Inverse Problem for Fractional Brownian Motion with Discrete Data

The problem of recovering information from measurement data has already been studied for a long time. In the beginning, the methods were mostly empirical, but already towards the end of the sixties Backus and Gilbert started the development of mathematical methods for the interpretation of geophysical data. The problem of recovering information about a physical phenomenon from measurement data is an inverse problem. Throughout this work, the statistical inversion method is used to obtain a solution. Assuming that the measurement vector is a realization of fractional Brownian motion, the goal is to retrieve the amplitude and the Hurst parameter. We prove that under some conditions, the solution of the discretized problem coincides with the solution of the corresponding continuous problem as the number of observations tends to infinity. The measurement data is usually noisy, and we assume the data to be the sum of two vectors: the trend and the noise. Both vectors are supposed to be realizations of fractional Brownian motions, and the goal is to retrieve their parameters using the statistical inversion method. We prove a partial uniqueness of the solution. Moreover, with the support of numerical simulations, we show that in certain cases the solution is reliable and the reconstruction of the trend vector is quite accurate.

Spin and relativistic motion of bound states

The wave functions of moving bound states may be expected to contract in the direction of motion, in analogy to a rigid rod in classical special relativity, when the constituents are at equal (ordinary) time. Indeed, the Lorentz contraction of wave functions is often appealed to in qualitative discussions. However, only few field theory studies exist of equal-time wave functions in motion. In this thesis I use the Bethe-Salpeter formalism to study the wave function of a weakly bound state such as a hydrogen atom or positronium in a general frame. The wave function of the e^-e^+ component of positronium indeed turns out to Lorentz contract both in 1+1 and in 3+1 dimensional quantum electrodynamics, whereas the next-to-leading e^-e^+\gamma Fock component of the 3+1 dimensional theory deviates from classical contraction. The second topic of this thesis concerns single spin asymmetries measured in scattering on polarized bound states. Such spin asymmetries have so far mainly been analyzed using the twist expansion of perturbative QCD. I note that QCD vacuum effects may give rise to a helicity flip in the soft rescattering of the struck quark, and that this would cause a nonvanishing spin asymmetry in \ell p^\uparrow -> \ell' + \pi + X in the Bjorken limit. An analogous asymmetry may arise in p p^\uparrow -> \pi + X from Pomeron-Odderon interference, if the Odderon has a helicity-flip coupling. Finally, I study the possibility that the large single spin asymmetry observed in p p^\uparrow -> \pi(x_F,k_\perp) + X when the pion carries a high momentum fraction x_F of the polarized proton momentum arises from coherent effects involving the entire polarized bound state.

Kalman Filter Algorithm for Rating and Prediction in Basketball

The Thesis presents a state-space model for a basketball league and a Kalman filter algorithm for the estimation of the state of the league. In the state-space model, each of the basketball teams is associated with a rating that represents its strength compared to the other teams. The ratings are assumed to evolve in time following a stochastic process with independent Gaussian increments. The estimation of the team ratings is based on the observed game scores that are assumed to depend linearly on the true strengths of the teams and independent Gaussian noise. The team ratings are estimated using a recursive Kalman filter algorithm that produces least squares optimal estimates for the team strengths and predictions for the scores of the future games. Additionally, if the Gaussianity assumption holds, the predictions given by the Kalman filter maximize the likelihood of the observed scores. The team ratings allow probabilistic inference about the ranking of the teams and their relative strengths as well as about the teams’ winning probabilities in future games. The predictions about the winners of the games are correct 65-70% of the time. The team ratings explain 16% of the random variation observed in the game scores. Furthermore, the winning probabilities given by the model are concurrent with the observed scores. The state-space model includes four independent parameters that involve the variances of noise terms and the home court advantage observed in the scores. The Thesis presents the estimation of these parameters using the maximum likelihood method as well as using other techniques. The Thesis also gives various example analyses related to the American professional basketball league, i.e., National Basketball Association (NBA), and regular seasons played in year 2005 through 2010. Additionally, the season 2009-2010 is discussed in full detail, including the playoffs.

Chemical Characterization of Urban Background Aerosol Using Online And Filter Methods

In order to evaluate the influence of ambient aerosol particles on cloud formation, climate and human health, detailed information about the concentration and composition of ambient aerosol particles is needed. The dura-tion of aerosol formation, growth and removal processes in the atmosphere range from minutes to hours, which highlights the need for high-time-resolution data in order to understand the underlying processes. This thesis focuses on characterization of ambient levels, size distributions and sources of water-soluble organic carbon (WSOC) in ambient aerosols. The results show that in the location of this study typically 50-60 % of organic carbon in fine particles is water-soluble. The amount of WSOC was observed to increase as aerosols age, likely due to further oxidation of organic compounds. In the boreal region the main sources of WSOC were biomass burning during the winter and secondary aerosol formation during the summer. WSOC was mainly attributed to a fine particle mode between 0.1 - 1 μm, although different size distributions were measured for different sources. The WSOC concentrations and size distributions had a clear seasonal variation. Another main focus of this thesis was to test and further develop the high-time-resolution methods for chemical characterization of ambient aerosol particles. The concentrations of the main chemical components (ions, OC, EC) of ambient aerosol particles were measured online during a year-long intensive measurement campaign conducted on the SMEAR III station in Southern Finland. The results were compared to the results of traditional filter collections in order to study sampling artifacts and limitations related to each method. To achieve better a time resolution for the WSOC and ion measurements, a particle-into-liquid sampler (PILS) was coupled with a total organic carbon analyzer (TOC) and two ion chromatographs (IC). The PILS-TOC-IC provided important data about diurnal variations and short-time plumes, which cannot be resolved from the filter samples. In summary, the measurements made for this thesis provide new information on the concentrations, size distribu-tions and sources of WSOC in ambient aerosol particles in the boreal region. The analytical and collection me-thods needed for the online characterization of aerosol chemical composition were further developed in order to provide more reliable high-time-resolution measurements.