Yield curve fitting

Wold proposed Eq. (6-18), finding that it yielded curve fits as good as Eq. (6-16), with D = 0. 

The traditional approach to yield curve fitting involves the calculation of a set of discount factors from market interest rates. From this, a spot yield curve can be estimated. The market data can be money market interest rates, futures and swap rates and bond yields. In general, though this approach tends to produce ragged spot rates and a forward rate curve with pronounced jagged knot points, due to the scarcity of data along the maturity structure. A refinement of this technique is to use polynomial approximation to the yield curve. 

FIGURE 5.3 Yield curves fitted using cubic spline method and Svensson parametric method, hypothetical bond yields. Reproduced with permission from the Bank of England Quarterly Bulletin, November 1999.)... 

Yield curve fitting techniques that use splines are often fitted using multiple regression methods. 

Part One, Introduction to Bonds, covers bond mathematics, including pricing and yield analytics. This includes modified duration and convexity. Chapters also cover the concept of spot (zero-coupon) and forward rates, and the rates implied by market bond prices and yields yield-curve fitting techniques an account of spline fitting using regression techniques and an introductory discussion of term structure models. 

Curve fitting to data is most successhil when the form of the equation used is based on a known theoretical relationship between the variables associated with the data points, eg, use of the Clausius-Clapeyron equation for vapor pressure. In the absence of known theoretical relationships, polynomials are one of the most usehil forms to describe a curve. Polynomials are easy to evaluate the coefficients are linear and the degree, ie, the highest power appearing in the equation, is a convenient measure of smoothness. Lower orders yield smoother fits. 

The method is quite effective, but is not widely used now because of the ubiquity of digital computers. Zuman and Patel - 36. show circuit designs for some kinetic schemes. Williams and Bruice made good use of the analog computer in their study of the reduction of pyruvate by 1,5-dihydroflavin. In this simulation eight rate constants were evaluated variations in these parameters of 5% yielded discemibly poorer curve fits. 

It must be appreciated that the selection of the best model—that is, the best equation having the form of Eq. (6-97)—may be a difficult problem, because the number of parameters is a priori unknown, and different models may yield comparable curve fits. A combination of statistical testing and chemical knowledge must be used, and it may be that the proton inventory technique is most valuable as an independent source capable of strengthening a mechanistic argument built on other grounds. 

The complexity of the integrated form of the second-order rate equation makes it difficult to apply in many practical applications. Nevertheless, one can combine this equation with modem computer-based curve-fitting programs to yield good estimates of reaction rate constants. Under some laboratory conditions, the form of Equation (A1.25) can be simplified in useful ways (Gutfreund, 1995). For example, this equation can be simplified considerably if the concentration of one of the reactants is held constant, as we will see below. 

Attempts to understand hardness from first principles have resulted in empirical equations that represent good curve fitting, but yield relatively little understanding (Gao, 2006). 

The data collected are subjected to Fourier transformation yielding a peak at the frequency of each sine wave component in the EXAFS. The sine wave frequencies are proportional to the absorber-scatterer (a-s) distance /7IS. Each peak in the display represents a particular shell of atoms. To answer the question of how many of what kind of atom, one must do curve fitting. This requires a reliance on chemical intuition, experience, and adherence to reasonable chemical bond distances expected for the molecule under study. In practice, two methods are used to determine what the back-scattered EXAFS data for a given system should look like. The first, an empirical method, compares the unknown system to known models the second, a theoretical method, calculates the expected behavior of the a-s pair. The empirical method depends on having information on a suitable model, whereas the theoretical method is dependent on having good wave function descriptions of both absorber and scatterer. 

Plots of y/yo — (1 — x) versus x are drawn in Figure 10.B.2 for b/a = 2 and various values of KSC. The sharpness of the maximum, and therefore the accuracy with which xmax is located, depends on the magnitude of this parameter. Curve fitting with Eq. (B.41) yields Ks. 

Under many circumstances, the behavior of a simple unireactant enzyme system cannot be described by the Michaelis-Menten equation, although a v versus [S] plot is still hyperbolic and can be described by a modified version of the equation. For example, as will be discussed later, when enzyme activity is measured in the presence of a competitive inhibitor, hyperbolic curve fitting with the Michaelis-Menten equation yields a perfectly acceptable hyperbola, but with a value for Km which is apparently different from that in the control curve O Figure 4-7). Of course, neither the affinity of the substrate for the active site nor the turnover number for that substrate is actually altered by the presence of a competitive... 

We will not concern ourselves here with problems associated with line broadening, overlapping peaks, and background subtraction. There are, however, examples discussed later where both deconvolution and curve fitting procedures are shown to be essential in unraveling the contributions of differently bonded species of the same molecule to the total photoelectron yield. Carley and Joyner (14) have discussed recently deconvolution procedures for photoelectron spectra. 

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