在無母數的迴歸當中，因為原始的函數類型未知，所以常用已知特定類型的函數來近似未知的函數，而spline函數也可以用來近似未知的函數，但是要估計spline函數就需要設定節點(knots)，越多的節點越能準確近似原始函數的內容，可是如果節點太多有較多的參數要估計, 就會變得比較不準確，所以選擇適合節點個數就變得很重要。 在本研究中，用交叉驗證的方式來尋找適合的節點個數， 考慮了幾種不同切割資料方式來決定訓練資料和測試資料, 並比較不同切割資料的方式下選擇節點的結果與函數估計的效果。 / In this thesis, I consider the problem of estimating an unknown regression function using spline approximation. Splines are piecewise polynomials jointed at knots. When using splines to approximate unknown functions, it is crucial to determine the number of knots and the knot locations. In this thesis, I determine the knot locations using least squares for given a given number of knots, and use cross-validation to find appropriate number of knots. I consider three methods to split the data into training data and testing data, and compare the estimation results.

The goal of this thesis is to analyze the mathematical apparatus of the most widespread methods used for the yield curves construction. It aims to introduce not only the various of construction models but also to describe the whole process of creation, while discussing the advantages and disadvantage of individual methods. The first chapter focus on the general theory and the use of the term structure of interest rates in practice. The second part deals with the construction process itself and describes the most frequently used methods. The last chapter then shows the real application of selected methods on given data set and the use of the constructed yield curves for interest rate derivative valuation too.

The goal of this thesis is to nd types of curves which would allow for the construction of a path that could be traversed by a vehicle. It seems that a minimal constraint for such a path is the continuity of curve's curvature. This leads to a closer look at the three types of curves: Clothoids, which are able to smoothly connect straights with arcs of a constant curvature, interpolation quintic splines, which are C2 smooth in the interpolation nodes and -splines, these belong to the family of quintic polynomial curves too, however, they are characterised by the vector of parameters which modies the shape of the curve. The thesis is accompanied by an application allowing for manual construction of the path composed of spline curves.

Cílem této diplomové práce je najít optimální profil mezery mazání pro turbodmychadlo. Cílem je minimalizovat tření, udržovat nosnost a nezvyšovat průtok maziva. Tato multiobjektivní optimalizace se provádí pomocí genetického algoritmu (GA) v MATLABu. Minimalizace třecí síly snižuje ztráty třecího výkonu turbodmychadla. Řešení Reynoldsovy rovnice je počítáno numericky pomocí MATLABu. Je zjištěna minimální tloušťka mazací mezery pro počáteční problém. Funkce spline se používá ke generování obecného profilu mazací mezery. Tento profil je poté optimalizován pomocí GA v MATLABu.

In dem aktuellen Beitrag wird eine kurze Einführung in die Mischung bzw. Entmischung hochviskoser inkompressibler Fluide gegeben. Hierzu wird die Methode der Phasenfeldmodellierung und ihre numerische Diskretisierung vorgestellt. Anhand eines konkreten Beispiels wird die technische Relevanz des vorgestellten Ansatzes demonstriert.

<div>Many practical applications benefit from the reconstruction of a smooth multivariate function from discrete data for purposes such as reducing file size or improving analytic and visualization performance. Among the different reconstruction methods, tensor product B-spline has a number of advantageous properties over alternative data representation. However, the problem of constructing a best-fit B-spline approximation effectively contains many roadblocks. Within the many free parameters in the B-spline model, the choice of the knot vectors, which defines the separation of each piecewise polynomial patch in a B-spline construction, has a major influence on the resulting reconstruction quality. Yet existing knot placement methods are still ineffective, computationally expensive, or impose limitations on the dataset format or the B-spline order. Moving beyond the 1D cases (curves) and onto higher dimensional datasets (surfaces, volumes, hypervolumes) introduces additional computational challenges as well. Further complications also arise in the case of undersampled data points where the approximation problem can become ill-posed and existing regularization proves unsatisfactory.</div><div><br></div><div>This dissertation is concerned with improving the efficiency and accuracy of the construction of a B-spline approximation on discrete data. Specifically, we present a novel B-splines knot placement approach for accurate reconstruction of discretely sampled data, first in 1D, then extended to higher dimensions for both structured and unstructured formats. Our knot placement methods take into account the feature or complexity of the input data by estimating its high-order derivatives such that the resulting approximation is highly accurate with a low number of control points. We demonstrate our method on various 1D to 3D structured and unstructured datasets, including synthetic, simulation, and captured data. We compare our method with state-of-the-art knot placement methods and show that our approach achieves higher accuracy while requiring fewer B-spline control points. We discuss a regression approach to the selection of the number of knots for multivariate data given a target error threshold. In the case of the reconstruction of irregularly sampled data, where the linear system often becomes ill-posed, we propose a locally varying regularization scheme to address cases for which a straightforward regularization fails to produce a satisfactory reconstruction.</div>

Dans cette thèse un système de synthèse sonore imitative sera présenté, applicable à la plupart des instruments de quasi-harmoniques. Le système se base sur les enregistrements d’une note unique qui représentent une version quantifiée de l'espace de timbre possible d'un instrument par rapport à sa hauteur et son intensité. Une méthode de transformation permet alors de générer des signaux sonores de valeurs continues des paramètres de contrôle d'expression qui sont perceptuellement cohérent avec ses équivalents acoustiques. Un modèle paramétrique de l'instrument se présente donc basé sur un modèle de filtre de source étendu avec des manipulations distinctes sur les harmoniques d’un signal et ses composantes résiduelles. Une procédure d'évaluation subjective sera présentée afin d’évaluer une variété de résultats de transformation par une comparaison directe avec des enregistrements non modifiés, afin de comparer la perception entre les résultats synthétiques et leurs équivalents acoustiques. / Within this thesis an imitative sound synthesis system will be introduced that is applicable to most quasi-harmonic instruments. The system bases upon single-note recordings that represent a quantized version of an instrument's possible timbre space with respect to its pitch and intensity dimension. A transformation method then allows to render sound signals with continuous values of the expressive control parameters which are perceptually coherent with its acoustic equivalents. A parametric instrument model is therefore presented based on an extended source-filter model with separate manipulations of a signal’s harmonic and residual components. A subjective evaluation procedure will be shown to assess a variety of transformation results by a direct comparison with unmodified recordings to determine how perceptually close the synthesis results are regarding their respective acoustic correlates.

Mass Spectrometry, especially matrix assisted laser desorption/ionization (MALDI) time of flight (TOF), is emerging as a leading technique in the proteomics revolution. It can be used to find disease-related protein patterns in mixtures of proteins derived from easily obtained samples. In this paper, a novel algorithm for MALDI-TOF MS data processing is developed. The software design includes the application of splines for data smoothing and baseline correction, wavelets for adaptive denoising, multivariable statistics techniques such as clustering analysis, and signal processing techniques to evaluate the complicated biological signals. A MatLab implementation shows the processing steps consecutively including step-interval unification, adaptive wavelet denoising, baseline correction, normalization, and peak detection and alignment for biomarker discovery.

This work focuses on generating approximations of complex T-spline surfaces with similar but less complex T-splines. Two approaches to simplifying T-splines are proposed: a bottom-up approach that iteratively refines an over-simple T-spline to approximate a complex one, and a top-down approach that evaluates existing control points for removal in producing an approximations. This thesis develops and compares the two simplification methods, determining the simplification tasks to which each is best suited. In addition, this thesis documents supporting developments made to T-spline research as simplification was developed.

The first part of my thesis is concerned with estimation for longitudinal data using generalized semi-parametric mixed models and multilevel generalized linear mixed models for a binary response. Likelihood based inferences are hindered by the lack of a closed form representation. Consequently, various integration approaches have been proposed. We propose a spherical radial integration based approach that takes advantage of the hierarchical structure of the data, which we call the 2 SR method. Compared to Pinheiro and Chao's multilevel Adaptive Gaussian quadrature, our proposed method has an improved time complexity with the number of functional evaluations scaling linearly in the number of subjects and in the dimension of random effects per level. Simulation studies show that our approach has similar to better accuracy compared to Gauss Hermite Quadrature (GHQ) and has better accuracy compared to PQL especially in the variance components. The second part of my thesis is concerned with identifying differentially expressed gene pathways/gene sets. We propose a logistic kernel machine to model the gene pathway effect with a binary response. Kernel machines were chosen since they account for gene interactions and clinical covariates. Furthermore, we established a connection between our logistic kernel machine with GLMMs allowing us to use ideas from the GLMM literature. For estimation and testing, we adopted Clarkson's spherical radial approach to perform the high dimensional integrations. For estimation, our performance in simulation studies is comparable to better than Bayesian approaches at a much lower computational cost. As for testing of the genetic pathway effect, our REML likelihood ratio test has increased power compared to a score test for simulated non-linear pathways. Additionally, our approach has three main advantages over previous methodologies: 1) our testing approach is self-contained rather than competitive, 2) our kernel machine approach can model complex pathway effects and gene-gene interactions, and 3) we test for the pathway effect adjusting for clinical covariates. Motivation for our work is the analysis of an Acute Lymphocytic Leukemia data set where we test for the genetic pathway effect and provide confidence intervals for the fixed effects.