Dans ce papier, la statistique de test CUSUM basée sur les résidus LASSO adaptatifs est proposée et étudiée pour détecter en temps réel si un changement a lieu dans un modèle linéaire qui a un nombre grand de variables explicatives.
Sous l’hypothèse nulle que le modèle ne subit pas de changements, la distribution asymptotique de la statistique de test est déterminée. Sous l’hypothèse alternative qu’un changement se produit dans le modèle à un instant inconnu, la statistique de test proposée converge en probabilité vers . Ces résultats permettent la construction d’une zone de rejet asymptotique. Ensuite, pour améliorer la performance de la statistique de test on propose une statistique de test modifiée.
Les résultats des simulations, par Monte Carlo, montrent la performance de la statistique de test proposée en la comparant aussi avec la statistique de test CUSUM classique.
In this paper, the CUSUM test statistic based on adaptive LASSO residuals is proposed and studied for detecting in real time a change-point in a linear model with a large number of explanatory variables.
Under null hypothesis that the model does not change, the asymptotic distribution of the test statistic is determined. Under alternative hypothesis that at some unknown observation there is a change in model, the proposed test statistic converges in probability to . These results allow to build an asymptotic critical region. Next, in order to improve the test statistic performance a modified test statistic is proposed.
Simulation results, using Monte Carlo technique, illustrate the performance of the proposed test statistic. We also compare it with the classical CUSUM test statistic.
@article{JSFS_2015__156_4_113_0, author = {Ciuperca, Gabriela}, title = {Real time change-point detection in a model by adaptive {LASSO} and {CUSUM}}, journal = {Journal de la soci\'et\'e fran\c{c}aise de statistique}, pages = {113--132}, publisher = {Soci\'et\'e fran\c{c}aise de statistique}, volume = {156}, number = {4}, year = {2015}, zbl = {1338.62129}, language = {en}, url = {http://www.numdam.org/item/JSFS_2015__156_4_113_0/} }
TY - JOUR AU - Ciuperca, Gabriela TI - Real time change-point detection in a model by adaptive LASSO and CUSUM JO - Journal de la société française de statistique PY - 2015 SP - 113 EP - 132 VL - 156 IS - 4 PB - Société française de statistique UR - http://www.numdam.org/item/JSFS_2015__156_4_113_0/ LA - en ID - JSFS_2015__156_4_113_0 ER -
%0 Journal Article %A Ciuperca, Gabriela %T Real time change-point detection in a model by adaptive LASSO and CUSUM %J Journal de la société française de statistique %D 2015 %P 113-132 %V 156 %N 4 %I Société française de statistique %U http://www.numdam.org/item/JSFS_2015__156_4_113_0/ %G en %F JSFS_2015__156_4_113_0
Ciuperca, Gabriela. Real time change-point detection in a model by adaptive LASSO and CUSUM. Journal de la société française de statistique, Tome 156 (2015) no. 4, pp. 113-132. http://www.numdam.org/item/JSFS_2015__156_4_113_0/
[Breiman, L., (1996)] Breiman, L., (1996). Heuristics of instability and stabilization in model selection, Annals of Statistics, 24(6), 2350-2383. | Zbl
[Ciuperca G., (2013)] Ciuperca G., (2013), Quantile regression in high-dimension with breaking, Journal of Statistical Theory and Applications, 12(3), 288-305.
[Ciuperca, G., (2013)] Ciuperca, G., (2013). Two tests for sequential detection of a change-point in a nonlinear model, Journal of Statistical Planning and Inference, 143(10), 1621-1834.
[Ciuperca G., (2014)] Ciuperca G., (2014), Model selection by LASSO methods in a change-point model, Statistical Papers, 55(2), 349-374. | Zbl
[Harchaoui, Z. and Lévy-Leduc, C.,(2010)] Harchaoui, Z. and Lévy-Leduc, C.,(2010). Multiple change-point estimation with a total variation penalty. Journal of the American Statistical Association, 105(492), 1480-1493.
[Horváth, L. et al., (2004)] Horváth, L., Hušková, M., Kokoszka, P., Steinebach, J.,(2004). Monitoring changes in linear models. Journal of Statistical Planning and Inference, 126, 225-251. | Zbl
[Lee S. et al., (2012)] Lee S., Seo M.H., Shin Y., (2012), The LASSO for high-dimensional regression with a possible change-point. arXiv:1209:4875v2.
[Lung-Yut-Fong A. et al., (2012)] Lung-Yut-Fong A., Lévy-Leduc C., Cappé O., (2012), Distributed detection/localization of change-points in high-dimensional network traffic data, Statistics and Computing, 22(2), 485-496.
[Siris V.A. and Papagalou F., (2006)] Siris V.A., Papagalou F., (2006), Application of anomaly detection algorithms for detecting SYN flooding attacks Computer Communications, 29(9), 1433-1442. iCON 2004–12th IEEE International Conference on Network 2004.
[Tibshirani, R.,(1996)] Tibshirani, R.,(1996). Regression shrinkage and selection via the LASSO. Journal of the Royal Statistical Society, Ser. B, 58, 267-288. | Zbl
[Wang H. et al., (2002)] Wang H., Zhang D., Shin K.G., (2002), Detecting SYN Flooding Attacks. Proceedings of IEEE INFOCOM, 3, 1530-1539.
[Zou, H.,(2006)] Zou, H.,(2006). The adaptive Lasso and its oracle properties. Journal of the American Statistical Association, 101(476), 1418-1428. | Zbl