Dissertations / Theses on the topic 'Local stationary'

We consider several related topics in the bifurcation theory of random dynamical systems: synchronisation by noise, noise-induced chaos, qualitative changes of finite-time behaviour and stability of systems surviving in a bounded domain. Firstly, we study the dynamics of a two-dimensional ordinary differential equation exhibiting a Hopf bifurcation subject to additive white noise. Depending on the deterministic Hopf bifurcation parameter and a phase-amplitude coupling parameter called shear, three dynamical phases can be identified: a random attractor with uniform synchronisation of trajectories, a random attractor with non-uniform synchronisation of trajectories and a random attractor without synchronisation of trajectories. We prove the existence of the first two phases which both exhibit a random equilibrium with negative top Lyapunov exponent but differ in terms of finite-time and uniform stability properties. We provide numerical results in support of the existence of the third phase which is characterised by a so-called random strange attractor with positive top Lyapunov exponent implying chaotic behaviour. Secondly, we reduce the model of the Hopf bifurcation to its linear components and study the dynamics of a stochastically driven limit cycle on the cylinder. In this case, we can prove the existence of a bifurcation from an attractive random equilibrium to a random strange attractor, indicated by a change of sign of the top Lyapunov exponent. By establishing the existence of a random strange attractor for a model with white noise, we extend results by Qiudong Wang and Lai-Sang Young on periodically kicked limit cycles to the stochastic context. Furthermore, we discuss a characterisation of the invariant measures associated with the random strange attractor and deduce positive measure-theoretic entropy for the random system. Finally, we study the bifurcation behaviour of unbounded noise systems in bounded domains, exhibiting the local character of random bifurcations which are usually hidden in the global analysis. The systems are analysed by being conditioned to trajectories which do not hit the boundary of the domain for asymptotically long times. The notion of a stationary distribution is replaced by the concept of a quasi-stationary distribution and the average limiting behaviour can be described by a so-called quasi-ergodic distribution. Based on the well-explored stochastic analysis of such distributions, we develop a dynamical stability theory for stochastic differential equations within this context. Most notably, we define conditioned average Lyapunov exponents and demonstrate that they measure the typical stability behaviour of surviving trajectories. We analyse typical examples of random bifurcation theory within this environment, in particular the Hopf bifurcation with additive noise, with reference to whom we also study (numerically) a spectrum of conditioned Lyapunov exponents. Furthermore, we discuss relations to dynamical systems with holes.

In this paper we prove structure results for the singular sets of stationary harmonic maps and mean curvature flows local to particular singularities. The original work is contained in Chapter 5 and Chapter 8. Chapters 1-5 are concerned with energy minimising maps and stationary harmonic maps. Chapters 6-8 are concerned with mean curvature flows and Brakke flows. In the case of stationary harmonic maps we consider a singularity at which the spine dimension is maximal, and such that the weak tangent map is homotopically non-trivial, and has minimal density amongst singularities of maximal spine dimen- sion. Local to such a singularity we show the singular set is a bi-Hölder continuous homeomorphism of the unit disk of dimension equal to the maximal spine dimension. A weak tangent map is translation invariant along a subspace, and invariant under dilations, so it completely defined by its values on a sphere. Such a map is said to be homotopically non-trivial if the mapping of a sphere into some target manifold cannot be deformed by a homotopy to a constant map. For an n-dimensional mean curvature flow we consider a singularity at which we can find a shrinking cylinder as a tangent flow, that collapses on an (n−1)-dimensional plane. Local to such a singularity we show that all singularities have such a cylindrical tangent, or else have lower Gaussian density than that of the shrinking cylinder. The subset of cylindrical singularities can be shown to be contained in a finite union of parabolic (n − 1)-dimensional Lipschitz submanifolds. In the case that the mean curvature flow arises from elliptic regularisation we can show that all singularities local to a cylindrical singularity with (n − 1)-dimensional spine are either cylindrical singularities with (n − 1)-dimensional spine, or contained in a parabolic Hausdorff (n − 2)-dimensional set.

The topic of this master's thesis is the time lapse from stationary camera images. Unwanted phenomena, which arise in time lapse, were analyzed and algorithms to overcome these limitations were designed. The algorithms were implemented and compared using the captured dataset. The resulting application creates time lapse from the video input and allows users to choose the processing technique including the setting of appropriate parameters.

Le monde produit 2,5 quintillions d’octets par jour, appelés mégadonnées. Le volume, la valeur, la variété, la vélocité et la véracité définissent les cinq caractéristiques du Big Data qui représentent une complexité fondamentale pour de nombreux algorithmes d’apprentissage automatique, tels que le clustering, la reconnaissance d’images et d’autres techniques d’apprentissage modernes. Avec ces données volumineuses, les estimations d’hyperparamètres ne prennent pas la forme de la moyenne de l’échantillon (non linéaire), mais celle de la forme de la moyenne sur m-tuples, appelée l’estimateur U-statistique. Nous considérons dans cette thèse la collection de U-statistiques, connue sous le nom de U-processus, pour deux types de variables dépendantes, les données Markoviennes et les variables aléatoires localement stationnaires. Ainsi, nous avons divisé notre travail en deux parties pour aborder chaque type indépendamment. Dans la première partie, nous considérons les variables Markoviennes. Nous nous concentrons particulièrement sur les développements de U-processus bootstrappés dans un cadre de Harris. L’idée fondamentale utilisée repose sur les méthodes régénératives consistant essentiellement à diviser l’échantillon en blocs de données indépendants et identiquement distribués (i.i.d.), où chaque bloc correspond à des segments de chemin entre deux visites à un atome appelé A formant une séquence de renouvellement. Nous caractérisons les propriétés limites pour les U-processus indexés par des classes de fonctions uniformément bornées et non bornées. Nous montrons la consistance du bootstrap dans ce cadre. L’approche du bootstrap permet de contourner les problèmes fréquemment rencontrés pour l’évaluation des lois limites dépendants d’une manière complexe de paramètres inconnus. La technique de bootstrap que nous utilisons dans cette thèse est le bootstrap de renouvellement, où l’échantillon bootstrap est formé par rééchantillonnage à partir des blocs. Comme les blocs non bootstrapés sont indépendants, une partie des preuves se réduit au cas i.i.d. Les principales difficultés sont liées à la taille aléatoire des blocs rééchantillonnés, ce qui crée un problème non trivial de temps d’arrêts aléatoires, constituant un des grands obstacles de la généralisation de la théorie dans notre contexte. Pour contourner cette difficulté, nous avons utilisé comme étape intermédiaire la substitution du temps d’arrêt aléatoire par son espérance. La convergence faible des U-processus bootstrappés est très délicate dans notre cadre, en particulier l’équicontinuité en utilisant la comparaison avec le U-processus initial. Nous avons étendu les résultats susmentionnés au cas où le degré du U-processus croît avec la taille de l’échantillon n, le noyau variant dans une classe de fonctions. Nous avons caractérisé la convergence faible pour le bootstrap de renouvellement pour le U-processus à degré infini en faisant usage de la technique de découplage combinée avec des techniques de symétrisation. Enfin, nous considérons un bootstrap pondéré échangeable des U-processus empiriques. Dans la deuxième partie de cette thèse, les données dépendantes sont représentées par des fonctions aléatoires localement stationnaires. Propulsés par la représentation croissante des séries temporelles par des données fonctionnelles ou courbes, et le comportement non stationnaire de ces dernières, nous nous sommes intéressés au U-processus conditionnel des séries temporelles fonctionnelles localement stationnaires. Plus précisément, nous avons étudié la convergence faible des U-processus conditionnels, indexée par des classes de fonctions, dans le cadre de données fonctionnelles localement stationnaires. Nous avons caractérisé la convergence faible dans les deux cas lorsque la classe de fonctions est bornée ou non bornée satisfaisant certaines conditions de moment
The world is producing 2.5 quintillion bytes daily, known as big data. Volume, value, variety, velocity, and veracity define the five characteristics of big data that represent a fundamental complexity for many machine learning algorithms, such as clustering, image recognition, and other modern learning techniques. With this large data, hyperparameter estimations do not take the form of the sample mean (not linear). Instead, they takethe form of average over m-tuples, known as the U-statistic estimator in probabilityand statistics. In this work, we treat the collection of U-statistics, known as the Uprocess,for two types of dependent variables, the Markovian data, and locally stationary random variables. Thus, we have divided our work into two parts to address each type independently.In the first part, we deal with Markovian data. The approach relies on regenerative methods, which essentially involve dividing the sample into independent and identically distributed (i.i.d.) blocks of data, where each block corresponds to the path segments between two visits of an atom called A, forming a renewal sequence. We derive the limiting theory for Harris recurrent Markov chain over uniformly bounded and unbounded classes of functions. We show that the results can be generalized also to the bootstrappe dU statistics. The bootstrap approach bypasses the problems faced with the asymptotic behavior due to the unknown parameters of limiting distribution. Furthermore, the bootstrap technique we use in this thesis is the renewal bootstrap, where the bootstrap samplevis formed by resampling the blocks. Since the non-bootstrapped blocks are independent, most proofs reduce to the i.i.d. case. The main difficulties are related to the randomsize of the resampled blocks, which creates a problem with random stopping times. This problem is degraded by replacing the random stopping time with their expectation. Also, since we resample from a random number of blocks, and the bootstrap equicontinuity can be verified by comparing with the initial process, the weak convergence of the bootstrap U-process must be treated very carefully. We successfully derive the results in the case of the k-Harris Markov chain. We extend all the above results to the case where the degreeof U-statistic grows with the sample size n, with the kernel varying in a class of functions. We provide the uniform limit theory for the renewal bootstrap for the infinite-degree U-process with the help of the decoupling technique combined with symmetrization techniques in addition to the chaining inequality. Remaining in the Markovian setting, we extend the weighted bootstrap empirical processes to a high-dimensional estimation. We consider an exchangeably weighted bootstrap of the general function-indexed empirical U-processes. In the second part of this thesis, dependent data are represented by locally stationary random variables. Propelled by the increasing representation of the data by functionalor curves time series and the non-stationary behavior of the latter, we are interested in the conditional U-process of locally stationary functional time series. More precisely, we investigate the weak convergence of the conditional U-processes in the locally stationary functional mixing data framework. We treat the weak convergence in both caseswhen the class of functions is bounded or unbounded, satisfying some moment conditions. Finally, we extend the asymptotic theory of conditional U-process to the locallystationary functional random field {Xs,An : s ∈ Rn} observed at irregular spaced locations in Rn = [0,An]d ∈ Rd, and include both pure increasing domain and mixed increasing domain. We treat the weak convergence in both cases when the class of functions is boundedor unbounded, satisfying some moment conditions. These results are established underfairly general structural conditions on the classes of functions and the underlying models

No contexto dos modelos fatoriais existem diferentes metodologias para abordar a modelagem de séries temporais multivariadas que exibem uma estrutura não estacionária de segunda ordem, co- movimentos e transições no tempo. Modelos com mudanças estruturais abruptas e restrições rigorosas (muitas vezes irreais) nas cargas fatoriais, quando elas são funções determinísticas no tempo, foram propostos na literatura para lidar com séries multivariadas que possuem essas características. Neste trabalho, apresentamos um modelo fatorial com cargas variando continuamente no tempo para modelar séries temporais não estacionárias e um procedimento para sua estimação que consiste em dois estágios. No primeiro, os fatores latentes são estimados empregando os componentes principais das séries observadas. Em um segundo estágio, tratamos estes componentes principais como co-variáveis e as cargas funcionais são estimadas através de funções de ondaletas e mínimos quadrados generalizados. Propriedades assintóticas dos estimadores de componentes principais e de mínimos quadrados dos coeficientes de ondaletas são apresentados. O desempenho da metodologia é ilustrado através de estudos de simulação. Uma aplicação do modelo proposto no mercado spot de energia do Nord Pool é apresentado.
In the context of the factor models there are different methodologies to modeling multivariate time series that exhibit a second order non-stationary structure, co-movements and transitions over time. Models with abrupt structural changes and strict restrictions (often unrealistic) in factor loadings, when they are deterministic functions of time, have been proposed in the literature to deal with multivariate series that have these characteristics. In this work, we present a factor model with time-varying loadings continuously to modeling non-stationary time series and a procedure for its estimation that consists of two stages. First, latent factors are estimated using the principal components of the observed series. Second, we treat principal components obtained in first stage as covariate and the functional loadings are estimated by wavelet functions and generalized least squares. Asymptotic properties of the principal components estimators and least squares estimators of the wavelet coefficients are presented. The per- formance of the methodology is illustrated by simulations. An application to the model proposed in the energy spot market of the Nord Pool is presented.

Am 14. Oktober 1999 wurde in Rossendorf die dritte Veranstaltung in einer Serie von Workshops über Meßtechnik für stationäre und transiente Mehrphasenströmungen durchgeführt. Dieses Jahr kann auf auf 11 interessante Vorträge zurückgeblickt werden. Besonders hervorzuheben sind die beiden Hauptvorträge, die von Herrn Professor Hetsroni aus Haifa und Herrn Dr. Sengpiel aus Karlsruhe gehalten wurden. Erneut lag ein wichtiger Schwerpunkt auf Meßverfahren, die räumliche Verteilungen von Phasenanteilen und Geschwindigkeiten sowie die Größe von Partikeln bzw. Blasen der dispersen Phase zugänglich machen. So wurde über einen dreidimensional arbeitenden Röntgentomographen, ein Verfahren zur Messung von Geschwindigkeitsprofilen mit Gittersensoren und eine Methode zur simultanen Messung von Blasengrößen sowie Feldern von Gas- und Flüssigkeitsgeschwindigkeit mit einer optischen Partikelverfolgungstechnik vorgetragen. Daneben wurden interessante Entwicklungen auf dem Gebiet der lokalen Sonden vorgestellt, wie z.B. eine Elektrodiffusionssonde. Neue meßtechnische Ansätze waren ebenfalls vertreten; hervorzuheben ist der Versuch, die Methode der optischen Tomographie für die Untersuchung von Zweiphasenströmungen nutzbar zu machen. Der Tagungsband enthält die folgenden Beiträge: S. John, R. Wilfer, N. Räbiger, Universität Bremen, Messung hydrodynamischer Parameter in Mehrphasenströmungen bei hohen Dispersphasengehalten mit Hilfe der Elektrodiffusionsmeßtechnik E. Krepper, A. Aszodi, Forschungszentrum Rossendorf, Temperatur- und Dampfgehaltsverteilungen bei Sieden in seitlich beheizten Tanks D. Hoppe, Forschungszentrum Rossendorf, Ein akustisches Resonanzverfahren zur Klassifizierung von Füllständen W. Sengpiel, V. Heinzel, M. Simon, Forschungszentrum Karlsruhe, Messungen der Eigenschaften von kontinuierlicher und disperser Phase in Luft-Wasser-Blasenströmungen R. Eschrich, VDI, Die Probestromentnahme zur Bestimmung der dispersen Phase einer Zweiphasenströmung U. Hampel, TU Dresden, Optische Tomographie O. Borchers, C. Busch, G. Eigenberger, Universität Stuttgart, Analyse der Hydrodynamik in Blasenströmungen mit einer Bildverarbeitungsmethode C. Zippe, Forschungszentrum Rossendorf, Beobachtung der Wechselwirkung von Blasen mit Gittersensoren mit einer Hochgeschwindigkeits-Videokamera H.-M. Prasser, Forschungszentrum Rossendorf, Geschwindigkeits- und Durchflußmessung mit Gittersensoren

Am 14. Oktober 1999 wurde in Rossendorf die dritte Veranstaltung in einer Serie von Workshops über Meßtechnik für stationäre und transiente Mehrphasenströmungen durchgeführt. Dieses Jahr kann auf auf 11 interessante Vorträge zurückgeblickt werden. Besonders hervorzuheben sind die beiden Hauptvorträge, die von Herrn Professor Hetsroni aus Haifa und Herrn Dr. Sengpiel aus Karlsruhe gehalten wurden. Erneut lag ein wichtiger Schwerpunkt auf Meßverfahren, die räumliche Verteilungen von Phasenanteilen und Geschwindigkeiten sowie die Größe von Partikeln bzw. Blasen der dispersen Phase zugänglich machen. So wurde über einen dreidimensional arbeitenden Röntgentomographen, ein Verfahren zur Messung von Geschwindigkeitsprofilen mit Gittersensoren und eine Methode zur simultanen Messung von Blasengrößen sowie Feldern von Gas- und Flüssigkeitsgeschwindigkeit mit einer optischen Partikelverfolgungstechnik vorgetragen. Daneben wurden interessante Entwicklungen auf dem Gebiet der lokalen Sonden vorgestellt, wie z.B. eine Elektrodiffusionssonde. Neue meßtechnische Ansätze waren ebenfalls vertreten; hervorzuheben ist der Versuch, die Methode der optischen Tomographie für die Untersuchung von Zweiphasenströmungen nutzbar zu machen. Der Tagungsband enthält die folgenden Beiträge: S. John, R. Wilfer, N. Räbiger, Universität Bremen, Messung hydrodynamischer Parameter in Mehrphasenströmungen bei hohen Dispersphasengehalten mit Hilfe der Elektrodiffusionsmeßtechnik E. Krepper, A. Aszodi, Forschungszentrum Rossendorf, Temperatur- und Dampfgehaltsverteilungen bei Sieden in seitlich beheizten Tanks D. Hoppe, Forschungszentrum Rossendorf, Ein akustisches Resonanzverfahren zur Klassifizierung von Füllständen W. Sengpiel, V. Heinzel, M. Simon, Forschungszentrum Karlsruhe, Messungen der Eigenschaften von kontinuierlicher und disperser Phase in Luft-Wasser-Blasenströmungen R. Eschrich, VDI, Die Probestromentnahme zur Bestimmung der dispersen Phase einer Zweiphasenströmung U. Hampel, TU Dresden, Optische Tomographie O. Borchers, C. Busch, G. Eigenberger, Universität Stuttgart, Analyse der Hydrodynamik in Blasenströmungen mit einer Bildverarbeitungsmethode C. Zippe, Forschungszentrum Rossendorf, Beobachtung der Wechselwirkung von Blasen mit Gittersensoren mit einer Hochgeschwindigkeits-Videokamera H.-M. Prasser, Forschungszentrum Rossendorf, Geschwindigkeits- und Durchflußmessung mit Gittersensoren

Cette thèse est une contribution à la construction de l'Approche Locale de la rupture à l'échelle microscopique à l'aide de la modélisation d'agrégats polycristallins. Elle consiste à prendre en compte la variabilité spatiale de la microstructure du matériau. Pour ce faire, la modélisation micromécanique du matériau est réalisée par la simulation d'agrégats polycristallins par éléments finis. Les champs aléatoires de contrainte (principale maximale et de clivage) dans le matériau qui représentent la variabilité spatiale de la microstructure sont ensuite modélisés par un champ aléatoire gaussien stationnaire ergodique. Les propriétés de variabilité spatiale de ces champs sont identifiés par une méthode d'identification, e.g. méthode du périodogramme, méthode du variogramme, méthode du maximum de vraisemblance. Des réalisations synthétiques des champs de contraintes sont ensuite simulées par une méthode de simulation, e.g. méthode Karhunen-Loève discrète, méthode "Circulant Embedding", méthode spectrale, sans nouveau calcul aux éléments finis. Enfin, le modèle d'Approche Locale de la rupture par simulation de champ de contrainte de clivage permettant d'y intégrer les réalisations simulées du champ est construit pour estimer la probabilité de rupture du matériau.

In this thesis, we study numerical analysis for random processes and fields. We investigate the behavior of the approximation accuracy for specific linear methods based on a finite number of observations. Furthermore, we propose techniques for optimizing performance of the methods for particular classes of random functions. The thesis consists of an introductory survey of the subject and related theory and four papers (A-D). In paper A, we study a Hermite spline approximation of quadratic mean continuous and differentiable random processes with an isolated point singularity. We consider a piecewise polynomial approximation combining two different Hermite interpolation splines for the interval adjacent to the singularity point and for the remaining part. For locally stationary random processes, sequences of sampling designs eliminating asymptotically the effect of the singularity are constructed. In Paper B, we focus on approximation of quadratic mean continuous real-valued random fields by a multivariate piecewise linear interpolator based on a finite number of observations placed on a hyperrectangular grid. We extend the concept of local stationarity to random fields and for the fields from this class, we provide an exact asymptotics for the approximation accuracy. Some asymptotic optimization results are also provided. In Paper C, we investigate numerical approximation of integrals (quadrature) of random functions over the unit hypercube. We study the asymptotics of a stratified Monte Carlo quadrature based on a finite number of randomly chosen observations in strata generated by a hyperrectangular grid. For the locally stationary random fields (introduced in Paper B), we derive exact asymptotic results together with some optimization methods. Moreover, for a certain class of random functions with an isolated singularity, we construct a sequence of designs eliminating the effect of the singularity. In Paper D, we consider a Monte Carlo pricing method for arithmetic Asian options. An estimator is constructed using a piecewise constant approximation of an underlying asset price process. For a wide class of Lévy market models, we provide upper bounds for the discretization error and the variance of the estimator. We construct an algorithm for accurate simulations with controlled discretization and Monte Carlo errors, andobtain the estimates of the option price with a predetermined accuracy at a given confidence level. Additionally, for the Black-Scholes model, we optimize the performance of the estimator by using a suitable variance reduction technique.

In Geostatistical modelling of the spatial distribution of rock attributes, the multivariate distribution of a Random Function defines the range of possible values and the spatial relationships among them. Under a decision of stationarity, the Random Function distribution and its statistics are inferred from data within a spatial domain deemed statistically homogenous. Assuming stationary multiGaussianity allows spatial prediction techniques to take advantage of this simple parametric distribution model. These techniques compute the local distributions with surrounding data and global spatially invariant statistics. They often fail to reproduce local changes in the mean, variability and, particularly, the spatial continuity, that are required for geologically realistic modelling of rock attributes. The proposed alternative is to build local Random Function models that are deemed stationary only in relation to the locations where they are defined. The corresponding location-dependent distributions and statistics are inferred by weighting the samples inversely proportional to their distance to anchor locations. These distributions are locally Gaussian transformed. The transformation models carry information on the local histogram. The distance weighted experimental measures of spatial correlation are able to adapt to local changes in the spatial continuity and are semi-automatically fitted by locally defined variogram models. The fields of local variogram and transformation parameters are used in locally stationary spatial prediction algorithms. The resulting attribute models are rich in non-stationary spatial features. This process implies a higher computational demand than the traditional techniques, but, if data is abundant enough to allow a reliable inference of the local statistics, the proposed locally stationary techniques outperform their stationary counterparts in terms of accuracy and precision. These improved models have the potential of providing better decision support for engineering design.
Mining Engineering

With an increasing number of large area data sets, many study areas exhibit spatial non-stationarity or spatial variation in mean and variance of observed phenomena. This poses issues for a number of spatial analysis methods which assume data are stationary. The Getis and Ord’s Gi* statistic is a popular measure that, like many others, is impacted by non stationarity. The Gi* is used for locating hot and cold spots in marked data through the detection of spatial autocorrelation in values that are extreme relative to the global mean value, or the mean entire study area. This thesis describes modifications of the Getis and Ord’s Gi* local measure of spatial association, in part to account for regional differences (spatial non-stationarity) in a dataset. Instead of using data from the entire study area to calculate the mean parameter, as is done for the standard Gi*, I capture points for calculation of the mean using a circular distance band centred on the pivot location, which I call the local region (similar to the Ord and Getis Oi statistic). This approach can be applied to a single instance of a local region or to multiple spatial scales of the local region. I explore both in this paper using simulated datasets and a case study on mountain pine beetle infestation data. I find that the local region, when of a similar size to a true region (homogeneous section of the study area where the mean is approximately the same across locations), obtains similar results to the standard Gi* calculated separately on distinct regions (simulated to be distinct), but has the advantage of not needing explicit delineation of regional boundaries or partitioning into separate subareas. The results of a probability score for a multi-scale approach include high and low scores that are more evenly distributed across the study area and that are thus able to pick out more subtle variations within different regions. Through the case study I demonstrate how the multi-scale approach may be applied to a real dataset.

碩士
國立中山大學
經濟學研究所
98
Jorda (2005) proposed the new method to estimate impulse response functions by local projection. The new method, local projection, can avoid the misspecification problem. That is, local projections are robust to misspecification of the data generating process (DGP). Wu, Lee, and Wang (2008) extended the Jorda’s local projection from stationary time series I(0) to non-stationary time series I(1). It makes the local projection be a more generally applicative method for the Macroeconomic. In the article, I relax the cointegration vector which assumed to be known in the Wu, Lee, and Wang (2008) and Lee(2010). From the inference of Johansen (1995) I can get the property of super-consistent between β and ˆ β in the cointegration vector. I use the above condition and OLS to estimate impulse response functions, and in the asymptotic theorem, the cointegration vectors which assumed to be known or estimated by Johansen MLE are both get the consistent coefficients of impulse responses.

For time-varying signals such as speech and audio, short-time analysis becomes necessary to compute specific signal attributes and to keep track of their evolution. The standard technique is the short-time Fourier transform (STFT), using which one decomposes a signal in terms of windowed Fourier bases. An advancement over STFT is the wavelet analysis in which a function is represented in terms of shifted and dilated versions of a localized function called the wavelet. A specific modeling approach particularly in the context of speech is based on short-time linear prediction or short-time Wiener filtering of noisy speech. In most nonstationary signal processing formalisms, the key idea is to analyze the properties of the signal locally, either by first truncating the signal and then performing a basis expansion (as in the case of STFT), or by choosing compactly-supported basis functions (as in the case of wavelets). We retain the same motivation as these approaches, but use polynomials to model the signal on a short-time basis (“short-time polynomial representation”). To emphasize the local nature of the modeling aspect, we refer to it as “local polynomial modeling (LPM).” We pursue two main threads of research in this thesis: (i) Short-time approaches for speech enhancement; and (ii) LPM for enhancing smooth signals, with applications to ECG, noisy nonuniformly-sampled signals, and voiced/unvoiced segmentation in noisy speech. Improved iterative Wiener filtering for speech enhancement A constrained iterative Wiener filter solution for speech enhancement was proposed by Hansen and Clements. Sreenivas and Kirnapure improved the performance of the technique by imposing codebook-based constraints in the process of parameter estimation. The key advantage is that the optimal parameter search space is confined to the codebook. The Nonstationary signal enhancement solutions assume stationary noise. However, in practical applications, noise is not stationary and hence updating the noise statistics becomes necessary. We present a new approach to perform reliable noise estimation based on spectral subtraction. We first estimate the signal spectrum and perform signal subtraction to estimate the noise power spectral density. We further smooth the estimated noise spectrum to ensure reliability. The key contributions are: (i) Adaptation of the technique for non-stationary noises; (ii) A new initialization procedure for faster convergence and higher accuracy; (iii) Experimental determination of the optimal LP-parameter space; and (iv) Objective criteria and speech recognition tests for performance comparison. Optimal local polynomial modeling and applications We next address the problem of fitting a piecewise-polynomial model to a smooth signal corrupted by additive noise. Since the signal is smooth, it can be represented using low-order polynomial functions provided that they are locally adapted to the signal. We choose the mean-square error as the criterion of optimality. Since the model is local, it preserves the temporal structure of the signal and can also handle nonstationary noise. We show that there is a trade-off between the adaptability of the model to local signal variations and robustness to noise (bias-variance trade-off), which we solve using a stochastic optimization technique known as the intersection of confidence intervals (ICI) technique. The key trade-off parameter is the duration of the window over which the optimum LPM is computed. Within the LPM framework, we address three problems: (i) Signal reconstruction from noisy uniform samples; (ii) Signal reconstruction from noisy nonuniform samples; and (iii) Classification of speech signals into voiced and unvoiced segments. The generic signal model is x(tn)=s(tn)+d(tn),0 ≤ n ≤ N - 1. In problems (i) and (iii) above, tn=nT(uniform sampling); in (ii) the samples are taken at nonuniform instants. The signal s(t)is assumed to be smooth; i.e., it should admit a local polynomial representation. The problem in (i) and (ii) is to estimate s(t)from x(tn); i.e., we are interested in optimal signal reconstruction on a continuous domain starting from uniform or nonuniform samples. We show that, in both cases, the bias and variance take the general form: The mean square error (MSE) is given by where L is the length of the window over which the polynomial fitting is performed, f is a function of s(t), which typically comprises the higher-order derivatives of s(t), the order itself dependent on the order of the polynomial, and g is a function of the noise variance. It is clear that the bias and variance have complementary characteristics with respect to L. Directly optimizing for the MSE would give a value of L, which involves the functions f and g. The function g may be estimated, but f is not known since s(t)is unknown. Hence, it is not practical to compute the minimum MSE (MMSE) solution. Therefore, we obtain an approximate result by solving the bias-variance trade-off in a probabilistic sense using the ICI technique. We also propose a new approach to optimally select the ICI technique parameters, based on a new cost function that is the sum of the probability of false alarm and the area covered over the confidence interval. In addition, we address issues related to optimal model-order selection, search space for window lengths, accuracy of noise estimation, etc. The next issue addressed is that of voiced/unvoiced segmentation of speech signal. Speech segments show different spectral and temporal characteristics based on whether the segment is voiced or unvoiced. Most speech processing techniques process the two segments differently. The challenge lies in making detection techniques offer robust performance in the presence of noise. We propose a new technique for voiced/unvoiced clas-sification by taking into account the fact that voiced segments have a certain degree of regularity, and that the unvoiced segments do not possess any smoothness. In order to capture the regularity in voiced regions, we employ the LPM. The key idea is that regions where the LPM is inaccurate are more likely to be unvoiced than voiced. Within this frame-work, we formulate a hypothesis testing problem based on the accuracy of the LPM fit and devise a test statistic for performing V/UV classification. Since the technique is based on LPM, it is capable of adapting to nonstationary noises. We present Monte Carlo results to demonstrate the accuracy of the proposed technique.

碩士
國立臺灣大學
企業管理碩士專班
102
This business plan uses the Chung Yuan Branch of Jinyuhtarng Stationery(金玉堂文具中原店) as the example to establish a sales growth plan of the local stationery store. I include analyses of the market, current problems of the store, the goals and the strategies of the store’s growth, and the operation and the marketing plan of the store in this business plan. Jinyuhtarng Stationery is one of the biggest stationery chain stores in Taiwan. The Chung Yuan Branch of Jinyuhtarng Stationery is around 320 ping坪 (or 11,387 sq.ft) with four floors and the elevator. Its current gross profit margin and revenue fall far below the expected performance due to its inefficient management. In this business plan aims for developing the strategies with new opearion methods in order to reach the expected goals.

This thesis includes solving the basic examples of differential calculus (especially two and three variables). The work covers examples of the solutions of the domain, the first and second partial derivatives, determining the functions of local extremes (both explicitly and implicitly given) and find the equation of the tangent plane to a point in the graphs of functions. The examples are sorted by difficulty.