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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-Minen_US
dc.contributor.author (Authors) 陳裕庭zh_TW
dc.contributor.author (Authors) Chen, Yu-Tingen_US
dc.creator (作者) 陳裕庭zh_TW
dc.creator (作者) Chen, Yu-Tingen_US
dc.date (日期) 2022en_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) G0104354501en_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) 104354501zh_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/#G0104354501en_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 problemen_US
dc.subject (關鍵詞) Functional principal component analysisen_US
dc.subject (關鍵詞) Covariance operatoren_US
dc.subject (關鍵詞) Hypothesis testingen_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 segmentationen_US
dc.type (資料類型) thesisen_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/NCCU202200419en_US