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"As this monograph shows, the purpose of cardinal spline interpolation is to bridge the gap between the linear spline and the cardinal series. The author explains cardinal spline functions, the basic properties of B-splines, including B- splines with equidistant knots and cardinal splines represented in terms of B-splines, and exponential Euler splines, leading to the most important case and central problem of the book-- cardinal spline interpolation, with main results, proofs, and some applications. Other topics discussed include cardinal Hermite interpolation, semi-cardinal interpolation, finite spline interpolation problems, extremum and limit properties, equidistant spline interpolation applied to approximations of Fourier transforms, and the smoothing of histograms."

"This book serves well as an introduction into the more theoretical aspects of the use of spline models. It develops a theory and practice for the estimation of functions from noisy data on functionals. The simplest example is the estimation of a smooth curve, given noisy observations on a finite number of its values. The estimate is a polynomial smoothing spline. By placing this smoothing problem in the setting of reproducing kernel Hilbert spaces, a theory is developed which includes univariate smoothing splines, thin plate splines in d dimensions, splines on the sphere, additive splines, and interaction splines in a single framework. A straightforward generalization allows the theory to encompass the very important area of (Tikhonov) regularization methods for ill-posed inverse problems. Convergence properties, data based smoothing parameter selection, confidence intervals, and numerical methods are established which are appropriate to a wide variety of problems which fall within this framework. Methods for including side conditions and other prior information in solving ill-posed inverse problems are included. Data which involves samples of random variables with Gaussian, Poisson, binomial, and other distributions are treated in a unified optimization context. Experimental design questions, i.e., which functionals should be observed, are studied in a general context. Extensions to distributed parameter system identification problems are made by considering implicitly defined functionals."

"Michael A. Matt constructs two trivariate local Lagrange interpolation methods which yield optimal approximation order and Cr macro-elements based on the Alfeld and the Worsey-Farin split of a tetrahedral partition. The first interpolation method is based on cubic C1 splines over type-4 cube partitions, for which numerical tests are given. The second is the first trivariate Lagrange interpolation method using C2 splines. It is based on arbitrary tetrahedral partitions using splines of degree nine. The author constructs trivariate macro-elements based on the Alfeld split, where each tetrahedron is divided into four subtetrahedra, and the Worsey-Farin split, where each tetrahedron is divided into twelve subtetrahedra, of a tetrahedral partition. In order to obtain the macro-elements based on the Worsey-Farin split minimal determining sets for Cr macro-elements are constructed over the Clough-Tocher split of a triangle, which are more variable than those in the literature."

"ABSTRAKPenaksiran parameter dalam model regresi memiliki dua pendekatan yaitu pendekatan regresi parametrik dan pendekatan regresi nonparametrik. Dalam regresi parametrik bentuk dari kurva hubungan antara variabel respon dan variabel prediktor sudah ditentukan berdasarkan plot data, sedangkan dalam regresi nonparametrik bentuk dari kurva tidak diketahui. Salah satu regresi nonparametrik yang dapat digunakan adalah regresi spline. Regresi spline adalah suatu piecewise polynomial yang dihubungkan oleh titik-titik bersama yang disebut dengan knot. Regresi spline yang menggunakan fungsi basis B Spline disebut dengan regresi B Spline. Pada umumnya estimasi parameter regresi B Spline dilakukan dengan menggunakan metode OLS Ordinary Least Square. Namun, dengan metode OLS akan menyebabkan plot taksiran kurva regresi menjadi fluktuatif apabila pemilihan jumlah knot terlalu banyak. Untuk itu diperlukan suatu tambahan kendala berupa penalty yang didalamnya mengandung smoothing parameter sehingga diperoleh taksiran ideal. Metode estimasi parameter ini dikenal dengan metode PLS Penalized Least Square . Metode PLS dengan penalty yang merupakan integral kuadrat derivatif kedua dari taksiran kurva disebut juga dengan metode o rsquo;sullivan penalized spline. Pada penerapan contoh data, didapat 23 buah knot dan smoothing parameter sebesar 0.68.

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ABSTRACTParameter estimation of regression model has two approaches, that is parametric and nonparametric regression approach. In parametric regression, the shape of regression curve is determined based on scatterplot of dependent variable vs independent variable, whereas in the nonparametric regression, the shape of the curve is unknown. One of the nonparametric regression is spline regression. Spline regression is piecewise polynomials that connected by the knots. Spline regression using B Spline basis function is B Spline regression. In B spline regression, parameter estimation were fitted by OLS Ordinary Least Square method. However, the OLS method will lead the plot of estimated regression curve be fluctuative when using too much knots. Therefore, it needs additional constraint of penalty that contain smoothing parameter to obtain ideal fit result. This parameter estimation method known as PLS Penalized Least Square method. The estimate PLS method used penalty which is the integral of the square of second derivative of the estimate curve that called o 39 sullivan penalized spline method. In the application of sample data, 23 is used knots and the smoothing parameters is 0.68. "

"The finite element method (FEM) has become the most widely accepted general-purpose technique for numerical simulations in engineering and applied mathematics. Principal applications arise in continuum mechanics, fluid flow, thermodynamics, and field theory. In these areas, computational methods are essential and benefit strongly from the enormous advances in computer technology. B-splines play an important role in approximation and geometric modeling. They are used in data fitting, computer-aided design (CAD), automated manufacturing (CAM), and computer graphics. Finite Element Methods with B-Splines describes new weighted approximation techniques, combining the computational advantages of B-splines and standard finite elements. In particular, no grid generation is necessary, which eliminates a difficult and often time-consuming preprocessing step. The meshless methods are very efficient and yield highly accurate solutions with relatively few parameters. This is illustrated for typical boundary value problems in fluid flow, heat conduction, and elasticity.Topics discussed by the author include basic finite element theory, algorithms for B-splines, weighted bases, stability and error estimates, multigrid techniques, applications, and numerical examples. The text is essentially self-contained and easily accessible to graduate students in mathematics, engineering, and computer sciences. The new concepts are also of interest to academic researchers and to scientists in industry who are developing finite element software. Some basic facts on functional analysis and partial differential equations, which are required, are listed in the appendix."

"ABSTRACTPemodelan regresi telah diterapkan dalam perbankan ritel karena kemampuannya dalam menganalisis data kontinu maupun diskrit. Hal tersebut merupakan alat yang penting dalam penilaian risiko kredit, stress testing, serta evaluasi aset kredit. Pada tugas akhir ini, pendekatan yang digunakan adalah dengan menggunakan model regresi logistik multinomial untuk mengetahui faktor-faktor yang memengaruhi terjadinya default dan attrition pada suatu kredit. Selain itu, pada tugas akhir ini juga akan diperkenalkan pendekatan regresi spline dengan menggunakan truncated power basis untuk memodelkan fungsi hazard. Fleksibilitas dari fungsi spline memberikan kemampuan untuk memodelkan fungsi hazard yang berbentuk nonlinier dan tidak beraturan. Kemudian, dengan menggunakan regresi spline dan regresi logistik multinomial, akan diperoleh sebuah hasil dan interpretasi yang lebih baik. Terdapat beberapa kelebihan dari penggunaan kedua model tersebut. Pertama, dengan menggunakan fungsi regresi spline yang fleksibel, dapat dimodelkan fungsi hazard yang berbentuk nonlinier dan tidak beraturan. Kedua, mudah dipahami dan diterapkan, dan bentuk parametrik model regresi logistik multinomial yang sederhana dapat memudahkan dalam interpretasi model. Ketiga, memiliki kemampuan untuk prediksi. Pada akhir pembahasan, dengan menggunakan sebuah data kartu kredit akan dilakukan pengaplikasian dari model regresi logistik multinomial dan regresi spline, dilengkapi dengan penjelasan secara statistika dan akurasi prediksi.

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ABSTRACTRegression modeling has been adapted in retail banking because of its capability to analyze the continuous and discrete data. It is an important tool for credit risk scoring, stress testing and credit asset evaluation. In this thesis, the approach used is multinomial logistic regression model to gain the information regarding the factors that affect the occurrence of default and attrition. In addition, this thesis will also introduce spline regression approach using truncated power basis to model the hazard function. The flexibility of spline function allows us to model the nonlinear and irregular shapes of the hazard functions. Then, by using spline regression and multinomial logistic regression model, there will be a better result and interpretation. There are several advantages by using those both models. First, by using the flexible spline regression function, it can model nonlinear and irregular shapes of the hazard functions. Second, it is easy to understand and implement, and its simple parametric form from multinomial logistic regression model can make it easy in model interpretation. Third, the model has the ability to do prediction. Furthermore, by using a credit card dataset, we will demonstrate how to build these model, and we also provide statistical explanatory and prediction accuracy."