分形拟合方法研究及其实证分析--分形插值函数及其积分拟合方法.pdf

摘 要 本文主要研究股票价格的变化规律及对其拟合的方法。 在介绍了分形理论，函数的分数阶积分以及一般分形插值法的要点之后， 进而重点介绍在传统的分形插值法的基础上做以改进： 首先，我们在理论上证明了一个线性分形插值函数的分数阶积分在一定情 况下仍然是线性分形插值函数，并且分数阶积分的阶数与两个分形插值函数的 盒维数之间存在线性关系，这也是本文的一个核心理论； 其次，我们从实证上对新的方法进行了分析和验证。选取中国上证综合指 数的股票价格作为原始数据，显然可将其盒维数看作1，通过一般的分形插值法， 我们可以得到其拟合图像，即分形插值函数的图像，继而求出分形插值函数的 盒维数（大于1）。对分形插值函数做积分后，即得到了最终的拟合结果，这里 积分的阶数为分形插值函数的盒维数减去1。 由理论证明和实证检验，得知对于大量数据的拟合,这种方法是合理的。 关键词：迭代函数系；分形插值函数；盒维数；分形插值函数的分数阶积 分 1    Abstract This paper studies the law of stock market price and the fitting method of the price. After introducing the main point of the fractal theories, fractional integral of functions and the general fractal interpolation method, the paper focus on explaining the improvement of the traditional fractal interpolation method: Firstly, we prove theoretically that the fractional integral of a linear FIF is also a linear FIF and in some cases, there exists a linear relationship between the order of fractional integral and box dimension of two linear FIFs. This is a key point of this paper; Secondly, we analyze and check this new method by practice. We choose the data of Shanghai Stock Index from Chinese Stock Market. Obviously, the box dimension of the data is 1. We can obtain the fitting graph, the graph of fractal interpolation function, of it by the general fractal interpolation method. Then the dimension of the FIF can be calculated (which is 1). After that, we can get the fractional integral of FIF with the order which equals to the dimension of FIF minus 1. And this is the final fitting result. This new method is proved to be effe