透過貪婪分割法偵測函數型資料序列的多重轉換點

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題名	透過貪婪分割法偵測函數型資料序列的多重轉換點 Multiple changepoints detection for functional data sequence through greedy segmentation
作者	陳裕庭 Chen, Yu-Ting
貢獻者	黃子銘<br>丘政民 Huang, Tzee-Ming<br>Chiou, Jeng-Min 陳裕庭 Chen, Yu-Ting
關鍵詞	多重轉換點問題函數型主成分分析共變函數算子假設檢定 Multiple changepoint problem Functional principal component analysis Covariance operator Hypothesis testing
日期	2022
上傳時間	1-Jun-2022 16:26:02 (UTC+8)
摘要	在本研究中，我們針對函數型資料序列中的多重轉換點偵測問題提出了適當的準則，透過此準則可將多重轉換點視作該準則下的$M$維最佳分割。但由於轉換點個數$M$為未知，在給定不同的$K$值的狀況下，我們進一步探討$K$維最佳分割與多重轉換點之間的關係，並且發現無論$K$相對於$M$大小，透過最佳分割作為轉換點的估計式皆展現了理論上的一致性。其中，應用$K同時，基於貪婪分割法估計式，我們同時提出了與之相關的檢定統計量，透過該檢定統計量在不同情境下的漸近分布來估計轉換點的個數，並給出具體的演算法。針對貪婪分割法在實務上的表現，我們透過一系列的模擬研究以及實例分析來加以驗證。 In this study, we propose a criterion for multiple changepoint detection in a functional data sequence. Using the proposed criterion, the set of multiple changepoints can be characterized as an optimal $M$-segmentation. However, because the number of changepoints $M$ is unknown, we further investigate the theoretical properties of the optimal $K$-segmentation with respect to different values of $K$. It turns out the optimal $K$-segmentation is always consistent no matter when $K\\geq M$ or when $K< M$. Using the consistency result when $K< M$, we propose Greedy Segmentation estimator, which is as efficient as Binary Segmentation and holds the consistency property without any assumption related to the at-most-one-changepoint assumption. Meanwhile, we also propose a test statistic based on the Greedy Segmentation estimator, whose asymptotic distribution is helpful in estimating the number of changepoints $M$. The whole procedure is integrated as an algorithm that is easy to apply. Finally, we study the finite-sample performance of Greedy Segmentation algorithm through simulation study and data applications
參考文獻	[1] Donald WK Andrews. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3):817–858, 1991. [2] John AD Aston and Claudia Kirch. Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis, 109(4):204–220, 2012. [3] Alexander Aue, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis, 100(10):2254–2269, 2009. [4] Alexander Aue, Gregory Rice, and Ozan Sönmez. Detecting and dating structural breaks in functional data without dimension reduction. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(3):509–529, 2018. [5] Alexander Aue, Gregory Rice, and Ozan Sönmez. Structural break analysis for spectrum and trace of covariance operators. Environmetrics, 31(1):e2617, 2020. [6] Ivan E Auger and Charles E Lawrence. Algorithms for the optimal identification of segment neighborhoods. Bulletin of mathematical biology, 51(1):39–54, 1989. [7] István Berkes, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(5):927–946, 2009. [8] Yu-Ting Chen, Jeng-Min Chiou, and Tzee-Ming Huang. Greedy segmentation for a functional data sequence. Journal of the American Statistical Association, DOI: 10.1080/01621459.2021.1963261, 2021. [9] Jeng-Min Chiou, Yu-Ting Chen, and Tailen Hsing. Identifying multiple changes for a functional data sequence with application to freeway traffic segmentation. The Annals of Applied Statistics, 13(3):1430–1463, 2019. [10] Holger Dette and Tim Kutta. Detecting structural breaks in eigensystems of functional time series. Electronic Journal of Statistics, 15(1):944–983, 2021. [11] Piotr Fryzlewicz. Wild binary segmentation for multiple change-point detection. The Annals of Statistics, 42(6):2243–2281, 2014. [12] Abdullah Gedikli, Hafzullah Aksoy, N Erdem Unal, and Athanasios Kehagias. Modified dynamic programming approach for offline segmentation of long hydrometeorological time series. Stochastic Environmental Research and Risk Assessment, 24(5):547–557, 2010. [13] Oleksandr Gromenko, Piotr Kokoszka, and Matthew Reimherr. Detection of change in the spatiotemporal mean function. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(1):29–50, 2017. [14] Zaıd Harchaoui and Céline Lévy-Leduc. Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association, 105(492):1480–1493, 2010. [15] Siegfried Hörmann, Lukasz Kidziński, and Marc Hallin. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(2):319–348, 2015. [16] Siegfried Hörmann and Piotr Kokoszka. Weakly dependent functional data. The Annals of Statistics, 38(3):1845–1884, 2010. [17] Lajos Horváth, Curtis Miller, and Gregory Rice. A new class of change point test statistics of rényi type. Journal of Business & Economic Statistics, 38(3):570–579, 2020. [18] Brad Jackson, Jeffrey D Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, Peter Gioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, and Tun Tao Tsai. An algorithm for optimal partitioning of data on an interval. IEEE Signal Processing Letters, 12(2):105–108, 2005. [19] Daniela Jarušková. Testing for a change in covariance operator. Journal of Statistical Planning and Inference, 143(9):1500–1511, 2013. [20] Rebecca Killick, Paul Fearnhead, and Idris A Eckley. Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500):1590–1598, 2012. [21] Robert Maidstone, Toby Hocking, Guillem Rigaill, and Paul Fearnhead. On optimal multiple changepoint algorithms for large data. Statistics and Computing, 27(2):519–533, 2017. [22] Adam B Olshen, ES Venkatraman, Robert Lucito, and Michael Wigler. Circular binary segmentation for the analysis of arraybased dna copy number data. Biostatistics, 5(4): 557–572, 2004. [23] ES Page. A test for a change in a parameter occurring at an unknown point. Biometrika, 42(3/4):523–527, 1955. [24] David E Parker, Tim P Legg, and Chris K Folland. A new daily central england temperature series, 1772–1991. International Journal of Climatology, 12(4):317–342, 1992. [25] Olimjon Sh Sharipov and Martin Wendler. Bootstrapping covariance operators of functional time series. Journal of Nonparametric Statistics, 32(3):648–666, 2020. [26] Ada W van der Vaart and Jon A Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, NewYork, 1996. [27] Tengyao Wang and Richard J Samworth. High dimensional change point estimation via sparse projection. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(1):57–83, 2018.
描述	博士國立政治大學統計學系 104354501
資料來源	http://thesis.lib.nccu.edu.tw/record/#G0104354501
資料類型	thesis

dc.contributor.advisor	黃子銘<br>丘政民	zh_TW
dc.contributor.advisor	Huang, Tzee-Ming<br>Chiou, Jeng-Min	en_US
dc.contributor.author (Authors)	陳裕庭	zh_TW
dc.contributor.author (Authors)	Chen, Yu-Ting	en_US
dc.creator (作者)	陳裕庭	zh_TW
dc.creator (作者)	Chen, Yu-Ting	en_US
dc.date (日期)	2022	en_US
dc.date.accessioned	1-Jun-2022 16:26:02 (UTC+8)	-
dc.date.available	1-Jun-2022 16:26:02 (UTC+8)	-
dc.date.issued (上傳時間)	1-Jun-2022 16:26:02 (UTC+8)	-
dc.identifier (Other Identifiers)	G0104354501	en_US
dc.identifier.uri (URI)	http://nccur.lib.nccu.edu.tw/handle/140.119/140200	-
dc.description (描述)	博士	zh_TW
dc.description (描述)	國立政治大學	zh_TW
dc.description (描述)	統計學系	zh_TW
dc.description (描述)	104354501	zh_TW
dc.description.abstract (摘要)	在本研究中，我們針對函數型資料序列中的多重轉換點偵測問題提出了適當的準則，透過此準則可將多重轉換點視作該準則下的$M$維最佳分割。但由於轉換點個數$M$為未知，在給定不同的$K$值的狀況下，我們進一步探討$K$維最佳分割與多重轉換點之間的關係，並且發現無論$K$相對於$M$大小，透過最佳分割作為轉換點的估計式皆展現了理論上的一致性。其中，應用$K同時，基於貪婪分割法估計式，我們同時提出了與之相關的檢定統計量，透過該檢定統計量在不同情境下的漸近分布來估計轉換點的個數，並給出具體的演算法。針對貪婪分割法在實務上的表現，我們透過一系列的模擬研究以及實例分析來加以驗證。	zh_TW
dc.description.abstract (摘要)	In this study, we propose a criterion for multiple changepoint detection in a functional data sequence. Using the proposed criterion, the set of multiple changepoints can be characterized as an optimal $M$-segmentation. However, because the number of changepoints $M$ is unknown, we further investigate the theoretical properties of the optimal $K$-segmentation with respect to different values of $K$. It turns out the optimal $K$-segmentation is always consistent no matter when $K\\geq M$ or when $K< M$. Using the consistency result when $K< M$, we propose Greedy Segmentation estimator, which is as efficient as Binary Segmentation and holds the consistency property without any assumption related to the at-most-one-changepoint assumption. Meanwhile, we also propose a test statistic based on the Greedy Segmentation estimator, whose asymptotic distribution is helpful in estimating the number of changepoints $M$. The whole procedure is integrated as an algorithm that is easy to apply. Finally, we study the finite-sample performance of Greedy Segmentation algorithm through simulation study and data applications	en_US
dc.description.tableofcontents	致謝 i 中文摘要 ii Abstract iii Contents iv List of Tables vi List of Figures vii 1 簡介 1 1.1 文獻回顧 1 1.2 研究方法簡介 4 2 函數型轉換點模型與轉換點估計式 6 2.1 多重轉換點模型 6 2.2 轉換點估計式與其相關性質 7 2.3 神諭子空間與函數型轉換點的可偵測性 11 3 貪婪分割估計式與其相關性質 13 3.1 轉換點之貪婪分割估計式 13 3.2 貪婪分割演算法 15 4 模擬研究 17 4.1 模擬設定 17 4.2 貪婪分割演算法對參數 h 的敏感性測試 18 4.3 方法比較 20 4.3.1 貪婪分割法與動態分割法之比較 20 4.3.2 貪婪分割演算法與二元分割法之比較 22 5 實際資料分析 27 5.1 英國中部氣溫資料集 27 5.2 澳洲氣象站資料 28 6 結語 31 A 理論證明 33 A.1 定理 2.2.1 證明 34 A.1.1 引理 A.1.1 證明 36 A.2 定理 2.3.1 證明 40 A.3 定理 3.1.1 證明 40 A.4 定理 3.2.1 證明 41 Bibliography 44	zh_TW
dc.format.extent	1057255 bytes	-
dc.format.mimetype	application/pdf	-
dc.source.uri (資料來源)	http://thesis.lib.nccu.edu.tw/record/#G0104354501	en_US
dc.subject (關鍵詞)	多重轉換點問題	zh_TW
dc.subject (關鍵詞)	函數型主成分分析	zh_TW
dc.subject (關鍵詞)	共變函數算子	zh_TW
dc.subject (關鍵詞)	假設檢定	zh_TW
dc.subject (關鍵詞)	Multiple changepoint problem	en_US
dc.subject (關鍵詞)	Functional principal component analysis	en_US
dc.subject (關鍵詞)	Covariance operator	en_US
dc.subject (關鍵詞)	Hypothesis testing	en_US
dc.title (題名)	透過貪婪分割法偵測函數型資料序列的多重轉換點	zh_TW
dc.title (題名)	Multiple changepoints detection for functional data sequence through greedy segmentation	en_US
dc.type (資料類型)	thesis	en_US
dc.relation.reference (參考文獻)	[1] Donald WK Andrews. Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica, 59(3):817–858, 1991. [2] John AD Aston and Claudia Kirch. Detecting and estimating changes in dependent functional data. Journal of Multivariate Analysis, 109(4):204–220, 2012. [3] Alexander Aue, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Estimation of a change-point in the mean function of functional data. Journal of Multivariate Analysis, 100(10):2254–2269, 2009. [4] Alexander Aue, Gregory Rice, and Ozan Sönmez. Detecting and dating structural breaks in functional data without dimension reduction. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(3):509–529, 2018. [5] Alexander Aue, Gregory Rice, and Ozan Sönmez. Structural break analysis for spectrum and trace of covariance operators. Environmetrics, 31(1):e2617, 2020. [6] Ivan E Auger and Charles E Lawrence. Algorithms for the optimal identification of segment neighborhoods. Bulletin of mathematical biology, 51(1):39–54, 1989. [7] István Berkes, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Detecting changes in the mean of functional observations. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 71(5):927–946, 2009. [8] Yu-Ting Chen, Jeng-Min Chiou, and Tzee-Ming Huang. Greedy segmentation for a functional data sequence. Journal of the American Statistical Association, DOI: 10.1080/01621459.2021.1963261, 2021. [9] Jeng-Min Chiou, Yu-Ting Chen, and Tailen Hsing. Identifying multiple changes for a functional data sequence with application to freeway traffic segmentation. The Annals of Applied Statistics, 13(3):1430–1463, 2019. [10] Holger Dette and Tim Kutta. Detecting structural breaks in eigensystems of functional time series. Electronic Journal of Statistics, 15(1):944–983, 2021. [11] Piotr Fryzlewicz. Wild binary segmentation for multiple change-point detection. The Annals of Statistics, 42(6):2243–2281, 2014. [12] Abdullah Gedikli, Hafzullah Aksoy, N Erdem Unal, and Athanasios Kehagias. Modified dynamic programming approach for offline segmentation of long hydrometeorological time series. Stochastic Environmental Research and Risk Assessment, 24(5):547–557, 2010. [13] Oleksandr Gromenko, Piotr Kokoszka, and Matthew Reimherr. Detection of change in the spatiotemporal mean function. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(1):29–50, 2017. [14] Zaıd Harchaoui and Céline Lévy-Leduc. Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association, 105(492):1480–1493, 2010. [15] Siegfried Hörmann, Lukasz Kidziński, and Marc Hallin. Dynamic functional principal components. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 77(2):319–348, 2015. [16] Siegfried Hörmann and Piotr Kokoszka. Weakly dependent functional data. The Annals of Statistics, 38(3):1845–1884, 2010. [17] Lajos Horváth, Curtis Miller, and Gregory Rice. A new class of change point test statistics of rényi type. Journal of Business & Economic Statistics, 38(3):570–579, 2020. [18] Brad Jackson, Jeffrey D Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, Peter Gioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, and Tun Tao Tsai. An algorithm for optimal partitioning of data on an interval. IEEE Signal Processing Letters, 12(2):105–108, 2005. [19] Daniela Jarušková. Testing for a change in covariance operator. Journal of Statistical Planning and Inference, 143(9):1500–1511, 2013. [20] Rebecca Killick, Paul Fearnhead, and Idris A Eckley. Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500):1590–1598, 2012. [21] Robert Maidstone, Toby Hocking, Guillem Rigaill, and Paul Fearnhead. On optimal multiple changepoint algorithms for large data. Statistics and Computing, 27(2):519–533, 2017. [22] Adam B Olshen, ES Venkatraman, Robert Lucito, and Michael Wigler. Circular binary segmentation for the analysis of arraybased dna copy number data. Biostatistics, 5(4): 557–572, 2004. [23] ES Page. A test for a change in a parameter occurring at an unknown point. Biometrika, 42(3/4):523–527, 1955. [24] David E Parker, Tim P Legg, and Chris K Folland. A new daily central england temperature series, 1772–1991. International Journal of Climatology, 12(4):317–342, 1992. [25] Olimjon Sh Sharipov and Martin Wendler. Bootstrapping covariance operators of functional time series. Journal of Nonparametric Statistics, 32(3):648–666, 2020. [26] Ada W van der Vaart and Jon A Wellner. Weak Convergence and Empirical Processes: With Applications to Statistics. Springer, NewYork, 1996. [27] Tengyao Wang and Richard J Samworth. High dimensional change point estimation via sparse projection. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 80(1):57–83, 2018.	zh_TW
dc.identifier.doi (DOI)	10.6814/NCCU202200419	en_US

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