Estimation Methods And Practical Applications Of The Fractal Dimensions Of Gaussian Processes

As a mathematical model, fractals are widely applied to in many specific areas nowadays,such as finance, geography, sociology and so on. The largest breakthrough to the traditionalgeometry by the proposal of fractal theory is that extends the dimension of a geometric objectfrom positive integers to positive real numbers. Hence, the calculation or estimation of ageometric object’s fractal dimension is an essential topic of fractal theory. For the theory itself,though it has already developed a relatively mature theoretical system, but because of its latebirth, there are still a lot of discussions need to be thoroughly done. So is for the study of fractaldimension.Based on recent researches, we will first introduce three different estimation methods offractal dimension and talk about their properties or extensions. These methods are called asbox-counting dimension estimation, level crossing estimation and variational dimensionestimation. After that, some accuracy comparison to these three methods will be held by thehelp of simulation. During the generation of sample paths, we proposed a new method ofmulti-dimensional simulation which is based on Cholesky decomposition.As a result, under the criterion of bias, box-counting dimension estimation always has theworst performance while the other two can work well under different situations. Also, under thecriterion of mean square error, box-counting dimension estimation always performs worse thanvariational dimension estimation while the level crossing estimation gets an unstableperformance. In summary, variational dimension estimation is the best choice for our stationaryGaussian processes.Besides that, since fractals are common in real financial markets, we will try using fractaldimensions to analyze the current status of Chinese stock market and predict the future trend.Of course, this heuristic method can still be largely improved in many detailed aspects.In general, the results of this paper have some theoretical significance and value ofapplication.