Exponential smoothing algorithms are very attractive for the practical world such as in industry. When considering bivariate exponential smoothing methods, in addition to the properties of univariate methods, additional properties give insight to relationships between the two components of a process, and also to the overall structure of the model. It is important to study these properties, but even with the merits the bivariate exponential smoothing algorithms have, exponential smoothing algorithms are nonstatistical/nonstochastic and to study the properties within exponential smoothing may be worthless. As an alternative approach, the (bivariate) ARIMA and the structural models which are classes of statistical models, are shown to generalize the exponential smoothing algorithms. We study these properties within these classes as they will have implications on exponential smoothing algorithms. Forecast properties are studied using the state space model and the Kalman filter. Comparison of ARIMA and structural model completes the study. / Mathematical Sciences / M. Sc. (Statistics)

Dans cette thèse nous proposons de nouveaux modèles paramétriques en traitement du signal et de l'image, fondés sur la décomposition de Wold des processus stochastiques. Les approches de modélisation font appel à l'analyse fonctionnelle et harmonique, l'analyse par ondelettes, ainsi qu'à la théorie des champs stochastiques. Le premier chapitre a un caractère introductif théorique et précise les éléments de base concernant le contexte de la prédiction linéaire des processus stochastiques stationnaires et la décomposition Wold, dans le cas 1-D et multi-D. On montre comment les différentes parties de la décomposition sont obtenues à partir de l'hypothèse de stationnarité, via la représentation du processus comme l'orbite d'un certain opérateur unitaire, l'isomorphisme canonique de Kolmogorov et les conséquences sur la prédiction linéaire du théorème de Szégö et de ses extensions multidimensionnelles. Le deuxième chapitre traite une approche de factorisation spectrale de la densité spectrale de puissance qu'on utilisera pour l'identification des modèles de type Moyenne Ajustée (MA), Autorégressif (AR) et ARMA. On utilise la représentation par le noyau reproduisant de Poisson d'une fonction extérieure pour construire un algorithme d'estimation d'un modèle MA avec une densité spectrale de puissance donnée. Cette méthode d'estimation est présentée dans le cadre de deux applications: - Dans la simulation de canaux sans fil de type Rayleigh (cas 1-D). - Dans le cadre d'une approche de décomposition de Wold des images texturées (cas 2-D). Dans le troisième chapitre nous abordons la représentation et la compression hybride d'images. Nous proposons une approche de compression d'images qui utilise conjointement : - les modèles issus de la décomposition de Wold pour la représentation des régions dites texturées de l'image; - une approche fondée sur les ondelettes pour le codage de la partie "cartoon" (ou non-texturée) de l' image. Dans ce cadre, nous proposons une nouvelle approche pour la décomposition d'une image dans une partie texturée et une partie non-texturée fondée sur la régularité locale. Chaque partie est ensuite codée à l'aide de sa représentation particulière. / In this thesis we propose new parametric models in signal and image processing based on the Wold decomposition of stationary stochastic processes. These models rely upon several theoretical results from functional and harmonic analysis, wavelet analysis and the theory of stochastic fields, The first chapter presents the theoretical background of the linear prediction for stationary processes and of the Wold decomposition theorems in 1-D and n-D. It is shown how the different parts of the decomposition are obtained and represented, by the means of the unitary orbit representation of stationary processes, the Kolmogorov canonical model and Szego-type extensions. The second chapter deals with a spectral factorisation approach of the power spectral density used for the parameter estimation of Moving Avergage (MA), AutoRegressif (AR) and ARMA models. The method uses the Poisson integral representation in Hardy spaces in order to estimate an outer transfer function from its power spectral density. - Simulators for Rayleigh fading channels (1-D). - A scheme for the Wold decomposition for texture images (2-D). In the third chapter we deal with hybrid models for image representation and compression. We propose a compression scheme which jointly uses, on one hand, Wold models for textured regions of the image, and on the other hand a wavelet-based approach for coding the 'cartoon' (or non-textured) part of the image. In this context, we propose a new algorithm for the decomposing images in a textured part and a non-textured part. The separate parts are then coded with the appropriate representation.

Södra Cell sells 1 900 000 ton pulp every year. Of this 490 000 tonne is sold with a contract system based on a pricing index called PIX NBSK. This index was started in 1996 and reflects the price of pulp from conferious forest. We study the NBSK PIX value of softwood from October 1996 to December 2006. People working in this branch known that there is strong periodicity in the prices. We use predictive analysis to see if clients can benefit from the periodicity and use the options in the contract system Södra offers today. We conclude that a drawback for the current contract system is that there are too many contracts in proportion to the duration time that is one year for all contracts. Using a time series model called ARMA we make successfull predictions the price difference between two contracts. Based on this prediction we change between these contracts, reducing the price with 0.81% in mean during 1997-2006. Due to the total turnover, if all clients would used such predictions during 1997-2006 Södra's income would have been reduced with 2.77 million USD a year in mean. The prices used before PIX are called list prices. The list prices seem to behave like the PIX index. Supposing that the same contract system we see in PIX today was used 1975-2006 with the list price as the base index I made a prediction of the list prices from 1986-2006. Thanks to my predictions, if I had been a client during this period and under mentioned considerations I would have been buying pulp to a price reduced with 0.57%. If clients had known the PIX between 1996-2006 in say 1995 Södra's contract system based on PIX would give them a price reduction that were 1.5% in mean during 1996-2006. Price reduction is not possible all years, but when it occurs it can be as big as 3% of the price. Suppose the clients always choose the contract with the lowest price and thereby get a reduced price over time. Then with 95% probability over a long period the price reduction is somewhere in between 0.4-2.7%. To strangle this price reduction possibility for the clients there are two ways to go: either reduce the number of contracts or extend the duration time of the contracts. To find a suitable duration time, we do spectral density estimation to get indications of which periods that are most important. From this we see that PIX index has a period of five years, wavelet approximated PIX index has 3.4 years and the list prices has a period of 5.6 years. This indicates that current duration time one year is too short. Therefore if it wouldn't effect Södra's clients, an extension of the duration time from one to five years would be good. If Södra don't extend the duration time of the contracts my recommendation is to have fewer contracts. The possibility to change between the contracts ''average last three months'' and ''average current month'' every other year is the weakest point of today's system. Therefore I recommend stop selling pulp to the contract ''average PIX last three months''. We can't prove any longterm difference between the contracts. If Södra chooses to have just one contract from this point of view it does not matter which one they choose. However, it seems like a good idea to follow the global market and therefore I recommend to choose ''average PIX current month'' rather than ''average PIX last three months'' which lags behind the market front. Since the price ''average current month'' is available at FOEX web page I think Södra should choose this contract if they decide to have only one contract. / Södra Cell säljer årligen 1 900 000 ton pappersmassa. Av denna mängd säljs 490 000 ton enligt ett kontraktsystem baserat på ett prisindex som heter PIX NBSK. Detta index introducerades 1996 och reflekterar priset på pappersmassa gjord av barrträd. Jag studerar priset på indexet från Oktober 1996 till December 2006. Dagens kontraktsystem är baserat på kontrakt med löptiden ett år. Jag undersöker om man kan prediktera prisskilllnaden mellan kontrakten, dra nytta att dagens löptid som bara är ett år och välja det kontrakt som ger det billigaste priset så ofta att priset över lång tid reduceras. När man predikterar gör man en uppskattningen av framtiden utifrån en modell av hur framtid beror på dåtid och nutid. Den modell jag har använt kallas ARMA. Denna tillsammans med priserna på pappersmassa från 1975 och framåt gav mig ett fruktbart sätt att förutsäga priserna. Resultatet blev ett pris reducerat med 0.81% i medel under perioden 1996-2006. Eftersom Södra ha så stor försäljningsvolym skulle de ha förlorat 2.27 miljoner dollar per ton i medel om alla kunder ha spekulerat utifrån den modellen jag använde. Om dagens kontraktsystem hade börjat användas 1975 med listpriserna som bas hade en kund som använt min prediktionsmetod fått ett pris reducerat med 0.57% under perioden 1986-2006. Om kunderna i förväg hade vetat priset under 1996-2001 gav det nuvarande systemet en reducerad medelintäkt med 1.5% av priset. Enskilda år reducerades intäkten med så mycket som 3%. Beräknar man konfidensintervall för prisreduktionerna så inser man att på lång sikt kommer dessa vara av storleksordningen 0.4-2.7% med sannolikheten 95%. Detta förutsatt att klienterna kan se in i framtiden. Siffran 2.7% alltså ett mått på hur stor risk man tar med dagens system. Jag tror inte att klienterna kommer reducera priset med 2.7% med nuvarande system, men det är en övre gräns. De gynsamma prediktionerna har sitt ursprung i att det finns periodicitet i priserna. Jag undersöker denna periodicitet med spektralanalys. Periodiciteten för PIX indexet är starkast kring 5 år. En wavelet-approximation av PIX-indexet hade störst periodicitet kring 3.4 år. Listpriserna hade starkast periodicitet kring 5.6 år. Detta indikerar att den nuvarande löptiden, ett år, är för kort. En lämpligare löptid för kontrakten är 5 år. Förmodligen är fem års löptid alltför lång tid att binda sig för många kunder. Därför föreslår jag att man reducerar antalet kontrakt istället. Den största svagheten i dagens system är den korta löptiden tillsammans med kontrakten ''average current month'' och ''average last three months''. Jag rekommenderar att man slutar erbjuda kontraktet ''average last three months''. Det allra säkraste är att endast erbjuda ett kontrakt. Vi har inte kunnat påvisa några skillnader över lång sikt mellan kontrakten såtillvida att något kontrakt skulle ge ett lägre medelpris än ett annat. Ur den aspekten är det godtyckligt vilket kontrakt man väljer, men det verkar vettigt att följa den globala marknaden. Därför är det eftersläpande kontraktet ''average PIX last three months'' inte att rekommendera, välj heller ''average PIX current month''. Ett annat argument för att välja ''average PIX current month'' är att dessa priser finns på FOEX hemsida och inga extra beräkningar behöver göras.

Uncertainty involved in the safe and comfort design of the structures is a major concern of civil engineers. Traditionally, the uncertainty has been overcome by utilizing various and relatively large safety factors for loads and structural properties. As a result in conventional design of for example tall buildings, the designed structural elements have unnecessary dimensions that sometimes are more than double of the ones needed to resist normal loads. On the other hand the requirements for strength and safety and comfort can be conflicting. Consequently, an alternative approach for design of the structures may be of great interest in design of safe and comfort structures that also offers economical advantages. Recently, there has been growing interest among the researchers in the concept of structural control as an alternative or complementary approach to the existing approaches of structural design. A few buildings have been designed and built based on this concept. The concept is to utilize a device for applying a force (known as control force) to encounter the effects of disturbing forces like earthquake force. However, the concept still has not found its rightful place among the practical engineers and more research is needed on the subject. One of the main problems in structural control is to find a proper algorithm for determining the optimum control force that should be applied to the structure. The investigation reported in this thesis is concerned with the application of active control to civil engineering structures. From the literature on control theory. (Particularly literature on the control of civil engineering structures) problems faced in application of control theory were identified and classified into two categories: 1) problems common to control of all dynamical systems, and 2) problems which are specially important in control of civil engineering structures. It was concluded that while many control algorithms are suitable for control of dynamical systems, considering the special problems in controlling civil structures and considering the unique future of structural control, many otherwise useful control algorithms face practical problems in application to civil structures. Consequently a set of criteria were set for judging the suitability of the control algorithms for use in control of civil engineering structures. Various types of existing control algorithms were investigated and finally it was concluded that predictive optimal control algorithms possess good characteristics for purpose of control of civil engineering structures. Among predictive control algorithms, those that use ARMA stochastic models for predicting the ground acceleration are better fitted to the structural control environment because all the past measured excitation is used to estimate the trends of the excitation for making qualified guesses about its coming values. However, existing ARMA based predictive algorithms are devised specially for earthquake and require on-line measurement of the external disturbing load which is not possible for dynamic loads like wind or blast. So, the algorithms are not suitable for tall buildings that experience both earthquake and wind loads during their life. Consequently, it was decided to establish a new closed loop predictive optimal control based on ARMA models as the first phase of the study. In this phase it was initially established that ARMA models are capable of predicting response of a linear SDOF system to the earthquake excitation a few steps ahead. The results of the predictions encouraged a search for finding a new closed loop optimal predictive control algorithm for linear SDOF structures based on prediction of the response by ARMA models. The second part of phase I, was devoted to developing and testing the proposed algorithm The new developed algorithm is different from other ARMA based optimal controls since it uses ARMA models for prediction of the structure response while existing algorithms predict the input excitation. Modeling the structure response as an AR or ARMA stochastic process is an effective mean for prediction of the structure response while avoiding measurement of the input excitation. ARMA models used in the algorithm enables it to avoid or reduce the time delay effect by predicting the structure response a few steps ahead. Being a closed loop control, the algorithm is suitable for all structural control conditions and can be used in a single control mechanism for vibration control of tall buildings against wind, earthquake or other random dynamic loads. Consequently the standby time is less than that for existing ARMA based algorithms devised only for earthquakes. This makes the control mechanism more reliable. The proposed algorithm utilizes and combines two different mathematical models. First model is an ARMA model representing the environment and the structure as a single system subjected to the unknown random excitation and the second model is a linear SDOF system which represents the structure subjected to a known past history of the applied control force only. The principle of superposition is then used to combine the results of these two models to predict the total response of the structure as a function of the control force. By using the predicted responses, the minimization of the performance index with respect to the control force is carried out for finding the optimal control force. As phase II, the proposed predictive control algorithm was extended to structures that are more complicated than linear SDOF structures. Initially, the algorithm was extended to linear MDOF structures. Although, the development of the algorithm for MDOF structures was relatively straightforward, during testing of the algorithm, it was found that prediction of the response by ARMA models can not be done as was done for SDOF case. In the SDOF case each of the two components of the state vector (i.e. displacement and velocity) was treated separately as an ARMA stochastic process. However, applying the same approach to each component of the state vector of a MDOF structure did not yield satisfactory results in prediction of the response. Considering the whole state vector as a multi-variable ARMA stochastic vector process yielded the desired results in predicting the response a few steps ahead. In the second part of this phase, the algorithm was extended to non-linear MDOF structures. Since the algorithm had been developed based on the principle of superposition, it was not possible to directly extend the algorithm to non-linear systems. Instead, some generalized response was defined. Then credibility of the ARMA models in predicting the generalized response was verified. Based on this credibility, the algorithm was extended for non-linear MDOF structures. Also in phase II, the stability of a controlled MDOF structure was proved. Both internal and external stability of the system were described and verified. In phase III, some problems of special interest, i.e. soil-structure interaction and control time delay, were investigated and compensated for in the framework of the developed predictive optimal control. In first part of phase III soil-structure interaction was studied. The half-space solution of the SSI effect leads to a frequency dependent representation of the structure-footing system, which is not fit for control purpose. Consequently an equivalent frequency independent system was proposed and defined as a system whose frequency response is equal to the original structure -footing system in the mean squares sense. This equivalent frequency independent system then was used in the control algorithm. In the second part of this phase, an analytical approach was used to tackle the time delay phenomenon in the context of the predictive algorithm described in previous chapters. A generalized performance index was defined considering time delay. Minimization of the generalized performance index resulted into a modified version of the algorithm in which time delay is compensated explicitly. Unlike the time delay compensation technique used in the previous phases of this investigation, which restricts time delay to be an integer multiplier of the sampling period, the modified algorithm allows time delay to be any non-negative number. However, the two approaches produce the same results if time delay is an integer multiplier of the sampling period. For evaluating the proposed algorithm and comparing it with other algorithms, several numerical simulations were carried during the research by using MATLAB and its toolboxes. A few interesting results of these simulations are enumerated below: ARM A models are able to predict the response of both linear and non-linear structures to random inputs such as earthquakes. The proposed predictive optimal control based on ARMA models has produced better results in the context of reducing velocity, displacement, total energy and operational cost compared to classic optimal control. Proposed active control algorithm is very effective in increasing safety and comfort. Its performance is not affected much by errors in the estimation of system parameters (e.g. damping). The effect of soil-structure interaction on the response to control force is considerable. Ignoring SSI will cause a significant change in the magnitude of the frequency response and a shift in the frequencies of the maximum response (resonant frequencies). Compensating the time delay effect by the modified version of the proposed algorithm will improve the performance of the control system in achieving the control goal and reduction of the structural response.

<p>Södra Cell sells 1 900 000 ton pulp every year. Of this 490 000 tonne is sold with a contract system based on a pricing index called PIX NBSK. This index was started in 1996 and reflects the price of pulp from conferious forest. We study the NBSK PIX value of softwood from October 1996 to December 2006.</p><p>People working in this branch known that there is strong periodicity in the prices. We use predictive analysis to see if clients can benefit from the periodicity and use the options in the contract system Södra offers today. We conclude that a drawback for the current contract system is that there are too many contracts in proportion to the duration time that is one year for all contracts. Using a time series model called ARMA we make successfull predictions the price difference between two contracts. Based on this prediction we change between these contracts, reducing the price with 0.81% in mean during 1997-2006. Due to the total turnover, if all clients would used such predictions during 1997-2006 Södra's income would have been reduced with 2.77 million USD a year in mean.</p><p>The prices used before PIX are called list prices. The list prices seem to behave like the PIX index. Supposing that the same contract system we see in PIX today was used 1975-2006 with the list price as the base index I made a prediction of the list prices from 1986-2006. Thanks to my predictions, if I had been a client during this period and under mentioned considerations I would have been buying pulp to a price reduced with 0.57%.</p><p>If clients had known the PIX between 1996-2006 in say 1995 Södra's contract system based on PIX would give them a price reduction that were 1.5% in mean during 1996-2006. Price reduction is not possible all years, but when it occurs it can be as big as 3% of the price. Suppose the clients always choose the contract with the lowest price and thereby get a reduced price over time. Then with 95% probability over a long period the price reduction is somewhere in between 0.4-2.7%.</p><p>To strangle this price reduction possibility for the clients there are two ways to go: either reduce the number of contracts or extend the duration time of the contracts.</p><p>To find a suitable duration time, we do spectral density estimation to get indications of which periods that are most important. From this we see that PIX index has a period of five years, wavelet approximated PIX index has 3.4 years and the list prices has a period of 5.6 years. This indicates that current duration time one year is too short. Therefore if it wouldn't effect Södra's clients, an extension of the duration time from one to five years would be good.</p><p>If Södra don't extend the duration time of the contracts my recommendation is to have fewer contracts. The possibility to change between the contracts ''average last three months'' and ''average current month'' every other year is the weakest point of today's system. Therefore I recommend stop selling pulp to the contract ''average PIX last three months''.</p><p>We can't prove any longterm difference between the contracts. If Södra chooses to have just one contract from this point of view it does not matter which one they choose. However, it seems like a good idea to follow the global market and therefore I recommend to choose ''average PIX current month'' rather than ''average PIX last three months'' which lags behind the market front. Since the price ''average current month'' is available at FOEX web page I think Södra should choose this contract if they decide to have only one contract.</p> / <p>Södra Cell säljer årligen 1 900 000 ton pappersmassa. Av denna mängd säljs 490 000 ton enligt ett kontraktsystem baserat på ett prisindex som heter PIX NBSK. Detta index introducerades 1996 och reflekterar priset på pappersmassa gjord av barrträd. Jag studerar priset på indexet från Oktober 1996 till December 2006.</p><p>Dagens kontraktsystem är baserat på kontrakt med löptiden ett år. Jag undersöker om man kan prediktera prisskilllnaden mellan kontrakten, dra nytta att dagens löptid som bara är ett år och välja det kontrakt som ger det billigaste priset så ofta att priset över lång tid reduceras. När man predikterar gör man en uppskattningen av framtiden utifrån en modell av hur framtid beror på dåtid och nutid. Den modell jag har använt kallas ARMA. Denna tillsammans med priserna på pappersmassa från 1975 och framåt gav mig ett fruktbart sätt att förutsäga priserna. Resultatet blev ett pris reducerat med 0.81% i medel under perioden 1996-2006. Eftersom Södra ha så stor försäljningsvolym skulle de ha förlorat 2.27 miljoner dollar per ton i medel om alla kunder ha spekulerat utifrån den modellen jag använde.</p><p>Om dagens kontraktsystem hade börjat användas 1975 med listpriserna som bas hade en kund som använt min prediktionsmetod fått ett pris reducerat med 0.57% under perioden 1986-2006.</p><p>Om kunderna i förväg hade vetat priset under 1996-2001 gav det nuvarande systemet en reducerad medelintäkt med 1.5% av priset. Enskilda år reducerades intäkten med så mycket som 3%. Beräknar man konfidensintervall för prisreduktionerna så inser man att på lång sikt kommer dessa vara av storleksordningen 0.4-2.7% med sannolikheten 95%. Detta förutsatt att klienterna kan se in i framtiden. Siffran 2.7% alltså ett mått på hur stor risk man tar med dagens system. Jag tror inte att klienterna kommer reducera priset med 2.7% med nuvarande system, men det är en övre gräns.</p><p>De gynsamma prediktionerna har sitt ursprung i att det finns periodicitet i priserna. Jag undersöker denna periodicitet med spektralanalys. Periodiciteten för PIX indexet är starkast kring 5 år. En wavelet-approximation av PIX-indexet hade störst periodicitet kring 3.4 år. Listpriserna hade starkast periodicitet kring 5.6 år. Detta indikerar att den nuvarande löptiden, ett år, är för kort. En lämpligare löptid för kontrakten är 5 år.</p><p>Förmodligen är fem års löptid alltför lång tid att binda sig för många kunder. Därför föreslår jag att man reducerar antalet kontrakt istället. Den största svagheten i dagens system är den korta löptiden tillsammans med kontrakten ''average current month'' och ''average last three months''. Jag rekommenderar att man slutar erbjuda kontraktet ''average last three months''. Det allra säkraste är att endast erbjuda ett kontrakt. Vi har inte kunnat påvisa några skillnader över lång sikt mellan kontrakten såtillvida att något kontrakt skulle ge ett lägre medelpris än ett annat. Ur den aspekten är det godtyckligt vilket kontrakt man väljer, men det verkar vettigt att följa den globala marknaden. Därför är det eftersläpande kontraktet ''average PIX last three months'' inte att rekommendera, välj heller ''average PIX current month''. Ett annat argument för att välja ''average PIX current month'' är att dessa priser finns på FOEX hemsida och inga extra beräkningar behöver göras.</p>

Non-stationary signals, that is signals with time-varying (TV) statistical properties, are commonly encountered in engineering practice. The vibration responses of structures, such as traffic-excited bridges, robotic devices, rotating machinery, and so on, constitute typical examples of non-stationary signals. Structures characterized by properties that vary with time are generally referred as TV structures and their vibration-based identification under normal operating conditions is a significant and challenging problem. An important class of parametric methods for the solution of this problem is based on Functional Series Time-dependent AutoRegressive Moving Average (FS-TARMA) models. These models have parameters that explicitly depend on time, with the dependence described by deterministic functions belonging to specific functional sub-spaces. The focus of the present thesis is on the development of complete and advanced FS-TARMA methods that will offer important improvements in overcoming drawbacks of existent methods and will further foster practical use and application of FS-TARMA models in non-stationary vibration analysis. The specific objectives of the thesis are: a) The introduction of a novel class of Adaptable FS-TARMA (AFS-TARMA) models and the development of a method for their effective identification. AFS-TARMA models are adaptable in the sense that they are not based on basis functions of a fixed form, but instead, they use basis functions with a-priori unknown properties that may adapt to the specific random signal characteristics. b) The postulation of a vector FS-TARMA method for output-only structural identification and the development of effective tools for both model parameter estimation and model structure selection. c) The introduction of a statistical method for vibration-based fault diagnosis in TV structures. d) The presentation of a thorough review on FS-TARMA models covering both theoretical and practical aspects of the model parameter estimation and structure selection problems with special emphasis being placed on promising recent methods. The methods that are developed in each chapter of this thesis are validated through their application in both numerical and experimental case studies and comparisons with currently available non-stationary signal identification methods. The results of the study demonstrate the new methods' applicability, effectiveness, and high potential for parsimonious and accurate identification and dynamic analysis of TV structures. / Μη-στάσιμα σήματα, δηλαδή σήματα με χρονικά μεταβαλλόμενες (ΧΜ) στατιστικές ιδιότητες, απαντώνται συχνά στην επιστήμη του μηχανικού. Τυπικά παραδείγματα αποτελούν οι ταλαντωτικές αποκρίσεις κατασκευών, όπως γέφυρες με κινούμενα οχήματα, ρομποτικές διατάξεις, περιστρεφόμενες μηχανές και άλλες. Κατασκευές που χαρακτηρίζονται από ιδιότητες οι οποίες μεταβάλλονται με τον χρόνο αναφέρονται ως ΧΜ κατασκευές και η δυναμική αναγνώριση και ανάλυση τους επί τη βάση ταλαντωτικών σημάτων απόκρισης αποτελεί σημαντικό και ταυτόχρονα δύσκολο πρόβλημα. Μια σημαντική τάξη παραμετρικών μεθόδων για την επίλυση αυτού του προβλήματος βασίζεται στα συναρτησιακά χρονικά μεταβαλλόμενα μοντέλα αυτοπαλινδρόμησης κινητού μέσου όρου (FS-TARMA, Functional Series Time-Dependent Auto-Regressive Moving Average). Τα μοντέλα αυτά χαρακτηρίζονται απο ΧΜ παραμέτρους οι οποίες ακολουθούν καθοριστικό πρότυπο και κατά συνέπεια μπορούν να προβληθούν σε κατάλληλα επιλεγμένους συναρτησιακούς υποχώρους. Ως βασικός στόχος της παρούσας διατριβής ορίζεται η ανάπτυξη εξελιγμένων μεθόδων μοντελοποίησης FS-TARMA οι οποίες θα προσφέρουν σημαντικές βελτιώσεις στις υπάρχουσες προσεγγίσεις και θα βοηθήσουν στην αντιμετώπιση πρακτικών προβλημάτων που σχετίζονται τόσο με την αναγνώριση των δυναμικών χαρακτηριστικών όσο και την διάγνωση βλαβών σε ΧΜ κατασκευές. Οι συγκεκριμένοι στόχοι της διατριβής μπορούν να περιγραφούν ως ακολούθως: α) Εισαγωγή καινοτόμων προσαρμόσιμων μοντέλων FS-TARMA και ανάπτυξη κατάλληλης μεθόδου για την αποτελεσματική εκτίμηση τους. Τα νέα μοντέλα είναι προσαρμόσιμα υπό την έννοια ότι δεν βασίζονται σε προκαθορισμένες συναρτήσεις βάσης, αλλά αντιθέτως χρησιμοποιούν συναρτήσεις βάσης με εκ των προτέρων άγνωστες ιδιότητες οι οποίες μπορούν να προσαρμοστούν στα χαρακτηριστικά συγκεκριμένου σήματος. β) Ανάπτυξη διανυσματικής μεθόδου εκτίμησης μοντέλων FS-TARMA για την αναγνώριση κατασκευών μέσα από διανυσματικά σήματα ταλαντωτικής απόκρισης. Ανάπτυξη αποδοτικών εργαλείων τόσο για το πρόβλημα εκτίμησης των παραμέτρων όσο και της επιλογής της δομής του μοντέλου. γ) Εισαγωγή στατιστικής μεθόδου για την διάγνωση βλαβών σε ΧΜ κατασκευές μέσω μοντέλων FS-TAR. δ) Παρουσίαση μιας διεξοδικής επισκόπησης των μοντέλων FS-TARMA η οποία καλύπτει τόσο θεωρητικά όσο και πρακτικά ζητήματα των προβλημάτων εκτίμησης των παραμέτρων και επιλογής της δομής των μοντέλων. Η αποτελεσματικότητα των μοντέλων και των μεθόδων που αναπτύσσονται σε κάθε κεφάλαιο αυτής της διατριβής διερευνάται µέσω της εφαρµογής τους τόσο σε αριθµητικές όσο και πειραµατικές µελέτες και συγκρίσεις µε υπάρχουσες µη-στάσιµες µεθόδους αναγνώρισης σηµάτων. Τα αποτελέσματα της εργασίας αυτής επιδεικνύουν την ικανότητα των νέων μοντέλων να παρέχουν εξαιρετικά ακριβείς αναπαραστάσεις ΧΜ κατασκευών κατάλληλων τόσο για την δυναμική ανάλυση όσο και για την διάγνωση βλαβών σε αυτές.

Exponential smoothing algorithms are very attractive for the practical world such as in industry. When considering bivariate exponential smoothing methods, in addition to the properties of univariate methods, additional properties give insight to relationships between the two components of a process, and also to the overall structure of the model. It is important to study these properties, but even with the merits the bivariate exponential smoothing algorithms have, exponential smoothing algorithms are nonstatistical/nonstochastic and to study the properties within exponential smoothing may be worthless. As an alternative approach, the (bivariate) ARIMA and the structural models which are classes of statistical models, are shown to generalize the exponential smoothing algorithms. We study these properties within these classes as they will have implications on exponential smoothing algorithms. Forecast properties are studied using the state space model and the Kalman filter. Comparison of ARIMA and structural model completes the study. / Mathematical Sciences / M. Sc. (Statistics)

Το αντικείμενο της διατριβής αφορά την ανάλυση και τον έλεγχο δυναμικών συστημάτων με περιορισμούς στο διάνυσμα της εισόδου ή/ και στις μεταβλητές κατάστασης. Τα θεωρητικά εργαλεία που χρησιμοποιήθηκαν για την εξαγωγή των αποτελεσμάτων προέρχονται από τη θεωρία ευστάθειας Lyapunov, την αρχή σύγκρισης συστημάτων και τη θεωρία συνόλων, και οδήγησαν στην εδραίωση συνθηκών ευστάθειας και την ανάπτυξη συστηματικών μεθόδων εύρεσης λύσης στο πρόβλημα ελέγχου συγκεκριμένων κατηγοριών δυναμικών συστημάτων με περιορισμούς. Πιο συγκεκριμένα, για την κατηγορία των γραμμικών συστημάτων συνεχούς και διακριτού χρόνου, προτάθηκε μια νέα μέθοδος επίλυσης του προβλήματος ευσταθειοποίησης συνόλου αρχικών συνθηκών και του υπολογισμού του μέγιστου θετικά αμετάβλητου ή αμετάβλητου με έλεγχο συνόλου παρουσία περιορισμών στις εισόδους ή/και στις καταστάσεις. Τα αποτελέσματα επεκτάθηκαν και στην κατηγορία των γραμμικών συστημάτων με πολυτοπικη αβεβαιότητα. Επίσης, μελετήθηκε η κατηγορία των αυτοανάδρομων μοντέλων κινούμενου μέσου όρου (ARMA models). Αρχικά εδραιώθηκαν συνθήκες που εγγυώνται ευστάθεια για ένα συγκεκριμένο σύνολο αρχικών συνθηκών παρουσία περιορισμών. Τα αποτελέσματα αυτά εφαρμόστηκαν στην κατηγορία των δικτυωμένων συστημάτων ελέγχου (NCS), όπου υπολογίστηκε ένας κοινός γραμμικός νόμος ελέγχου ανατροφοδότησης κατάστασης για όλο το εύρος της καθυστέρησης της εισόδου. Τέλος, μελετήθηκε η κατηγορία των διγραμμικών συστημάτων συνεχούς και διακριτού χρόνου. Αρχικά διατυπώθηκαν ικανές συνθήκες ύπαρξης πολυεδρικών συναρτήσεων Lyapunov για αυτήν την κατηγορία συστημάτων. Το πρόβλημα που μελετήθηκε είναι η ευσταθειοποίηση μιας συγκεκριμένης περιοχής του χώρου κατάστασης παρουσία περιορισμών στις εισόδους και τις καταστάσεις και προτάθηκε μια υποβέλτιστη λύση που οδηγεί στον υπολογισμό γραμμικού νόμου ελέγχου ανατροφοδότησης κατάστασης. Όλα τα αποτελέσματα προκύπτουν από την επιλογή πολυεδρικών συναρτήσεων Lyapunov οι οποίες οδηγούν στο χαρακτηρισμό πολυεδρικών εκτιμήσεων της περιοχής ελκτικότητας και θετικά αμετάβλητων συνόλων. Τα κυριότερα οφέλη της επιλογής τέτοιων συναρτήσεων είναι η μη συντηρητική εκτίμησης της περιοχή ευστάθειας και η εδράιωση συνθηκών που οδηγούν σε συστηματικές μεθόδους επίλυσης των προβλημάτων ανάλυσης και ελέγχου, η λύση των οποίων προκύπτει από τη λύση γραμμικών προβλημάτων βελτιστοποίησης. / This dissertation considers the problem of stability analysis and control of dynamical systems under constraints in the input and/or state vector. The theoretical tools used arise from Lyapunov stability theory, comparison systems theory and set theoretic methods and lead to the determination of stability conditions and development of systematic methods that solve the control problem of constrained systems of particular type. In specific, for linear discrete or continuous time systems, a novel method that leads to the solution of the initial condition set stabilization problem as well as the maximal controlled invariant set computation problem is presented. These results have been extended for the case of linear systems with polytopic uncertainty. Also, the category of auto regressive moving average (ARMA) models is investigated. First, conditions that guarantee stability for a preassigned initial conditions set for constrained ARMA models are established. These results are applied to the category of networked control systems (NCS), were a single linear state feedback control law is computed for the whole range of the input delay. Finally, the category of bilinear discrete-time or continuous-time systems is investigated. Initially, sufficient conditions which guarantee existence of polyhedral Lyapunov functions are presented. The problem studied here is the stabilization of an initial condition set in the presence of input and state constraints. The solution proposed is suboptimal and leads to the determination of a linear state feedback control law. The choice of Lyapunov functions leads to the determination of a polyhedral approximation of the domain of attraction as well as polyhedral positively invariant sets. The main benefits of choosing this type of functions is the nonconservative estimation of the domain of attraction and the establishment of stability conditions that lead to systematic control design methods through the solution of linear programming problems.