Weakly dependent functional data. (English) Zbl 1189.62141
Summary: Functional data often arise from measurements on fine time grids and are obtained by separating an almost continuous time record into natural consecutive intervals, for example, days. The functions thus obtained form a functional time series, and the central issue in the analysis of such data consists in taking into account the temporal dependence of these functional observations. Examples include daily curves of financial transaction data and daily patterns of geophysical and environmental data. For scalar and vector valued stochastic processes, a large number of dependence notions have been proposed, mostly involving mixing type distances between \(\sigma \)-algebras.
In time series analysis, measures of dependence based on moments have proven most useful (autocovariances and cumulants). We introduce a moment-based notion of dependence for functional time series which involves \(m\)-dependence. We show that it is applicable to linear as well as nonlinear functional time series. Then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data. We study the estimation of the functional principal components, the long-run covariance matrix, change point detection and the functional linear model. We explain when temporal dependence affects the results obtained for i.i.d. functional observations and when these results are robust to weak dependence.
In time series analysis, measures of dependence based on moments have proven most useful (autocovariances and cumulants). We introduce a moment-based notion of dependence for functional time series which involves \(m\)-dependence. We show that it is applicable to linear as well as nonlinear functional time series. Then we investigate the impact of dependence thus quantified on several important statistical procedures for functional data. We study the estimation of the functional principal components, the long-run covariance matrix, change point detection and the functional linear model. We explain when temporal dependence affects the results obtained for i.i.d. functional observations and when these results are robust to weak dependence.
MSC:
| 62M10 | Time series, auto-correlation, regression, etc. in statistics (GARCH) |
| 62G08 | Nonparametric regression and quantile regression |
| 62H25 | Factor analysis and principal components; correspondence analysis |
| 60G10 | Stationary stochastic processes |
| 62G05 | Nonparametric estimation |
Keywords:
asymptotics; change points; eigenfunctions; functional principal components; functional time series; long-run variance; weak dependenceSoftware:
fda (R)References:
| [1] | Anderson, T. W. (1994). The Statistical Analysis of Time Series . Wiley, New York. · Zbl 0835.62074 |
| [2] | Andrews, D. W. K. (1984). Nonstrong mixing autoregressive processes. J. Appl. Probab. 21 930-934. JSTOR: · Zbl 0552.60049 · doi:10.2307/3213710 |
| [3] | Andrews, D. W. K. (1991). Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 59 817-858. JSTOR: · Zbl 0732.62052 · doi:10.2307/2938229 |
| [4] | Antoniadis, A. and Sapatinas, T. (2003). Wavelet methods for continuous time prediction using Hilbert-valued autoregressive processes. J. Multivariate Anal. 87 133-158. · Zbl 1030.62075 · doi:10.1016/S0047-259X(03)00028-9 |
| [5] | Aue, A., Horváth, S., Hörmann L. and Reimherr, M. (2009). Detecting changes in the covariance structure of multivariate time series models. Ann. Statist. 37 4046-4087. · Zbl 1191.62143 · doi:10.1214/09-AOS707 |
| [6] | Benko, M., Härdle, W. and Kneip, A. (2009). Common functional principal components. Ann. Statist. 37 1-34. · Zbl 1169.62057 · doi:10.1214/07-AOS516 |
| [7] | Berkes, I., Gabrys, R., Horváth, L. and Kokoszka, P. (2009). Detecting changes in the mean of functional observations. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 927-946. |
| [8] | Berkes, I., Hörmann, S. and Schauer, J. (2009). Asymptotic results for the empirical process of stationary sequences. Stochastic Process. Appl. 119 1298-1324. · Zbl 1161.60312 · doi:10.1016/j.spa.2008.06.010 |
| [9] | Berkes, I., Horváth, L., Kokoszka, P. and Shao, Q.-M. (2005). Almost sure convergence of the Bartlett estimator. Period. Math. Hungar. 51 11-25. · Zbl 1092.62030 · doi:10.1007/s10998-005-0017-5 |
| [10] | Berkes, I., Horváth, L., Kokoszka, P. and Shao, Q.-M. (2006). On discriminating between long-range dependence and changes in mean. Ann. Statist. 34 1140-1165. · Zbl 1112.62085 · doi:10.1214/009053606000000254 |
| [11] | Besse, P., Cardot, H. and Stephenson, D. (2000). Autoregressive forecasting of some functional climatic variations. Scand. J. Statist. 27 673-687. · Zbl 0962.62089 · doi:10.1111/1467-9469.00215 |
| [12] | Billingsley, P. (1968). Convergence of Probability Measures . Wiley, New York. · Zbl 0172.21201 |
| [13] | Billingsley, P. (1999). Convergence of Probability Measures , 2nd ed. Wiley, New York. · Zbl 0944.60003 |
| [14] | Bosq, D. (2000). Linear Processes in Function Spaces . Springer, New York. · Zbl 0962.60004 · doi:10.1007/978-1-4612-1154-9 |
| [15] | Bosq, D. and Blanke, D. (2007). Inference and Prediction in Large Dimensions . Wiley, Chichester. · Zbl 1183.62157 |
| [16] | Bradley, R. C. (2007). Introduction to Strong Mixing Conditions 1 , 2 , 3 . Kendrick Press, Heber City, UT. · Zbl 1134.60004 |
| [17] | Cai, T. and Hall, P. (2006). Prediction in functional linear regression. Ann. Statist. 34 2159-2179. · Zbl 1106.62036 · doi:10.1214/009053606000000830 |
| [18] | Cardot, H., Ferraty, F., Mas, A. and Sarda, P. (2003). Testing hypothesis in the functional linear model. Scand. J. Statist. 30 241-255. · Zbl 1034.62037 · doi:10.1111/1467-9469.00329 |
| [19] | Cardot, H., Ferraty, F. and Sarda, P. (2003). Spline estimators for the functional linear model. Statist. Sinica 13 571-591. · Zbl 1050.62041 |
| [20] | Chiou, J.-M. and Müller, H.-G. (2007). Diagnostics for functional regression via residual processes. Comput. Statist. Data Anal. 15 4849-4863. · Zbl 1162.62394 · doi:10.1016/j.csda.2006.07.042 |
| [21] | Chiou, J.-M., Müller, H.-G. and Wang, J.-L. (2004). Functional response models. Statist. Sinica 14 675-693. · Zbl 1073.62098 |
| [22] | Cuevas, A., Febrero, M. and Fraiman, R. (2002). Linear functional regression: The case of fixed design and functional response. Canad. J. Statist. 30 285-300. JSTOR: · Zbl 1012.62039 · doi:10.2307/3315952 |
| [23] | Damon, J. and Guillas, S. (2002). The inclusion of exogenous variables in functional autoregressive ozone forecasting. Environmetrics 13 759-774. |
| [24] | Dauxois, J., Pousse, A. and Romain, Y. (1982). Asymptotic theory for principal component analysis of a vector random function. J. Multivariate Anal. 12 136-154. · Zbl 0539.62064 · doi:10.1016/0047-259X(82)90088-4 |
| [25] | Doukhan, P. (1994). Mixing: Properties and Examples . Springer, New York. · Zbl 0801.60027 |
| [26] | Doukhan, P. and Louhichi, S. (1999). A new weak dependence and applications to moment inequalities. Stochastic Process. Appl. 84 313-343. · Zbl 0996.60020 · doi:10.1016/S0304-4149(99)00055-1 |
| [27] | Engle, R. F. (1982). Autoregressive conditional heteroskedasticity with estimates of the variance of United Kingdom inflation. Econometrica 50 987-1007. JSTOR: · Zbl 0491.62099 · doi:10.2307/1912773 |
| [28] | Giraitis, L., Kokoszka, P. S. and Leipus, R. (2000). Stationary ARCH models: Dependence structure and Central Limit Theorem. Econometric Theory 16 3-22. JSTOR: · Zbl 0986.60030 · doi:10.1017/S0266466600161018 |
| [29] | Giraitis, L., Kokoszka, P. S., Leipus, R. and Teyssière, G. (2003). Rescaled variance and related tests for long memory in volatility and levels. J. Econometrics 112 265-294. · Zbl 1027.62064 · doi:10.1016/S0304-4076(02)00197-5 |
| [30] | Gohberg, I., Golberg, S. and Kaashoek, M. A. (1990). Classes of Linear Operators, vol. 1. Operator Theory: Advances and Applications 49 . Birkhaüser, Basel. · Zbl 0745.47002 |
| [31] | Gordin, M. (1969). The central limit theorem for stationary processes. Dokl. Akad. Nauk SSSR 188 739-741. · Zbl 0212.50005 |
| [32] | Granger, C. W. J. and Anderson, A. P. (1979). An Introduction to Bilinear Time Series Models . Vandenhoeck and Ruprecht, Göttingen. |
| [33] | Hall, P. and Hosseini-Nasab, M. (2006). On properties of functional principal components. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 109-126. · Zbl 1141.62048 · doi:10.1111/j.1467-9868.2005.00535.x |
| [34] | Hamilton, J. D. (1994). Time Series Analysis . Princeton Univ. Press, Princeton, NJ. · Zbl 0831.62061 |
| [35] | Hörmann, S. (2008). Augmented GARCH sequences: Dependence structure and asymptotics. Bernoulli 14 543-561. · Zbl 1155.62066 · doi:10.3150/07-BEJ120 |
| [36] | Hörmann, S. (2009). Berry-Esseen bounds for econometric time series. ALEA 6 377-397. · Zbl 1276.60024 |
| [37] | Ibragimov, I. A. (1962). Some limit theorems for stationary processes. Teor. Veroyatn. Primen. 7 361-392. · Zbl 0119.14204 · doi:10.1137/1107036 |
| [38] | Kargin, V. and Onatski, A. (2008). Curve forecasting by functional autoregression. J. Multivariate Anal. 99 2508-2506. · Zbl 1151.62073 · doi:10.1016/j.jmva.2008.03.001 |
| [39] | Kiefer, J. (1959). K -sample analogues of the Kolmogorov-Smirnov and Cramér-von Mises tests. Ann. Math. Statist. 30 420-447. · Zbl 0134.36707 · doi:10.1214/aoms/1177706261 |
| [40] | Li, Y. and Hsing, T. (2007). On rates of convergence in functional linear regression. J. Multivariate Anal. 98 1782-1804. · Zbl 1130.62035 · doi:10.1016/j.jmva.2006.10.004 |
| [41] | Malfait, N. and Ramsay, J. O. (2003). The historical functional model. Canad. J. Statist. 31 115-128. JSTOR: · Zbl 1039.62035 · doi:10.2307/3316063 |
| [42] | Mas, A. and Menneteau, L. (2003). Perturbation approach applied to the asymptotic study of random operators. Progr. Probab. 55 127-134. · Zbl 1053.60002 |
| [43] | Maslova, I., Kokoszka, P., Sojka, J. and Zhu, L. (2009). Removal of nonconstant daily variation by means of wavelet and functional data analysis. Journal of Geophysical Research 114 A03202. |
| [44] | Maslova, I., Kokoszka, P., Sojka, J. and Zhu, L. (2009). Estimation of Sq variation by means of multiresolution and principal component analyses. Journal of Atmospheric and Solar-Terrestrial Physics . |
| [45] | Müller, H.-G. and Yao, F. (2008). Functional additive models. J. Amer. Statist. Assoc. 103 1534-1544. · Zbl 1286.62040 |
| [46] | Müller, H.-G. and Stadtmüller, U. (2005). Generalized functional linear models. Ann. Statist. 33 774-805. · Zbl 1068.62048 · doi:10.1214/009053604000001156 |
| [47] | Philipp, W. and Stout, W. F. (1975). Almost Sure Invariance Principles for Partial Sums of Weakly Dependent Random Variables. Mem. Amer. Math. Soc. 161 . Amer. Math. Soc., Providence, RI. · Zbl 0361.60007 |
| [48] | Pötscher, B. and Prucha, I. (1997). Dynamic Non-linear Econonometric Models. Asymptotic Theory . Springer, Berlin. |
| [49] | Priestley, M. (1988). Nonlinear and Nonstationary Time Series Analysis . Academic Press, London. · Zbl 0667.62068 |
| [50] | Ramsay, J. O. and Silverman, B. W. (2005). Functional Data Analysis . Springer, New York. · Zbl 1079.62006 |
| [51] | Reiss, P. T. and Ogden, R. T. (2007). Functional principal component regression and functional partial least squares. J. Amer. Statist. Assoc. 102 984-996. · Zbl 1469.62237 · doi:10.1198/016214507000000527 |
| [52] | Reiss, P. T. and Ogden, R. T. (2009). Smoothing parameter selection for a class of semiparametric linear models. J. R. Stat. Soc. Ser. B Stat. Methodol. 71 505-523. · Zbl 1248.62057 |
| [53] | Reiss, P. T. and Ogden, R. T. (2010). Functional generalized linear models with images as predictors. Biometrics . · Zbl 1187.62112 |
| [54] | Rosenblatt, M. (1956). A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 43-47. JSTOR: · Zbl 0070.13804 · doi:10.1073/pnas.42.1.43 |
| [55] | Rosenblatt, M. (1959). Stationary processes as shifts of functions of independent random variables. J. Math. Mech. 8 665-681. · Zbl 0092.33601 |
| [56] | Rosenblatt, M. (1961). Independence and dependence. In Proc. 4th Berkeley Sympos. Math. Statist. Probab. 2 431-443. Univ. California Press, Berkeley, CA. · Zbl 0105.11802 |
| [57] | Rosenblatt, M. (1971). Markov Processes. Structure and Asymptotic Behavior . Springer, New York. · Zbl 0236.60002 |
| [58] | Stadtlober, E., Hörmann, S. and Pfeiler, B. (2008). Qualiy and performance of a PM10 daily forecasting model. Athmospheric Environment 42 1098-1109. |
| [59] | Stine, R. (1997). Nonlinear time series. In Encyclopedia of Statistical Sciences (S. Kotz, C. Y. Read and D. L. Banks, eds.) 430-437. Wiley, New York. |
| [60] | Tong, H. (1990). Non-linear Time Series: A Dynamical System Approach . Oxford Univ. Press, New York. · Zbl 0716.62085 |
| [61] | Vostrikova, L. J. (1981). Detection of “disorder” in multidimensional random processes. Sov. Dokl. Math. 24 55-59. · Zbl 0487.62072 |
| [62] | Wu, W. (2005). Nonlinear system theory: Another look at dependence. Proc. Natl. Acad. Sci. USA 102 14150-14154. · Zbl 1135.62075 · doi:10.1073/pnas.0506715102 |
| [63] | Wu, W. (2007). Strong invariance principles for dependent random variables. Ann. Probab. 35 2294-2320. · Zbl 1166.60307 · doi:10.1214/009117907000000060 |
| [64] | Yao, F. and Lee, T. (2006). Penalized spline models for functional principal component analysis. J. R. Stat. Soc. Ser. B Stat. Methodol. 68 3-25. · Zbl 1141.62050 · doi:10.1111/j.1467-9868.2005.00530.x |
| [65] | Yao, F., Müller, H.-G. and Wang., J.-L. (2005). Functional linear regression analysis for longitudinal data. Ann. Statist. 33 2873-2903. · Zbl 1084.62096 · doi:10.1214/009053605000000660 |
| [66] | Yao, F., Müller, H.-G. and Wang., J.-L. (2005). Functional data analysis for sparse longitudinal data. J. Amer. Statist. Assoc. 100 577-590. · Zbl 1117.62451 · doi:10.1198/016214504000001745 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
