Spline-Based Methods

Beyond the cubic polynomial, there are two main approaches to fitting the term structure parametric and non-parametric curves. Parametric curves are based on term-structure models such as those discussed in chapter 4. As such, they need not be discussed here. Non-parametric curves, which are constructed employing spline-based methods, are not derived from any interest rate models. Instead, they are general approaches, described using sets of parameters. They are fitted using econometric principles rather than stochastic calculus, and are suitable for most purposes. 

A spline is a type of linear interpolation. It takes several forms. The spline function fitted using regression is the most straightforward and easiest to understand. Unfortunately, as illustrated in James and Webber (2000), section 15.3, when applied to yield-curve construction, this method can be overly sensitive to changes in parameters, causing curves to jump wildly. 

An wth-order spline is a piecewise polynomial approximation usir w-degree polynomials that are differentiable n- times, i.e., they have n- derivatives. Piecewise signifies that the different polynomials are connected 

The x-axis in the regression is divided into segments at the knot points, at each of which the slopes of adjoining curves on either side of the point must match, as must the curvatures. FIGURE 5.4 shows a cubic spline with knot points at 0, 2, 5, 10, and 25 years, at each of which the curve is a cubic polynomial. This function permits a high and low to be accommodated in each space bounded by the knot points. The values of the curve can be adjoined at the knot point in a smooth function. 

Cubic spline interpolation assumes that there is a cubic polynomial that can estimate the yield curve at each maturity gap. A spline can be thought of as a number of separate polynomials of the form y = f(X), where X is the complete range of the maturity term divided into user-specified segments that are joined smoothly at the knot points. Given a set 

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Vedam, H., Venkatasubramanian, V., and Bhalodia, M A B-spline base method for data compression, process monitoring and diagnosis. Comput. Chem. Eng. 22(13), S827-S830 (1998). 

The results of these calculations for monthly averaged intervals are shown in Fig. 8. The values obtained differed slightly from the estimates that were reported by other authors who used visual or linear regression criteria for estimating of the onset of hydrogen sulfide [17,19,39,40]. The Akima spline-based method we used [84] should be better because it is nonlinear and based on an objective approach for every station, which is necessary in analysis of the large data arrays. 

The Bank of England uses a variation of the Svensson yield curve model, a one-dimensional paranetric yield curve model. This is similar to the Nelson and Siegel model and defines the forward rate curve/(/n) as a function of a set of unknown parameters, which are related to the short-term interest rate and the slope of the yield curve. The model is summarised in Appendix B. Anderson and Sleath (1999) assess parametric models, including the Svensson model, against spline-based methods such as those described by Waggoner (1997), and we summarise their results later in this chapter. 

Anderson and Sleath presented a model in the Bank of England Quarterly Bulletin in November 1999. The main objective of this work was to evaluate the relative efficacy of parametric versus spline-based methods. In fact, different applications call for different methods the main advantage of spline methods is that individual functions in between knot points may move in fairly independent fashion, which makes the resulting curve more flexible than that possible using parametric techniques. In Section 5.5.1 we reproduce their results with permission, which shows that a shock introduced at one end of the curve produces xmsatisfactory results in the parametric curve. 

This reflects the properties of the spline curve, including the fact that forward rates are described by a series of segments that are in effect connected together. This has the effect of localising the influence of individual yield movements to only the relevant part of the yield curve it also allows the curve to match more closely the observed yield curve. The goodness of the spline-based method is measured using Equation (5.23) ... 

Linear interpolation is simple but not accurate enough to be recommended. Market analysts use multiple regression or spline-based methods instead. One technique is to assume that the discount factors represent a functional form—that is, a higher-order function that takes... 

Models of the form y =f(x) or v =/(x1, x2,..., xm) can be linear or nonlinear they can be formulated as a relatively simple equation or can be implemented as a less evident algorithmic structure, for instance in artificial neural networks (ANN), tree-based methods (CART), local estimations of y by radial basis functions (RBF), k-NN like methods, or splines. This book focuses on linear models of the form... 

T.K. Fang, T.N. Chang, B-spline-based multichannel K-matrix method for atomic photo ionization, Phys. Rev. A 61 (2000) 062704. 

Kim, K Lee, K.S., and Lee, J.H. (2010a) Bilevel optimizing control structure for a simulated moving bed process based on a reduced-order model using the cubic spline collocation method. Ind. Eng. Chem. Res., 49, 3689-3699. 

In meshless methods, the choice of the interpolation kernel is the core of the method. Various types of kernel functions are used in the literature the Gaussian kernel and spline-based kernels such as the cubic-spline, quartic, or quintic kernels are among the most frequently used kernels. 

The acidic groups have different pK values depending on their location on the carbon surface relative to the location of nonacidic groups that can exert an inductive effect on them [46], Then, by a potentiometric titration method it is assumed that the system under study consists of acidic sites characterized by their acidity constants K. The site population can thus be described by a continuous pK, distribution function f(pKJ [54-57]. The experimental titration data are thus transformed into a proton-binding curve from which the distribution of acidity constants is obtained by using, for example, the splines-based numerical procedure SAIEUS suggested by Jagiello [58]. 

Oberlin D, Scheraga HA (1998) B-spline method for energy minimization in grid-based molecular mechanics calculations. J Comp Chem 19 71-85... 

Ying, Xugeng Ye Ruqiang Hu ying Phase Equilibria for Complex Mixtures. Continuous-thermodynamics Method Based on Spline Fit. Fluid Phase Equilibria, 53, 407-414(1989). 

Ying, X., Ye, R., and Hu, Y., Phase equilibria for complex mixtures. Continuous thermodynamics method based on spline fit. Fluid Phase Eq. 53,407 (1989). 

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