http://swrc.ontoware.org/ontology#Thesis
Nonlinear time series analysis by the generalized exponentiol autoregressive model and its applications
en
施 招雲
セ ショウウン
SHI Zhaoyun
総研大乙第61号
Due to the complexity and nonlinear variety of the real world, nonlinear time series analysis has become one topics of the most popular interest. Accordingly, there are several nonlinear modeling approaches provided to tackle the complexity of nonlinear time series, the approaches can be generally divided into three categories: parametric, semi-parametric, and non-parametric model. Actually nonlinear statistical time series analysis started its research on parametric modeling about two decades ago for the motivation to overcome the limitations of the linear models in revealing nonlinear phenomena such as limit cycle. Two typical parametric nonlinear time series models are the exponential autoregressive (ExpAR) model and the threshold autoregressive model, they have been well-known in describing several typical nonlinear behaviors. However, it has been argued recently that the parametric models are hard to justify a priori the universal description appropriateness in most real applications to complex dynamic systems analysis. Moreover, the parametric modeling approach as the traditional statistical nonlinear time series analysis methodology has been faced with the serious challenges from neural network techniques and deterministic dynamic theory. In this situation, the non-parametric identification such as non- parametric density estimation of nonlinear time series became popular being an alternative paradigm of modeling in recent years. The non-parametric models allow for great flexibility without confining oneself to a special parametric model, however, this approach still has some drawbacks such as the curse of dimensionality, which hinder its wide application. Therefore the semi-parametric modeling is introduced in order to evade the limitations of non-parametric approach, although some of the semi-parametric models lose their generality simultaneously so that they seem limited to be applicable for some particular cases, and the boundary between parametric and some semi-parametric models become not sharp.<br /> Since the parametric approach can be easily interpreted, and in turn the dynamic mechanism of the underlying systems can be looked for, this study focuses on nonlinear time series analysis by one of the parametric model, the exponential autoregressive model and its general version, with comparison to other approaches. The thesis gives a deep investigation into the properties of the model, and shows the performance of the ExpAR modeling in identification and prediction as two important issues in the statistical analysis of time series. The details of the thesis are summarized as follows.<br /> First, the classic exponential autoregressive model is introduced to nonlinear time series analysis. The ExpAR model looks very simple in structure, it is basically the AR model but the state-dependent coefficients. Then the dynamics of the ExpAR model is investigated. By calculating the Lyapunov exponent and the bifurcation of the ExpAR model, it is found that even very simple ExpAR may behave very complicated dynamics such as fixed point, limit cycle and chaos. Several typical chaotic phenomena like period-double bifurcation, intermittency and synchronization occur in the simple ExpAR model too. So we can let the ExpAR model reveal rich different nonlinear behaviors just by controlling the parameters and the delay dimension. This can be a solid evidence to accept the ExpAR model to identify some complex data since the model is originally provided to model time series instead of as an analytical model like Logistic equation and others. However, from the standout of practitioner, the model estimate, as a nonlinear optimization problem, becomes obviously important since minor difference in the values of parameters may result in large different dynamics. Actually the ExpAR model estimate has not been yet well solved. In this study, the genetic algorithm is applied to solve this problem, and several procedures are provided. The procedures take mutual estimation, in which the first one is to use the genetic algorithm to determine the scaling parameter in the model, and then other linear weights are estimated by linear least squares. In another procedure, the genetic algorithm is used to optimize both the model structure and the nonlinear scaling parameter. We therefore name the procedures as the self-organization modeling. As a by-product of the self-organizing procedure, we can get a subset ExpAR model that is of importance to dimensionality reduction. The efficiency of the estimates is shown by the numerical examples. <br /> Secondly, the investigation of applying the ExpAR model to machine tool chatter monitoring system design is carried out. According to the fact that machine tool chatter is a nonlinear self-excited oscillation of limit cycle type, we provide the limit cycle behavior to be the intrinsic index of chatter occurrence, and the ExpAR model is proposed to detect the index from on-line cutting signal. Consider the actual implementation of the proposal, a real-time estimation of the ExpAR model is provided to satisfy the needs of on-line monitoring system design. Actual cutting signal analysis shows the effectiveness of the proposal.<br /> Thirdly, the generalized exponential autoregressive modeling is considered to complex time series analysis. The classic ExpAR model is actually a kind of state-dependent AR model in which only the amplitude is taken as the state. However, it can be imagined that the coefficients of the model are not necessarily dependent on one state, for some complex data, the coefficients of ExpAR model should be dependent on several states, say state vector. Therefore we introduce state vector-based ExpAR model, namely generalized ExpAR model, to nonlinear time series analysis, in which the coefficients are Gaussian-product functions (Gaussian radial basis function) of state vector. We prove that the geometric ergodic condition of the generalized exponential autoregressive model is still the same as that of the classic Expel model since factorizable property of the Gaussian radial basis function. Thus the stability of the model can be expected. For the estimate of the generalized ExpAR model, we provide a mutual estimating procedure which hybridizes the multi-agent random optimization algorithm namely evolutionary programming with the ordinary least squares to estimate the unknown parameters in the model. Simulations show the efficient ability of the generalized ExpAR model to dynamics reconstruction and prediction error reduction.<br /> Finally, empirical investigation of statistical identification of several actual nonlinear time series is given. We introduce three typical models respectively belonged to non-parametric conditional mean estimator (Nadaraya-Watson estimator), semi-parametric model (radial basis function network) and local linearization model (generalized exponential autoregressive) to analyze the epilepsy EEG (spike and wave) and human pulse wave (quasi-periodicity). Empirical comparisons show that the non-parametric estimator is of advantage to dynamics reconstruction, the local linearization model is good at forecasting for short term horizon, and the semi-parametric model has its site between the above models.