Image generating system is usually considered as a noncausal system. The Kalman filter and the Wiener filter are two important linear filters for signal estimation. They are developed for the causal signal and noncausal signal respectively. However, the Kalman filter can also be applied to the noncausal system by rewriting the signal generating equation. In this thesis, we study the performance of the Wiener filter and the Kalman filter applied to image restoration. Our experiments have demonstrated that the rank of list for error performance is: the full order Winner filter, the Kalman filter, the reduced Kalman filter, the three-order Wiener filter. This performance is consisted with the amount of data used in the linear estimation. On the other hand the list for computation performance is as following: the reduced Kalman filter, the three-order Wiener filter, the Kalman filter, the full order Wiener filter. The efficiency of the reduced Kalman filter can be understood by the computation saving of huge updating procedures. It should be noted that the efficiency of applying the regular Kalman filter in this thesis is achieved by fully employed the special form of system matrix involved. In addition to the above noncausal image model, a causal image model can also be built if the central pixel is assumed to be affected only by the left and the upper pixels. The second model is not natural but is obviously advantageous in computation efficiency compared to the first model. However, the first model is much better than the second model error performance. Therefore, it is suggested that the natural image should be modeled as a noncausal model.

The causal system is more practical then the noncausal system in the world. Causality implies only the past input can effect the future output. As a consequence, noncausal system is seldom investigation. The purpose of this thesis is to study the signal recury for a noncausal system. The principle of signal estimation is based upon the Wiener-Hopf equation. Therefore, the correlation computation is very important. By transforming the noncausal dynamic equations to a causal equation, we achieve a partial recursive computation structure for correlation computation. However the current input is not independent of the past signal in the noncausal system. Hence, the Mason Rule is applied to solved this problem to make the above recursive structure complete. Furthermore, a recursive computation of Mason Rule for stage propagation is developed in this thesis to accelerating the processing speed. Our algorithm is applied to image restoration. We first segment the image to find the required generating input ponen for each correlated region. Secondly, we extend our 1-D algorithms to 2-D algorithm to restore the image. Our method is compared with the method developed base upon the Gaussian Markov model. The experiments results demonstrate the advantage of method in both visual quailty and numerical results.

本研究主要針對 Breidt et al.(1991) 等多位學者所建構的 Mixed causal-noncausal model,探討其假設與可拆解特性,並仔細討論相關資料模擬估計及預測的方法,最後將其實際應用於隱含波動率指數 (Volatility Index)的估計及預測上。根據本研究的實證結果,我們發現隱含波動率指數確實包含非因果的特性,並可進一步對其拆解及預測。另外 , 我們也以移動窗格的方式觀察係數估計結果的變化,發現 Mixed Causal-Noncausal Model 的確能夠捕捉到泡沫或危機正在生成的過程。 / This paper first focuses on Mixed causal-noncausal model constructed by Breidt et al.(1991) and then conducts empirical research on the CBOE Volatility Index. The assumptions, simulation, estimation and prediction methods of Mixed causal-noncausal model are introduced in great detail. Our empirical results show that the CBOE Volatility Index really contains non-causal parts, such that we can filter this part from the index and then further predict it. Moreover, by employing the rolling window estimation scheme the resulting coefficients of Mixed causal-noncausal model really could detect a bubble or a crisis which is going to happen.

過去實證研究多以時間序列模型搭配 GARCH 模型針對台灣股價指數進行分析。然而，Gourieroux and Zakoian(2017) 提出，當一時間序列具有泡沫現象時，noncausal Cauchy AR(1) process 是可能的優選模型。此外，Sarno and Taylor(1999) 的研究認為，台灣股價指數具有泡沫現象，故我們以 noncausal Cauchy AR(1) with Gaussian component 分析台灣股價指數，進而判斷其泡沫效果係來自 noncausal linear process 之 local explosive，並根據 noncausal Cauchy AR(1) 與 Gaussian component 之係數變動，捕捉泡沫效果之形成與來源。 / Most of the previous studies focused on analyzing Taiwan Stock Price Index using time series models with GARCH effects. However, Gourieroux and Zakoian (2017) have demonstrated that noncausal Cauchy AR(1) process may be a possible model in which the bubbles are observed. Besides, according to the studies of Sarno and Taylor (1991), some bubbles exactly existed in Taiwan Stock Price Index before 1990. Accordingly, this study aims at investigating the possible bubbles in Taiwan Stock Price Index from 2005 to 2015 by employing noncausal Cauchy AR(1) with Gaussian component method. As a result, we find out he bubbles which modeled by the noncausal linear process are local explosive. And based on the changes of the coefficients from noncausal Cauchy AR(1) and Gaussian component, this study successfully captures the form of bubbles.