In this dissertation empirical Bayes estimators for the coefficients in time series regression models are presented. Due to the uncontrollability of time series observations, explanatory variables in each stage do not remain unchanged. A generalization of the results of O'Bryan and Susarla is established and shown to be an extension of the results of Martz and Krutchkoff. Alternatively, as the distribution function of sample observations is hard to obtain except asymptotically, the results of Griffin and Krutchkoff on empirical linear Bayes estimation are extended and then applied to estimating the coefficients in time series regression models. Comparisons between the performance of these two approaches are also made. Finally, predictions in time series regression models using empirical Bayes estimators and empirical linear Bayes estimators are discussed. / Ph. D.

A method for determining the specific acoustic admittance of the inlet and outlet ports of a combustion chamber is presented. Parameter estimation techniques of Gauss linearization and genetic algorithms are employed to recover the acoustic boundary conditions. These techniques are used with a combination of two resources: dynamic pressure measurements obtained during combustor operation, and a finite element method-based model of the combustion chamber. Results of a theoretical analysis are presented which show that the method is capable of accomplishing its mission. An observation of particular significance is the lack of sensitivity of the pressure mode shape to relatively large changes in the acoustic boundary conditions. / Master of Science

This thesis essentially takes two separate paths to solve the same problem, namely that of obtaining an estimator, a parameter of the negative binomial distribution, for which we can show that such properties as bias and variance of this estimator are "better" than corresponding properties of the simple moment estimator, the latter being the estimator which is used most often in practice. We first consider two moment estimators involving the third sample moment. In the case of both of these estimators, for a restricted range of the parameters and of sample size, these estimators are not an improvement over the simple moment estimator. In fact, for the range considered, the bias and variance of the simple moment estimator was always smaller. We then considered an estimator which was defined as the simple moment estimator for part of the sample space and defined as a constant elsewhere. This was primarily done to remove a "singularity" in the moment estimator. It was felt that this singularity was causing the large bias and variance which seemed to exist for certain values of the parameters. For n=100, the bias and variance were approximated in a range of interest of the parameters. The results indicate an improvement over the sample moment estimator. / Ph. D.

Statistics of tensor forms appear in various contexts and provide a useful way to model dependence, which naturally arises in network data. In this thesis, we study the statistics of two different tensor forms. In the first part of the thesis, we derive central limit theorem results which exhibit a fourth moment phenomenon. That is, the fourth moment converging to 3 implies the convergence of the statistic to a normal distribution. We also establish the other direction, which provides us with an if and only if condition for asymptotic normality. The settings and the results are very easily applied to the monochromatic subgraph count in the problem of graph coloring. The second part of the thesis compares the relative efficiency of the maximum likelihood estimator (MLE) and the maximum pseudolikelihood estimator (MPLE) for particular p-tensor Ising models. Specifically, we show that in the graph case, i.e. when p = 2, the MLE and MPLE are equally Bahadur efficient. For high-order tensors, i.e. p ≥ 3, we show a two-layer phase transition in which the MPLE is less Bahadur efficient than the MLE in certain regimes of the parameter space, depending also on the magnitudes of the null and alternate parameters.

This study is concerned with the simultaneous detection and least squares estimation of vector random processes. The problem is formulated in the following context: A random process, out of a countably infinite collection of (not necessarily Gaussian) vector random processes with known distributions, is observed with additive white Gaussian noise. The <i>a priori</i> probability, that a specific random process will be observed, is specified for each one in the collection. The least squares estimate of the random process that is being observed is to be found in terms of the hypothesis conditioned estimates. It is shown that the best estimate is the linear combination of the hypothesis conditioned estimates weighted by the <i>a posteriori</i> probabilities of the hypotheses conditioned on the observations. A Radon-Nikodym derivative representation is derived for the <i>a posteriori</i> probability by using the specific structure of the product probability measure for this problem. It is shown that this Radon-Nikodym derivative can be expressed in terms of the Radon-Nikodym derivatives of measures induced by the random processes in the collection with respect to Wiener measure. By using the recent results on likelihood functions, an expression for the <i>a posteriori</i> probability is found in terms of the conditioned estimates. In this connection, an extended version of the partition theorem of parameter adaptive estimation is proved. The unique stochastic differential equation, that each <i>a posteriori</i> probability satisfies with its associated <i>a posteriori</i> probability as the initial condition, is derived for the case of finitely many hypotheses along with an expression for the conditional error covariance in terms of the hypothesis conditioned error covariances. The results are applied to the parameter adaptive estimation problem in linear continuous and discrete stochastic dynamic systems. In the continuous case, the solution is also obtained through an alternate approach using nonlinear filtering theory. An application of the theory to the design of a digital flight control system which is tolerant of sensor failures is presented with real-time hybrid computer simulation results. A review of random processes and statistical decision theory is also included. / Ph. D.

Signal acquisition, signal reconstruction and analysis of spectrum of the signal are the three most important steps in signal processing and they are found in almost all of the modern day hardware. In most of the signal processing hardware, the signal of interest is sampled at uniform intervals satisfying some conditions like Nyquist rate. However, in some cases the privilege of having uniformly sampled data is lost due to some constraints on the hardware resources. In this thesis an important problem of signal reconstruction and spectral analysis from nonuniformly sampled data is addressed and a variety of methods are presented. The proposed methods are tested via numerical experiments on both artificial and real-life data sets. The thesis starts with a brief review of methods available in the literature for signal reconstruction and spectral analysis from non uniformly sampled data. The methods discussed in the thesis are classified into two broad categories - dense and sparse methods, the classification is based on the kind of spectra for which they are applicable. Under dense spectral methods the main contribution of the thesis is a non-parametric approach named LIMES, which recovers the smooth spectrum from non uniformly sampled data. Apart from recovering the spectrum, LIMES also gives an estimate of the covariance matrix. Under sparse methods the two main contributions are methods named SPICE and LIKES - both of them are user parameter free sparse estimation methods applicable for line spectral estimation. The other important contributions are extensions of SPICE and LIKES to multivariate time series and array processing models, and a solution to the grid selection problem in sparse estimation of spectral-line parameters. The third and final part of the thesis contains applications of the methods discussed in the thesis to the problem of radial velocity data analysis for exoplanet detection. Apart from the exoplanet application, an application based on Sudoku, which is related to sparse parameter estimation, is also discussed.

Thesis (MSc)--Stellenbosch University, 2004 / ENGLISH ABSTRACT: The main aim of this study is to establish a standard procedure for all sedimentary particle size analysis methods specifically for clay minerals and mixtures thereof. Not only will it improve accuracy and reproducibility during clay size analysis, it will also secure comparability between different operators. As a start, all the apparatus-related parameters that can affect the accuracy and reproducibility were determined for the apparatus used, viz. the Sedigraph SOOOD. Thereafter, these parameters were kept constant, and the effects of potential material-related parameters were investigated one-by-one. First to be investigated were those parameters relating specifically to sample preparation. They were: grinding intensity, chemical dissolution of cementing materials, duration of prior soaking, salt content, centrifugal washing with polar organic liquids, deflocculant type and concentration, the effect of pH, ultrasonic time, and stirring during ultrasonic treatment. Then, the influence on accuracy and reproducibility of the physical and chemical parameters related to the suspension was determined. They were: the use of the viscosity and density of water to calibrate the apparatus in stead of those of the suspension liquid, hydrolysis of the deflocculant with suspension-ageing, and the effect of solid concentration on hindered settling. During this investigation a novel method was developed to enable faster and more accurate pycnometric density determinations. Next, the unique characteristics of clays, which can influence the results of sedimentary particle size analyses, were examined. Serious problems are encountered with the accuracy of the analyses of some clay types abundantly found in nature, viz. the smectites and mixed-layered clay minerals. Due to their swelling in water, and variations in the amounts of their crystal layers, they experience unpredictable changes in particle size. The latter is caused by the following external factors: clay type, humidity, type of exchange cation, electrolyte concentration, clay concentration, pH, deflocculant type and concentration, pressure history of the swell-clay suspension, and ageing of the suspension. The effect of each of them on the accuracy and reproducibility of the sedimentary particle size analysis of clays are investigated in detail. Another problem that influences the accuracy of the sedimentary methods is that owing to swelling, the densities of smectites and mixed-layered clays change by varying degrees when suspended in water. It is, however, impossible to pycnometrically determine the density of a swell-clay since it absorbs a part of the water used for its volume determination. To solve this problem, a novel method was devised to calculate swell-clay density. This method makes use of existing Monte Carlo simulations of the swelling mechanism of montmorillonite. During all sedimentary methods, an average clay density is normally used to calculate the particle size distribution of clay mixtures. However, if there is a large enough difference between the calculated average density and that of a component, then inaccurate results will be recorded. The magnitude of this effect was investigated for a few self-made clay mixtures, which consisted of different proportions of kaolinite, illite, and montmorillonite. Based on all the above results, a practical approach to, and a standard methodology for all the sedimentary methods of particle size analysis of clay minerals are presented. Additionally, a condensed summary is provided in table-form, which contains the magnitudes of the errors associated with each of the parameters that were examined. / AFRIKAANSE OPSOMMING: Die doel van hierdie studie is om 'n standaard prosedure daar te stel vir alle sedimentêre metodes van partikelgrootte analise, spesifiek vir gebruik met kleiminerale en mengsels daarvan. So 'n standaard prosedure sal die akkuraatheid en herhaalbaarheid van klei-analises verbeter, en die vergelykbaarheid tussen verskillende operateurs verseker. Aanvanklik is slegs die parameters bepaal wat die akkuraatheid en herhaalbaarheid van die gekose apparaat (Sedigraph 5000D) kan beïnvloed. Daarna is al hierdie parameters konstant gehou, en is die potensiële effekte van die moontlike materiaal-verwante parameters een na die ander ondersoek. Eerstens is die invloed van monstervoorbereiding op akkuraatheid en herhaalbaarheid bepaal. Verskillende parameters nl. maal-intensiteit, chemiese oplossing van sementerende materiale, sentrifugale wassing met polêre organiese vloeistowwe, tipe ontvlokker en konsentrasie, die effek van pH, ultrasoniese tyd en die effek van roer tydens ultrasonikasie is ondersoek. Vervolgens is die invloed op die akkuraatheid en herhaalbaarheid van die fisiese en chemiese parameters verwant aan die suspensie bepaal. Hierdie parameters was nl. die gebruik van die viskositeit en digtheid van water in plaas van dié van die suspensievloeistof, hidrolise van die ontvlokker tydens suspensieveroudering, asook die effek van vastestof-konsentrasie op belemmerde uitsakking. Gedurende hierdie ondersoek is ook 'n nuwe metode ontwikkel wat vinniger, en meer akkurate piknometriese digtheidsbepalings moontlik maak. Die unieke eienskappe van kleie wat die resultate van sedimentêre metodes van partikelgrootte analises kan beïnvloed, is volgende ondersoek. Tydens die analises van party kleie wat baie volop in die natuur voorkom, nl. die smektiete en menglaag-kleie, word ernstige akkuraatheids-probleme ondervind. Hul swelling in water, tesame met variasies in hul aantal kristal-lagies, veroorsaak onvoorspelbare verandering van hul partikelgroottes. Laasgenoemde word deur die volgende eksterne faktore veroorsaak: klei tipe, humiditeit, tipe uitruil-katioon, elektrolietkonsentrasie, kleikonsentrasie, pH, ontvlokker-tipe en konsentrasie, drukgeskiedenis van 'n swelklei-suspensie, en veroudering van die suspensie. Die effek van elk op die akkuraatheid en herhaalbaarheid van die sedimentêre partikelgrootte analises van kleie word in detail bespreek. 'n Verdere probleem wat die akkuraatheid van sedimentêre metodes beïnvloed, is dat wanneer smektiete en menglaag-kleie in water gesuspendeer word, hulle digthede in verskillende mates weens swelling verander. Dit is egter onmoontlik om die digtheid van swelkleie in water piknometries te bepaal, omdat swelklei 'n gedeelte van die water absorbeer wat gebruik moet word om die kleivolume mee te bepaal. Om hierdie probleem op te los, is 'n nuwe metode ontwikkelom die digtheid van swelkleie mee te bereken. Die metode maak gebruik van reedsbestaande Monte Carlo simulasies van die swelling van montmorillonite. Tydens alle sedimentêre metodes word normaalweg van 'n gemiddelde kleidigtheid gebruik gemaak om die partikelgrootte-verspreiding van kleimengsels mee te bereken. Indien die berekende gemiddelde digtheid egter genoegsaam met dié van 'n kleikomponent verskil, sal onakkurate resultate verkry word. Hierdie effek is ondersoek vir 'n paar selfgemaakte kleimengsels wat uit verskillende hoeveelhede kaoliniet, illiet, en montmorilloniet bestaan het. Laastens word 'n praktiese benadering en 'n standaard metode vir alle sedimentêre metodes voorgestel, wat gebaseer is op al die bogenoemde resultate. 'n Verkorte opsomming, met die groottes van die foute geassosieer met elke parameter wat ondersoek is, word laastens in tabelvorm verskaf.

Στις ημέρες μας, η εμφάνιση των ασύρματων δικτύων σε πολλές πτυχές της καθημερινότητας, είναι συνεχώς αυξανομενη. Το γεγονός αυτό, έχει ως συνέπεια να υπάρχει μεγάλη ερευνητική δραστηριότητα γύρω από τα ασύρματα δίκτυα, η οποία αφορά όχι μόνο το σχεδιασμό τους και την ανάπτυξη διάφορων πρωτοκόλλων, αλλά και άλλες εφαρμογές, όπως είναι για παράδειγμα η εκτίμηση παραμέτρων. Στα πλαίσια της εργασίας αυτής, μελετάται η ανάπτυξη των αλγορίθμων φλυαρίας, οι οποίοι αφορούν μία κατανεμημένη προσέγγιση του προβλήματος της εκτίμησης παραμέτρων σε ένα δίκτυο. Πιο συγκεκριμένα, σε αντίθεση με τις κλασσικές μεθόδους στις οποίες αναλαμβάνει ένας κεντρικός κόμβος με μεγάλη υπολογιστική ισχύ να λύσει το πρόβλημα της εκτίμησης της παραμέτρου ενδιαφέροντος, με τους αλγόριθμους φλυαρίας αναιρείται η έννοια του κεντρικού κόμβου και η εκτίμηση στηρίζεται στη συνεχή ανταλλαγή πληροφοριών μεταξύ των κόμβων του δικτύου. Με τις προσομοιώσεις που έγιναν στα πλαίσια αυτής της εργασίας, αποδεικνύεται ότι οι εν λόγω αλγόριθμοι εξασφαλίζουν επιτυχημένη προσέγγιση του προβλήματος που καλούνται να επιλύσουν παρότι οι αλγόριθμοι φλυαρίας στηρίζονται σε υποβέλτιστες τεχνικές εκτίμησης παραμέτρων οι οποίες βασίζονται σε αναδρομικούς προσαρμοστικούς αλγορίθμους. Τέλος, αντιμετωπίζεται το πρόβλημα της εκτίμησης της θέσης ενός στόχου που κινείται στην περιοχή ενός δικτύου με βάση τη διαδικασία της φλυαρίας. / In recent years, the emergence of wireless networks in many aspects of daily life, is increasingly growing. This fact has as consequence a strong research activity around various types of wireless networks, not only in the design and development of various protocols, but also in other applications such as parameter estimation. In this thesis, we study the development of gossip algorithms that are related to a distributed approach to the problem of parameter estimation in a network. More specifically, in contrast with classical methods that assume a central node with high computational power to solve the problem of estimation of the parameter of interest, the use of gossip algorithms negates this concept and the estimation process is based on continuing exchange of information between network nodes. Additionally, despite the fact that gossip algorithms belong to suboptimal parameter estimation techniques, that are based on recursive adaptive algorithms, the simulation results presented show that these algorithms ensure successful approach to the problem they have to solve. Finally, the process of gossiping deals with the problem of estimating the position of a moving target in the region of a wireless network.

Thesis (MScEng)--Stellenbosch University, 2012 / ENGLISH ABSTRACT: Wind energy is a very current topic, both locally and internationally. It is one of the most rapidly growing renewable energy sources with installed capacity doubling every three years. South Africa's installed wind energy currently accounts for only 10 MW of the 197 GW worldwide installed capacity. With a 10 TWh renewable energy production target set for 2013 by the South African government, renewable energy projects have gained momentum in recent years. This target, together with data from case studies and reports on resource planning and technical requirements, shows that South Africa is well positioned for the implementation of wind energy sources. All this development in the local wind generation market creates a need for local knowledge in the field of wind energy as well as a need to efficiently model and analyse wind turbine systems and grid interactions for local operating conditions. Although the relevant model topologies are well established, obtaining or deriving appropriate parameter values from first principles remains problematic. Some parameters are also dependent on operating conditions and are best determined from site measurements using parameter estimation methodologies. One of the objectives of this project is to investigate whether the system parameter values can be obtained by performing parameter estimation on the model of a wind turbine system. The models used for parameter estimation processes require fast simulation times. Therefore, basic C-code S-function models of the wind turbine system components, i.e., the wind turbine blade, gearbox and generator, were developed and compiled as a Simulink library. These library components were then used for the parameter estimation process. The developed models, as well as the complete wind turbine system model, were validated and their performance evaluated, by comparing them to existing Simulink block models. These models all proved to be accurate and all showed reductions in simulation times. The principle of performing parameter estimation on C-code S-function models is proven by case studies performed on the individual models and the complete wind turbine system. The power coefficient matrix parameter values of the individual turbine blade model estimated with 100% accuracy for the excited elements. The individual gearbox parameter values all estimated accurately with errors below 2.5%. The parameter values of the individual generator models were estimated accurately for the ABC model, with errors below 4%, and less accurately for the DQ model with errors below 13%. The estimation results obtained for the complete wind turbine system model showed that the parameter values of the gearbox model and generator model were estimated accurately when the system model was excited through a step in angular velocity and steps in amplitude of the stator voltages respectively. A final estimation showed that a combination of gearbox and generator parameter values were accurately estimated when the model was excited through both a step in angular velocity and steps in the amplitude of the stator voltages. / AFRIKAANSE OPSOMMING: Windenergie is 'n baie aktuele onderwerp beide plaaslik en internasionaal. Windenergie is een van die vinnigste groeiende hernubare energie bronne met die geïnstalleerde kapasiteit wat driejaarliks verdubbel. Suid-Afrika se geïnstalleerde windenergie maak tans slegs 10 MW uit van die wêreldwye geïnstalleerde kapasiteit van 197 GW. Die Suid-Afrikaanse regering het ’n 10 TWh hernubare-energie produksie teiken gestel vir 2013. As gevolg hiervan het hernubare-energie projekte die laaste paar jaar momentum gekry. Hierdie teiken, tesame met die data van gevallestudies en verslae oor hulpbronbeplanning en tegniese vereistes, toon dat Suid-Afrika goed geposisioneer is vir die implementering van windenergiebronne. Hierdie ontwikkelinge in die plaaslike windenergie mark skep ’n behoefte aan plaaslike kennis op die gebied van windenergie, asook die behoefte vir ’n doeltreffende wyse vir die modellering en analisering van windturbine stelsels en netwerk integrasie vir plaaslike werkskondisies. Alhoewel die betrokke model topologieë reeds goed gevestig is, is die verkryging van toepaslike parameter waardes vanuit eerste beginsels steeds problematies. Sommige parameters is ook afhanklik van die werkskondisies en kan die beste bepaal word deur gebruik te maak van parameter estimasie metodologieë vanaf terrein metings. Een van die doelwitte van die projek is om ondersoek in te stel na die moontlikheid om die stelsel parameter waardes te verkry deur parameter estimasie toe te pas op ’n windturbine stelsel. Die modelle wat gebruik word vir die parameter estimasie prosesse benodig vinnige simulasie tye. Daarom is basiese C-kode S-funksie modelle vir die komponente van windturbine stelsels, d.w.s., die wind turbine lemme, ratkas en generator, ontwikkel en saamgestel as ’n Simulink biblioteek. Die komponente in hierdie biblioteek was toe gebruik vir die parameter estimasie proses. Die ontwikkelde modelle sowel as die hele windturbine stelsel model was gevalideer en hul werksverrigting geëvalueer, deur dit te vergelyk met bestaande Simulink blok modelle. Hierdie modelle het almal getoon dat hulle akkuraat is en het almal ’n vermindering in simulasie tyd getoon. Die beginsel van parameter estimasie wat uitgevoer word op C-kode S-funksie modelle, is bewys deur gevallestudies wat op die individuele modelle en die hele windturbine stelsel model uitgevoer was. Die geperturbeerde elemente van die kragkoëffisiënt-matriks arameter van die individuele turbine lemme model se waardes het 100% akkuraatheid geëstimeer. Die individuele ratkas model se parameter waardes was almal akkuraat geëstimeer, met foute kleiner as 2.5%. Die individuele generator modelle se parameter waardes was akkuraat geëstimeer vir die ABC model, met foute kleiner as 4%, en minder akkuraat vir die DQ model, met foute kleiner as 13%. Die resultate wat verkry is van die estimasie wat uitgevoer is op die volledige windturbine stelsel model, het getoon dat die parameter waardes van die ratkas model en die generator model akkuraat geëstimeer word, wanneer die stelsel model onderskeidelik deur ’n trap in die hoeksnelheid en trappe in die amplitude van die stator spannings geperturbeer word. ’n Finale estimasie het getoon dat ’n kombinasie van ratkas en generator parameter waardes akkuraat geëstimeer kan word as die model deur beide die trap in hoeksnelheid en die trappe in die amplitude van die stator spannings geperturbeer word.

<p>In order to find effective measures that meet the requirements for proper groundwater quality and quantity management, there is a need to develop a Decision Support System (DSS) and a suitable modelling tool. Central components of a DSS for groundwater management are thought to be models for surface- and groundwater flow and solute transport. The most feasible approach seems to be integration of available mathematical models, and development of a strategy for evaluation of the uncertainty propagation through these models. The physically distributed hydrological model ECOMAG has been integrated with the groundwater model MODFLOW to form a new integrated watershed modelling system - ECOFLOW. The modelling system ECOFLOW has been developed and embedded in Arc View. The multiple-scale modelling principle, combines a more detailed representation of the groundwater flow conditions with lumped watershed modelling, characterised by simplicity in model use, and a minimised number of model parameters. A Bayesian statistical downscaling procedure has also been developed and implemented in the model. This algorithm implies downscaling of the parameters used in the model, and leads to decreasing of the uncertainty level in the modelling results. The integrated model ECOFLOW has been applied to the Vemmenhög catchment, in Southern Sweden, and the Örsundaån catchment, in central Sweden. The applications demonstrated that the model is capable of simulating, with reasonable accuracy, the hydrological processes within both the agriculturally dominated watershed (Vemmenhög) and the forest dominated catchment area (Örsundaån). The results show that the ECOFLOW model adequately predicts the stream and groundwater flow distribution in these watersheds, and that the model can be used as a possible tool for simulation of surface– and groundwater processes on both local and regional scales. A chemical module ECOMAG-N has been created and tested on the Vemmenhög watershed with a highly dense drainage system and intensive fertilisation practises. The chemical module appeared to provide reliable estimates of spatial nitrate loads in the watershed. The observed and simulated nitrogen concentration values were found to be in close agreement at most of the reference points. The proposed future research includes further development of this model for contaminant transport in the surface- and ground water for point and non-point source contamination modelling. Further development of the model will be oriented towards integration of the ECOFLOW model system into a planned Decision Support System.</p>