Spline estimation method

In practice, the term structure of coupon bonds is not complete, so the coefficients in (3.33) cannot be identified. To address this problem, McCulloch (1971, 1975) prescribes a spline estimation method that assumes zero-coupon bond prices vary smoothly with term to maturity. This approach defines price as a discount function of maturity, P N), which is a given by (3.34). 

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. 

McCulloch (1971) proposes a more practical approach, using polynomial splines. This method produces a fimction that is both continuous and linear, so the ordinary least squares regression technique can be employed. A 1981 study by James Langetieg and Wilson Smoot, cited in Vasicek and Fong (1982), describes an extended McCulloch method that fits cubic splines to zero-coupon rates instead of the discount fimction and uses nonlinear methods of estimation. 

Summary. In this chapter, we are concerned with the problem of multivariate data interpolation. The main focus hes on the concept of minimizing a quadratic form which, in practice, emerges from a physical model, subject to the interpolation constraints. The approach is a natural extension of the one-dimensional polynomial spline interpolation. Besides giving a basic outline of the mathematical framework, we design a fast numerical scheme and analyze the performance quality. We finally show that optimal interpolation is closely related to standard hnear stochastic estimation methods. 

In a particle implementation of transported PDF methods (see Chapter 7), it will be necessary to estimate go(f) using, for example, smoothing splines, gi (f) will then be found by differentiating the splines. Note that this implies that estimates for the conditional moments (i.e., go) are found only in regions of composition space where the mixture fraction occurs with non-negligible probability. 

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... 

Differences in calibration graph results were found in amount and amount interval estimations in the use of three common data sets of the chemical pesticide fenvalerate by the individual methods of three researchers. Differences in the methods included constant variance treatments by weighting or transforming response values. Linear single and multiple curve functions and cubic spline functions were used to fit the data. Amount differences were found between three hand plotted methods and between the hand plotted and three different statistical regression line methods. Significant differences in the calculated amount interval estimates were found with the cubic spline function due to its limited scope of inference. Smaller differences were produced by the use of local versus global variance estimators and a simple Bonferroni adjustment. 

Wegscheider fitted a cubic spline function to the logarithmically transformed sample means of each level. This method obviates any lack of fit, and so it is not possible to calculate a confidence band about the fitted curve. Instead, the variance in response was estimated from the deviations of the calibration standards from their means at an Ot of 0.05. The intersection of this response interval with the fitted calibration line determined the estimated amount interval. 

First, amount error estimations in Wegscheider s work were the result of only the response uncertainty with no regression (confidence band) uncertainty about the spline. His spline function knots were found from the means of the individual values at each level. Hence the spline exactly followed the points and there was no lack of fit in this method. Confidence intervals around spline functions have not been calculated in the past but are currently being explored ( 5 ). 

Table V includes the same comparison as Table IV between Kurtz transformed method and Wegscheider s spline method for Dataset B and shows the actual amount units. It gives the reader a clear idea of the actual sizes of estimated amount ranges resulting from the uncertainty in response data.

In spite of its simplicity the direct integral method has relatively good statistical properties and it may be even superior to the traditional indirect approach in ill-conditioned estimation problems (ref. 18). Good performance, however, can be expected only if the sampling is sufficiently dense and the measurement errors are moderate, since otherwise spline interpolation may lead to severely biased estimates. 

A. Yermakova, S. Vajda and P. Valkd, Direct integral method via spline approximation for estimating rate constants. Applied Catalysis,... 

A method for interpolation of calculated vapor compositions obtained from U-T-x data is described. Barkers method and the Wilson equation, which requires a fit of raw T-x data, are used. This fit is achieved by dividing the T-x data into three groups by means of the miscibility gap. After the mean of the middle group has been determined, the other two groups are subjected to a modified cubic spline procedure. Input is the estimated errors in temperature and a smoothing parameter. The procedure is tested on two ethanol- and five 1-propanol-water systems saturated with salt and found to be satisfactory for six systems. A comparison of the use of raw and smoothed data revealed no significant difference in calculated vapor composition. 

In what follows, some comments will be made on the commonly used functional approaches to estimating C t)dt and t -CiOdt (i.e., the trapezoidal rule, or a combination of the trapezoidal and log-trapezoidal rule) (15, 16). Other methods such as splines and Lagrangians will not be discussed. The interested reader is referred to Yeh and Kwan (14) and Purves (15). 

Numerically, the LSA approach may be implemented by calcnlations that involve estimation by (general) linear regression, e.g., nse of cnbic spline functions." The intrinsic problems of nonlinear estimation common in more structured methods can thereby be avoided or significantly reduced. 

In this section, on the one hand, methods that are used to estimate intrinsically nonlinear parameters by means of nonlinear regression (NLR) analysis will be introduced. On the other hand, we will learn about methods that are based on nonpara-metric, nonlinear modeling. Among those are nonlinear partial least squares (NPLS), the method of alternating conditional expectations (ACE), and multivariate adaptive regression splines (MARS). 

In the Svensson model, there are six coefficients Pq, fii, 2. 3. 1 and T2 that must be estimated. The model was adopted by central monetary authorities such as the Swedish Riksbank and the Bank of England (who subsequently adopted a modified version of this model, which we describe shortly, following the publication of the Waggoner paper by the Federal Reserve Bank of England). In their 1999 paper, Anderson and Sleath evaluate the two parametric techniques we have described, in an effort to improve their flexibUity, based on the spline methods presented by Fisher et al. (1995) and Waggoner (1997). 

THE CUBIC SPLINE METHOD FOR ESTIMATING AND FITTING THE YIELD CURVE... 

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