Knots, spline functions

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 ). 

A typical curve-fitting problem is illustrated in Figure 8. A spline function S(x) is required which interpolates a set of observed values (represented by f, f etc.) in an interval a x b. If we choose a set of knots kj, k, ..., in the interval, then Curry and Schoenberg (12) show that S(xf has a unique representation in a s x b of the form... 

Figure 7. Part of a typical set of cubic B-spline functions calculated using equi-spaced knots...

Figure 8. Example of a typical curvefitting problem in one variable illustrating the use of spline functions ((9) knots (- -) observed data)...

The only requirement for the knot sequence is that it must be a non-decreasing sequence of numbers. When ti = ti+i it indicates a multiple knot and the segment Qj is reduced to a point. This is one of the great advantages with non-uniform B-splines since it offers great flexibility in the representation ol functions. For example [0, 0, 1, 1, 1, 1, 2, 3, 4, 4] is a valid sequence of knots. The knot value 0 has multiplicity of 2, knot value 1 has multiplicity of 4 and so on. The multiplicity is used to control the continuity of a point. The higher the multiplicity, the less smooth the spline function at this point becomes. A curve segment Qj in cubic B-splines is defined by four control points... 

As with polynomial splines, the knots will be one count less than the number of sphnes. It is also important to keep the splines to a minimum. This author prefers to use a linear transformation of the original data and then, if required, use a knot to connect two spline functions. 

Problems, however, arise if the intervals between the knots are not narrow enough and the spline begins to oscillate (cf. Figure 3.13). Also, in comparison to polynomial filters, many more coefficients are to be estimated and stored, since in each interval, different coefficients apply. An additional disadvantage is valid for smoothing splines, where the parameter estimates are not expectation-true. The statistical properties of spline functions are, therefore, more difficult to describe than in the case of linear regression (cf. Section 6.1). ... 

The first set of n - 1 constraints require that the spline function join perfectly at the knot points. The second and third set of 2m - 2 constraints require that first and second derivative constraints match adjacent splines. Finally, the last two constraints are end-point constraints that set the derivative equal to zero at both ends. 

As an example, the above procedure is applied to Ar and Xe clusters. The DIM models for all clusters of a particular noble gas are fully defined only when the atomic and diatomic fragment matrices are specified. The former were fixed by taking I(Ar) to be 15.76 eV and I(Xe) to be 12.13 eV [14] the latter were defined by taking the points on the curves U,G,U and G for A from the ab initio computations of Bdhmer and Peyerimhoff [15]. The corresponding points for X were taken from Wadt [16]. The Ai2( I ) and Xe2( IJ) interactions were taken from Watts [17]. For each diatomic curve, the points were fitted to a cubic spline function in the asymptotic region, the interaction was represented by A/Rtt, where A and n were chosen to match the spline function at its largest knot. Some relevant input data is collected in Table 1. 

An important feature of the method of lines is selection of the basis functions i (co), which determines the precision of (spatial) curve fitting. The piecewise polynomials known as B splines meet the requirements. Curve fitting by means of spline functions entails division of the solution space into subintervals by means of a series of points called knots. Knots may be either single or multiple, a multiple knot being formed by the coincidence of two or more such points. They are numbered in nondecreasing order of location Si, S2,..., 5i,. A normalized B spline of order k takes nonzero values only over a range of k subintervals between knots, and, for example, Bij (co), the ith normalized B spline of order k for the knot sequence s, is zero outside the interval + nonnegative at = s, and w = Si + j, and strictly... 

A spline function of order k with knot sequence s is then defined as any linear combination of splines of order k for the particular knot sequence. 

We now return to Eqs. (6.1) and (6.2). If the right-hand sides of these equations are taken to be B spline functions of order k, then it can be shown (Margolis, 1978) that the spatial truncation error, that is, the amount by which the approximate solution fails to solve the partial differential equations (4.12) and (4.13) at time t, is 0((5 ). Next, if the number of continuity conditions to be satisfied at each interior (i = 2. .. /) breakpoint is v, then only the first m = kl — v l — 1) knots of the sequence form the origins of fresh polynomial pieces. It follows that the number of pp coefficients and hence the... 

Sphnes are functions that match given values at the points X, . . . , x t and have continuous derivatives up to some order at the knots, or the points X9,. . . , x vr-i-Cubic sphnes are most common see Ref. 38. The function is represented by a cubic polynomial within each interval Xj, X, +1) and has continuous first and second derivatives at the knots. Two more conditions can be specified arbitrarily. These are usually the second derivatives at the two end points, which are commonly taken as zero this gives the natural cubic splines. 

Here the B-spline Bim(zf, Xj) is the ith B-spline basis function on the extended partition Xj (which contains locations of the knots in the Zj direction), and is a coefficient. We use cubic splines and sufficient numbers of uniformly spaced knots so that the estimation problem is not affected by the partition. The estimation problem now involves determining the set of B-spline coefficients that minimizes Eq. (4.1.26), subject to the state equations [Eqs. (4.1.24 and 4.1.25)], for a suitable value of the regularization parameter. At this point, the minimization problem corresponds to a nonlinear programming problem. 

Spline Fitting in Two Variables. The methods described in the previous section may be extended to functions of two variables (9). The problem now is to find a surface S(x,y) which interpolates data values in a rectangular region a xsb, c ysd. By analogy with the previous section, we specify h "interior" knots in the x direction and l in the y direction. These knots, k, k2 ", kh and nl n2 ni divi< e... 

The spline surface S(x,y) consists of a set of bicubic polynomials, one in each panel, joined together with continuity up to the second derivative across the panel boundaries. Because each B-spline only extends over four adjacent knot intervals, the functions B.(x)C.(y) are each non-zero only over a rectangle composed of 16 adjacent panels in a 4 x 4 arrangement. The amount of calculation required to evaluate the coefficients y may be reduced by making use of this property. As before, least-squares methods may be used if the number of data exceeds (h+4)(jJ+4), which is usually the case. 

The quantitative method in Section 2.2 is used to determine the intrinsic magnetization intensity for each voxel. Cubic B-spline basis functions with a partition of 60 interior knots logarithmically spaced between 1 x 10 5 and 10 s are used to represent the relaxation distribution within each voxel. The optimal regularization parameter, A, of each voxel is found within the range between 1 x 10 5 and 5 x 10"18 s by using the UBPR9 criterion. 

A spline is a piecewise polynomial2 whose pieces meet with continuity as high as possible given the degree. The abscissa values at which consecutive pieces meet are called the knots. A B-spline is a spline expressed with respect to a particular basis, in which the basis functions are each non-zero over as small a number of consecutive pieces as possible, given the degree and the continuity, and the basis functions sum to unity. 

When we take the nth derivative of a degree n spline, we get a piecewise constant. If we try to take the (n+ l)th derivative we get zero within all of the pieces, and, in order to be able to recover the nth derivative by integration, there have to be Dirac delta-functions (infinite spikes of zero width) at all of the knots where the pieces meet. 

The refined curve has a refined basis, and needs a corresponding sequence of control points. In fact it is the process of finding these new control points which is actually called knot insertion or subdivision. If we ignore end-conditions, which is a sensible way to start, there are twice as many of them. We can determine them by looking at the way in which a coarse basis function can be expressed in terms of the refined ones. Let the coarse basis be and the finer one 5"(t) where the subscripts r indicate the position in abscissa space of the central maximum4 of the particular basis function, and the superscript n the degree of the functions. Consider first B-splines of degree zero. 

Another structure for expressing a nonlinear relationship between X and Y is splines [333] or smoothing functions [75]. Splines are piecewise polynomials joined at knots (denoted by Zj) with continuity constraints on the function and all its derivatives except the highest. Splines have good approximation power, high flexibility and smooth appearance as a result of continuity constraints. For example, if cubic splines are used for representing the inner relation ... 

A new type of covariate screening method is to use partially linear mixed effects models (Bonate, 2005). Briefly, the time component in a structural model is modeled using a penalized spline basis function with knots at usually equally spaced time intervals. Under this approach, the knots are treated as random effects and linear mixed effects models can be used to find the optimal smoothing parameter. Further, covariates can be introduced into the model to improve the goodness of fit. The LRT between a full and reduced model with and without the covariate of interest can be used to test for the inclusion of a covariate in a model. The advantage of this method is that the exact structural model (i.e., a 1-compartment or 2-compartment model with absorption) does not have to be determined and it is fast and efficient at covariate identification. 

Here the 7 , (r) are B-splines [41], functions that are polynomials (typically of fifth or sixth order in our applications) in certain regions, but which vanish for most values of r, which provide great flexibility in representing arbitrary functions. They are defined between knot points, which can be... 

---