S-NURBS模型：混沌数据的高精度几何插值方法

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"Series-NonUniform Rational B-Spline (S-NURBS)模型：用于混沌数据的几何插值框架" 在混沌理论的研究中，时间序列分析是揭示复杂系统内在特性的关键工具。现有的混沌建模方法通常依赖于最小化误差策略或设定可接受的误差阈值来完成模型构建，这可能导致模型的精度不足。为了解决这个问题，一种名为Series-NonUniform有理B样条（S-NURBS）模型的新框架被提出，它专注于几何理论以提高建模精度，并能有效地处理任意维度的时间序列。 S-NURBS模型是一种高级的几何插值方法，它通过精巧地结合非均匀有理B样条（NURBS）的概念，能够在保持较高精度的同时，处理混沌数据的复杂性。NURBS是一种强大的数学工具，常用于计算机图形学和几何建模，因其灵活性和控制点的概念而受到青睐。在S-NURBS模型中，这些优点被扩展到处理混沌数据的插值问题，以减少建模误差。文章指出，与传统的误差最小化方法不同，S-NURBS模型利用几何插值理论，使得模型更接近原始数据，特别是在处理高维混沌序列时。通过调整权重序列，该模型可以随着更多信息的提供逐步优化，进一步降低内插误差。这种适应性调整的能力使得S-NURBS模型在处理混沌数据时展现出更高的准确性和鲁棒性。实验部分通过在Musa数据集上的应用验证了S-NURBS模型的有效性和可靠性。Musa数据集是一种实际混沌数据的实例，实验结果证明了该模型在混沌数据插值方面的优越性能。此外，这些实验还表明，从几何角度出发研究物理系统不仅是可能的，而且可能是提高建模精度的一个有力途径。该研究论文由Chenxi Shao、Qingqing Liu、Tingting Wang、Peifeng Yin和Binghong Wang共同撰写，发表在AIP Publishing LLC旗下的《Chaos》期刊上，卷23，期3，文章编号033132（2013）。文章链接为http://dx.doi.org/10.1063/1.4819479，感兴趣的研究者可以查阅更多细节。 S-NURBS模型提供了一个创新的几何插值框架，用于混沌数据的建模，通过几何方法减少了误差，提高了模型的精确度，特别是在高维混沌序列的处理中。这一方法的引入为混沌理论和复杂系统的研究开辟了新的可能性，为未来的数据分析和建模提供了有价值的工具。

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chemical reaction system by Kim et al.

but also used by

Molkov et al.

to determine embedding dimension from

noisy time series. In the same year, Smith

applied the

radial basis function to model chaotic time series by mini-

mizing the error of linear ﬁtting of all basis functions and

explaining nonlinearity by basis functions. This method was

used by Pilgram et al.

to model the dynamics of nonlinear

time series and also used by Suzuki et al.

to examine some

measures used for evaluating the validity of nonlinear mod-

eling. In 1994 Gouesbet and Letellier

designed a multivari-

able polynomial for global vector-ﬁeld reconst ruction aimed

at ﬁnding the relationship between the current state and its

derivatives from one single time series. In the same year,

Hiew et al.

built chaotic time series with fuzzy logic by

turning numerical variables into qualitative description. This

method is used to demonstrate strong nonlinearity via imitat-

ing the human’s nondeterministic conceptual judgment and

logical thinking. However, people seldom model in this way

due to the large error in numerical transformation. In 1995

Cao

present wavelet-based model to model chaotic time

series which is like a radial basis function (RBF) model. The

wavelet functions coming from the expansion and conver-

sion of wavelet prototype provide more nonlinear expression

and selection. Degaudenzi and Arizmendi

practiced model-

ing in this way to forecast an airborne pollen time series, and

Wei and Billings

investigated many structure selection

algorithms for wavelet models of chaotic systems. In 1998

Bagarinao et al.

introduced discrete-time polynomials into

chaotic time series, which could better show nonlinearity

compared with RBF model. It is permitted to select proper

basis functions according to the properties of the modeling

data and to model by exploiting the linear combination

among the basis functions. Aguirre et al.

had ever forecast

that way the time series of sunspot numbers. Then we intro-

duce rational model. Rational model was ﬁrst analytically

proposed by Gouesbet

in 1991, but it had not been prac-

ticed until 2000 when Correa et al.

modeled time series

derived from electronic oscillator.

Fuzzy logic method may suffer unpredictable error due

to its transformation between qualitative and quantitative

representation. All other methods above use either error min-

imization strategy or an acceptable error to end the modeling

process, making it hard to recover original data from these

models. This means the models lose some information of the

time series. Moreover, these methods have other potential

problems, such as the ability to explain nonlinearity, the

numerical stability, the selection of polynomials and parame-

ters, and the model runtime. All these shortcomings may

cause uncertainty for exhibiting the original system.

Interpolation is very useful for solving differential equa-

tions with a small modeling error. Therefore, based on the

idea of studying the physical phenomenon from geometric as-

pect, we make use of non-uniform rational B-spline (NURBS)

as the tool and develop a high-precision interpolation frame-

work to deal with arbitrary-dimensional time series by intro-

ducing the time parameter into the time-insensitive NURBS.

There are good reasons to choose NURBS. First a

NURBS curve is shown as a uniﬁed mathematical expression

and able to represent a continuous curve at any precision.

In theory, once we have enough information, we can

reconstruct the system’s real motion behavior trajectory,

described as normalized uniﬁed mathematical expression.

Furthermore, each data point in NURBS curve is a weighted

sum of nearby control points, and the basic formula of

NURBS is a division by two polynomials of control points

and weights, making it capable of explaining the nonlinearity

of time series. Finally, the calculation of a NURBS curve is

numerically stable and time saving (the running time is

O(NlogN)). All these advantages indicate that NURBS

should be the best carrier to interpolate chaotic time series.

In Sec. II, we introduce how to interpolate arbitrary-

dimensional data to reconstruct a ﬁnite trajectory by S-NURBS

model. Section III presents the validation of S-NURBS method

by ﬁve benchmark chaotic systems. After validation, the

S-NURBS framework is applied to Musa standard dataset

(Musa dataset is very classic in the area of software reliability)

in Sec. IV. Finally, the main conclusions about the S-NURBS

framework are drawn in Sec. V.

II. S-NURBS FRAMEWORK

How to add time parameter into geometrical tool is the

key factor to implement the idea. Shao and Xiao

did some

pioneering work for this issue. As an intensive study, the

method is optimized for better performance. We adopt more

methods to calculate the knot vector and develop a more

accurate algorithm to map time parameter. In this section,

we introduce the improved method to calculate the parame-

ters in NURBS method from chaotic data and to add time

parameter into NURBS expression. Generally, we conduct

sector mapping between occurred time and knot vector (knot

vector is a parameter in calculating NURBS, see below) to

get a continuous smooth curve which indicates a possible

trajectory of the system.

Given a short piece of time series X ¼fx

; x

; …; x

two procedures are used to build an S-NURBS model. First,

we construct a K-order NURBS expression to describe the

movement showing by the time series. Then we add the

occurred time into the NURBS expression. The smooth

curve with occurred time is the so-called S-NURBS model,

which passes every point in X.

A. Constructing a K-order NURBS expression

A complete K-order NURBS expression is shown in

Eq. (1)

cðuÞ¼

i¼0

i;k

ðuÞ

i¼0

i;k

ðuÞ

: (1)

In this expression, p

is the control point, x

is the

weight, N(u) is the B-spine basis function, and u is a continu-

ous value bounded by knot vector and widely deﬁned to

range in [0,1]. It reﬂects the changing process of the data

points on the curve. The knot vector is used as a general

frame to reﬂect the distribution of data points.

033132-2 Shao et al. Chaos 23, 033132 (2013)

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