Parameterschätzung in zeitdiskreten ergodischen Markov-Prozessen am Beispiel des Cox-Ingersoll-Ross-Modells

In dieser Arbeit geht es um die Schätzung von Parametern in zeitdiskreten ergodischen Markov-Prozessen im allgemeinen und im CIR-Modell im besonderen. Beim CIR-Modell handelt es sich um eine stochastische Differentialgleichung, die von Cox, Ingersoll und Ross (1985) zur Beschreibung der Dynamik von Zinsraten vorgeschlagen wurde. Problemstellung ist die Schätzung der Parameter des Drift- und des Diffusionskoeffizienten aufgrund von äquidistanten diskreten Beobachtungen des CIR-Prozesses. Nach einer kurzen Einführung in das CIR-Modell verwenden wir die insbesondere von Bibby und Sørensen untersuchte Methode der Martingal-Schätzfunktionen und -Schätzgleichungen, um das Problem der Parameterschätzung in ergodischen Markov-Prozessen zunächst ganz allgemein zu untersuchen. Im Anschluss an Untersuchungen von Sørensen (1999) werden hinreichende Bedingungen (im Sinne von Regularitätsvoraussetzungen an die Schätzfunktion) für die Existenz, starke Konsistenz und asymptotische Normalität von Lösungen einer Martingal-Schätzgleichung angegeben. Angewandt auf den Spezialfall der Likelihood-Schätzung stellen diese Bedingungen zugleich lokal-asymptotische Normalität des Modells sicher. Ferner wird ein einfaches Kriterium für Godambe-Heyde-Optimalität von Schätzfunktionen angegeben und skizziert, wie dies in wichtigen Spezialfällen zur expliziten Konstruktion optimaler Schätzfunktionen verwendet werden kann. Die allgemeinen Resultate werden anschließend auf das diskretisierte CIR-Modell angewendet. Wir analysieren einige von Overbeck und Rydén (1997) vorgeschlagene Schätzer für den Drift- und den Diffusionskoeffizienten, welche als Lösungen quadratischer Martingal-Schätzfunktionen definiert sind, und berechnen das optimale Element in dieser Klasse. Abschließend verallgemeinern wir Ergebnisse von Overbeck und Rydén (1997), indem wir die Existenz einer stark konsistenten und asymptotisch normalen Lösung der Likelihood-Gleichung zeigen und lokal-asymptotische Normalität für das CIR-Modell ohne Einschränkungen an den Parameterraum beweisen.

Helix-coil-transition in solid polypeptides

Abstract Poly(L-glutamic acid) (PLGA) was synthesized by living anionic ring-opening polymerization of the NCA monomer, which was obtained by reacting diphosgene with an amino acid derivative. The chemical structures and thermal properties were characterized by 1H-NMR, 13C-NMR, TGA and DSC. XRD powder patterns found to be amorphous for all polymers obtained. The molecular weights could be determined under severe limitations due to low solubility and high aggregation tendency. The secondary structure of the PLGA films was analyzed in the solid state by IR spectroscopy; the order was determined mainly by XRD. Uniform bulk films (1-5 µm) were produced by drop-casting of PLGA solutions in TFA on silica. The XRD film analysis indicated the absence of a long range order or an orientation even if a helical microstructure was confirmed by IR spectroscopy. The coil solvent TFA delivered constantly a helical or a β-sheet structure in the solid state depending on the water content of the solvent which was observed for the first time to exhibit a high influence on the crystallization process for PLGA. Temperature dependent in-situ IR measurements were examined to analyze if a helix-coil transition occurs, but there could be no solvent system determined, which resulted in a disordered coil structure in the solid state. General parameters like solvent systems, evaporation conditions, concentration, substrates etc. were analyzed. New crystallizations were obtained on silica prepared by drop-casting of solutions of PLGA in DMF, DMA, TMU, NMP, and pyridine/water mixtures, respectively. PSCBC in DMF, CDCl3/TFA-d, and PSBC in CDCl3/TFA-d exhibited the same crystalline diffraction patterns like PLGA. The long range order in the X-ray diffraction pattern is proven by extremely sharp crystalline signals, which are not changing the shape or the position of the peak by increasing the temperature up to 160°C. The substrate seems to play a decisive role because the crystalline structures were not obtainable on glass. The crystal structure consists probably of two different layered structures based on the intensity ratios of the two series of crystalline signals in the X-ray diffraction patterns. The source of the layered structure remains unclear and needs further studies to investigate the spatial arrangement of the chains in more detail. The secondary structure was still not changing upon heating even if a highly crystalline diffraction pattern occurs. Concluding that even the newly investigated crystallization did not show a helix-coil transition in the solid state by annealing, the phenomenon known in solution has to be claimed as unachievable in the solid state based on the results of this work. A remaining open question represents the observation that the same crystalline pattern can be reproducibly prepared with exhibiting two different ordered secondary structures (helix and β-sheet). After the investigation that the evaporation time cannot be decisive for the crystal growth, the choice of a strong hydrogen bonding interrupting solvent is most probably the key to support and induce the crystallization process.

Non-parametric estimation of the diffusion coefficient of a branching diffusion with immigration

Wir betrachten Systeme von endlich vielen Partikeln, wobei die Partikel sich unabhängig voneinander gemäß eindimensionaler Diffusionen [dX_t = b(X_t),dt + sigma(X_t),dW_t] bewegen. Die Partikel sterben mit positionsabhängigen Raten und hinterlassen eine zufällige Anzahl an Nachkommen, die sich gemäß eines Übergangskerns im Raum verteilen. Zudem immigrieren neue Partikel mit einer konstanten Rate. Ein Prozess mit diesen Eigenschaften wird Verzweigungsprozess mit Immigration genannt. Beobachten wir einen solchen Prozess zu diskreten Zeitpunkten, so ist zunächst nicht offensichtlich, welche diskret beobachteten Punkte zu welchem Pfad gehören. Daher entwickeln wir einen Algorithmus, um den zugrundeliegenden Pfad zu rekonstruieren. Mit Hilfe dieses Algorithmus konstruieren wir einen nichtparametrischen Schätzer für den quadrierten Diffusionskoeffizienten $sigma^2(cdot),$ wobei die Konstruktion im Wesentlichen auf dem Auffüllen eines klassischen Regressionsschemas beruht. Wir beweisen Konsistenz und einen zentralen Grenzwertsatz.

Partial melting within the lower crust – Constraints from bulk rock geochemical data and trace element mineral analyses of granulites and migmatites from Central Finland

This PhD thesis concerns geochemical constraints on recycling and partial melting of Archean continental crust. A natural example of such processes was found in the Iisalmi area of Central Finland. The rocks from this area are Middle to Late Archean in age and experienced metamorphism and partial melting between 2.7-2.63 Ga. The work is based on extensive field work. It is furthermore founded on bulk rock geochemical data as well as in-situ analyses of minerals. All geochemical data were obtained at the Institute of Geosciences, University of Mainz using X-ray fluorescence, solution ICP-MS and laser ablation-ICP-MS for bulk rock geochemical analyses. Mineral analyses were accomplished by electron microprobe and laser ablation ICP-MS. Fluid inclusions were studied by microscope on a heating-freezing-stage at the Geoscience Center, University Göttingen. Part I focuses on the development of a new analytical method for bulk rock trace element determination by laser ablation-ICP-MS using homogeneous glasses fused from rock powder on an Iridium strip heater. This method is applicable for mafic rock samples whose melts have low viscosities and homogenize quickly at temperatures of ~1200°C. Highly viscous melts of felsic samples prevent melting and homogenization at comparable temperatures. Fusion of felsic samples can be enabled by addition of MgO to the rock powder and adjustment of melting temperature and melting duration to the rock composition. Advantages of the fusion method are low detection limits compared to XRF analyses and avoidance of wet-chemical processing and use of strong acids as in solution ICP-MS as well as smaller sample volumes compared to the other methods. Part II of the thesis uses bulk rock geochemical data and results from fluid inclusion studies for discrimination of melting processes observed in different rock types. Fluid inclusion studies demonstrate a major change in fluid composition from CO2-dominated fluids in granulites to aqueous fluids in TTG gneisses and amphibolites. Partial melts were generated in the dry, CO2-rich environment by dehydration melting reactions of amphibole which in addition to tonalitic melts produced the anhydrous mineral assemblages of granulites (grt + cpx + pl ± amph or opx + cpx + pl + amph). Trace element modeling showed that mafic granulites are residues of 10-30 % melt extraction from amphibolitic precursor rocks. The maximum degree of melting in intermediate granulites was ~10 % as inferred from modal abundances of amphibole, clinopyroxene and orthopyroxene. Carbonic inclusions are absent in upper-amphibolite facies migmatites whereas aqueous inclusion with up to 20 wt% NaCl are abundant. This suggests that melting within TTG gneisses and amphibolites took place in the presence of an aqueous fluid phase that enabled melting at the wet solidus at temperatures of 700-750°C. The strong disruption of pre-metamorphic structures in some outcrops suggests that the maximum amount of melt in TTG gneisses was ~25 vol%. The presence of leucosomes in all rock types is taken as the principle evidence for melt formation. However, mineralogical appearance as well as major and trace element composition of many leucosomes imply that leucosomes seldom represent frozen in-situ melts. They are better considered as remnants of the melt channel network, e.g. ways on which melts escaped from the system. Part III of the thesis describes how analyses of minerals from a specific rock type (granulite) can be used to determine partition coefficients between different minerals and between minerals and melt suitable for lower crustal conditions. The trace element analyses by laser ablation-ICP-MS show coherent distribution among the principal mineral phases independent of rock composition. REE contents in amphibole are about 3 times higher than REE contents in clinopyroxene from the same sample. This consistency has to be taken into consideration in models of lower crustal melting where amphibole is replaced by clinopyroxene in the course of melting. A lack of equilibrium is observed between matrix clinopyroxene / amphibole and garnet porphyroblasts which suggests a late stage growth of garnet and slow diffusion and equilibration of the REE during metamorphism. The data provide a first set of distribution coefficients of the transition metals (Sc, V, Cr, Ni) in the lower crust. In addition, analyses of ilmenite and apatite demonstrate the strong influence of accessory phases on trace element distribution. Apatite contains high amounts of REE and Sr while ilmenite incorporates about 20-30 times higher amounts of Nb and Ta than amphibole. Furthermore, trace element mineral analyses provide evidence for magmatic processes such as melt depletion, melt segregation, accumulation and fractionation as well as metasomatism having operated in this high-grade anatectic area.

Reliability and consistency affect the integration of visual depth cues

Abstract Originalsprache (englisch) Visual perception relies on a two-dimensional projection of the viewed scene on the retinas of both eyes. Thus, visual depth has to be reconstructed from a number of different cues that are subsequently integrated to obtain robust depth percepts. Existing models of sensory integration are mainly based on the reliabilities of individual cues and disregard potential cue interactions. In the current study, an extended Bayesian model is proposed that takes into account both cue reliability and consistency. Four experiments were carried out to test this model's predictions. Observers had to judge visual displays of hemi-cylinders with an elliptical cross section, which were constructed to allow for an orthogonal variation of several competing depth cues. In Experiment 1 and 2, observers estimated the cylinder's depth as defined by shading, texture, and motion gradients. The degree of consistency among these cues was systematically varied. It turned out that the extended Bayesian model provided a better fit to the empirical data compared to the traditional model which disregards covariations among cues. To circumvent the potentially problematic assessment of single-cue reliabilities, Experiment 3 used a multiple-observation task, which allowed for estimating perceptual weights from multiple-cue stimuli. Using the same multiple-observation task, the integration of stereoscopic disparity, shading, and texture gradients was examined in Experiment 4. It turned out that less reliable cues were downweighted in the combined percept. Moreover, a specific influence of cue consistency was revealed. Shading and disparity seemed to be processed interactively while other cue combinations could be well described by additive integration rules. These results suggest that cue combination in visual depth perception is highly flexible and depends on single-cue properties as well as on interrelations among cues. The extension of the traditional cue combination model is defended in terms of the necessity for robust perception in ecologically valid environments and the current findings are discussed in the light of emerging computational theories and neuroscientific approaches.