學術產出-Theses
Article View/Open
Publication Export
-
題名 記憶效應下馬可夫鏈的尺度不變性初探
A Study of Scaling Behavior in Binary Markov Chains caused by Memory Effect作者 馮信堯
Feng, Hsin-Yao貢獻者 馬文忠
Ma, Wen-Jong
馮信堯
Feng, Hsin-Yao關鍵詞 二位元序列
滾動視窗
埃倫費斯特模型
馬可夫鏈
縮放行為
Binary sequence
Rolling window
Ehrenfest urn model
Markov chain
Scaling behavior日期 2023 上傳時間 1-Sep-2023 16:27:34 (UTC+8) 摘要 透過滾動視窗逐步生成的方式，我們產生一組"0"與"1"的二位元序列，其每一步新位元由前幾步的位元以機率性的方式決定，最終序列會收斂至特定狀態，此過程可以透過廣義的埃倫費斯特模型的物理機制來理解。此機制決定新的位元係由由"0"與"1"個數的比例決定，其中較少的具有較高的機率出現（既少數決定規則），此程序會趨向一種穩定狀態，即"0"與"1"出現的機會相等的狀態，此過程為一種負回饋機制，相當於是一種擴散過程。若將此機制改成正回饋機制（既多數決定規則），此序列將收斂至其中一個全為"0"或全為"1"的狀態。這種過程中的自我放大的機制能夠捕捉金融市場不穩定性的主要特徵。我們特別考慮到的是，兩種反饋機制控制的收斂過程均顯示出尺度不變性的特性。
A series of binary digits generated sequentially with its next bit determined probabilistically by the preceding bits within a rolling window, will converge to specific sets of states, that can be comprehended by the physics of Urn models. A sequence with its new bit determined by a probabilistic minority rule converges to the states with equal probability, to have either of the two binary digits ("0" and "1"). Such a process, with its new bit produced by a procedure of negative feedback, is equivalent to that of diffusion. The process to produce a sequence via a probabilistic majority rule (a procedure of positive feedback), on the other hand, leads the sequence to converge to the states, with probability exact one, to have exclusive only one of the two binary digits. The self-amplified effect in the process captures the major feature in an unstable situation in financal markets. The convergences, governed by either of the two kinds of feedback procedures, are found to show scaling behavior.參考文獻 [Bachelier, 1900] Bachelier, L. (1900). Theorie de la sp ´ eculation. In ´ Annales scientifiquesde l’Ecole normale sup ´ erieure ´ , volume 17, pages 21–86.[Black and Scholes, 1973] Black, F. and Scholes, M. (1973). The pricing of options andcorporate liabilities. Journal of political economy, 81(3):637–654.[Chen, 2015] Chen, C.-W. (2015). Time characteristic in cross correlation of stock fluctuations. Master’s thesis, National Chengchi University.[Hod and Keshet, 2004] Hod, S. and Keshet, U. (2004). Phase transition in random walkswith long-range correlations. Physical Review E, 70(1):015104.[Kac, 1947] Kac, M. (1947). Random walk and the theory of brownian motion. TheAmerican Mathematical Monthly, 54(7P1):369–391.[Keshet and Hod, 2005] Keshet, U. and Hod, S. (2005). Survival probabilities of historydependent random walks. Physical Review E, 72(4):046144.[Laloux et al., 1999] Laloux, L., Cizeau, P., Bouchaud, J.-P., and Potters, M. (1999).Noise dressing of financial correlation matrices. Physical review letters, 83(7):1467.[Lay et al., 2016] Lay, D. C., Lay, S. R., and McDonald, J. J. (2016). Linear algebra andits applications. Pearson.[Li, 2021] Li, Y.-C. (2021). A primitive study of memory effect of run of ones in binarysequences. Master’s thesis, National Chengchi University[Lutz and Ciliberto, 2015] Lutz, E. and Ciliberto, S. (2015). Information: Frommaxwell’s demon to landauer’s eraser. Physics Today, 68(9):30–35.[Ma et al., 2004] Ma, W.-J., Hu, C.-K., and Amritkar, R. E. (2004). Stochastic dynamicalmodel for stock-stock correlations. Physical Review E, 70(2):026101.[Ma et al., 2013] Ma, W.-J., Wang, S.-C., Chen, C.-N., and Hu, C.-K. (2013). Crossoverbehavior of stock returns and mean square displacements of particles governed by thelangevin equation. Europhysics Letters, 102(6):66003.[Mandelbrot, 1960] Mandelbrot, B. (1960). The pareto-levy law and the distribution ofincome. International economic review, 1(2):79–106.[Mandelbrot, 1963] Mandelbrot, B. (1963). New methods in statistical economics. Journal of political economy, 71(5):421–440.[Mandelbrot, 2013] Mandelbrot, B. B. (2013). Fractals and scaling in finance: Discontinuity, concentration, risk. Selecta volume E. Springer Science & Business Media.[Mantegna and Stanley, 1995] Mantegna, R. N. and Stanley, H. E. (1995). Scaling behaviour in the dynamics of an economic index. Nature, 376(6535):46–49.[Osborne, 1967] Osborne, M. (1967). Some quantitative tests for stock price generating models and trading folklore. Journal of the American Statistical Association,62(318):321–340.[Osborne, 1959] Osborne, M. F. (1959). Brownian motion in the stock market. Operationsresearch, 7(2):145–173.[Papoulis and Unnikrishna Pillai, 2002] Papoulis, A. and Unnikrishna Pillai, S. (2002).Probability, random variables and stochastic processes.[Plerou et al., 1999] Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L. A. N., andStanley, H. E. (1999). Universal and nonuniversal properties of cross correlations infinancial time series. Physical review letters, 83(7):1471.[Sagawa, 2012] Sagawa, T. (2012). Thermodynamics of information processing in smallsystems. Progress of theoretical physics, 127(1):1–56.[Shih, 2017] Shih, Y.-F. (2017). Scaling of event-occurrence in stock market. Master’sthesis, Shih, Yi-Fu.[Sornette, 2009] Sornette, D. (2009). Why stock markets crash: critical events in complexfinancial systems. Princeton university press.[Strang, 2006] Strang, G. (2006). Linear algebra and its applications. Belmont, CA:Thomson, Brooks/Cole.[West, 2018] West, G. (2018). Scale: The universal laws of life, growth, and death inorganisms, cities, and companies. Penguin.[Wu, 2018] Wu, P.-Y. (2018). Some hydrodynamic signatures in fluctuations of stockprice changes–crossing day effect. Master’s thesis, National Chengchi University.[Zivot et al., 2003] Zivot, E., Wang, J., Zivot, E., and Wang, J. (2003). Rolling analysisof time series. Modeling financial time series with S-Plus®, pages 299–346.[李政道, 2006] 李政道 (2006). 李政道講義：統計力學. 上海科學技術出版社.[林清涼, 2014] 林清涼 (2014). 從物理學切入的線性代數導論. 五南圖書. 描述 碩士
國立政治大學
應用物理研究所
109755008資料來源 http://thesis.lib.nccu.edu.tw/record/#G0109755008 資料類型 thesis dc.contributor.advisor 馬文忠 zh_TW dc.contributor.advisor Ma, Wen-Jong en_US dc.contributor.author (Authors) 馮信堯 zh_TW dc.contributor.author (Authors) Feng, Hsin-Yao en_US dc.creator (作者) 馮信堯 zh_TW dc.creator (作者) Feng, Hsin-Yao en_US dc.date (日期) 2023 en_US dc.date.accessioned 1-Sep-2023 16:27:34 (UTC+8) - dc.date.available 1-Sep-2023 16:27:34 (UTC+8) - dc.date.issued (上傳時間) 1-Sep-2023 16:27:34 (UTC+8) - dc.identifier (Other Identifiers) G0109755008 en_US dc.identifier.uri (URI) http://nccur.lib.nccu.edu.tw/handle/140.119/147294 - dc.description (描述) 碩士 zh_TW dc.description (描述) 國立政治大學 zh_TW dc.description (描述) 應用物理研究所 zh_TW dc.description (描述) 109755008 zh_TW dc.description.abstract (摘要) 透過滾動視窗逐步生成的方式，我們產生一組"0"與"1"的二位元序列，其每一步新位元由前幾步的位元以機率性的方式決定，最終序列會收斂至特定狀態，此過程可以透過廣義的埃倫費斯特模型的物理機制來理解。此機制決定新的位元係由由"0"與"1"個數的比例決定，其中較少的具有較高的機率出現（既少數決定規則），此程序會趨向一種穩定狀態，即"0"與"1"出現的機會相等的狀態，此過程為一種負回饋機制，相當於是一種擴散過程。若將此機制改成正回饋機制（既多數決定規則），此序列將收斂至其中一個全為"0"或全為"1"的狀態。這種過程中的自我放大的機制能夠捕捉金融市場不穩定性的主要特徵。我們特別考慮到的是，兩種反饋機制控制的收斂過程均顯示出尺度不變性的特性。 zh_TW dc.description.abstract (摘要) A series of binary digits generated sequentially with its next bit determined probabilistically by the preceding bits within a rolling window, will converge to specific sets of states, that can be comprehended by the physics of Urn models. A sequence with its new bit determined by a probabilistic minority rule converges to the states with equal probability, to have either of the two binary digits ("0" and "1"). Such a process, with its new bit produced by a procedure of negative feedback, is equivalent to that of diffusion. The process to produce a sequence via a probabilistic majority rule (a procedure of positive feedback), on the other hand, leads the sequence to converge to the states, with probability exact one, to have exclusive only one of the two binary digits. The self-amplified effect in the process captures the major feature in an unstable situation in financal markets. The convergences, governed by either of the two kinds of feedback procedures, are found to show scaling behavior. en_US dc.description.tableofcontents 1 Introduction 11.1 History 11.2 Motivation 31.3 Method 42 Models 82.1 Scenario in Financial Market 82.2 Urn Model 92.2.1 Analyzing Feedback Mechanism an Urn Model using Markov Chain 112.3 Binary Sequence 152.3.1 Rolling Window (Shift Window) 152.3.2 Modeling Feedback in a Rolling Window Model using Markov Chains 162.3.3 Bi-state Urn Model 213 Simulation Results Analysis 233.1 Memory Effects in the Urn Model: Dynamics and Scaling 233.1.1 Urn Model with Positive Feedback 243.1.2 Urn Model with Negative Feedback 273.2 Memory Effects in the Rolling Window Model: Dynamics and Scaling 303.2.1 Rolling Window with Positive Feedback 313.2.2 Rolling Window with Negative Feedback 344 Discussion 394.1 Understanding the Concept of Scaling 394.2 Evolution of Dynamical Systems in the Context of Markov Chains 424.2.1 Urn Model 424.2.2 Binary Sequence 464.2.3 Graphing the Eigenvalues 564.3 Exploring Combinations of Various Feedback Mechanisms and Dynamical Systems 575 Concluding Remarks 60Appendix 63A.1 Markov Chain 63A.2 Ehrenfest Urn Model 66References 67 zh_TW dc.format.extent 32545714 bytes - dc.format.mimetype application/pdf - dc.source.uri (資料來源) http://thesis.lib.nccu.edu.tw/record/#G0109755008 en_US dc.subject (關鍵詞) 二位元序列 zh_TW dc.subject (關鍵詞) 滾動視窗 zh_TW dc.subject (關鍵詞) 埃倫費斯特模型 zh_TW dc.subject (關鍵詞) 馬可夫鏈 zh_TW dc.subject (關鍵詞) 縮放行為 zh_TW dc.subject (關鍵詞) Binary sequence en_US dc.subject (關鍵詞) Rolling window en_US dc.subject (關鍵詞) Ehrenfest urn model en_US dc.subject (關鍵詞) Markov chain en_US dc.subject (關鍵詞) Scaling behavior en_US dc.title (題名) 記憶效應下馬可夫鏈的尺度不變性初探 zh_TW dc.title (題名) A Study of Scaling Behavior in Binary Markov Chains caused by Memory Effect en_US dc.type (資料類型) thesis en_US dc.relation.reference (參考文獻) [Bachelier, 1900] Bachelier, L. (1900). Theorie de la sp ´ eculation. In ´ Annales scientifiquesde l’Ecole normale sup ´ erieure ´ , volume 17, pages 21–86.[Black and Scholes, 1973] Black, F. and Scholes, M. (1973). The pricing of options andcorporate liabilities. Journal of political economy, 81(3):637–654.[Chen, 2015] Chen, C.-W. (2015). Time characteristic in cross correlation of stock fluctuations. Master’s thesis, National Chengchi University.[Hod and Keshet, 2004] Hod, S. and Keshet, U. (2004). Phase transition in random walkswith long-range correlations. Physical Review E, 70(1):015104.[Kac, 1947] Kac, M. (1947). Random walk and the theory of brownian motion. TheAmerican Mathematical Monthly, 54(7P1):369–391.[Keshet and Hod, 2005] Keshet, U. and Hod, S. (2005). Survival probabilities of historydependent random walks. Physical Review E, 72(4):046144.[Laloux et al., 1999] Laloux, L., Cizeau, P., Bouchaud, J.-P., and Potters, M. (1999).Noise dressing of financial correlation matrices. Physical review letters, 83(7):1467.[Lay et al., 2016] Lay, D. C., Lay, S. R., and McDonald, J. J. (2016). Linear algebra andits applications. Pearson.[Li, 2021] Li, Y.-C. (2021). A primitive study of memory effect of run of ones in binarysequences. Master’s thesis, National Chengchi University[Lutz and Ciliberto, 2015] Lutz, E. and Ciliberto, S. (2015). Information: Frommaxwell’s demon to landauer’s eraser. Physics Today, 68(9):30–35.[Ma et al., 2004] Ma, W.-J., Hu, C.-K., and Amritkar, R. E. (2004). Stochastic dynamicalmodel for stock-stock correlations. Physical Review E, 70(2):026101.[Ma et al., 2013] Ma, W.-J., Wang, S.-C., Chen, C.-N., and Hu, C.-K. (2013). Crossoverbehavior of stock returns and mean square displacements of particles governed by thelangevin equation. Europhysics Letters, 102(6):66003.[Mandelbrot, 1960] Mandelbrot, B. (1960). The pareto-levy law and the distribution ofincome. International economic review, 1(2):79–106.[Mandelbrot, 1963] Mandelbrot, B. (1963). New methods in statistical economics. Journal of political economy, 71(5):421–440.[Mandelbrot, 2013] Mandelbrot, B. B. (2013). Fractals and scaling in finance: Discontinuity, concentration, risk. Selecta volume E. Springer Science & Business Media.[Mantegna and Stanley, 1995] Mantegna, R. N. and Stanley, H. E. (1995). Scaling behaviour in the dynamics of an economic index. Nature, 376(6535):46–49.[Osborne, 1967] Osborne, M. (1967). Some quantitative tests for stock price generating models and trading folklore. Journal of the American Statistical Association,62(318):321–340.[Osborne, 1959] Osborne, M. F. (1959). Brownian motion in the stock market. Operationsresearch, 7(2):145–173.[Papoulis and Unnikrishna Pillai, 2002] Papoulis, A. and Unnikrishna Pillai, S. (2002).Probability, random variables and stochastic processes.[Plerou et al., 1999] Plerou, V., Gopikrishnan, P., Rosenow, B., Amaral, L. A. N., andStanley, H. E. (1999). Universal and nonuniversal properties of cross correlations infinancial time series. Physical review letters, 83(7):1471.[Sagawa, 2012] Sagawa, T. (2012). Thermodynamics of information processing in smallsystems. Progress of theoretical physics, 127(1):1–56.[Shih, 2017] Shih, Y.-F. (2017). Scaling of event-occurrence in stock market. Master’sthesis, Shih, Yi-Fu.[Sornette, 2009] Sornette, D. (2009). Why stock markets crash: critical events in complexfinancial systems. Princeton university press.[Strang, 2006] Strang, G. (2006). Linear algebra and its applications. Belmont, CA:Thomson, Brooks/Cole.[West, 2018] West, G. (2018). Scale: The universal laws of life, growth, and death inorganisms, cities, and companies. Penguin.[Wu, 2018] Wu, P.-Y. (2018). Some hydrodynamic signatures in fluctuations of stockprice changes–crossing day effect. Master’s thesis, National Chengchi University.[Zivot et al., 2003] Zivot, E., Wang, J., Zivot, E., and Wang, J. (2003). Rolling analysisof time series. Modeling financial time series with S-Plus®, pages 299–346.[李政道, 2006] 李政道 (2006). 李政道講義：統計力學. 上海科學技術出版社.[林清涼, 2014] 林清涼 (2014). 從物理學切入的線性代數導論. 五南圖書. zh_TW
