傳統的時間序列方法大部分都是用來分析點值的資料。但是實務上有很多的時間序列是區間值的資料，區間觀察值一般會比點觀察值包含更多的信息。所以，如何對區間時間序列建立模型及預測是一個重要議題。
在本文中我們介紹區間資料的平穩性和與區間相關的統計量，並探討區間時間序列的分析與建模。我們使用向量自回歸模型及向量誤差修正模型對區間的統計量包含：中位數、半徑、上下界建立時間序列模型 並進行區間預測。我們將所提的模型和古典濾波技術：指數平滑和無母數技術：K-近鄰算法比較區間預測的表現。最後，我們使用股票和指數作為例子，對本文所提出之區間時間序列模型進行區間預測及效率評估。

Conventional time series methods are developed for analyzing point-valued data. However, in practice there are many interval-valued time series data, which usually contain more information than point-valued data. It is thus important to develop time series modeling and forecasting techniques for interval-valued data.

In this paper, we introduce concepts of interval stationarity and related interval statistics and investigate methodology for interval time series analysis. We use vector autoregressive (VAR) and vector error correction (VEC) models to build time series models for interval statistics including : medium, radius, upper and lower bounds, and obtain interval forecasts. We compare the forecast performance of the proposed methods with classical filtering technique :
the exponential smoothing method, and nonparametric technique : the k-Nearest
Neighbors (k-NN) algorithm. Finally, in the empirical study, we use stock and index data to evaluate the forecast performance and efficiency of the proposed interval time series models.

誌謝. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . ii
摘要. . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . iii
Abstract. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . iv
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Denition and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 Basic Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.3 Descriptive Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.4 Error Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3 Interval Time Series Models . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.1 Vector Autoregressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3.2 Vector error correction models . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 Model variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . 8
4 Forecasting method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.1 VAR and VEC models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4.2 Smoothing Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.3 k-NN Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .... . . 13
5 Empirical Study. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . 14
6 Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . 17
7 References. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..... . . . . 18

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