Multi-scale Algorithms For Time Series Analysis

As one of the branches of statistic, time series analysis focus on the variation characters and trend of discrete ordered data series mainly. For a long time, time series analysis have been applying in many fields successfully, such as economics, finance, management, chronometer, aerograph, oceanography, physical geography, biology, mechanics, electronic engineering etc.It displays the developing transformation about the researched object during a interval time and finds the character trend through analyzing the old measured data, so we can predict the object state in future in order to make decision. Therefore, modeling theory about time series plays the important role in data analysis field.In summary, we can offer three arguments as to why the time series analysis is of interest.1. Whether researching model depicts observed object correctly with testing parameter model and measured data.2. Realizing and master the phenomena rule through analyzing the measured data.3. We can utilize the rule to control and predict the future development.But traditional tools ignore the influence between observations that are widely separated in time, in other word, without considering the memory of the distant past. In addition, former researchers just only study one dimension sequence and be short of multi-dimension.In nature and engineering practices, many phenomena or processes have the multi-scale characters or multi-scale effects. While people often observe or measure phenomena and processes at different scales. So it is natural to describe and analysis these phenomena and processes according to multi-scale system theory which can show the essence of these phenomena and processes perfectly. Besides, as a time-frequency method, multi-scale analysis can solve many practical problems in a conceit way with the low computation complexity. Wavelet transform introduce the model information into multi-scale system theory and gained the new research as the bridge combing the model and signal at different scales.In this paper, we apply the multi-scale theory to time series analysis, especial long memory time series. The main contributions of this paper are as follows:1. We firstly use discrete wavelet transform to analyze stochastic processes of time series scale-by-scale, then study the variance and statistical properties of the series over a range of different scales. Subsequently, apply Least Squares Estimation to parameter estimation according to the property that log variance is approximately simple linear equation of log scale; finally a new method named multi-scale Least Squares Estimation is put forward.2. Utilizing the decomposition property of the wavelet, we apply it to multi-dimension sequences, so the correlation between variants will be weakened and the computation complexity will be decreased .Then we combine proper estimation rule to estimate parameters.