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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 1-六月-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
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 matrixestimation. Econometrica, 59(3):817–858, 1991.[2] John AD Aston and Claudia Kirch. Detecting and estimating changes in dependentfunctional data. Journal of Multivariate Analysis, 109(4):204–220, 2012.[3] Alexander Aue, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Estimation of achange-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 breaksin 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 spectrumand trace of covariance operators. Environmetrics, 31(1):e2617, 2020.[6] Ivan E Auger and Charles E Lawrence. Algorithms for the optimal identification ofsegment neighborhoods. Bulletin of mathematical biology, 51(1):39–54, 1989.[7] István Berkes, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Detecting changesin 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 fora 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 afunctional data sequence with application to freeway traffic segmentation. The Annals ofApplied Statistics, 13(3):1430–1463, 2019.[10] Holger Dette and Tim Kutta. Detecting structural breaks in eigensystems of functionaltime series. Electronic Journal of Statistics, 15(1):944–983, 2021.[11] Piotr Fryzlewicz. Wild binary segmentation for multiple change-point detection. TheAnnals of Statistics, 42(6):2243–2281, 2014.[12] Abdullah Gedikli, Hafzullah Aksoy, N Erdem Unal, and Athanasios Kehagias. Modifieddynamic programming approach for offline segmentation of long hydrometeorologicaltime series. Stochastic Environmental Research and Risk Assessment, 24(5):547–557,2010.[13] Oleksandr Gromenko, Piotr Kokoszka, and Matthew Reimherr. Detection of change in thespatiotemporal mean function. Journal of the Royal Statistical Society: Series B (StatisticalMethodology), 79(1):29–50, 2017.[14] Zaıd Harchaoui and Céline Lévy-Leduc. Multiple change-point estimation with a totalvariation penalty. Journal of the American Statistical Association, 105(492):1480–1493,2010.[15] Siegfried Hörmann, Lukasz Kidziński, and Marc Hallin. Dynamic functional principalcomponents. 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 Annalsof Statistics, 38(3):1845–1884, 2010.[17] Lajos Horváth, Curtis Miller, and Gregory Rice. A new class of change point test statisticsof rényi type. Journal of Business & Economic Statistics, 38(3):570–579, 2020.[18] Brad Jackson, Jeffrey D Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, PeterGioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, and Tun Tao Tsai.An algorithm for optimal partitioning of data on an interval. IEEE Signal ProcessingLetters, 12(2):105–108, 2005.[19] Daniela Jarušková. Testing for a change in covariance operator. Journal of StatisticalPlanning and Inference, 143(9):1500–1511, 2013.[20] Rebecca Killick, Paul Fearnhead, and Idris A Eckley. Optimal detection of changepointswith 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 optimalmultiple 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 binarysegmentation 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 temperatureseries, 1772–1991. International Journal of Climatology, 12(4):317–342, 1992.[25] Olimjon Sh Sharipov and Martin Wendler. Bootstrapping covariance operators offunctional 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 estimationvia sparse projection. Journal of the Royal Statistical Society: Series B (StatisticalMethodology), 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 (作者) 陳裕庭 zh_TW dc.contributor.author (作者) 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 1-六月-2022 16:26:02 (UTC+8) - dc.date.available 1-六月-2022 16:26:02 (UTC+8) - dc.date.issued (上傳時間) 1-六月-2022 16:26:02 (UTC+8) - dc.identifier (其他 識別碼) 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 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 iiAbstract iiiContents ivList of Tables viList of Figures vii1 簡介 11.1 文獻回顧 11.2 研究方法簡介 42 函數型轉換點模型與轉換點估計式 62.1 多重轉換點模型 62.2 轉換點估計式與其相關性質 72.3 神諭子空間與函數型轉換點的可偵測性 113 貪婪分割估計式與其相關性質 133.1 轉換點之貪婪分割估計式 133.2 貪婪分割演算法 154 模擬研究 174.1 模擬設定 174.2 貪婪分割演算法對參數 h 的敏感性測試 184.3 方法比較 204.3.1 貪婪分割法與動態分割法之比較 204.3.2 貪婪分割演算法與二元分割法之比較 225 實際資料分析 275.1 英國中部氣溫資料集 275.2 澳洲氣象站資料 286 結語 31A 理論證明 33A.1 定理 2.2.1 證明 34A.1.1 引理 A.1.1 證明 36A.2 定理 2.3.1 證明 40A.3 定理 3.1.1 證明 40A.4 定理 3.2.1 證明 41Bibliography 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 matrixestimation. Econometrica, 59(3):817–858, 1991.[2] John AD Aston and Claudia Kirch. Detecting and estimating changes in dependentfunctional data. Journal of Multivariate Analysis, 109(4):204–220, 2012.[3] Alexander Aue, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Estimation of achange-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 breaksin 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 spectrumand trace of covariance operators. Environmetrics, 31(1):e2617, 2020.[6] Ivan E Auger and Charles E Lawrence. Algorithms for the optimal identification ofsegment neighborhoods. Bulletin of mathematical biology, 51(1):39–54, 1989.[7] István Berkes, Robertas Gabrys, Lajos Horváth, and Piotr Kokoszka. Detecting changesin 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 fora 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 afunctional data sequence with application to freeway traffic segmentation. The Annals ofApplied Statistics, 13(3):1430–1463, 2019.[10] Holger Dette and Tim Kutta. Detecting structural breaks in eigensystems of functionaltime series. Electronic Journal of Statistics, 15(1):944–983, 2021.[11] Piotr Fryzlewicz. Wild binary segmentation for multiple change-point detection. TheAnnals of Statistics, 42(6):2243–2281, 2014.[12] Abdullah Gedikli, Hafzullah Aksoy, N Erdem Unal, and Athanasios Kehagias. Modifieddynamic programming approach for offline segmentation of long hydrometeorologicaltime series. Stochastic Environmental Research and Risk Assessment, 24(5):547–557,2010.[13] Oleksandr Gromenko, Piotr Kokoszka, and Matthew Reimherr. Detection of change in thespatiotemporal mean function. Journal of the Royal Statistical Society: Series B (StatisticalMethodology), 79(1):29–50, 2017.[14] Zaıd Harchaoui and Céline Lévy-Leduc. Multiple change-point estimation with a totalvariation penalty. Journal of the American Statistical Association, 105(492):1480–1493,2010.[15] Siegfried Hörmann, Lukasz Kidziński, and Marc Hallin. Dynamic functional principalcomponents. 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 Annalsof Statistics, 38(3):1845–1884, 2010.[17] Lajos Horváth, Curtis Miller, and Gregory Rice. A new class of change point test statisticsof rényi type. Journal of Business & Economic Statistics, 38(3):570–579, 2020.[18] Brad Jackson, Jeffrey D Scargle, David Barnes, Sundararajan Arabhi, Alina Alt, PeterGioumousis, Elyus Gwin, Paungkaew Sangtrakulcharoen, Linda Tan, and Tun Tao Tsai.An algorithm for optimal partitioning of data on an interval. IEEE Signal ProcessingLetters, 12(2):105–108, 2005.[19] Daniela Jarušková. Testing for a change in covariance operator. Journal of StatisticalPlanning and Inference, 143(9):1500–1511, 2013.[20] Rebecca Killick, Paul Fearnhead, and Idris A Eckley. Optimal detection of changepointswith 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 optimalmultiple 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 binarysegmentation 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 temperatureseries, 1772–1991. International Journal of Climatology, 12(4):317–342, 1992.[25] Olimjon Sh Sharipov and Martin Wendler. Bootstrapping covariance operators offunctional 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 estimationvia sparse projection. Journal of the Royal Statistical Society: Series B (StatisticalMethodology), 80(1):57–83, 2018. zh_TW dc.identifier.doi (DOI) 10.6814/NCCU202200419 en_US
