Estimation of noisy cubic spline using a natural basis

We define a new basis of cubic splines such that the coordinates of a natural cubic spline are sparse. We use it to analyse and to extend the classical Schoenberg and Reinsch result and to estimate a noisy cubic spline. We also discuss the choice of the smoothing parameter. All our results are illustrated graphically.Comment: 29 pages, 6 figure

Interpolation algorithms and image data artifacts

Interpolation, or resampling coefficients, which are generated from low pass filter Fourier transforms yield more accurate resampled values than those obtained using cubic spline techniques. This is due to the utilization of six data points rather than four as currently used in cubic spline analysis. After resampling functions are applied to image data, artifacts which are similar to ringing may become pronounced. These effects are often present in the original data and the interpolation merely enhances them

Comparison and Evaluation of Didactic Methods in Numerical Analysis for the Teaching of Cubic Spline Interpolation

In mathematical education it is crucial to have a good teaching plan and to execute it correctly. In particular, this is true in the field of numerical analysis. Every teacher has a different style of teaching. This thesis studies how the basic material of a particular topic in numerical analysis was developed in four different textbooks. We compare and evaluate this process in order to achieve a good teaching strategy. The topic we chose for this research is cubic spline interpolation. Although this topic is a basic one in numerical analysis it may be complicated for students to understand. The aim of the thesis is to analyze the effectiveness of different approaches of teaching cubic spline interpolation and then use this insight to write our own chapter. We intend to channel every-day thinking into a more technical/practical presentation of a topic in numerical analysis. The didactic methodology that we use here can be extended to cover other topics in numerical analysis.Methods of teaching mathematics are different for several reasons, for example, the presentation style of teacher of a particular topic. In several books we can observe a different approach of presentation material of a topic, and at the end we can produce a unique way of teaching but in a different way. In our thesis we study different approaches to teaching in a several numerical analysis books in the topic of cubic spline interpolation. What is cubic spline interpolation? Cubic spline interpolation is a type of interpolation of data points. Interpolation is a method of constructing a curve between some data points. We chose cubic spline interpolation because it is better than other kinds of interpolation. Cubic spline interpolation has a smaller curvature compared with other types of interpolation. Therefore, cubic spline interpolation produces a smooth curve. In this research we study different approaches of teaching cubic spline interpolation to find a good way for presenting the cubic spline interpolation topic, because this topic may be complicated for students to understand. To reach a good process of presentation of cubic spline interpolation we compare each part of different approaches in the books we have studied for teaching cubic spline interpolation by asking questions and then answering those questions. In this way we will show how we can evaluate each answer. Evaluating each answer we will obtain a good result which will prepare us for writing our own chapter in order to present cubic spline interpolation in our way

A rational cubic spline with tension

A rational cubic spline curve is described which has tension control parameters for manipulating the shape of the curve. The spline is presented in both interpolatory and rational B-spline forms, and the behaviour of the resulting representations is analysed with respect to variation of the control parameters

Cubic spline population density functions and subcentre delimitation. The case of Barcelona

The presence of subcentres cannot be captured by an exponential function. Cubic spline functions seem more appropriate to depict the polycentricity pattern of modern urban systems. Using data from Barcelona Metropolitan Region, two possible population subcentre delimitation procedures are discussed. One, taking an estimated derivative equal to zero, the other, a density gradient equal to zero. It is argued that, in using a cubic spline function, a delimitation strategy based on derivatives is more appropriate than one based on gradients because the estimated density can be negative in sections with very low densities and few observations, leading to sudden changes in estimated gradients. It is also argued that using as a criteria for subcentre delimitation a second derivative with value zero allow us to capture a more restricted subcentre area than using as a criteria a first derivative zero. This methodology can also be used for intermediate ring delimitation.polycentrism, cubic spline functions

Rational-spline approximation with automatic tension adjustment

An algorithm for weighted least-squares approximation with rational splines is presented. A rational spline is a cubic function containing a distinct tension parameter for each interval defined by two consecutive knots. For zero tension, the rational spline is identical to a cubic spline; for very large tension, the rational spline is a linear function. The approximation algorithm incorporates an algorithm which automatically adjusts the tension on each interval to fulfill a user-specified criterion. Finally, an example is presented comparing results of the rational spline with those of the cubic spline

A comparison of some numerical methods for the advection-diffusion equation

This paper describes a comparison of some numerical methods for solving the advection-diffusion (AD) equation which may be used to describe transport of a pollutant. The one-dimensional advection-diffusion equation is solved by using cubic splines (the natural cubic spline and a ”special” AD cubic spline) to estimate first and second derivatives, and also by solving the same problem using two standard finite difference schemes (the FTCS and Crank-Nicolson methods). Two examples are used for comparison; the numerical results are compared with analytical solutions. It is found that, for the examples studied, the finite difference methods give better point-wise solutions than the spline methods

Cubic spline interpolation of harmonic functions

It is shown that for the two dimensional Laplace equation a univariate cubic spline approximation in either space direction together with a difference approximation in the other leads to the well-known nine-point finite-difference formula. For harmonic problems defined in rectangular regions this property provides a means of determining with ease accurate approximations at any point in the region

A Spline LR Test for Goodness-of-Fit

Goodness-of-Fit tests, nuisance parameters, cubic spline, Neyman smooth test, Lagrange Multiplier test, stable distributions, student t distributions