具有函数尺度因子的有理分形曲线

带有常数尺度因子的分形插值,是描述具有明显自相似事物的一个有效工具,然而,它却难以精确地刻画自相似较弱的不规则数据.为此,提出一种具有函数尺度因子的有理样条分形插值方法.首先,在迭代函数系统中引入函数尺度因子,构造了一种仅仅基于函数值的带有形状参数的有理分形插值曲线;然后讨论了分形曲线的分析性质,包括分形曲线在尺度因子满足适当条件下的光滑性、分形曲线对插值数据扰动的稳定性以及分形插值函数的收敛性;最后,研究了分形曲线的计盒维数,给出了计盒维数的上下界.数值算例验证了该分形曲线造型的可控性和对噪声的鲁棒性;对海岸线数据插值时,该方法相比B样条、Bézier曲线和三次样条能更好地还原海岸线的粗糙程度;处理股票时序数据时,相比ARIMA和SVM方法,在RMSE等多项指标下更优.

Rational Fractal Curves with Function Scaling Factors

Fractal interpolation with constant vertical scaling factors is an effective tool for describing things with obvious self-similarity,and yet,which is difficult to accurately characterize irregular data with weak self-similarity.Therefore,an interpolation method of rational spline fractal with function scaling factors is proposed.Firstly,a type of rational fractal interpolation curves with shape parameters is constructed by introducing function scaling factors into the iterated function system.And then,analytical properties of fractal curves are discussed,including smoothness of fractal curves under the appropriate condition of scaling factors,stability of fractal curves to perturbation of interpolation data,and convergence of fractal interpolation functions.Finally,the box-counting dimension of fractal curves is studied,and the upper and lower bounds of box-counting dimension are given.The numerical examples verify the curve controllability and robustness against noise;for interpolating the coastline data,this algorithm performs better in restoring the coastline coarse appearance than B Spline,Bézier curve and cubic spline method;for processing the time series data in the stock market,this algorithm performs better than ARIMA and SVM method under multiple indexes such as RMSE.