The Non-stationarity Measure Of Time Series And Its Application

The non-stationarity is an important feature of time-series, how to measure the non-stationarity level of a time series is a challenge problem. In this thesis, by integration views from ergodic the-ory, coarse-grained method and information theory, we provide a comprehensive study on the non-stationarity measure problem in time series and establish a general research framework of the non-stationarity measure. We present an index—NS, to measure the non-stationarity level of time series, and provide effective approximation algorithms to compute it. Furthermore, we apply NS in model selection, empirical analysis of stock market, lottery data analysis.The main contributions of the thesis are following:1. We introduce the concept—stable set (SS) of a time series, and provide effective criterion to check if a set is stable set or not under the finite samples based on the central limit theorem and the law of large numbers.2. We introduce the concept of stable information structure (SIS) of a time series and give four approximation algorithms to obtain the SIS under finite samples.3. We propose a non-stationarity measure index—NS of a time series and provide four effective approximation algorithms to compute it. The rationality and validity of NS are demonstrated via Monte Carlo simulations.4. We apply the NS to model selection, empirical analysis of stock market data and the lottery market data:· A new model selection criterion—minimize the non-stationarity measure of the residual sequence of a model, is proposed, which is validated by two simulated examples; NS can be used to distinguish trend stationary series from difference stationary series, with an accuracy rate≥98%; NS can be used to compare the non-stationrity level of different series.· Based on the empirical analysis of the returns of13stock indexes from1991to2010, we find that the13stock markets can be divided into three groups by NS,which is consistent to the relationship between the markets, and the large value of NS is often accompanied by higher fluctuations in stock market.· Three lotteries from China or USA are analyzed using ideas of stable set and non-stationarity measure, we find that each digit in "0,…,9" appeared with a stable probability but not with equal probability1/10, which can be used to test the fairness of a lotteries.We summarize the full text, and give the prospect of non-stationarity measure on its farther theory development as well as its possible applications.