Multiresolution Analysis And Forecasting Methods Using Wavelet Analysis

Multiresolution analysis (MRA) which is an important part of wavelet analysis is developing very fast in the whole field of mathematics. It is widely used both in mathematics and application of other fields, such as forecasting of finance, digital signal processing and so on. Mlutiresolution analysis was first established by the famous mathematician S.Mallat, and then a vigorous development started. During the 20 years when the MRA theory developed, the theoretic part of MRA obtained a gradual consummation, and these theories are all focusing on the basic concept of MRA, such as decomposition, restructuring, multiresolution and so on. MRA is also used in finance forecast which bring a new method to the field of forecasting. This article have three chapters, the first part outlines the multiresolution analysis, and give a concise definition form of it as below:4. there exists aφ(x)∈V₀ which makes that {φ(x—k):k∈Z}is a Standard orthogonal basis of V₀.A scaling relation and a wavelet relation are founded by MRA:Including double scaling relation:φ(x) =∑_k∈zp_kφ(2x- k),And the formula of wavelet function:ψ(x) =∑_k∈z(-1)^k(?)φ(2z—k).And introduce the decomposition and restructuring method which comes from MRA:Decomposition formula: Restructuring formula:The second chapter summary the development in theoretic mathematics of resolution analysis theory, including that the Mallat algorithm which is based on the decomposition and restructuring, the decomposition algorithm is:The restructuring algorithm is:and an important function of MRA-constructing wavelet functions, and then point out the three wavelet functions which was established before the MRA existed, finally, introduce the construction of two kinds of wavelets, Daubechies wavelet and spline wavelet.Multiwavelet theory based on dimension r-multiresolution analysis:From formula (2.34), there exists a MRA in dimension r based onφ(x), if there exists thatforms a Riesz basis of V_j.We also constructed double scale orthogonal multiwavelet, and point out that a-scale orthogonal multiwavelet is not easy to be constructed.In chapter 3, the forecasting method based on MRA is have an epoch-making meaning that it contacts the wavelet analysis and time sequence analysis closely. We should make a AR model and then make forecasting.With the changing condition, we can not complete all forecasting project only in on way. So we show another way of wavelet forecasting-wavelet packet analysis, and then give an example of China stock market.Wavelet packet:we call is a wavelet packet based on W₀=φ..process of forecasting the chaos system using wavelet packet:(1).use wavelet packet method to decompose the time sequence;(2).estimate whether the data in each frequency is chaos;(3).make the chaos model and then get the forecasting result;(4).restructure the result of forecasting with wavelet packet method.At last, let the wavelet function be the kernel function, and then contacts it with support vector machine method-the more and more popular method for forecasting and we show the result that wavelet support vector machine is better than the neural net forecasting model.In this article, we summarize the MRA theory and show some methods of wavelet forecasting. The idea of this summary is helping we students working on the field of forecasting to make a further research.