Spline Methodology

We assume that the practitioner has already calculated a set of nodes using a yield curve construction technique such as bootstrapping. A zero curve is then fitted using the cubic spline methodology by interpolating between nodes using individual cubic polynomials. Each polynomial has its own parameters but is constructed in such a way that its ends touch each node at the start and end. The set of splines, which touch at the nodes, therefore form a continuous curve. Our objective is to 

Requirement 1 the value of each polynomial is equal at tenor points 

Requirement 4 the second differential of each polynomial is continuous between tenor points 

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Fitting the Term Structure of Interest Rates The Practical Implementation of Cubic Spline Methodology ... 

We have presented an accessible account of how the cubic spline methodology of term structure estimation could be implemented by users involved in any area of the debt capital markets. The technique is straightforward and quick, and is valid for a number of applications, most of which are normal or conventional yield curves. For example, users are recommended to use it when curves are positively sloping, or when the long end of the curve is not downward sloping. The existence of humps along the short or medium terms of the curve can cause excessive oscillation in the forward curve, but the zero curve may still be used for valuation or relative value purposes. 

Oscillation is a natural effect of the cubic spline methodology, and its existence does not impair its effectiveness under many conditions. If observed rates produce very humped curves, the fitted zero-curve using cubic spline does not produce usable results. For policy-making purposes, for example, as used in central banks, and also for certain market valuation purposes, users require forward rates with minimal oscillation. In such cases, however, the Waggoner or Anderson—Sleath models will overcome this problem. We therefore recommend the cubic spline approach under most market conditions. 

A practical demonstration of cubic spline methodology, useful in constructing yield curves. 

It is the aim of this contribution to review the principles of the very flexible curve fitting approach by cubic spline functions for construction of concentration/response curves, to demonstrate the applicability of this methodology for several data sets relevant in pesticide residue analysis... 

Gu and Wahba (1993) used a smoothing-spline approach with some similarities to the method described in this chapter, albeit in a context where random error is present. They approximated main effects and some specified two-variable interaction effects by spline functions. Their example had only three explanatory variables, so screening was not an issue. Nonetheless, their approach parallels the methodology we describe in this chapter, with a decomposition of a function into effects due to small numbers of variables, visualization of the effects, and an analysis of variance (ANOVA) decomposition of the total function variability. 

The nondimensional B-Spline Impulse Response Function for the soil-tie system is independent of the actual soil conditions and the moving loads. This leads to an efficient implementation of the BEM solver of the proposed methodology. 

In 1968, Sing introduced the as-analysis comparison plot methodology for specific surface area determination [2] the method found a wide application to identify the presence of porosity and evaluate (micro)pore volumes in test adsorbents. No detailed uncertainty analyses exist for pore volumes in porous materials no internationally recognized standard porous materials exist. The comparison of the amount adsorbed by standard and test adsorbents leads to a complex interplay of the combined standard uncertainty (m ) in the amount adsorbed and in the clamped cubic spline functions employed to interpolate common relative pressures and their dependent amounts adsorbed. 

Equations (4) and (5) are not evaluated explicitly in the minimization program, but are fit using a combination of spline [17] methods, which provide stability, the ability to filter noise easily, and the flexibility to describe an arbitrarily shaped potential curve. Moreover, the final functional form is inexpensive to evaluate, making it amenable to global minimization. The initial step in our methodology is to fit the statistical pair data for each amino acid and for the density profile to Bezier splines [17]. In contrast to local representations such as cubic splines, the Bezier spline imposes global as well as local smoothness and hence effectively eliminates the random oscillatory behavior observed in our data. 

This section provides a discussion of piecewise cubic spline interpolation methodology and its application to the term stmcture. Our intent is to provide an accessible approach to cubic spline interpolation for... 

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