This study investigated the application of Bayesian multiple change-point detection techniques in the context of piecewise polynomial signals. Given the limited number of existing methodologies for identifying change-points in such signals, we proposed an objective Bayesian change-point detection approach that accommodated heterogeneous error distributions. Our methodology was grounded in a piecewise polynomial regression framework and employed binary segmentation. Initially, we identified change-points across various signals using Bayesian binary segmentation. Subsequently, we applied Bayesian model selection to ascertain the most suitable polynomial order for the identified segments. This approach facilitated a change-point detection method that minimized reliance on subjective inputs. We incorporated intrinsic priors that allowed for the formulation of Bayes factors and model selection probabilities. To evaluate the efficacy of the proposed change-point detection techniques, we conducted a simulation study alongside two empirical case studies: one involving the Goddard Institute for space studies surface temperature analysis and the other concerning the daily closing stock prices of Samsung Electronics Co.
Citation: Sang Gil Kang, Woo Dong Lee, Yongku Kim. Bayesian multiple changing-points detection[J]. AIMS Mathematics, 2025, 10(3): 4662-4708. doi: 10.3934/math.2025216
This study investigated the application of Bayesian multiple change-point detection techniques in the context of piecewise polynomial signals. Given the limited number of existing methodologies for identifying change-points in such signals, we proposed an objective Bayesian change-point detection approach that accommodated heterogeneous error distributions. Our methodology was grounded in a piecewise polynomial regression framework and employed binary segmentation. Initially, we identified change-points across various signals using Bayesian binary segmentation. Subsequently, we applied Bayesian model selection to ascertain the most suitable polynomial order for the identified segments. This approach facilitated a change-point detection method that minimized reliance on subjective inputs. We incorporated intrinsic priors that allowed for the formulation of Bayes factors and model selection probabilities. To evaluate the efficacy of the proposed change-point detection techniques, we conducted a simulation study alongside two empirical case studies: one involving the Goddard Institute for space studies surface temperature analysis and the other concerning the daily closing stock prices of Samsung Electronics Co.
| [1] |
J. Bai, P. Perron, Computation and analysis of multiple structural change models, J. Appl. Econometrics, 18 (2003), 1–22. https://doi.org/10.1002/jae.659 doi: 10.1002/jae.659
|
| [2] |
R. Killick, I. A. Eckley, P. Jonathan, A wavelet-based approach for detecting changes in second order structure within nonstationary time series, Electron. J. Stat., 7 (2013), 1167–1183. https://doi.org/10.1214/13-EJS799 doi: 10.1214/13-EJS799
|
| [3] |
A. Futschik, T. Hotz, A. Munk, H. Sieling, Multiscale DNA partitioning: statistical evidence for segments, Bioinformatics, 30 (2014), 2255–2262. https://doi.org/10.1093/bioinformatics/btu180 doi: 10.1093/bioinformatics/btu180
|
| [4] |
E. Galceran, A. G. Cunningham, R. M. Eustice, E. Olson, Multipolicy decisionmaking for autonomous driving via changepoint-based behavior prediction: theory and experiment, Auton. Robot., 41 (2017), 1367–1382. https://doi.org/10.1007/s10514-017-9619-z doi: 10.1007/s10514-017-9619-z
|
| [5] |
Y. C. Yao, Estimating the number of change-points via Schwarz criterion, Stat. Probab. Lett., 6 (1988), 181–189. https://doi.org/10.1016/0167-7152(88)90118-6 doi: 10.1016/0167-7152(88)90118-6
|
| [6] | Y. C. Yao, S. T. Au, Least-squares estimation of a step function, Sankhya A, 51 (1989), 370–381. |
| [7] |
N. R. Zhang, D. O. Siegmund, A modified Bayes information criterion with applications to the analysis of comparative genomic hybridization data, Biometrics, 63 (2007), 22–32. https://doi.org/10.1111/j.1541-0420.2006.00662.x doi: 10.1111/j.1541-0420.2006.00662.x
|
| [8] |
N. R. Zhang, D. O. Siegmund, Model selection for high-dimensional, multi-sequence change-point problems, Stat. Sin., 22 (2012), 1057–1538. https://doi.org/10.5705/ss.2010.257 doi: 10.5705/ss.2010.257
|
| [9] |
J. Bai, P. Perron, Estimating and testing linear models with multiple structural changes, Econometrica, 66 (1998), 57–78. https://doi.org/10.2307/2998540 doi: 10.2307/2998540
|
| [10] |
J. Braun, R. K. Braun, H. G. M$\rm\ddot{u}$ller, Multiple changepoint fitting via quasilikelihood with application to DNA sequence segmentation, Biometrika, 87 (2000), 301–304. https://doi.org/10.1093/biomet/87.2.301 doi: 10.1093/biomet/87.2.301
|
| [11] |
L. Boysen, A. Kempe, V. Liebsher, A. Munk, O. Wittich, Consistencies and rates of convergence of jump-penalized least squares estimator, Ann. Stat., 37 (2009), 157–183. https://doi.org/10.1214/07-AOS558 doi: 10.1214/07-AOS558
|
| [12] |
R. Tibshirani, P. Wang, Spatial smoothing and hot spot detection for CGH data using the fused lasso, Biostatistics, 9 (2008), 18–29. https://doi.org/10.1093/biostatistics/kxm013 doi: 10.1093/biostatistics/kxm013
|
| [13] |
Z. Harchaoui, C. L$\rm\acute{e}$vy-Leduc, Multiple change-point estimation with a total variation penalty, J. Amer. Stat. Assoc., 105 (2010), 1480–1493. https://doi.org/10.1198/jasa.2010.tm09181 doi: 10.1198/jasa.2010.tm09181
|
| [14] |
X. J. Jeng, T. T. Cai, H. Li, Optimal sparse segment identification with application in copy number variation analysis, J. Amer. Stat. Assoc., 105 (2010), 1156–1166. https://doi.org/10.1198/jasa.2010.tm10083 doi: 10.1198/jasa.2010.tm10083
|
| [15] |
K. Frick, A. Munk, H. Sieling, Multiscale change-point inference, J. R. Stat. Soc. Ser. B, 76 (2014), 495–580. https://doi.org/10.1111/rssb.12047 doi: 10.1111/rssb.12047
|
| [16] |
R. Killick, P. Fearnhead, I. A. Eckley, Optimal detection of changepoints with a linear computational cost, J. Amer. Stat. Assoc., 107 (2012), 1590–1598. https://doi.org/10.1080/01621459.2012.737745 doi: 10.1080/01621459.2012.737745
|
| [17] |
R. Maidstone, T. Hocking, G. Rigaill, P. Fearnhead, On optimal multiple changepoint algorithms for large data, Stat. Comput., 27 (2017), 519–533. https://doi.org/10.1007/s11222-016-9636-3 doi: 10.1007/s11222-016-9636-3
|
| [18] |
A. B. Olshen, E. S. Venkatraman, R. Lucito, M. Wigler, Circular binary segmentation for the analysis of array-based DNA copy number data, Biostatistics, 5 (2004), 557–572. https://doi.org/10.1093/biostatistics/kxh008 doi: 10.1093/biostatistics/kxh008
|
| [19] |
E. Venkatraman, A. B. Olshen, A faster circular binary segmentation algorithm for the analysis of array CGH data, Bioinformatics, 23 (2007), 657–663. https://doi.org/10.1093/bioinformatics/btl646 doi: 10.1093/bioinformatics/btl646
|
| [20] |
B. Jackson, J. D. Scargle, D. Barnes, S. Arabhi, P. Gioumousis, E. Gwin, et al., An algorithm for optimal partitioning of data on an interval, EEE Signal Process. Lett., 12 (2005), 105–108. https://doi.org/10.1109/LSP.2001.838216 doi: 10.1109/LSP.2001.838216
|
| [21] |
K. Haynes, I. Eckley, P. Fearnhead, Computationally efficient changepoint detection for a range of penalties, J. Comput. Graph. Stat., 26 (2017), 134–143. https://doi.org/10.1080/10618600.2015.1116445 doi: 10.1080/10618600.2015.1116445
|
| [22] |
N. Hao, Y. S. Niu, H. Zhang, Multiple change-point detection via a screening and ranking algorithm, Stat. Sin., 23 (2013), 1553–1572. https://doi.org/10.5705/ss.2012.018s doi: 10.5705/ss.2012.018s
|
| [23] |
D. Barry, J. Hartigan, A Bayesian analysis for change-point problems, J. Amer. Stat. Assoc., 88 (1993), 309–319. https://doi.org/10.1080/01621459.1993.10594323 doi: 10.1080/01621459.1993.10594323
|
| [24] |
S. Chib, Estimation and comparison of multiple change-point models, J. Econom., 86 (1998), 221–241. https://doi.org/10.1016/S0304-4076(97)00115-2 doi: 10.1016/S0304-4076(97)00115-2
|
| [25] |
M. Lavielle, E. Lebarbier, An application of MCMC methods for the multiple change-points problem, Digit. Signal Process., 81 (2001), 39–53. https://doi.org/10.1016/S0165-1684(00)00189-4 doi: 10.1016/S0165-1684(00)00189-4
|
| [26] |
P. Fearnhead, P. Clifford, On‐line inference for hidden Markov models via particle filters, J. R. Stat. Soc. Ser. B, 65 (2003), 887–899. https://doi.org/10.1111/1467-9868.00421 doi: 10.1111/1467-9868.00421
|
| [27] |
G. Koop, S. M. Potter, Estimation and forecasting in models with multiple breaks, J. Econ. Stud., 74 (2007), 763–789. https://doi.org/10.1111/j.1467-937X.2007.00436.x doi: 10.1111/j.1467-937X.2007.00436.x
|
| [28] |
P. Fearnhead, Z. Liu, On-line inference for multiple changepoint problems, J. R. Stat. Soc. Ser. B, 69 (2007), 589–605. https://doi.org/10.1111/j.1467-9868.2007.00601.x doi: 10.1111/j.1467-9868.2007.00601.x
|
| [29] |
P. Giordani, R. Kohn, Efficient Bayesian inference for multiple change-point and mixture innovation models, J. Bus. Econ. Stat., 26 (2008), 66–77. https://doi.org/10.1198/073500107000000241 doi: 10.1198/073500107000000241
|
| [30] |
A. F. Martinez, R. H. Mena, On a nonparametric change point detection model in Markovian regimes, Bayesian Anal., 9 (2014), 823–858. https://doi.org/ 10.1214/14-BA878 doi: 10.1214/14-BA878
|
| [31] |
P. Fearnhead, Exact and efficient Bayesian inference for multiple changepoint problems, Stat. Comput., 16 (2006), 203–213. https://doi.org/10.1007/s11222-006-8450-8 doi: 10.1007/s11222-006-8450-8
|
| [32] |
E. Ruggieri, A Bayesian approach to detecting change points in climatic records, Int. J. Climatol., 33 (2013), 520–528. https://doi.org/10.1002/joc.3447 doi: 10.1002/joc.3447
|
| [33] |
S. Aminikhanghahi, D. J. Cook, A survey of methods for time series change point detection, Knowl. Inf. Syst., 51 (2017), 339–367. https://doi.org/10.1007/s10115-016-0987-z doi: 10.1007/s10115-016-0987-z
|
| [34] |
M. Bianchi, M. Boyle, D. Hollingsworth, A comparison of methods for trend estimation, Appl. Econ. Lett., 6 (1999), 103–109. https://doi.org/10.1080/135048599353726 doi: 10.1080/135048599353726
|
| [35] |
M. W. Robbins, R. B. Lund, C. M. Gallagher, Q. Lu, Changepoints in the north atlantic tropical cyclone record, J. Am. Stat. Assoc., 106 (2011), 89–99. https://doi.org/10.1198/jasa.2011.ap10023 doi: 10.1198/jasa.2011.ap10023
|
| [36] | S. J. Kim, K. Koh, S. Boyd, D. Gorinevsky, $l_1$ trend filtering, SIAM Rev., 51 (2009), 339–360. https://doi.org/10.1137/070690274 |
| [37] |
R. Baranowski, Y. Chen, P. Fryzlewicz, Narrowest-over-threshold detection of multiple change points and change-point-like features, J. R. Stat. Soc. B, 81 (2019), 649–672. https://doi.org/10.1111/rssb.12322 doi: 10.1111/rssb.12322
|
| [38] |
A. Anastasiou, P. Fryzlewicz, Detecting multiple generalized change-points by isolating single ones, Metrika, 85 (2022), 141–174. https://doi.org/10.1007/s00184-021-00821-6 doi: 10.1007/s00184-021-00821-6
|
| [39] |
H. Maeng, P. Fryzlewicz, Detecting linear trend changes in data sequences, Stat. Papers, 65 (2024), 1645–1675. https://doi.org/10.1007/s00362-023-01458-5 doi: 10.1007/s00362-023-01458-5
|
| [40] | R. V. Mehrizi, S. Chenouri, Detection of change points in piecewise polynomial signals using trend filtering, arXiv, 2020. https://doi.org/10.48550/arXiv.2009.08573 |
| [41] | S. Gavioli-Akilagun, P. Fryzlewicz, Fast and optimal inference for change points in piecewise polynomials via differencing, arXiv, 2023. https://doi.org/10.48550/arXiv.2307.03639 |
| [42] |
J. R. Faulkner, V. N. Minin, Locally adaptive smoothing with Markov random fields and shrinkage priors, Bayesian Anal., 13 (2018), 225–252. https://doi.org/10.1214/17-BA1050 doi: 10.1214/17-BA1050
|
| [43] | D. R. Kowal, D. S. Matteson, D. Ruppert, Dynamic shrinkage processes, J. R. Stat. Soc. Ser. B, 81 (2019), 781–804. https://doi.org/10.1111/rssb.12325 |
| [44] |
Y. Zhao, X. Liao, X. He, M. Zhou, C. Li, Accelerated primal-dual mirror dynamics for centralized and distributed constrained convex optimization problems, J. Mach. Learn. Res., 24 (2023), 16413–16471. https://doi.org/10.5555/3648699.3649042 doi: 10.5555/3648699.3649042
|
| [45] |
A. Shirilord, M. Dehghan, Stationary Landweber method with momentum acceleration for solving least squares problems, Appl. Math. Lett., 157 (2024), 109174. https://doi.org/10.1016/j.aml.2024.109174 doi: 10.1016/j.aml.2024.109174
|
| [46] |
S. Arlot, A. Celisse, Segmentation of the mean of heteroscedastic data via cross-validation, Stat. Comput., 21 (2011), 613–632. https://doi.org/10.1007/s11222-010-9196-x doi: 10.1007/s11222-010-9196-x
|
| [47] |
V. M. R. Muggeo, G. Adelfio, Efficient change point detection for genomic sequences of continuous measurements, Bioinformatics, 27 (2011), 161–166. https://doi.org/10.1093/bioinformatics/btq647 doi: 10.1093/bioinformatics/btq647
|
| [48] | F. Pein, H. Sieling, A. Munk, Heterogeneous change point inference. J. R. Stat. Soc. Ser. B, 79 (2017), 1207–1227. https://doi.org/10.1111/rssb.12202 |
| [49] |
J. O. Berger, L. R. Pericchi, The intrinsic Bayes factor for model selection and prediction, J. Amer. Stat. Assoc., 91 (1996), 109–122. https://doi.org/10.2307/2291387 doi: 10.2307/2291387
|
| [50] |
E. Moreno, F. Bertolino, W. Racugno, An intrinsic limiting procedure for model selection and hypotheses testing, J. Amer. Stat. Assoc., 93 (1998), 1451–1460. https://doi.org/10.2307/2670059 doi: 10.2307/2670059
|
| [51] | J. O. Berger, L. R. Pericchi, On criticism and comparison of default Bayes factor for model selection and hypothesis testing, Proceedings of the Workshop on Model Selection, 1998. |
| [52] |
E. Moreno, Bayes factor for intrinsic and fractional priors in nested models. Bayesian robustness, IMS Lecture Notes Monogr. Ser., 1997 (1997), 257–270. https://doi.org/10.1214/LNMS/1215454142 doi: 10.1214/LNMS/1215454142
|
| [53] |
A. J. Scott, M. Knott, A cluster analysis method for grouping means in the analysis of variance, Biometrics, 30 (1974), 507–512. https://doi.org/10.2307/2529204 doi: 10.2307/2529204
|
| [54] | L. J. Vostrikova, Detecting "disorder" in multidimensional random processes, Sov. Math. Dokl., 24 (1981), 55–59. |
| [55] |
P. Fryziewicz, Wild binary segmentation for multiple change-point detection, Ann. Stat., 42 (2014), 2243–2281. https://doi.org/10.1214/14-AOS1245 doi: 10.1214/14-AOS1245
|
| [56] |
G. Jensen, Closed-form estimation of multiple change-point models, PeerJ Preprint, 3 (2013), e90v3. https://doi.org/10.7287/peerj.preprints.90v3 doi: 10.7287/peerj.preprints.90v3
|
| [57] | R. E. Kass, A. Raftery, Bayes factors, J. Amer. Stat. Assoc., 90 (1995), 773–795. https://doi.org/10.2307/2291091 |
| [58] |
G. Casella, E. Moreno, Objective Bayesian variable selection, J. Amer. Stat. Assoc., 101 (2009), 157–167. https://doi.org/10.1198/016214505000000646 doi: 10.1198/016214505000000646
|
| [59] |
F. J. Gir$\rm \acute{o}$n, L. L. Mart$\rm\acute{i}$nez, E. Moreno, F. Torres, Objective testing procedures in linear models, Scand. J. Stat., 33 (2006), 765–784. https://doi.org/10.1111/j.1467-9469.2006.00514.x doi: 10.1111/j.1467-9469.2006.00514.x
|
| [60] | E. Keogh, S. Chu, D. Hart, M. Pazzani, Segmenting time series: a survey and novel approach, Series in Machine Perception and Artificial Intelligence, 2004. https://doi.org/10.1142/9789812565402_0001 |
| [61] |
J. Hansen, R. Ruedy, M. Sato, K. Lo, Global surface temperature change, Rev. Geophys., 48 (2010), RG4004. https://doi.org/10.1029/2010RG000345 doi: 10.1029/2010RG000345
|
| [62] |
N. J. Lenssen, G. A. Schmidt, J. E. Hansen, M. J. Menne, A. Persin, R. Ruedy, et al., Improvements in the GISTEMP uncertainty model, J. Geophys. Res. Atmos., 124 (2019), 6307–6326. https://doi.org/10.1029/2018JD029522 doi: 10.1029/2018JD029522
|
| [63] | N. A. James, D. S. Matteson, Change points via probabilistically pruned objectives, arXiv, 2015. https://doi.org/10.48550/arXiv.1505.04302 |
Figures(1) / Tables(10)
Sang Gil Kang, Woo Dong Lee, Yongku Kim. Bayesian multiple changing-points detection[J]. AIMS Mathematics, 2025, 10(3): 4662-4708. doi: 10.3934/math.2025216
DownLoad: