Integrals Of A Class Of Fractal Interpolation Functions And Applications Of Fractal Models

Since 1986,Barnsley proposed the concept of Fractal Interpolation Function(FIF)based on the theory of Iterated Function System(IFS),this theory and method has attracted much attention.Because the fractal interpolation function is usually continuous but non-differential,it is difficult to describe their analysis properties with the classic calculus.Owing to the relation between the fractional calculus and fractal dimension,the fractional calculus becomes a powerful tool to research fractal interpolation function.On the other hand,fractal interpolation has big advantages in analysis and forecast of non-smooth,irregular and rough curves,so the method has been extensively applied in the field of the traffic data analysis and the financial time series modeling.This dissertation proceeds in two aspects.Firstly,we study the integrals of a class of FIFs,and estimate the perturbation error of the FIFs cased by the perturbation of vertical scaling factors.Secondly the forecast models of fractal interpolation with wavelet are applied to analyze AQI.This dissertation consists of five chapters and its structure is as follows:The first chapter is the introduction,in which we introduce the backgrounds,significance,research status at home and abroad and the research content of this article.In addition,the innovations of this article are also pointed out.The second chapter introduces some of the basic concepts and theorems of fractal geometry and wavelet transformation,including the definitions of dimensions of fractal sets,fractional integrals and the methods of IFS,FIF and discrete wavelet transformation.In the third chapter,we discuss the integral characteristics of a class of fractal interpolation functions with function vertical scaling factors.It is proved that the integrals of the class of FIFs are also a class of FIFs in some situations.At the same time,the perturbation errors of both the FIFs and their fractional integrals cased by the perturbation of the vertical scaling factors are investigated.The results show that the class of FIFs and their fractional integrals are not sensitive to the slight perturbations of the parameters of FIFs.In the fourth chapter,the method of wavelet transformation is firstly applied to analyze the self-similarity of the AQI,then the vertical scaling factors and other parameters can be calculated by the Hurst exponent.Finally,the extrapolation methodof fractal interpolation and a new model of fractal interpolation in the statistical sense are applied to forecast AQI.The results show that the construction the new model of fractal interpolation has higher accuracy than that of extrapolation method by comparison of the average standard error.The fifth chapter is summery and outlook.