B-Splines and Smoothing Splines

In this toolbox, the definition of a B-spline with knotst_j, ...,t_j+kis given by

$B_{j, k} (x) = B (x | t_{j}, ..., t_{j + k}) = (t_{j + k} - t_{j}) [t_{j}, ..., t_{j + k}] {(x - \cdot)}_{+}^{k - 1} .$

This is only one of several reasonable normalizations of the B-spline. It is chosen so that

$\sum_{j = 1}^{n} B_{j, k} (x) = 1, t_{k} \leq x \leq t_{n + 1} .$

但是，请查看GUI的参考页面，而不是试图理解上述公式bspliguifor some of the basic properties of the B-spline, and use that GUI to gain some firsthand experience with this intriguing function. Its most important property for the purposes of this toolbox is also the reason the letter B is in its name:

Every space of (univariate) piecewise-polynomials of a given order has aBasis consisting ofB-splines (hence the “B” in B-spline).

B-Spline Properties

BecauseB_J，kis nonzero only on the interval (t_j..t_j₊_k), the linear system for the B-spline coefficients of the spline to be determined, by interpolation or least squares approximation, or even as the approximate solution of some differential equation, isbanded, making the solving of that linear system particularly easy. For example, to construct a splinesof orderkwith knot sequencet₁≤t₂≤···≤t_n₊_kso thats(x_i)=y_ifori=1, ...,n，使用线性系统

$\begin{matrix} \sum_{j = 1}^{n} B_{j, k} (x_{i}) a_{j} = y_{i} & i = 1 : n \end{matrix}$

对于未知B-Spline系数a_j每个方程最多都有knonzero entries.

Also, many theoretical facts concerning splines are most easily stated and/or proved in terms of B-splines. For example, it is possible to match arbitrary data at sites $x_{1} < \dots < x_{n}$ uniquely by a spline of orderkwith knot sequence(t₁, ..., t_n+k)if and only ifB_J，k(x_j）≠0for allj(Schoenberg-Whitney Conditions). Computations with B-splines are facilitated by stablerecurrence relations

$B_{j, k} (x) = \frac{x - t_{j}}{t_{j + k - 1} - t_{j}} B_{j, k - 1} (x) + \frac{t_{j + k} - x}{t_{j + k} - t_{j + 1}} B_{j + 1, k - 1} (x)$

这也有帮助conversion from B-form to ppform. The双功能

$a_{j} (s) : = \sum_{i < k} {(- D)}^{k - i - 1} ψ_{j} (τ) D^{i} s (τ)$

provides a useful expression for thejth B-spline coefficient of the splinesin terms of its value and derivatives at an arbitrary site τ betweent_j和t_J+K。，与ψ_j(t):=(t_j+1–t）··（t_J+K。–1–t）/（k–1)!. It can be used to show thata_j(s) is closely related tos在间隔[t_j..t_J+K。], and seems the most efficient means for converting from ppform to B-form.

Variational Approach and Smoothing Splines

以上constructiveapproachis not the only avenue to splines. In thevariational方法，将样条作为一个最好的插值, e.g., as the function with smallestm在某些站点在所有匹配的规定函数值中的衍生物。事实证明，在许多可用的细条中，只有那些是分段 - 多项式的，或者也许是分段指数的人才发现了很多用途。特别有趣的是smoothing splines=s_p,对于给定数据(x_i，y_i）和x∊ [a..b], alli, and given corresponding positive weightsw_i，以及给定的smoothing parameterp,minimizes

$p \sum_{i} w_{i} {| y_{i} - f (x_{i}) |}^{2} + (1 - p) \int_{a}^{b} {| D^{m} f (t) |}^{2} d t$

over all functionsfwithmderivatives. It turns out that the smoothing splinesis a spline of order2mwith a break at every data site. The smoothing parameter,p, is chosen artfully to strike the right balance between wanting theerror measure

$E (s) = \sum_{i} w_{i} {| y_{i} - s (x_{i}) |}^{2}$

small and wanting theroughness measure

$F (D^{m} s) = \int_{a}^{b} {| D^{m} s (t) |}^{2} d t$

小的。希望是scontains as much of the information, and as little of the supposednoise, in the data as possible. One approach to this (used inspaps) is to makef（d^mf)尽可能小的条件E(f)be no bigger than a prescribed tolerance. For computational reasons,spapsuses the (equivalent) smoothing parameterρ= p/（1 – p）, i.e., minimizesρE(f） +F(D^mf). Also, it is useful at times to use the more flexible roughness measure

$F (D^{m} s) = \int_{a}^{b} λ (t) {| D^{m} s (t) |}^{2} d t$

with λ a suitable positive weight function.