The huge volume of time series data generated in many applications poses new challenges in the techniques of data storage, transmission, and computation. Further more, when the time series are in the form of streaming data, new problems emerge and new techniques are required because of the streaming characteristics, e.g. high volume, high speed and continuous flowing. Approximate representation is one of the most efficient and effective solutions to address the large-volume-high-speed problem. In this thesis, we propose a dynamic representation scheme for streaming time series. Existing methods use a unitary function form for the entire approximation task. In contrast, our method adopts a set of function candidates such as linear function, polynomial function(degree ≥ 2), and exponential function. We provide a novel segmenting strategy to generate subsequences and dynamically choose candidate functions to approximate the subsequences. / Since we are dealing with streaming time series, the segmenting points and the corresponding approximate functions are incrementally produced. For a certain function form, we use a buffer window to find the local farthest possible segmenting point under a user specified error tolerance threshold. To achieve this goal, we define a feasible space for the coefficients of the function and show that we can indirectly find the local best segmenting point by the calculation in the coefficient space. Given the error tolerance threshold, the candidate function representing more information by unit parameter is chosen as the approximate function. Therefore, our representation scheme is more flexible and compact. We provide two dynamic algorithms, PLQS and PLQES, which involve two and three candidate functions, respectively. We also present the general strategy of function selection when more candidate functions are considered. In the experimental test, we examine the effectiveness of our algorithms with synthetic and real time series data sets. We compare our method with the piecewise linear approximation method and the experimental results demonstrate the evident superiority of our dynamic approach under the same error tolerance threshold.

Linear factor models have attracted considerable interest over recent years especially in the econometrics literature. The intuitively appealing idea to explain a panel of economic variables by a few common factors is one of the reasons for their popularity. From a statistical viewpoint, the need to reduce the cross-section dimension to a much smaller factor space dimension is obvious considering the large data sets available in economics and finance. One of the characteristics of the traditional factor model is that the process is stationary in the time dimension. This appears restrictive, given the fact that over long time periods it is unlikely that e.g. factor loadings remain constant. For example, in the capital asset pricing model (CAPM) of Sharpe (1964) and Lintner (1965), typical empirical results show that factor loadings are time-varying, which in the CAPM is caused by time-varying second moments. In this thesis we generalize the tools of factor analysis for the study of stochastic processes whose behavior evolves over time. In particular, we introduce a new class of factor models with loadings that are allowed to be smooth functions of time. To estimate the resulting nonstationary factor model we generalize the properties of the principal components technique to the time-varying framework. We mainly consider separately two classes of Evolutionary Factor Models: Evolutionary Static Factor Models (Chapter 2) and Evolutionary Dynamic Factor Models (Chapter 3). In Chapter 2 we propose a new approximate factor model where the common components are static but nonstationary. The nonstationarity is introduced by the time-varying factor loadings, that are estimated by the eigenvectors of a nonparametrically estimated covariance matrix. Under simultaneous asymptotics (cross-section and time dimension go to infinity simultaneously), we give conditions for consistency of our estimators of the time varying covariance matrix, the loadings and the factors. This paper generalizes to the locally stationary case the results given by Bai (2003) in the stationary framework. A simulation study illustrates the performance of these estimators. The estimators proposed in Chapter 2 are based on a nonparametric estimator of the covariance matrix whose entries are computed with the same moothing parameter. This approach has the advantage of guaranteeing a positive definite estimator but it does not adapt to the different degree of smoothness of the different entries of the covariance matrix. In Chapter 5 we give an additional theoretical result which explains how to construct a positive definite estimate of the covariance matrix while while permitting different smoothing parameters. This estimator is based on the Cholesky decomposition of a pre-estimator of the covariance matrix. In Chapter 3 we introduce the dynamics in our modeling. This model generalizes the dynamic (but stationary) factor model of Forni et al. (2000), as well as the nonstationary (but static) factor model of Chapter 2. In the stationary (dynamic) case, Forni et al. (2000) show that the common components are estimated by the eigenvectors of a consistent estimator of the spectral density matrix, which is a matrix depending only on the frequency. In the evolutionary framework the dynamics of the model is explained by a time-varying spectral density matrix. This operator is a function of time as well as of the frequency. In this chapter we show that the common components of a locally stationary dynamic factor model can be estimated consistently by the eigenvectors of a consistent estimator of the time-varying spectral density matrix. In Chapter 4 we apply our theoretical results to real data and compare the performance of our approach with that based on standard techniques. Chapter 6 concludes and mention the main questions for future research.

The present paper is related to the problem of approximating the exact solution to the magnetohydrodynamic equations (MHD). The behaviour of a viscous, incompressible and resistive fluid is exemined for a long period of time. Contents: 1 The magnetohydrodynamic equations 2 Notations and precise functional setting of the problem 3 Existence, uniqueness and regularity results 4 Statement and Proof of the main theorem 5 The approximate inertial manifold 6 Summary

The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.

In this paper we consider optimal design of experiments in the case of correlated observations, when no replications are possible. This situation is typical when observing a random process or random field with known covariance structure. We present a theorem which demonstrates that the computation of optimum exact designs corresponds to solving minimization problems in terms of design measures. (author's abstract) / Series: Forschungsberichte / Institut für Statistik

The capillary surface formed within a symmetric annular tube is analyzed. Assuming identical contact angles along each boundary, we consider surfaces u(x,y) that satisfy the capillary problem on an annular region. Several qualitative properties of u are determined and in particular, the behaviour of u is examined in the limiting cases of the annular domain approaching a disk as well as a thin ring. The iterative method of Siegel is also applied to the boundary value problem and convergence is demonstrated under conditions which include a contact angle of zero. Moreover, some geometries still yield interleaving iterates, allowing for upper and lower bounds to be placed on the boundary values of u. However, the interleaving properties no longer hold universally and for other geometries, another more complex behaviour is described. Finally, a numerical method is designed to approximate the iterative scheme.

In this study we will consider the construction of approximate optimal design for one-dimensional regression by exchange algorithm. Sufficient conditions under which an optimal design must have the minimal support points are known in Theorem 2.3.2 of Fedorov (1972). However, there are only a few cases which the analytic optimal designs are known. The exchange procedure for computing optimal designs is easily adopted to most criteria. We describe implementations for constructing the well-known special cases D-, A-, and c-optimal designs with the minimum number of support points. Examples which illustrate how the algorithm can be used to obtain these optimal designs and the performance of the algorithm are discussed. The commonly used D-, A-, and c-optimal criteria will be employed to study the convergence properties of the exchange algorithm for regression model which the set of the product of regression functions forms a Chebyshev system.

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Geostatistical techniques are used to estimate recoverable reserves at unsampled locations and to quantify uncertainty. Several variables are often measured and important for reserve evaluation. Using more variables improves the quality of modeling, but quantifying the relationships between the variables is difficult. The traditional linear model of coregionalization has been used to quantify the relationship between multiple variables, but ensuring the mathematical validity of the model is cumbersome. This research proposes an approximate method that improves the speed and practicality of the numerical modeling process by easily modeling multiple regionalized variables. The proposed algorithm is based on block LU simulation and takes local transformation into consideration. Application to a nickel deposit demonstrates the proposed methodology. / Mining Engineering

Providing a logical characterisation of rational agent reasoning has been a long standing challenge in artificial intelligence (AI) research. It is a challenge that is not only of interest for the construction of AI agents, but is of equal importance in the modelling of agent behaviour. The goal of this thesis is to contribute to the formalisation of agent reasoning by showing that the computational limitations of agents is a vital component of modelling rational behaviour. To achieve this aim, both motivational and formal aspects of resource-bounded agents are examined. It is a central argument of this thesis that accounting for computational limitations is critical to the success of agent reasoning, yet has received only limited attention from the broader research community. Consequently, an important contribution of this thesis is in its advancing of motivational arguments in support of the need to account for computational limitations in agent reasoning research. As a natural progression from the motivational arguments, the majority of this thesis is devoted to an examination of propositional approximate logics. These logics represent a step towards the development of resource-bounded agents, but are also applicable to other areas of automated reasoning. This thesis makes a number of contributions in mapping the space of approximate logics. In particular, it draws a connection between approximate logics and knowledge compilation, by developing an approximate knowledge compilation method based on Cadoli and Schaerf??s S-3 family of approximate logics. This method allows for the incremental compilation of a knowledge base, thus reducing the need for a costly recompilation process. Furthermore, each approximate compilation has well-defined logical properties due to its correspondence to a particular S-3 logic. Important contributions are also made in the examination of approximate logics for clausal reasoning. Clausal reasoning is of particular interest due to the efficiency of modern clausal satisfiability solvers and the related research into problem hardness. In particular, Finger's Logics of Limited Bivalence are shown to be applicable to clausal reasoning. This is subsequently shown to logically characterise the behaviour of the well-known DPLL algorithm for determining boolean satisfiability, when subjected to restricted branching.