Hybrid systems control is a huge task, the processes need to be guided from the Starting point to the desired End point, passing a through of different specific states and points in the trajectory. The system can be structured in different levels of abstraction and the control in three layers for the Hybrid systems from planning the process to produce the actions, these are the planning, the process and control layer. In this case the algorithms will be applied to robotics ¡V a domain where improvements are well accepted ¡V it is expected to find a simple repetitive processes for which the extra effort in complexity can be compensated by some cost reductions. It may be also interesting to implement some control optimisation to processes such as fuel injection, DC-DC converters etc. In order to apply the RW theory of discrete event systems on a Hybrid system, we must abstract the continuous signals and to project the events generated for these signals, to obtain new sets of observable and controllable events. Ramadge & Wonham¡¦s theory along with the TCT software give a Controllable Sublanguage of the legal language generated for a Discrete Event System (DES). Continuous abstraction transforms predicates over continuous variables into controllable or uncontrollable events, and modifies the set of uncontrollable, controllable observable and unobservable events. Continuous signals produce into the system virtual events, when this crosses the bound limits. If this event is deterministic, they can be projected. It is necessary to determine the controllability of this event, in order to assign this to the corresponding set, , controllable, uncontrollable, observable and unobservable set of events. Find optimal trajectories in order to minimise some cost function is the goal of the modelling procedure. Mathematical model for the system allows the user to apply mathematical techniques over this expression. These possibilities are, to minimise a specific cost function, to obtain optimal controllers and to approximate a specific trajectory. The combination of the Dynamic Programming with Bellman Principle of optimality, give us the procedure to solve the minimum time trajectory for Hybrid systems. The problem is greater when there exists interaction between adjacent states. In Hybrid systems the problem is to determine the partial set points to be applied at the local models. Optimal controller can be implemented in each local model in order to assure the minimisation of the local costs. The solution of this problem needs to give us the trajectory to follow the system. Trajectory marked by a set of set points to force the system to passing over them. Several ways are possible to drive the system from the Starting point Xi to the End point Xf. Different ways are interesting in: dynamic sense, minimum states, approximation at set points, etc. These ways need to be safe and viable and RchW. And only one of them must to be applied, normally the best, which minimises the proposed cost function. A Reachable Way, this means the controllable way and safe, will be evaluated in order to obtain which one minimises the cost function. Contribution of this work is a complete framework to work with the majority Hybrid systems, the procedures to model, control and supervise are defined and explained and its use is demonstrated. Also explained is the procedure to model the systems to be analysed for automatic verification. Great improvements were obtained by using this methodology in comparison to using other piecewise linear approximations. It is demonstrated in particular cases this methodology can provide best approximation. The most important contribution of this work, is the Alpha approximation for non-linear systems with high dynamics While this kind of process is not typical, but in this case the Alpha approximation is the best linear approximation to use, and give a compact representation.