Rational Fractal Interpolation With Variable Parameters And Its Applications In Stock Index Analysis

Financial market varies in random situation every day.Forecasting finan-cial time series has been regarded as one of the most challenging applications of modern financial market.The difficulties in forecasting derives from the in-herent non-linearity,high volatility,fractal and other features of the financial time series.Traditional linear statistical models are not sufficient for finan-cial time series prediction,such as the autoregressive moving average model(ARIMA).Machine learning methods involving artificial neural networks(AN-N)and support vector machines(SVM)require a large amount,of data w,hen forecasting financial time series.Since fractals are capable to provide a good deterministic representation for complex real world,fractal interpolation is a powerful tool for handling non-linear,non-stationary,and highly irregular da-ta.In this paper,on basis of the existent relevant researches,a new type of fractal interpolation functions are investigated,namely,rational fractal inter-polation functions with variable parameters,and investigate its applications in stock index analysis.The main contents are summarized as follows:In Chapter one,the research background and the preliminaries of this article are briefly introduced.In Chapter two,using a constructed iterative function system with func-tion vertical scaling factors,a type of rational fractal interpolation functions is first constructed with the help of the classical rational spline interpolation.And then,some analytical properties of rational fractal interpolation function-s are explored,including smoothness,stability and convergence.Further,the box-counting dimension of rational fractal interpolation curves is investigated,and the upper and lower bounds of the dimension are given.Finally,numerical examples aregiven to demonstrate the effectiveness of the proposed rational fractal interpolation model.In Chapter three,based on the presented rational fractal model,a forecast-ing algorithm of stock index time series is provided.First,a new method for determining the function scaling factors is proposed by utilizing feed-forward neural network(FNN).And then,using the epitaxial method and the support vector regression(SVR),a rational fractal interpolation forecasting algorithm is designed to analyze and predict the fluctuation of the stock indexChapter four gives empirical analysis and discussion on the forecast re-sults of stock index.Taking the Chinese SSE Composite Index data and Dow Jones index data in the US as examples,the prediction performance of the proposed algorithm is compared with traditional interpolation methods,clas-sical statistical models and advanced machine learning approaches.The results show that the proposed algorithm has the smallest forecasting error in terms of the three objective evaluation criteria MAE,RMSE and MAPE compared with other methods.Moreover,by comparing the actual values with predicted values,it implies that the developed model can predict more accurate trends.Thus,the rational fractal model with function scaling factors proposed in this paper is feasible and effective for the prediction of stock index time series,and more accurate forecast results can be obtained.