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.

Asteroid identification using statistical orbital inversion methods

An efficient and statistically robust solution for the identification of asteroids among numerous sets of astrometry is presented. In particular, numerical methods have been developed for the short-term identification of asteroids at discovery, and for the long-term identification of scarcely observed asteroids over apparitions, a task which has been lacking a robust method until now. The methods are based on the solid foundation of statistical orbital inversion properly taking into account the observational uncertainties, which allows for the detection of practically all correct identifications. Through the use of dimensionality-reduction techniques and efficient data structures, the exact methods have a loglinear, that is, O(nlog(n)), computational complexity, where n is the number of included observation sets. The methods developed are thus suitable for future large-scale surveys which anticipate a substantial increase in the astrometric data rate. Due to the discontinuous nature of asteroid astrometry, separate sets of astrometry must be linked to a common asteroid from the very first discovery detections onwards. The reason for the discontinuity in the observed positions is the rotation of the observer with the Earth as well as the motion of the asteroid and the observer about the Sun. Therefore, the aim of identification is to find a set of orbital elements that reproduce the observed positions with residuals similar to the inevitable observational uncertainty. Unless the astrometric observation sets are linked, the corresponding asteroid is eventually lost as the uncertainty of the predicted positions grows too large to allow successful follow-up. Whereas the presented identification theory and the numerical comparison algorithm are generally applicable, that is, also in fields other than astronomy (e.g., in the identification of space debris), the numerical methods developed for asteroid identification can immediately be applied to all objects on heliocentric orbits with negligible effects due to non-gravitational forces in the time frame of the analysis. The methods developed have been successfully applied to various identification problems. Simulations have shown that the methods developed are able to find virtually all correct linkages despite challenges such as numerous scarce observation sets, astrometric uncertainty, numerous objects confined to a limited region on the celestial sphere, long linking intervals, and substantial parallaxes. Tens of previously unknown main-belt asteroids have been identified with the short-term method in a preliminary study to locate asteroids among numerous unidentified sets of single-night astrometry of moving objects, and scarce astrometry obtained nearly simultaneously with Earth-based and space-based telescopes has been successfully linked despite a substantial parallax. Using the long-term method, thousands of realistic 3-linkages typically spanning several apparitions have so far been found among designated observation sets each spanning less than 48 hours.

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.

Almost Stable Matchings by Truncating the Gale–Shapley Algorithm

We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + ε)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.

A local 2-approximation algorithm for the vertex cover problem

We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in (Δ + 1)2 synchronous communication rounds, where Δ is the maximum degree of the graph. For Δ = 3, we give a 2-approximation algorithm also for the weighted version of the problem.

A simple local 3-approximation algorithm for vertex cover

We present a local algorithm (constant-time distributed algorithm) for finding a 3-approximate vertex cover in bounded-degree graphs. The algorithm is deterministic, and no auxiliary information besides port numbering is required. (c) 2009 Elsevier B.V. All rights reserved.

A local 2-approximation algorithm for the vertex cover problem

We present a distributed 2-approximation algorithm for the minimum vertex cover problem. The algorithm is deterministic, and it runs in (Δ + 1)2 synchronous communication rounds, where Δ is the maximum degree of the graph. For Δ = 3, we give a 2-approximation algorithm also for the weighted version of the problem.

An optimal local approximation algorithm for max-min linear programs

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.