臺灣博碩士論文加值系統

由於碎形幾何與渾沌動力是非線性科學中的兩個重要組成單元，而它們是目前解析複雜水文系統的熱門工具。因此，本研究旨在利用此二學門於水科學相關領域的探討及應用，其中有八項主題是文中所欲解析者，分別是：(1)水文歷程的點碎形維度值計算與變異性解析；(2)區域性碎形特徵量之面狀解析與群集性探討；(3)水文變量之時間尺度律驗證與頻率分析之碎形趨近法；(4)驗證日流量資料的吸子存在性；(5)分析各種變數轉換下之系統複雜性的變遷程度；(6)以相空間觀點來解析部份階次模式之物理機制；(7)流量歷程之非線性模式建立與單超前期距預測；(8)非線性模式在集水區面積雨量之多超前期距預測。
首先，是利用碎形幾何來解析水文變量的時空變化，計有三個主體。在水文歷程的點碎形維度值計算與變異性解析方面：經計算得容積維度、資訊維度及相關維度等數據，顯示降雨及逕流歷程表現出多重碎形結構，又前者的複雜性遠比後者為高。至於兩者的間隙性係隨著觀測尺度的增加而下降，惟遞減型態有程度上的差異，同時亦均顯示出時間分佈上的強烈非均勻性。在區域性碎形特徵量之面狀解析與群集性探討方面：雨量的時間性維度在空間上之變異及解析能力，與一般所理解者是相互一致的；集水區流量歷程的時間性維度與河系網路分佈的空間性維度之間，具有一定程度的關聯性。又間隙性除可作為區分相同維度體系之辨識功能外，亦可作為偵測或檢測其是否受到外在衝擊的一項解析工具。在水文變量之時間尺度律驗證與頻率分析之碎形趨近法方面：降雨量具有時間的尺度不變性及隨門限值升高而導致群集性的下降，此外亦驗證得不同的門限值變動下，可獲得相同的一致性尺度不變區最大值。又利用超越門限值所得之機率-尺度律，進而建立起飽和尺度(重現期)與門限值(設計水文量)之相依關係，其有別於傳統的水文頻率分析法。當所關注的尺度為年單位以上時，暴雨的碎形尺度律公式可順利地予以建立。
接著，是利用渾沌動力來解析水文變量的時間變化，亦計有三個主體。在驗證日流量資料的吸子存在性方面，經計算得日流量的相關維度與最大Lyapunov指數，結果顯示了渾沌吸子之存在證據，而所需解析的變數為3個或4個，因此可利用該等資訊來重建系統的運動軌跡。在不同變數轉換下之系統複雜性的變遷程度方面：以所選用的四種資料別為例，當未有任何轉換時，則複雜度的大小順序是白色干擾、日降雨、日流量及Lorenz吸子。除白色干擾是明顯的無序之外，其餘均含有某種程度的型態或結構，而且該測度展現出辨識能力。又若對資料進行積分或移動平均之轉換處理，則可有效降低其複雜度，此隱喻能以較簡單的方式來建立模式且具有更長期距的預報能力。在使用相空間觀點來解析部份階次模式之物理機制方面：部份階次具有完備的數學理論架構及物理性之闡明機制，同時經由實際案例之計算與分析而給予證實。不僅如此，多維非線性部份階次模式更擴展成具有時間-空間關聯性並存之完整模式。
最後，是採用非線性模式來進行時間序列的模擬與預測，計有兩個主題。在流量歷程之非線性模式建立與單超前期距預測方面：於所比較的眾多模式當中，以結合相空間及類神經網路之時序建模方式，具有較佳的整體效能，顯見類神經網路所展現在學習非線性行為的適應能力。又在非線性模式於集水區面積雨量之多超前期距預測方面：經實證分析得具備渾沌動力性的非線性模式優於時間稽延式的線性模式，而當預報的超前期距增加時，前者明顯地展現出較佳的特性，同時亦維持一定的穩健性。因此，若驗證得一時間序列為渾沌過程，則應以非線性觀點為出發，而不應再簡化成線性問題。
雖然整個研究採用了較為新穎的觀念及工具來解析複雜水文系統，然大多數的研究本體亦納入了傳統分析方法之成果，使得不同方法論能有所參照及比較的機會，同時藉由其間的差異來截長補拙，使之能完善整個複雜系統之描述能力。

The fractal geometry and the chaotic dynamics are two important components in nonlinear sciences. They are powerful tools to analyze the complex hydrologic systems currently. In this study, we focus the attention on hydrosciences. Eight topics are concerned and described as follows: (1) estimation of point-fractal dimension and its variability for hydrologic process, (2) analysis and clustering for areal fractal characteristics, (3) identification of time-scale law and fractal approach of frequency analysis for hydrologic variable, (4) existence identification of attractor for daily streamflow data, (5) complexity comparison of a system with different transformation, (6) mechanism description of subset regression model by phase-space concept, (7) nonlinear modeling of streamflow process and its one-lead time forecasting and (8) nonlinear model for multi-lead time forecasting in areal rainfall of watershed.
Firstly, the fractal geometry is used to analyze the tempo-spatial variation for hydrologic variables and three parts are included. On the estimation of point-fractal dimension and its variability for hydrologic process, the capacity dimension, information dimension and correlation dimension are calculated. The results show that the rainfall and runoff processes are both multi-fractals. The former is far more complex than the latter. When the observation time-scale increases, the lacunarity decreases. The decreasing patterns are different in degree and have a strong non-uniform tendency on the temporal distribution. On the analysis and clustering for areal fractal characteristics, the spatial variation and explanation with the temporal dimension of rainfall are the same of our understanding from experience. As for the relation between the temporal dimension of watershed runoff process and the spatial dimension of river basin network, they are correlative in degree. Not only the lacunarity measure can be served as the verification tool, once the systems have the same dimensions, but also is a powerful tool to detect or test whether the external impact or not. On the identification of time-scale law and fractal approach of frequency analysis for hydrologic variable, the rainfall is evident that scale invariance exists in time and clustering will decrease in accord with the increase of threshold. Under the variation of threshold, it can be verified that the maximum values of the homogenous scale-invariant interval are just the same. In addition, taking the probability-scale law on different levels of threshold, the relation can be established between the saturation scale (return period) and the threshold (design hydrologic variable). The methodology is different from the traditional one, i.e. frequency analysis method. When the time-scale we concerned is increasing up to one-year, the fractal scaling law of storm can be established readily.
Secondly, the chaotic dynamics is used to analyze the temporal variation of hydrologic variables and three parts are also included. On the existence identification of attractor for the daily streamflow data, the correlation dimension and maximum Lyapunov exponent are calculated. The results show that the chaotic attractor exists and suggests that it may be described by 3 or 4 variables. This information can be used to reconstruct the motion trajectory of a system. On the complexity comparison of a system with different transformation, four types of data are selected and the complexity is white noise, daily rainfall, daily streamflow and Lorenz attractor in descending order. Only the white noise is apparently in disorder, the others have some kinds of pattern or structure. This measure shows the power on the verification. Once the data are transformed by the integration and moving average, the complexity will be decreased. It implies that the model can be established with a simpler way and have a longer lead-time predictability. On the mechanism description of subset regression model by phase-space concept, the subset-order model has the dual properties, the framework of mathematical theory and the mechanism of the physical explanation, and this result is proved by the case studies. Moreover, the multi-dimensional nonlinear subset-order model is extended to the tempo-spatial related model.
Finally, the nonlinear model is used for the time series simulation and forecasting and two aspects are concerned: (1)On the nonlinear modeling of streamflow process and its one-lead time forecasting, many models are compared. The results shows that the combined phase-space theory with the artificial neural network framework is excellent and it is powerful on the learning of nonlinear behavior for the artificial neural network. (2)On the nonlinear model for multi-lead time forecasting in areal rainfall of watershed, it is verified that the nonlinear model based on the chaotic dynamics is superior to the linear one based on the time-lag correlation. When the lead-time for forecasting is increased, the former is apparently better than the latter on the overall performance and it is still robust. If a time series is proved a chaotic process, it belongs a nonlinear problem. It must come from the nonlinear point of view and not treated as a linear one.
Many modern concepts and tools are used to analyze the complex hydrologic systems in this study, but much of them are also introduced the results from the traditional one. The purpose is to provide a reference and a comparison with different methodologies. Viewing the differences, we overcome the shortcomings, instead of the strong points of the other. The description of complex system will be improved in advance.

封面
中文摘要
英文摘要
謝誌
目錄
表目錄
圖目錄
符號說明
第一章 導論
1-1 研究動機與目的
1-2 文獻回顧與探討
1-3 本文內容與組織
第二章 碎形幾何及其動力理論
2-1 碎形幾何學
2-1.1 何謂碎形
2-1.2 碎形維度之定義
2-1.3 尺度不變性與尺度律
2-2 碎形特性之解析
2-2.1 容積維度之計算
2-2.2 資訊維度之計算
2-2.3 相關維度之計算
2-2.4 間隙性測度
2-3 頻率分析之碎形趨近法
2-3.1 Cantor 集觀念
2.3.2 碎形分佈與機率之變換
2.3.3 時間軸上之點事件解析
2.3.4 極端水丈量之碎形推估法
第三章 渾沌動力理論
3-1 渾沌學
3-1.1 何謂渾沌
3-1.2 渾沌的研究方式
3-1.3 相空間及其重建
3-2 渾沌特徵之解析
3-2.1 吸子維度
3-2.2 Lyapunov指數
3-2.3 Kolmogorov熵
3-2.4 LZ複雜度
3-3 變數轉換之複雜性變化分析
3-3.1 非線性轉換
3-3.2 差分與積分之轉換
3-3.3 移動平均轉換
第四章 非線性模式之構建
4-1 非線性預報模式
4-1.1 線性時序方法及其考量
4-1.2 為何使用非線性方法
4-1.3 導入渾沌動力性之模性建立概念
4-1.4 非線性函數之選定與處理
4-2 最佳化預測-類神經網路學習法
4-2.1 類神經網路概述
4-2.2 類神經網路架構
4-2.3 傳輸函數之選取
4-2-4 學習演算法
4-2.5 資料常態化或正規化處理
4-3 模式效庇能比較準則
4-3.1 評鑑指標-統計類與水文類
4-3.2 總體效能分數評鑑體系
第五章 碎形幾何之水文時空應用
5-1 區域性水文變量之碎形特徵計算與解析
5-1.1 概要
5-1.2 資料選用之說明
5-1.3 水文變量之碎形特徵與變異性解析
5-1.4 水文變量之間隙性解性
5-1.5 結語
5-2 水文變量之時間尺度律與頻率分析
5-2.1 概要
5-2.2 資料選用之說明
5-2.3 水文變量之時間尺度律驗證
5-2.4 設計水文量之碎形推估
5-2.5 結語
第六章 渾沌理論在水文時序分析之應用
6-1 水文歷程之渾沌吸子探尋與特徵解析
6-1.1 概要
6-1.2 資料選定與相關資訊
6-1.3 目流量渾沌特徵之診斷
6-1.4 結語
6-2 水文歷程之變數轉換及其影響性初探
6-2.1 概要
6-2.2 變數轉換之影響性
6-2.3 水文資料之變數轉換對系統複雜度之影響
6-2.4 結語
6-3 以相空間觀點驗證部份自迴歸過程之物理性的存在
6-3.1 概要
6-3.2 自迴歸與部份自迴歸過程
6-3.3 存在性之佐證與綜合解說
6-3.4 實際資料之驗證
6-3.5 結語
第七章 非線性動力模式在水文時序之建模
7-1 河川逕流歷程之非線性模擬與預測
7-1.1 概要
7-1.2 資料選用與各模式介紹
7-1.3 日流量時間序列之模擬與預測
7-1.4 結語
7-2 集水區雨量歷程之非線性模擬與預測
7-2.1 概要
7-2.2 研究對象之背景說明
7-2.3 轉移機率之預報函數與預測值
7-2.4 多超前時期之日雨量模擬與預測
7-2.5 結語
第八章 結論與建議
8-1 結論
8-2 建議與展望
參考文獻
著作
簡歷
授權書

丁晶、鄧育仁、吳伯賢、楊榮富 (1993)，洪水渾沌分析，成都科技大學學報，總第73期，第1-5; 20頁。
王如意、李鴻源、許銘熙 (1996)，「台北防洪整體檢討計畫(一)」，經濟部水利司研究計畫，計畫編號85EC2A043003，台北。
王如意、陳弘正 (1996)，水文系統混沌動力現象之研究及其在降雨與逕流時間序列之預測，行政院農業委員會研究計畫報告，計畫編號85科技-1.11-林-03-1(1)，台北。
平建軍 (1993)，分形幾何學在地震綜合預報中的應用，於：「分形理論及其應用」，主編：辛厚文，中國科學技術大學出版社，合肥，第397-401頁。
吳明進 (1997)，臺灣地區長期預報評述，於：「長期水資源預測研討會論文彙編」，國立臺灣大學全球變遷研究中心，台北，第1-15頁。
吳瑞賢、朱佳仁、林永敏、蘇文瑞 (1994)，台灣北部地區溫度、雨量變遷之初步研究，八十三年度農業工程研討會論文集，高雄，第401-415頁。
李宗仰 (1995)，「地文水文時空變化之碎形結構及其應用」，國立成功大學水利及海洋工程研究所博士論文，台南。
李宗仰、林淑真 (1998)，非常態週期性水文序列之預處理及其在預測之應用，台灣水利，第46卷，第2期，第71-84頁。
李宗仰、劉長齡 (1989)，月流量之季節消除性ARMA模式，土木水利，第16卷，第1期，第23-38頁。
林振山 (1993)，「非線性力學與大氣科學」，南京大學出版社，南京。
林淑真 (1993)，「機率分佈在水文頻率分析上之效能評估」，國立成功大學水利及海洋工程研究所碩士論文，台南。
林淑真 (1995)，遞歸倒傳遞神經網路於時流量歷程之預測，未發表，共14頁。
林淑真 (1997)，結合相空間理論與類神經網路架構之水文時序預報，八十六年電子計算機於土木水利工程應用研討會論文集，新竹，第1607-1618頁。
林淑真 (1998)，以Cantor集觀念解析水文變量在時間尺度上之行為，台灣水利，第46卷，第2期，第85-96頁。
林淑真、李宗仰 (1996)，神經元傳輸函數在水文時序建模之分析，第八屆水利工程研討會論文集，台北，第211-218頁。
林淑真、劉長齡、游保杉 (1996)，日流量時間序列之渾沌動力探求，台灣水利，第44卷，第2期，第20-30頁。
林淑真、劉長齡、李宗仰、游保杉 (1996)，河川流量歷程之渾沌性與非線性預測，第八屆水利工程研討會論文集，台北，第219-226頁。
林鴻溢、李映雪 (1992)，「分形論－奇異性探索」，北京理工大學出版社，北京，第262-285頁。
易任、黃文政 (1984)，擴展式卡門濾波理論應用於降雨逕流模式之研究，台灣水利季刊，第三十二卷，第四期，第12-46頁。
馬鏡嫻、羅哲賢 (1997)，乾旱、半乾旱地區降水趨勢可預報期限的初步研究，於：「中國西北乾旱氣候研究」，孫國武主編，氣象出版社，北京，第216-219頁。
夏愛玲、羅哲賢 (1997)，近500年旱澇等級序列分維特徵的初步分析，於：「中國西北乾旱氣候研究」，孫國武主編，氣象出版社，北京，第3-6頁。
許正芳 (1995)，「台灣地區降雨量之碎形分析」，國立台灣大學農業工程研究所碩士論文，台北。
黃潤生、胡家元 (1998)，再探武漢地區雨強分維數隨時間變化，武漢大學學報(自然科學版)，第44卷，第1期，第133-136頁。
郭明哲 (1983)，「預測方法－理論與實例」，中興管理顧問公司，台北。
陳樹群、黃美君 (1995)，水文混沌現象之研究，中華水土保持學報，第26卷，第1期，第31-42頁。
陳耀茂 (1996)，「探索式數據解析法」，育友圖書有限公司，台北。
楊傳愚、楊培才 (1992)，近百年氣候系統的複雜性和可預報性，於：「氣候變化若干問題研究」，李崇銀主編，科學出版社，北京，第39-46頁。
劉式達、鄭祖光、林振山 (1992)，氣候層次動力學初探，於：「氣候變化若干問題研究」，李崇銀主編，科學出版社，北京，第1-6頁。
謝惠民 (1994)，「複雜性與動力系統」，上海科技教育出版社，上海。
Abraham, N. B., A. M. Albano, A. Passamante, P. E. Rapp and R. Gilmore /Editors (1993). Complexity and Chaos, Proceedings of the Second Bryn Mawr Workshop on Measures of Complexity and Chaos, World Scientific, Singapore.
Abraham, N. B., A. M. Albano, B. Das, G. De Guzman, S. Young, R. S. Gioggia, G. P. Puccioni and J. R. Tredicce (1986), Calculating the Dimension of Attractors from Small Data Sets, Physics Letter A, 114, 217-221.
Allain, C. and M. Cloitre (1991). Scaling Rules in Rock Fracture and Possible Implications for Earthquake Prediction, Nature, 297, 47-49.
Auerbach, D. and I. Procaccia (1990). Grammatical Complexity of Strange Sets, Physical Review A, 41, 6602-6614.
Azoff, E. M. (1994). Neural Network Time Series Forecasting of Financial Markets, John Wiley & Sons, Ltd., New York.
Badii, R., G. Broggi, B. Derighetti, M. Ravani, S. Ciliberto, A. Politi and M. A. Rubio (1988). Dimension Increase in Filtered Chaotic Signals, Physical Review Letters, 60(11), 979-982.
Baldo, S., F. Normant and C. Tricot /Editors (1995). Fractals in Engineering, World Scientific, Singapore.
Barndorff-Nielsen, O. E., J. L. Jensen and W. S. Kendall /Editors (1993). Networks and Chaos - Statistical and Probabilistic Aspects, Chapman & Hall, London.
Bate, A. K. (1990). Climate in Crisis: The Greenhouse Effect and What We Can Do, The Book Publishing Company.
Bianchi, M. M., C. M. Arizmendi and J. R. Sanchez (1992), Detection of Chaos: New Approach to Atmospheric Pollen Time-Series Analysis, International Journal of Biometeorology, 36, 172-175.
Biardi, G., M. Giona and A. R. Giona /Editors (1995). Chaos and Fractals in Chemical Engineering, Proceedings of the First National Conference, Rome, Italy, 25-27 May 1994, World Scientific, Singapore.
Box, G. E. P. and D. R. Cox (1964). An Analysis of Transformation, Journal of the Royal Statistical Society, B(26), 211-252.
Box, G. E. P. and G. W. Jenkins (1976). Time Series Analysis: Forecasting and Control, Revised Edition, Holden-Day, San Francisco.
Boxian, W. and L. M. Lye (1994). Identification of Temporal Scaling Behaviour of Flood: A Study of Fractals, Fractals, 2(2), 283-286.
Bras, R. L. and I. Rodriguez-Iturbe (1985). Random Functions and Hydrology, Addison-Wesley Publishing Company, Massachusetts.
Briggs, J. and F. D. Peat (1989). Turbulent Mirror: An Illustrated Guide to Chaos Theory and the Science of Wholeness, Harper & Row, Publishers, New York.
Cambel, A. B. (1993). Applied Chaos Theory: A Paradigm for Complexity, Academic Press, Inc., Boston.
Casdagli, M., D. D. Jardins, S. Eubank, J. D. Farmer, J. Gibson and J. Theiler (1992). Nonlinear Modeling of Chaotic Time Series: Theory and Applications; In: Applied Chaos, Edited by J. H. Kim and J. Stringer, John Wiley & Sons, Inc., New York, 335-380.
Casdagli, M. and S. Eubank /Editors (1992). Nonlinear Modeling and Forecasting, Proceedings of the Workshop on Nonlinear Modeling and Forecasting, Held September, 1990 in Santa Fe, New Mexico, Addison-Wesley Publishing Company, Redwood City, California.
Chen, F.-G., K.-X. Xue and W.-K. Cai (1998). The Chaotic Attractor of the Sediment Movement, Fractals, 6(2), 191-196.
Creedy, J. and V. L. Martin /Editors (1994). Chaos and Non-linear Models in Economics: Theory and Applications, Edward Elgar Publishing Limited, Hants.
Crilly A. J., R. A. Earnshaw and H. Jones (1993). Applications of Fractals and Chaos: The Shape of Things, Springer-Verlag, Berlin.
Crownover, R. M. (1995). Introduction to Fractals and Chaos, Jones and Bartlett Publishers, Boston.
Crutchfield, J. P. and B. S. McNamara (1987). Equations of Motion from a Data Series, Complex Systems, 1, 417-452.
Crutchfield, J. P. and K. Young (1989). Inferring Statistical Complexity, Physical Review Letters, 63(2), 105-108.
D''Alessandro, G. and A. Politi (1990). Hierachical Approach to Complexity with Applications to Dynamical Systems, Physical Review Letters, 64(14), 1609-1612.
Day, R. H. and P. Chen / Editors (1993). Nonlinear Dynamics & Evolutionary Economics, Oxford University Press, New York.
Devaney, R. L. (1992). A First Course in Chaotic Dynamical Systems, Addision-Wesley Publishing Company, Reading, Massachusetts.
Dwyer, I. J. and D. W. Reed (1994). Effective Fractal Dimension and Correction to the Mean of Annual Maxima, Journal of Hydrology, 157, 13-34.
Eckmann, J.-P., S. O. Kamphorst, D. Ruelle and S. Ciliberto (1986). Liapunov Exponents from Time Series, Physical Review A, 34(6), 4971-4979.
Elms, D. (1994). Forecasting in Financial Markets, In: Chaos and Non-linear Models in Economics: Theory and Applications, Edited by J. Creedy and V. L. Martin, Edward Elgar Publishing Limited, Hants, 169-186.
Ensley, D. and D. E. Nelson (1992). Extrapolation of Mackey-Glass Data Using Cascade Correlation, Simulation, 58(5), 333-339.
Eubank, S. G. and J. D. Farmer (1997). Introduction to Dynamical Systems (Chapter 5), In: Introduction to Nonlinear Physics, Edited by L. Lam, Springer-Veralg, New York, 55-105.
Evertsz, C. J. G., H.-O. Peitgen and R. F. Voss /Editors (1996). Fractal Geometry and Analysis: The Mandelbrrrot Festschrift, Curacao 1995, Curacao, Netherlands Antilles, World Scientific, Signapore.
Falconer, K. (1990). Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, Ltd., Chichester.
Falconer, K. (1997). Techniques in Fractal Geometry, John Wiley & Sons Ltd, Chichester.
Fan, L. T., D. Neogi and M. Yashima (1991). Elementary Introduction to Spatial and Temporal Fractals, Springer-Verlag Berlin Heidelberg.
Farmer, J. D. (1982a). Dimension, Fractal Measure and Chaotic Dynamics, In: Evolution of Order and Chaos, Edited by H. Haken, Springer-Verlag, Heidelberg, ???-???.
Farmer, J. D. (1982b). Information Dimension and the Probabilistic Structure of Chaos, Zeitschrift fur Naturforschung, 37a, 1304-1325.
Farmer, J. D. and J. J. Sidorowich (1987). Predicting Chaotic Time Series, Physics Review Letters, 59(8), 845-848.
Farmer, J. D. and J. J. Sidorowich (1988). Predicting Chaotic Dynamics, In: Dynamic Patterns in Complex Systems, Edited by J. A. S. Kelso, A. J. Mandell and M. F. Schlesinger, World Scientific, Singapore, 248-264.
Fausett, L. (1994). Fundamentals of Neural Networks: Architectures, Algorithms, and Applications, Prentice-Hall, Inc., Englewood Cliffs, New Jersey.
Feder, J. (1988). Fractals, Plenum Press, New York.
Finkenstadt, B. (1995). Nonlinear Dynamics in Economics: A Theoretical and Statistical Approach to Agricultural Markets, Lecture Notes in Economics and Mathematical System, 426, Springer-Verlag, Berlin.
Ford, J. (1986). Chaos: Solving the Unsolvable, Predicting the Unpredictable!, In: Chaotic Dynamics and Fractals, Edited by M. F. Barnsley and S. G. Demko, Academic Press, Inc., San Diego, 1-52.
Gefen, Y., Y. Meir, B. B. Mandelbrot and A. Aharony (1983). A Geometric Implementation of Hypercubic Lattices with Noninteger Dimensionality by Use of Low Lacunarity Fractal Lattices, Physics Review Letters, 50, 145-148.
Gefen, Y., A. Aharony and B. B. Mandelbrot (1984). Phase Transition on Fractals: III. Infinitely Ramified Lattices, J. Phys., A17, 1277-1289.
Georgakakos, K. P., M. B. Sharifi and P. L. Sturdevant (1995). Aanalysis of High-Resolution Rainfall Data, In: New Uncertainity Concepts in Hydrology and Water Resources, Edited by Z. W. Kundzewicz, Cambridge University Press, Cambridge, 114-120.
Ghilardi, P. and R. Rosso (1990). Comment on "Chaos in Rainfall" by I. Rodriguez-Iturbe et al., Water Resources Research, 26(8), 1837-1839.
Gouyet, J. F. (1996). Physics and Fractal Structures, Springer-Verlag, Berlin.
Granger, C. W. J. and T. Terasvirta (1993). Modelling Nonlinear Economic Relationships, Oxford University Press, Inc., New York.
Grassberger, P. and I. Procaccia (1983a). Characterization of Strange Attractors, Physical Review Letters, 50(5), 346-349.
Grassberger, P. and I. Procaccia (1983b). Measuring the Strangeness of Strange Attractors, Physical D, 9, 189-208.
Grassberger, P. and I. Procaccia (1984). Dimensions and Entropies of Strange Attractors from Fluctuating Dynamics Approach, Physica D, 13, 34-54.
Greenside, H. S., A Wolf, J. Swift and T. Pignataro (1982). Impracticality of a Box-Counting Algorithm for Calculating the Dimensionality of Strange Attractors, Physical Review A, 25(6), 3453-3456.
Gupta, V. K. and E. Waymire (1987). On Taylor''s Hypothesis and Dissipation in Rainfall, Journal of Geophysical Research, 92(D8), 9657-9660.
Gupta, V. K. and E. Waymire (1990). Multiscaling Properties of Spatial Rainfall and River Flow Distributions, Water Resources Research, 95(D3), 1999-2009.
Haan, C. T. (1977). Statistical Methods in Hydrology, Iowa State University Press, Iowa, 289-312.
Haastrup, P. and S. Funtowicz (1992). Accident Generating Systems and Chaos: A Dynamic Study of Accident Time Series, Reliability Engineering and System Safety, 35(1), 31-37.
Hao, B. L. (1991). Symbolic Dynamics and Characterization of Complexity, Physica D, 51, 161-176.
Hastings, H. M. and G. Sugihara (1993). Fractals: A User''s Guide for the Natural Sciences, Oxford University Press, Inc., New York.
Havstad, J. W. and C. L. Ehlers (1989). Attractor Dimension of Nonstationary Dynamical Systems from Small Data Sets, Physics Review A, 39, 845-853.
Hipel, K. W., A. I. McLeod, and W. C. Lennox (1977). Advances in Box-Jenkins Modeling, 1. Model Construction, Water Resources Research, 13(3), 567-575.
Hubert, P. (1995). Fractals Multifractals Appliques a L''etude De La Variabilite Temporelle Des Precipitations, In: Space and Time Scale Variability and Interdependencies in Hydrological Processes, Edited by R. A. Feddes, Cambridge University Press, Cambridge, 175-181. (In French)
Hubert, P., F. Friggit and J. P. Carbonnel (1995). Multifractal Structure of Rainfall Occurrence in West Africa, In: New Uncertainity Concepts in Hydrology and Water Resources, Edited by Z. W. Kundzewicz, Cambridge University Press, Cambridge, 109-113.
Hubert, P., Y. Tessier, S. Lovejoy, D. Schertzer, F. Schmitt, P. Ladoy, J. P. Carbonnel, S. Violette and I. Desurosne (1993). Multifractals and Extreme Rainfall Events, Geophysical Research Letters, 20(10), 931-934.
Hudson, J. L., M. Kube, R. A. Adomaitis, I. G. Kevrekidis, A. S. Lapedes and R. M. Farber (1990). Nonlinear Signal Processing and System Identification: Application to Time Series from Electrochemical Reactions, Chemical Engineering Science, 45(8), 2075-2081.
Itagaki, K., G. E. Lemieux and N. Ji (1995). Laser Scanning of Natural and Artificial Snow Packs, In: Fractal Aspects of Materials, Symposium Held November 28 - December 1, 1994, Boston, Massachusetts, USA, Edited by F. Family, P. Meakin, B. Sapoval and R. Wool, Materials Research Society, Pittdburgh, Pennsylvania, 379-384.
Jayawardena, A. W. and F. Lai (1994). Analysis and Prediction of Chaos in Rainfall Stream Flow Time Series, Journal of Hydrology, 153, 23-52.
Jeong, G. D. and A. R. Rao (1996). Chaos Characteristics of Tree Ring Series, Journal of Hydrology, 182, 239-257.
Kantz, H. and T. Schreiber (1997). Nonlinear Time Series Analysis, Cambridge University Press, Cambridge.
Kaplan, D. and L. Glass (1995). Understanding Nonlinear Dynamics, Springer-Verlag, New York.
Karlsson, M. and S. Yakowitz (1987). Nearest Neighbor Methods for Nonparametric Rainfall-Runoff Forecasting, Water Resources Research, 23(7), 1300-1308.
Kaspar, F. and H. G. Schuster (1987). Easily Calculable Measure for the Complexity of Spatiotemporal Patterns, Physical Review A, 36(2), 842-848.
Kedem, B. and L. S. Chiu (1987). Are Rain Rate Processes Self-Similar?, Water Resources Research, 23(10), 1816-1818.
Kember, G., A. C. Flower and J. Holubeshen (1993). Forecasting River Flow Using Nonlinear Dynamics, Stochastic Hydrology and Hydraulics, 7, 205-212.
Kim, J. H. and J. Stringer /Editors (1992). Applied Chaos, John Wiley & Sons, Inc., New York.
Kolmogorov, A. V. (1965). Three Approaches to the Quantitative Definition of Information, Problems on Information Transmission, 1, 1-7.
Korvin, G. (1992). Fractal Models in the Earth Sciences, Elsevier Science Publishers B.V., Amsterdam, 144-170.
Kottegoda, N. T. (1980). Stochastic Water Resources Technology, John Wiley & Sons, New York.
Kruhl, J. H. /Editor (1994). Fractals and Dynamic Systems in Geoscience, Springer-Berlag, Berlin.
Krzysztofowicz, R., G. Vachaud and P. van Cappellean /Editors (1996). Fractals, Scaling and Nonlinear Variability in Hydrology, Special Issue, 187(1-2), Journal of Hydrology, Elsevier, Amsterdam.
Lachtermacher, G. and J. D. Fuller (1994). Backpropagation in Hydrological Time Series Forecasting, In: Stochastic and Statistical Methods in Hydrology and Environmental Engineering, Volume 3, Time Series Analysis in Hydrology and Environmental Engineering, Edited by K. W. Hipel, A. I. McLeod, U. S. Panu and V. P. Singh, Kluwer Academic Publishers, Dordrecht, 229-242.
Lall, U., T. Sangoyomi and H. D. Abarbanel (1996). Nonlinear Dynamics of the Lake: Nonparametric Short-Term Forecasting, Water Resources Research, 32(4), 975-985.
Lam, L (1997). Introduction (Chapter 1), In: Introduction to the Nonlinear Physcis, Edited by L. Lam, Springer-Verlag, New York, 1-11.
LeFranc, M., D. Hennequin and P. Glorieux (1992). Improved Correlation Dimension Estimates Through Change of Variables, Physics Letter A, 163(4), 269-274.
Lempel, A. and J. Ziv (1976). On the Complexity of Finite Sequences, IEEE Transcations on Information Theory, IT-22(1), 75-81.
Lettenmaier, D. P. and E. F. Wood (1993). Hydrologic Forecasting (Chapter 26), In: Handbook of Hydrology, Edited by D. R. Maidment, McGraw-Hall, Inc., New York, 26.1-26.30.
Li, T.-Y. and Yorke, J. (1975). Period Three Implies Chaos, American Mathematics Monthly, 82, 985-992.
Lin, S.-C. and T.-Y. Lee (1998). Nonlinear Dynamics of Reservoir Watershed System: Inflow Simulation and Forecasting, The 3rd International Conference on Hydroscience and Engineering, August 31 - September 3, 1998, Cottbus/Berlin, Germany.
Lin, Y.-D., F.-C. Chong, S.-M. Sung, T.-S. Huo and C.-H. Liu (1998). The Calculation of Complexity in Normal and Apoplectic EEG Singals, Journal of the Chinese Institute of Engineers, 21(5), 585-594.
Liu, Q., S. Islam, I. Rodriguez-Iturbe and Y. Le (1998). Phase-Space Analysis of Daily Streamflow: Characterization and Prediction, Advances in Water Resources, 21(6), 463-476.
Lorenz, E. N. (1963). Deterministic Nonperiodic Flow, Journal of Atmosphere Sciences, 20, 130-141.
Lorenz, E. N. (1993). The Essence of Chaos, The University of Washington Press, Seattle.
Lovejoy, S. and D. Schertzer (1990). Multifractals, Universality Classes and Satellite and Radar Measurements of Cloud and Rain Fields, Journal of Geophysical Research, 95(D3), 2021-2034.
Lovejoy, S. and D. Schertzer (1985). Generalized Scale Invariance in the Atmosphere and Fractal Models of Rain, Water Resources Research, 21(8), 1233-1250.
Lovejoy, S., D. Schertzer and A. A. Tsonis (1987). Functional Box-Counting and Multiple Elliptical Dimensions in Rain, Science, 235, 1036-1038.
Mandelbrot, B. B. (1967). How Long is the Coast of Britain? Statistical Self-Similarity and Fractal Dimension, Science, 155, 636-638.
Mandelbrot, B. B. (1977). Fractals: Form, Chance, and Dimension, W. H. Freeman, San Fransisco.
Mandelbrot, B. B. (1982). The Fractal Geometry of Nature, W. H. Freeman and Company, New York.
Mayer, L. (1992). Fractal Characteristics of Desert Storm Sequences and Implications for Geomorphic Studies, Geomorphology, 5, 167-183.
McGuire, M. (1991). An Eye for Fractals: A Graphic & Photographic Essay, Addison-Wesley Publishing Company, Inc., Redwood City, California.
Merceron, T. and B. Velde (1991). Application of Cantor''s Method for Fractal Analysis of Fractures in the Toyaha Mine, Hokkaido, Japan, Journal of Geophysical Research, 96(B10), 16641-16650.
Minsky, M. and S. Rapert (1969). Perceptrons, MIT Press, Cambridge, MA.
Miyazima, S. (1996). Future of Fractals: Proceedings of the International Conference, Nagoya, Japan, World Scientific, Signapore.
Mullin, T. /Editor (1993). The Nature of Chaos, Clarendon Press, Oxford.
Nemec, J. (1986). Hydrological Forecasting: Design and Operation of Hydrological Forecasting Systems, D. Reidel Publishing Company, Dordrecht.
Nicolis, C. (1995). Predictability of the Atmosphere and Climate: Towards a Dynamical View, In: Space and Time Scale Variability and Interdependencies in Hydrological Processes, Edited by R. A. Feddes, Cambridge University Press, Cambridge, 145-152.
Nonnenmacher, T. F., G. A. Losa and E. R. Weibel /Editors (1994). Fractals in Biology and Medicine, Birkhauser Verlag, Basel.
Novak, M. M. and T. G. Dewey /Editors (1997). Fractal Frontiers, World Scientific, Singapore.
Novak, M. M. /Editor (1998). Fractals and Beyond: Complexities in the Sciences, World Scientific, Singapore.
Olsson, J., J. Niemczynowicz, R. Berndtsson and M. Larson (1992). An Analysis of the Rainfall Time Structure by Box Counting - Some Practical Implications, Journal of Hydrology, 137, 261-277.
Olsson, J., J. Niemczynowicz and R. Berndtsson (1993). Fractal Analysis of High-Resolution Rainfall Time Series, Journal of Geophysical Research - Atmospheres, 98(D12), 23265-23274.
Olsson, J. and J. Niemczynowicz (1994). On the Possible Use of Fractal Theory in Rainfall Allpications, Water Science and Technology, 29(1-2), 47-52.
Packard, N. H., J. P. Crutchfield, J. D. Farmer and R. S. Shaw (1980). Geometry from a Time Series, Physical Review Letters, 45(9), 712-715.
Pandey, G., S. Lovejoy and D. Schertzer (1998). Multifractal Analysis of Daily River Flows Including Extremes for Basins of Five to Two Million Square Kilometers, One Day to 75 Years, Journal of Hydrology, 208, 62-81.
Parker, D. B. (1985). Learning Logic, Technical Report TR-47,Center for Computational Research in Economics and Management Science, Massachusetts Institute of Technology, Cambridge, MA.
Peitgen, H.-O., H. Jurgens and D. Saupe (1992). Chaos and Fractals: New Frontiers of Science, Springer-Verlag New York, Inc.
Pesaran, M. H. and S. Potter /Editors (1993). Nonlinear Dynamics, Chaos and Econometrics, John Wiley & Sons Ltd., Chichester.
Peters, E. E. (1994). Fractal Market Analysis: Applying Chaos Theory to Investment and Economics, John Wiley & Sons Ltd, Chichester.
Peters, E. E. (1996). Chaos and Order in the Capital Markets: A New View of Cycles, Prices, and Market Volatility, Second Edition, John Wiley & Sons, Inc., New York.
Pfeifer, P. J. and M. Obert (1989). Fractals: Basic Concepts and Terminology, In: The Fractal Approach to Heterogeneous Chemistry, Edited by D. Avnir, John Wiley and Sons, Chichester, 11-52.
Porporato, A. and L. Ridolfi (1996). Clues to the Existence of Deterministic Chaos in River Flow, International Journal Modern Physics B, 10(15), 1821-1862.
Porporato, A. and L. Ridolfi (1997). Nonlinear Analysis of River Flow Time Sequences, Water Resources Research, 33(6), 1353-1367.
Provenzale, A., L. A. Smith, R. Vio and G. Murante (1992). Distinguishing Between Low-Dimensional Dynamics and Randomness in Measured Time Series, Physica D, 58, 31-49.
Radziejewski, M. and Z. W. Kundzewicz (1997). Fractal Analysis of Flow of the River Warta, Journal of Hydrology, 200, 280-294.
Rasband, S. N. (1990). Chaotic Dynamics of Nonlinear Systems, John Wiley & Sons, Inc., New York.
Rodriguez-Iturbe, I., B. F. De Power, M. B. Sharifi and K. P. Georgakakos (1989). Chaos in Rainfall, Water Resources Research, 25(7), 1667-1675.
Romanelli, L., M. A. Figliola and F. A. Hirsch (1988). Deterministic Chaos and Natural Phenomena, Journal of Statistical Physics, 53(3/4), 991-994.
Rumelhart, D. E., G. E. Hinton and R. J. Williams (1986a). Learning Internal Representations by Error Propagation, In: Parallel Distributed Processing: Explorations in the Microstructure of Cognition, Volume 1: Foundations, Edited by D. E. Rumelhart, J. L. McClelland and the PDP Research Group, The MIT Press, Cambridge, Massachusetts, 318-364.
Rumelhart, D. E., G. E. Hinton and R. J. Williams (1986b). Learning Representations by Error-Propagation Error, Nature, 323, 533-536.
Rzempoluck, E. J. (1998). Neural Network Data Analysis Using Simunet, Springer-Verlag New York, Inc., New Yoek.
Sangoyomi, T. B., U. Lall and H. D. I. Abarbanel (1996). Nonlinear Dynamics of the Great Salt Lake: Dimension Estimation, Water Resources Research, 32(1), 149-159.
Sarker, N. (1985). Box-Cox Transformation and the Problem of Heteroscedasticity, Comm, Statist.-Theor. Meth, 14(2), 363-379.
Saupe, D. and J. C. Hart /Organizers (1996). Fractal Models for Image Synthesis, Compression and Analysis, SIGGARPH 96, Course Notes, Course #27, 23rd International Conference on Computer Graphics and Interactive Techniques, New Orleans, Louisiana, USA.
Schaffer, W. M. and C. W. Tidd (1991). NLF: Nonlinear Forecasting for Dynamical Systems, Dynamical Systems, Inc., Tucson, Arizona.
Schertzer, D. and S. Lovejoy (1995). From Scalar Cascades to Lie Cascades: Joint Multifractal Analysis of Rain and Cloud Processes, In: Space and Time Scale Variability and Interdependencies in Hydrological Processes, Edited by R. A. Feddes, Cambridge University Press, Cambridge, 153-173.
Schertzer, D. and S. Lovejoy (1988). Multifractal Simulations and Analysis of Clouds by Multiplicative Processes, Atmospheric Research, 21, 337-361.
Schuster, H. G. (1995). Deterministic Chaos: An Introduction, Third Augmented Edition, VCH Verlagsgesellschaft mbH, D-6940 Weinheim.
Scott, S. K. (1994). Oscillations, Waves, and Chaos in Chemical Kinetics, Oxford University Press, Oxford.
Sevruk, B. and H. Geier (1981). Selection of Distribution Types for Extremes of Precipitation, Operational Hydrology Report No. 15, Secretariat of the World Meteorological Organization, Geneva.
Shamseldin, A. Y. and K. M. O''Connor (1996). A Nearest Neighbour Linear Perturbation Model for River Flow Forecasting, Journal of Hydrology, 179, 353-375.
Smalley, R. F., J. L. Chatelain, D. L. Turcotte and R. Prevot (1987). A Fractal Approach to the Clustering of Earthquakes: Applications to the Seismicity of the New Hebrides, Seis. Soc. Am. Bull., 77, 1368-1381.
Smith, L. A. (1988). Intrinstic Limits on Dimension Calculations, Physics Letters A, 133(6), 283-288.
Smith, M. (1993). Neural Networks for Statistical Modeling, Van Nostrand Reinhold, New York.
Snow, R. S. and L. Mayer (1992). Fractals in Geomorphology, Special Issue, Geomorphology, 5, Elsevier, Amsterdam.
Stoyan, D. and H. Stoyan (1994). Fractals, Random Shapes and Point Fields: Methods of Geometrical Statistics, John Wiley & Sons Ltd, Chichester.
Sugihara, G. and R. M. May (1990). Nonlinear Forecasting as a Way of Distinguishing Chaos from Measurement Error in a Data Series, Nature, 344, 734-741.
Sulis, W. and A. Combs /Editors (1996). Nonlinear Dynamics in Human Behavior, World Scientific, Signapore.
Svensson, C., J. Olsson and R. Berndtsson (1996). Multifractal Properties of Daily Rainfall in Two Different Climates, Water Resources Research, 32(8), 2463-2472.
Takens, F. (1981). Detecting Strange Attractors in Turbulence, In: Dynamical Systems and Turbulence, Warwick 1980, Lecture Notes in Mathematics, No. 898, Edited by D. A. Rand and L.-S. Young, Springer-Verlag, Berlin, 366-381.
Thompson, J. M. T. and S. R. Bishop /Editors (1994). Nonlinear Dynamics and Chaos in Engineering Dynamics, John Wiley & Sons Ltd., Chichester.
Tong, H. (1983). Threshold Models in Nonlinear Time Series, Lecture Notes in Statistics, Vol.21, Springer-Verlag, New York.
Tong, H. (1990). Nonlinear Time Series Analysis: A Dynamical Systems Approach, Oxford University Press.
Tong, H. /Editor (1995). Chaos and Forecasting, Proceedings of the Royal Society Discussion Meeting, London, 2-3 March 1994, World Scientific, Singapore.
Tsonis, A. A. (1992). Chaos: From Theory to Applications, Plenum Press, New York.
Turcotte, D. L. (1992). Fractals and Chaos in Geology and Geophyscis, Cambridge University Press, Cambridge.
Turcotte, D. L. (1994). Fractal Theory and the Estimation of Extreme Floods, Journal of Research of the National Institute of Standards and Technology, 99(4), 377-389.
Turcotte, D. L. (1997). Fractals and Chaos in Geology and Geophyscis, Second Edition, Cambridge University Press, Camdridge.
Turcotte, D. L. and L. Greene (1993). A Scale-Invariant Approach to Flood-Frequency Analysis, Stochastic Hydrology and Hydraulics, 7, 33-40.
Vallejo, L. E. /Editor (1997). Fractals in Engineering Geology, Special Issue, Engineering Geology, 48(3-4), Elsevier Science B.V.
Vassilicos, J. C., A. Demos and F. Tata (1993). No Evidence of Chaos But Some Evidence of Multifractals in the Foregin Exchange and the Stock Markets, In: Applications of Fractals and Chaos - The Shape of Things, Edited by A. J. Crilly, R. A. Earnshaw and H. Jones, Springer-Verlag, Berlin, pp. 249-265.
Vemuri, V. R. and R. D. Rogers (1994). Artificial Neural Networks: Forecasting Time Series, IEEE Computer Society Press, Los Alamitos, California.
Voss, R. F. (1988). Fractals in Nature: From Characterization to Simulation (Chapter 1), In: The Science of Fractal Images, Edited by M. F. Barnsley, R. L. Devaney, B. B. Mandelbrot, H.-O. Peitgen, D. Saupe and R. F. Voss, Springer-Verlag, Heidelberg, 21-70.
Waldrop, M. M. (1992). Complexity: The Emerging Science at the Edge of Order and Chaos, Penguin Books Ltd, London.
Wang, L. and D. L. Alkon (1993). Artificial Neural Networks: Oscillations, Chaos, and Sequence Processing, IEEE Computer Society Press, Los Alamitos, California.
Wang, Q. and T. Y. Gan (1998). Biases of Correlation Dimension Estimates od Streamflow Data in the Canadian Prairies, Water Resources Research, 34(9), 2329-2339.
Waymire, E. (1985). Scaling Limits and Self-similarity in Precipitation Fields, Water Resources Research, 21(8), 1271-1281.
Wegner, T. and B. Tyler (1993). Fractals Creations, Second Edition, The Waite Group, Inc. Corte Madera, CA.
Weigend, A. S. and N. A. Gershenfeld /Editors (1994). Time Series Prediction: Forecasting the Future and Understanding the Past, Proceedings of the NATO Advanced Research Workshop on Comparative Time Series Analysis, Santa Fe, New Mexico, May 1992, Addison-Wesley Publishing Company, Reading, Massachusetts.
Werbos, P. J. (1974). Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences, Ph.D. Thesis, Harvard University, Cambridge, MA.
Werbos, P. J. (1994). The Roots of Backpropagation: From Ordered Derivatives to Neural Networks and Ploitical Forecasting, John Wiley & Sons, Inc., New York.
Wilcox, B. P., M. S. Seyfried and T. H. Matison (1991). Searching for Chaotic Dynamics in Snowmelt Runoff, Water Resources Research, 27(6), pp. 1005-1010.
Wolf, A. (1986). Quantifying Chaos with Lyapunov Exponents, In: Chaos, Edited by A. V. Holden, Manchester University Press, Manchester, 273-290.
W