Yield curve fitting comparing curves

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 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. 

There are instances where data are compared to models that predict linear relationships between ordinates and abscissae. Before the widespread availability of computer programs allowing nonlinear fitting techniques, linearizing data was a common practice because it yielded simple algebraic functions and calculations. However, as noted in discussions of Scatchard analysis (Chapter 4) and double reciprocal analysis (Chapter 5), such procedures produce compression of data points, abnormal emphasis on certain data points, and other unwanted aberrations of data. For these reasons, nonlinear curve fitting is... 

A similar approach was used with a mercury pool substrate. In this case the small tip can be pushed into the mercury pool to trap a thin layer of solution between tip and Hg (Fig. 12). This allows very small ri-values to be obtained and decreases the danger of breaking the tip with close approach to substrate (73). In this case the value of d can be obtained from the limiting r-value, compared to iT/ . This approach was used to determine the rate constant for the reduction of C60 in 1,2-dichlorobenzene (ODCB) and in benzonitrile (PhCN) solution (74). Typical voltammograms are shown in Figure 13. Fitting these yielded k°-values of 0.46 (ODCB) and 0.12 (PhCN) cm/s. Details about the procedure used for curve fitting by the voltammetric approach can be found in Refs. 11, 72, and 74. 

Making comparison between bonds could be difficult and several aspects must be considered. One of these is the bond s maturity. For instance, we know that the yield for a bond that matures in 10 years is not the same compared to the one that matures in 30 years. Therefore, it is important to have a reference yield curve and smooth that for comparison purposes. However, there are other features that affect the bond s comparison such as coupon size and structure, liquidity, embedded options and others. These other features increase the curve fitting and the bond s comparison analysis. In this case, the swap curve represents an objective tool to understand the richness and cheapness in bond market. According to O Kane and Sen (2005), the asset-swap spread is calculated as the difference between the bond s value on the par swap curve and the bond s market value, divided by the sensitivity of 1 bp over the par swap. 

In order to calculate the range of implied forward rates, we require the term stmcture of spot rates for all periods along the continuous discount function. This is not possible in practice, because a bond market will only contain a finite number of coupon-bearing bonds maturing on discrete dates. While the coupon yield curve can be observed, we are then required to fit the observed curve to a continuous term structure. Note that in the United Kingdom gilt market, for example there is a zero-coupon bond market, so that it is possible to observe spot rates directly, but for reasons of liquidity, analysts prefer to use a fitted yield curve (the theoretical curve) and compare this to the observed curve. 

We can compare fitted yield curves to an actual spot rate curve wherever there is an active government (risk-free) zero-couprai market in operation. In the United Kingdom, a zero-couprai bmid market was introduced in December... 

The kinetic parameters determined from the analysis of DMA data yields expected differences and re-emphasizes the differences in the properties monitored by the DMA as compared to DSC and FT-IR. A comparison of the kinetics parameters determined by DMA for the uncatalyzed and catalyzed resins reveals a drop in reaction order from 2.1 to 0.7 a decrease in activation energy from 28.7 kcal/mole to jj1.7 kcal/mole and a reducl ion of InA from a value of 33.0(sec ) to a value of 10.9(sec ). Calculated degree of cure curves fit reasonably well with experimentally determined... 

The comparison with the Rouse model thus requires an estimation of the maximum time. That should be, in pritKiple, the Rouse time M. In fact, the calculated curve is compared and, if the fit is possible, one extracts from that a maximum time. The data of set I have been compared as shown in Fig. 11. This yields ... 

This model tends to approach a zero probability rapidly at low doses (although it never reaches zero) and thus is compatible with the threshold hypothesis. Mantel and Bryan, in applying the model, recommend setting the slope parameter b equal to 1, since this appears to yield conservative results for most substances. Nevertheless, the slope of the fitted curve is extremely steep compared to other extrapolation methods, and it will generally yield lower risk estimates than any of the polynomial models as the dose approaches zero. 

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