时间序列分析理论与发展趋势

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摘要:时间序列分析提供的理论和方法是进行高难度综合课题的研究工具之一。近几年来已有很多的学者对时间序列的研究取得了丰硕的成果,有的在已有时间序列分析方法的基础上进行创新,研究出了新的预测方法。该文从基本理论和方法等方面对时间序列分析进行了综述,同时阐述了其研究动态和发展趋势。

关键词:时间序列分析;预测;模型

中图分类号:TP391文献标识码:A文章编号:1009-3044(2010)02-257-02

The Theory And Development Trend of Time Series Analysis

LIU Ying-hui, CAO Jia-lian

(Dalian Jiaotong University, Dalian 116028, China)

Abstract: The theory and method which were provided by the time series analysis is one of the tools to carry out lage-scal sophisticdted research projects .In later years some scholars have achieved a lot of significant results in the study of time series analysis,and some made innovationsbased on the original methods of time series analysis and obtained new forecasting methods. This paper is the summary of the basic theories, methods, and illuminate its dynamic research and trend of development about time series analysis.

Key words: time series analysis; forecasting; model

1 Introduction

Time series analysis provide a method which is used to process the dynamic data. Our task then, is to identify an appropriate subclass of mathematical models that may be employed to represent a given time series, from the models we can learn the inherent structure and complex character of the data, on the one hand, and reach the goal of forecasting the future state of the system and make necessary control, on the other.

2 Time series analysis

2.1 Time series analysis concept and background

A great deal of data in business, economics, engineering and the natural sciences occur in the form of time series where observations are dependent and where the nature of this dependence is of interest in itself. The body of techniques available for the analysis of such series of dependent observations is called time series analysis.[1] Statistical analysis of time series data started a long time ago and forecasting has an even longer history. In 1927 mathematician Yuel who introduced the AR(autoregressive) model which used to manipulate economic data which were taken from observations recorded over time and forecasting, and this is the original method of time series analysis. Then based on the AR model another mathematician established MA(moving average) model. In 1970, the publication of Time Series Analysis: Forecasting and Control by BOX and Jenkins was an important milestone for time series analysis. It provided moving systematic approach that enables practitioners to apply time series methods in forecasting. After that the method of time series analysis make a new step, it was widely applied in engineering domains. In recent years the theory and method of time series analysis are further improved with the development of computing technology and signal processing, on the one hand, algorithm parameter estimate, pattern recognition, and the method of defining orders, etc. Are combined with intelligent computing and achieved significant results, on the other.

2.2 Time series analysis theory progress

Theoretical progress of time series analysis is mainly manifested in two aspects: nonlinear model theory and unit root theory. The progress of nonlinear model theory is focused on both the problem of geometric traversal and nonlinear process stationary. Chen, Tsay(1991), Petruccelli and Woolford who drew significant conclusions for the simple TAR(1) model[2].

Unit root theory is developed faster in the time series analysis theories, later years. This theory is used to study the asymmetric of random walks statistics, more and more contemporaneous econometricians and statisticians devote to unit root theory. This theory provides formal test methods to define the difference order of ARIMA model, as well as, opens up new fields for some statistics tests. Unit root test was extended to pluralistic by Tsay and Tiao (1990) which is called cointegration test.

2.3 Time series model

Our objective will be to derive models possessing maximum simplicity and the minimum number of parameters consonant with representational adequacy. The obtaining of such models is important. Because: 1) They may tell us something about the nature of the system generating the time series, 2) They can be used for obtaining optimal forecasts of future values of the series, 3) They can be used in the derivation of optimal control policies. 4) When two or more related time series are under study, they can be extended to represent dynamic relationships between the variables and hence to estimate transfer function. The general models are: AR(autoregressive) model, MA(moving average) model, ARMA(autoregressive moving average)model, and ARIMA (autoregressive integrated moving average)model.

1) AR(p) model: This model may be written :zt =?准1zt-1+?准2zt-2…+?准pzt-p+at (1)

Where we now use the symbols ?准1?准2…?准p for the finite set of weight parameters and at for white noise .The process defined by(1) is called an autoregressive processive process of order p , or more succinctly ,an AR(p) process.

2) MA (q) model: The process: zt=at-θ1at-1-θ2at-2…-θqat-q (2)

Where we now use the symbols -θ1,-θ2,…-θq for the finite set of weight parameters. The process defined by(2) is called a moving average process of order q, which we sometimes abbreviate to MA(q). Using the linear combination of random perturbation and forecasting error from past period to express the current forecast.

3) ARMA (p, q) model: The ARMA model ,the forecasting methods is also called Box-Jenkins(BJ) model . If the time series ztis equal to its current and previous error and random items ,as well as its linear function of the preliminary value , ARMA(p, q) model is expressed as : zt =?准1zt-1+?准2zt-2…+?准pzt-p+at-θ1at-1-θ2at-2…-θqat-q (3)

This model is called the (p, q) -order autoregressive moving average model. Parameters ?准1?准2…?准p are the autoregressive parameters; θ1θ2…θq are the average parameters, is the estimated parameters of the model

4) ARIMA(p, d, q) model: The most general form of an ARIMA process is:

zt=?准1zt-1+?准2zt-2…+?准p+dzt-p-d + at-θ1at-1-θ2at-2…-θqat-q (4)

Where p, q and d indicate the autoregressive, moving average and difference orders of the process respectively. This model capable of representing time series which, although not necessarily stationary, are homogeneous and in statistical equilibrium. The relating of a model of this kind to data is usually best achieved by a three stage iterative procedure based on identification, estimation, and diagnostic checking. The first three are linear stationary models and the last one is the nonstationary model.

3 Development trend

Although the research of time series analysis has achieved much progress both in theory and method in these years which also been employed into the prediction and control in many fields, and the results are satisfactory. As we all know the model and the method of data processing are not perfect so the forecasting results are not very accurate. Therefore, in this area also have many issues worth to exploring and the work in the future will focus on the following aspects:

1) Multivariable time series

The multivariable time series (MTS)dataset is a common data type in various scientific domains. An MTS is usually very high dimensional with its main distinguishing characteristic being the inter-correlations among its variables and these variables can supply more effective information, thus obtaining better prediction results. So it is significant to do some research on the analysis and modeling of the multivariable time series.

2) Neural networks

Although a large number of forecasting techniques have been put forward in recent years, the information of time series data is incomplete and various factors, so the forecasting system with an ability of intelligent information process is necessary, the use of neural network may be a attempt in this field. Fuzzy logic and genetic algorithms will be incorporated to neural network in order to achieve more accurate prediction.

3) Date preprocessing

With the coming of the information age, we are confronted with increasing data and information in many fields. However, as we known, there are many issues in database, such as redundant data, missing data, uncertain data, inconsistent data, etc. they are the barriers to knowledge discovery and some times they will affect the accuracy of the prediction. Therefore, in order to improve the efficient of data mining and reduce the size of data processing, it is necessary to process the initial data before data mining. The method of how to process large scale data efficiently will play an important role in the future research.

4) Time interval

Not only study the common time series data, but the time interval of different observations may be a development trend. Therefore, the time when events occurred will play a key role in time series analysis and forecasting.

4 Conclusion

Time series analysis has become increasingly important in many fields, such as: economic, engineering and natural sciences, etc. This paper mainly summary the basic theories and methods of time series analysis, several generally models and the tendency in the future. In this area there are also many issues that worth to exploring and in the future the work will focus on data processing, multivariable time series analysis and so on. Therefore, in order to study the time series analysis in the deeper level, lots of work still need to be completed.

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