轉折點與趨勢的研究在時間數列分析、經濟與財務領域裡一直是重要的研究主題。隨著所欲研究的物件之結構複雜性日益增加,再加上人類的知識語言因人類本身的主觀意識、不同時間、環境的變遷與研判事件的角度等條件下，可能使得所蒐集到的時間數列資料具某種程度的模糊性。為此,Zadeh[1965]提出了模糊理論,專門解決這一類的問題。在討論時間數列分析中的轉折點與趨勢問題時，常常會遇到時間數列的轉折過程緩慢且不明顯的情況。因此傳統的轉折點研究方法在這種情形下便顯得不足。對此，許多學者提出了轉折區間的概念。然而轉折區間的概念仍然存在一個潛在的困擾：在一個小的時間區間下，一個被認定的轉折區間可能在時間區間拉得很長的情況下，被視為是一個不重要的擾動或雜訊。本文嘗試藉由模糊統計量，提出一個轉折區間與趨勢的偵測方法。與其他轉折區間偵測法不同的是我們所提的方法能藉由控制參數，偵測到合乎使用者需求的轉折區間，進而找到一個趨勢的起點與終點。藉此避免把雜訊當成轉折區間或把轉折區間當成雜訊的困擾。因為使用了模糊統計量,同時也解決了資料的模糊性問題。 / Because the structural change of a time series from one pattern to another may not switch at once but rather experience a period of adjustment time, conventional change points detection may be inappropriate to apply under this circumstance. Furthermore, changes in time series often occur gradually so that there is a certain amount of fuzziness in the change point. For this, many research have focused on the theory of change periods detection for a better model to fit. However, a change period in some small observation time interval may seem a neglectable noise in a larger observation time interval. In this paper, we propose an approach to detect trends and change periods with fuzzy statistics through using partial cumulative sum. By controlling the parameters, we can filter the noises and find out suitable change periods. With the change periods, we can further find the trends in a time series. Finally, some simulated data and empirical examples are studied to test our approach. Simulation and empirical results show that the performance of our approach is satisfactorily successful.

本文研究台灣貨幣需求結構改變，我們研究「變數」值(Piecewise in Variable)的結構轉折而非「時間」值(Piecewise in Time)，因為轉折點只是轉折區間的特例，所以本文建立一「變數模糊時間序列」(Fuzzy Time Series in Variable)模型來探討「變數的轉折區間」，相較於傳統時間序列研究方法如：時間序列模型、門檻轉折點模型與模糊時間序列模型等，本文所建立的變數模糊時間序列模型，所求取的股價轉折區間，不僅可改善對稱模型殘差項的非隨機現象，同時也改善了門檻轉折模型之轉折點股價指數太低的現象，並且有效地將轉折點變更為較一般化的轉折區間，足見本文所提出變數模糊時間序列模型在結構轉折的偵測上具有相對優勢，詳述如下： （一）、相較於對稱模型方面：變數模糊時間序列模型可避免對稱模型估計貨幣需求函數所產生的偏差，並且有效改善其殘差項具有非白噪音現象。 （二）、相較於門檻轉折模型方面：1.變數模糊時間序列模型較能有效驗證以下假說：貨幣需求的股價指數彈性在高股價區時較大，以及貨幣需求的所得彈性在高股價區時較小。2.變數模糊時間序列模型所求出的股價指數轉折區間水準值，對央行目前及未來貨幣政策較具實用性，3. 變數模糊時間序列模型再預測貨幣需求時，未如門檻轉折模型產生高估的偏誤。 （三）、相較於傳統模糊時間序列模型方面：變數模糊時間序列模型已改善傳統模糊時間序列模型的結構轉折區間太長之不合理現象。 （四）、相較於以「時間」為轉折的傳統時間序列模型方面：當貨幣需求函數的重要解釋變數在短時間持續發生較大幅度變化時，傳統時間序列模型可能無法診斷出結構轉變的缺失，本文的變數模糊時間序列模型可避免此一缺失。 （五）、在政策的應用上： 1. 中央銀行若未將資料，區分高低股價指數來分段估計貨幣需求函數，將使貨幣需求的所得彈性抑或是狹義貨幣需求的股價指數彈性的估計，產生頗大的偏誤。 2. 經建會在計算台灣地區的景氣對策信號中，其金融面指標同時包括有M1B貨幣供給的增加率與股價指數變動率，如此將造成在高股價指數下，股價指數上揚時高估了台灣地區的景氣狀況，而在股價指數下降時，則反之。 另外，由於台灣欠缺貨幣需求函數的重要解釋變數「所得」的月資料，以往文獻以工業生產指數等為替代變數以估計月貨幣需求函數，本文不僅證明這些方法的缺失，並提出「模糊距離權數法」來估計出月國內生產毛額資料，此一資料不僅可避免月工業生產指數等方法的三項缺失，而且在貨幣需求的估計上與預測上均有較佳的表現。 / Whether the ”money demand function” makes “structural change” happened or not ,that is crucial research for the monetary theory field. Therefore, many foreign and domestic papers have ever made studies on this. There have two major methods of study structural change. The first method is piecewise in time that is so popular and so many lecture study by it e.g. Juda and Scadding(1982), Shen(1999) ,Lin and Huang(1999),etc . Tsay(1989) had proposed a new methoed that is piecewise in variable . Distinct situation is suitable in using the two methods .We have two reasons to use the new method to study the structural change of Taiwan’s money demand function. First one is that Friedman(1988,Paul（1992）,Wu and Shea（1993）and Shen(1996) find the trade-volume of stock market or stock price are the important factors of money demand function. TSE is 12495 in February of 1990 and 2573 in October of 1990. TSE is changing so huge but all the Papers of piecewise in time can’t detect the structural change of Taiwan money demand. The second reason is that to detect the ” interval of financial variable” of structural change of Taiwan money demand is more benefit to the Central Bank than to detect the ” past time point” of structural change.　To detect the ” interval of financial variable” of structural change of Taiwan money demand is much convenient matters for monetary policy of Center Bank from now and future. Our research propose “fuzzy time series in variable” try to find the ” smoothing interval of financial variable” of structural change of money demand . Our method has two major benefits as follow: 1. Difference to TAR model: The TAR model find out the ” point of financial variable” of structural change. It seems metaphorically money demand function’s structural change suddenly. Our method find out the ” interval of financial variable” of structural change .It’s more reasonable that structural change of money demand function is gradually. 2. Difference to STAR model: So many STAR（Smooth Transition Autoregressive ）papers also find out the Gradual Transition Interval .For example: Terasvirta and Anderson(1992), Sarantis(1999) etc. But those lectures have the following point on why our method can improve it (a).STAR is piecewise in time. (b). STAR investigate structural change by just one variable AR process. But economists concern the structural change of variables. (c). The power of STAR to detect structural change is too weak. 3. We propose new summation average entropy formula that can improve the interval of structural change too longer.

由於電腦革命的成功，在短暫的幾年之間，更加速了經濟的成長，而金融的投資分析，是社會經濟發展的原動力，因此研究這方向的財務數學也相對的提高了專家、學者的研究熱潮。就以股票、匯率市場來說，如果能比别人早一步掌握行情走勢，就能獲得較高的利潤。但影響股價、匯率波動的因素很多，尤其是在複雜多變及不確定性的資訊下。因此；如何進行更精確的趨勢分析與預測，是本文研究的主題。由於，傳統的期望值是二元的邏輯思考(非1即0)，比較無法符合多變與不確定的財金問題，因此本文考慮以模糊統計方法，以模糊期望值的方法來作趨勢分析與預測，期望能對複雜多變的財金體系提共一套更精確合理的投資分析方法，可以提供投資者更多的訊息，做出明確的抉擇。最後；以我國集中市場加權股票指數、台幣對美元匯率及台積電股價為例，做一實例上的詳細探討。 / Based on computer revolutionary coming off, economics grows fast in previous several years, then the investment analyze of finance is the impetus of development of society economic. Therefore, many experts and scholars are interested in the research of financial mathematics. Taking stock market and exchange market for example, if you can predict the future trend of market, you obtain more profit. However, there are many factors that act on stock prices and exchange rate. Especially, the market information is complicated and incomplete. How to go along accurate trend analysis and divination is the important point of the text research. Because traditional expectation value is dibasic logic thought (either 1 or 0), that can’t conform to the highly changeable and uncertain finance problems. For this reason, in this research we propose an integrated procedure for fuzzy expectation value modeling and forecasting through fuzzy relation equations. We apply this technique to construct a fuzzy expectation value model for Taiwan Weighted Stock Index and exchange rate and forecast future trend. We strongly believe that this model will be profound of meaning in forecasting future trend of financial market.

由於許多經濟指標的定義不明確，或是因為資料蒐集的時間不一，導致代表經濟景氣的數值，實際上即具有相當大的的不確定性。傳統的方法多不考慮這樣的模糊性,而傾向尋找一準確的模式轉折點。本文則以模糊數學的方法,運用模糊分類法以及模糊熵，訂定一個評判的準則。藉以找出一時間數列模式發生變化的轉折區間。最後以台灣經濟景氣指標為例，說明此方法可不需對資料的模式有任何事先的認知，即可得出與傳統方法相近，甚至更為合理的預測結果。 / Unlike conventional change points detecting, which seeks to find a decision boundary between classes for certain structural changed time series, the purpose of this research is to investigate a new approach about fuzzy change period identification. Based on the concept of fuzzy theory, we propose a procedure for the - level of fuzzy change period detecting and prove some useful properties for a fuzzy time series. We use some numerical examples to demonstrate how these procedures can be applied. Finally, experimental results show that the proposed detecting approach for structure change of fuzzy time series is available and practical in identifying the alpha-level of fuzzy change period.

動態資料往往隨著時間區間取法或測量工具的不同而有差異,此種不確定的特質我們稱為模糊性。但是傳統的時間數列仍是以確定的觀察值來記錄具有模糊性的動態資料。為了更完整的表示一個動態過程,我們考慮模糊時間數列(fuzzy time series)以具有不確定性的模糊集合來取代明確的數值,保持原來的模糊性。 本文探討模糊時間數列中模糊自我迴歸模式(fuzzy autoregressive model簡寫為 FAR 模式)的建構過程,並分別利用此模式來預測中央政府總預算和匯率。FAR 模式乃根據Box-Jenkins(1970)所提出的 ARMA 三階段模式建立的流程並推廣Zadeh(1965)所提出的模糊集合理論而來。在這過程中 ,我們考慮人類思維方法,使FAR 模式更具有彈性且適合未來預測時的需要。而對於所討論的動態過程,也不需要任何模式上的假設(例如:線性或穩定 ),因此 FAR 模式的適用範圍極為廣泛,更不會因為模式的誤判而導致預測時的嚴重錯誤。最後,我們將 FAR 模式的預測結果與傳統 ARMA 模式做比較。 文中關於模糊時間數列的一些性質,例如:模糊趨勢(fuzzy trend)和模糊穩定(fuzzy stationary),由於傳統文獻中沒有加以討論,本文亦提出定義和新的看法。 / Representations of dynamic data are always different as the time interval or measuring tool change. We call these characteristics of uncertainty fuzziness. But traditional time series use crisp observations to record a fuzzy dynamic process. To completely represent, we consider fuzzy time series replacing the crisp numbers with fuzzy sets and preserve original fuzziness. In this paper, the fuzzy autoregressive model (FAR model) of fuzzy time series is studied and used to forecast the Central government expenditure and exchange rates, respectively. The modeling process is according to Box- Jenkins' (1970) method of ARMA model and merged with the fuzzy set theory proposed by Zadeh (1965). Reasonable human judgements and ways of thinking are taken into consideration throughout the modeling process to make the FAR model more elastic and appropriate for forecasting. Unlike certain incorrectly identified models which lead to inaccurate forecasts, the FAR model can be widely applied due to its not having any assumptions on the original time series (e.g., linearity and stationarity). Finally, the performances of the FAR model to Central government expenditure and exchange rates are compared with that of the traditional ARMA model. Additionally, some properties about fuzzy time series, e.g., fuzzy trend and fuzzy stationary, have not been studied in the literature, and we propose definitions and new opinions.

本文嘗試以模糊理論的觀念，應用到時間數列分析上。研究重點包括模糊關係、模糊規則庫和模糊時間數列模式建構與預測等。我們首先給定模糊時間數列模式的概念與一些重要性質。接著提出模糊規則庫的定義，以及模式建構的流程，並以模糊關係方程式的推導，提出模糊時間數列模式建構方法。最後，利用提出的方法，對台灣地區加權股票指數建立模糊時間數列模式，並對未來進行預測，且考慮以平均預測準確度來做預測效果之比較。這對於財務金融的未來走勢分析將深具意義。 / The paper has attempted to apply the concept of fuzzy method on the analysis of time series. This reserch is also to include fuzzy relation, fuzzy rule base, fuzzy time series model constructed and forecasting. First, we'll define the concept of fuzzy time series model and some important properties. Next, the definition of fuzzy rule base will also be put forward, along with procedure of model constructed, the formation of fuzzy relation polynomial, and the methods to construct fuzzy time series model. At last, with the above methods, we'll build up fuzzy time series model on Taiwan Weighted Index and predict future trend while examine the predictive results with average forecasting accuracy. This shall carry profund signifigornce on the analysis of future trend in terms of financialism.

時間序列是用來預測未來趨勢的一種重要技術，然而在實務上建構時間序列模型時，參數很難有效估計。原因可能來自於時間序列本身的模糊性質，而導致參數的不確定性使得預測結果產生極大誤差。如果將參數模糊化引進時間序列的模型中，往往過於複雜。本論文提出相對移動率為新的模糊時間序列建構方法，讓原本具有模糊性質的時間序列經由反模糊化(defuzzification)後，以點估計的方式估計起始中心點，經由適當的修正調整為較佳的中心點以及半徑，建立有效的區間時間序列。並將相對移動率引進門檻自廻規模型中，取代原有之門檻值設定，並建立區間時間序列。最後，我們使用台灣加權股價指數為例，以本論文所提出之方法進行區間預測及效率評估。 / The time series is an important technology that is used to predict future trends, however in the real world, parameter is difficult to estimate effectively when we construct a time series model due to the of the fuzzy property of the times series data. The estimated parameters in the time series will cause a big error due to the uncertainty of fuzzy data. It is too complex to introduce the fuzzy parameters into the time series model. In this thesis, we propose relative moving ratio as a new criteria in constructing procedure of an interval time series. We defuzzify a fuzzy data and use point estimation to obtain an initial center, then we adjust the center and radius making it more appropriately. The resulting center and radius is then become an interval time series that can be use to forecast an interval data. We also apply relative moving ratio in threshold autoregressive models by replacing the threshold in constructing interval time series. Finally, in empirical studies chapter, we use Taiwan weighted Stock Index as examples to evaluate the performance of the proposed two methods in building the interval time series.

Submitted by Izabel Franco (izabel-franco@ufscar.br) on 2016-09-06T18:59:08Z No. of bitstreams: 1 TeseFJJS.pdf: 3277696 bytes, checksum: 0a34a4499fb5e482fa95ea8925603968 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T14:12:50Z (GMT) No. of bitstreams: 1 TeseFJJS.pdf: 3277696 bytes, checksum: 0a34a4499fb5e482fa95ea8925603968 (MD5) / Approved for entry into archive by Marina Freitas (marinapf@ufscar.br) on 2016-09-12T14:13:02Z (GMT) No. of bitstreams: 1 TeseFJJS.pdf: 3277696 bytes, checksum: 0a34a4499fb5e482fa95ea8925603968 (MD5) / Made available in DSpace on 2016-09-12T14:13:13Z (GMT). No. of bitstreams: 1 TeseFJJS.pdf: 3277696 bytes, checksum: 0a34a4499fb5e482fa95ea8925603968 (MD5) Previous issue date: 2015-04-30 / Não recebi financiamento / The good results obtained by the fuzzy approaches applied in the analysis of time series (TS) has contributed significantly to the growth of the area. Although there are satisfactory results in TS analysis with methods that use the classic concepts of TS and with the recent concepts of fuzzy time series (FTS), there is a lack of models combining both areas. Face of this context, the contributions of this thesis are associated with the development of models for TS analysis combining the concepts of FTS with statistical methods aiming at the improvement in accuracy of forecasts and in identification of behavioral changes in the TS. In order to allow a suitable fuzzy representation of crisp values observed, the approaches developed in this thesis were combined with a new proposal for pre-processing of the data. The prediction value is calculated from a new smoothing technique combined with an extension of the fuzzy logic relationships. This combination allow to be considered in value computed different degrees of influence to the most recent behavior and to the oldest behavior of the series. In situations where the model does not have the necessary knowledge to calculate the predicted value, the concepts of simple linear regression are combined with the concepts of the FTS to identify the most recent trend in the TS. The approach developed for the behavioral analysis of the TS aims to identify changes in behavior from the definition of prototypes that represent the groups of the TS and from the segmentation of the series that will be analyzed. In this new approach, the dissimilarity between a segment of a TS and the corresponding interval of a given prototype is defined by metric Fuzzy Dynamic Time Warping weighted by a new smoothing technique applied to the distance matrix between the observed data. The accuracy obtained by the forecast model not only demonstrates the effectiveness of the developed approach, but also shows the evolution of model throughout the research and the importance of preprocessing in the forecast. The analysis of segmented TS identifies satisfactorily the behavioral changes of the series by calculating the membership functions of these segments in the respective groups represented by the prototypes. / Os bons resultados obtidos pelas abordagens fuzzy utilizadas para a análise de séries temporais (ST) tem contribuído significativamente para o crescimento da área. Embora haja resultados satisfatórios na análise de ST com métodos que utilizam os conceitos clássicos de ST e também com os conceitos recentes de séries temporais fuzzy (STF), há uma carência de modelos que combinem ambas as áreas. Diante deste contexto, as contribuições deste trabalho estão associadas ao desenvolvimento de modelos para a análise de ST combinando os conceitos de STF e métodos estatísticos visando a melhora na acurácia das previsões e a identificação de alterações comportamentais nas séries. Com o objetivo de permitir uma melhor representação fuzzy dos valores crisp observados, as abordagens desenvolvidas nesta tese foram associadas a uma nova proposta de pré-processamento dos dados. A previsão de valores é calculada a partir de uma nova técnica de suavização combinada a uma extensão das relações lógicas fuzzy. Essa combinação permite que sejam considerados no cálculo do valor previsto diferentes graus de influência para o comportamento mais recente e para o comportamento mais antigo da série. Em ocasiões onde o modelo não dispõe do conhecimento necessário para o cálculo do valor previsto, os conceitos de regressão linear simples são associados aos conceitos das STF para identificar a tendência mais recente da ST. A abordagem desenvolvida para a análise comportamental das séries tem como objetivo identificar mudanças no comportamento a partir da definição de protótipos que representam um grupo de ST e da segmentação das séries a serem analisadas. Nesta nova abordagem, a dissimilaridade entre um segmento de uma ST e o intervalo correspondente de um determinado protótipo é definida por meio da métrica Dynamic Time Warping (DTW) Fuzzy, ponderada por uma nova técnica de suavização aplicada à matriz de distâncias entre os dados observados. A acurácia obtida pelo modelo de previsão não só comprova a eficácia da abordagem desenvolvida, como também demonstra a evolução do modelo ao longo da pesquisa e a importância do pré-processamento nas previsões. A análise das ST segmentadas identifica satisfatoriamente as alterações comportamentais das séries por meio do cálculo da pertinência dos segmentos nos respectivos grupos representados pelos protótipos.

對於一個具有結構性改變性質的非線性時間數列，通常很難判斷何處為轉 捩點，或者何處為所謂的轉型期。雖然長久以來已有不少偵查轉捩點的方 法被提出，但是對於轉捩區間以及對於一些語言性的時間數列資料問題（ 例如：景氣指標的紅綠燈時間數列），都很少被提出來。本論文中，我們 首先引用Zadeh於1965年提出來的模糊理論的觀念來介紹糢糊時間數列（ FTS）。進而定義出在□水準下的模糊點（FP）和模糊轉捩區間（FCP）， 並且證明了一些有用的性質。最後再以台灣地區出生率資料為例，說明□ 水準的模糊轉捩區間的判定方法，並列出了詳細的執行步驟。實驗結果更 證明出我們的模糊檢驗法非常具有實用性及有效性。 / As far as structural change of a non-linear time series is concerned, it is hard to tell when the change point or the fuzzy change period occurs. Though many methods are used for the task of detecting, most of them primarily deal with the case of change point, and few examine the problem of fuzzy change period and linguistic time series ( for example, the index of prosperity represented by red or green light ). In this article, we adopt the theory of fuzzy which is proposed by Zedeh ( 1965 ) to introduce the concept of fuzzy time series ( FTS ). Furthermore, we define the □level of fuzzy point (FP) as well as fuzzy change period (FCP), and prove some useful properties. Finally we explain the method we proposed in detecting the □level of fuzzy change period in terms of the data of Taiwan birth rate and provide step-by-step procedures. Experimental results show that the proposed method of fuzzy detecting is available and practical in detecting the □level of fuzzy change period.

近年來，隨著科技的進步與工商業的發展，預測技術的創新與改進愈來愈受到重視，同樣地，對於預測準確度的要求也愈來愈高。尤其在經濟建設、人口政策、經營規畫、管理控制等問題上，預測更是決策過程中不可或缺的重要資訊。有鑑於此，本論文嘗試應用模糊關係方程式，提出多變量模糊時間數列建構過程及轉折區間檢測模式理論架構。另一方面，多變量模糊時間數列模式建構過程，研究者曾提出很多轉折點之偵測與檢定方法，然而在實際的例子中，時間數列之結構改變所呈現出來的是一種緩慢的改變過程，即轉折點本身就是模糊不確定。這個概念在建構不同模式分析各國經濟活動數據時更顯重要。本論文針對轉折區間之檢測提出一個完整的認定程序。多變量時間數列系統中的隸屬度函數等於在計算成果指標群時的群集中心。應用本論文提出的方法，我們以德國、法國及希臘之總體經濟指標GDP為例，考慮通貨膨脹率、GDP成長率及投資率來進行轉折區間的檢測。 / In recent years, along with the technological advancement and commercial development, the creation and improvement of forecasting techniques have more and more attention. Especially at the economic developments, population policy, management planning and control, forecasting gives necessary and important information in the decision-making process. Regarding stock market as the example, these numerals of closing price are uncertain and indistinct. Again, the factors of influence on quantity are numerous, such as turnover, exchange rate etc. Therefore, if we consider merely the closing price of front day to build and forecast, we will not only misestimate the future trend, but also will cause unnecessary damage. Owing to this reason, we propose the procedure of multivariate fuzzy time series model constructed and theory structure by fuzzy relation equation. Combining closing price with turnover, we apply our methods to build up multivariate fuzzy time series model on Taiwan Weighted Index and predict future trend while examine the predictive results with average forecasting accuracy. A fuzzy time series is defined on averages of cumulative fuzzy entropies of the tree time series. Finally, an empirical study about change periods identification for Germany, France and Greece major macroeconomic indicators are demonstrated.