On projection methods for functional time series forecasting. (English) Zbl 1520.62425

Summary: Two nonparametric methods are presented for forecasting functional time series (FTS). The FTS we observe is a curve at a discrete-time point. We address both one-step-ahead forecasting and dynamic updating. Dynamic updating is a forward prediction of the unobserved segment of the most recent curve. Among the two proposed methods, the first one is a straightforward adaptation to FTS of the \(k\)-nearest neighbors methods for univariate time series forecasting. The second one is based on a selection of curves, termed the curve envelope, that aims to be representative in shape and magnitude of the most recent functional observation, either a whole curve or the observed part of a partially observed curve. In a similar fashion to \(k\)-nearest neighbors and other projection methods successfully used for time series forecasting, we “project” the \(k\)-nearest neighbors and the curves in the envelope for forecasting. In doing so, we keep track of the next period evolution of the curves. The methods are applied to simulated data, daily electricity demand, and NOx emissions and provide competitive results with and often superior to several benchmark predictions. The approach offers a model-free alternative to statistical methods based on FTS modeling to study the cyclic or seasonal behavior of many FTS.


62R10 Functional data analysis
62G08 Nonparametric regression and quantile regression
62G30 Order statistics; empirical distribution functions
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
62M20 Inference from stochastic processes and prediction
Full Text: DOI arXiv


[1] Aneiros, G.; Cao, R.; Fraiman, R.; Genest, C.; Vieu, P., Recent advances in functional data analysis and high-dimensional statistics, J. Multivariate Anal., 170, 3-9 (2019), Special Issue on Functional Data Analysis and Related Topics · Zbl 1415.62043
[2] Aneiros, G.; Cao, R.; Vilar, J. M., Functional methods for time series prediction: a nonparametric approach, J. Forecast., 30, 4, 377-392 (2011) · Zbl 1217.91138
[3] Aneiros, G.; Vieu, P., Nonparametric time series prediction: A semi-functional partial linear modeling, J. Multivariate Anal., 99, 5, 834-857 (2008) · Zbl 1133.62075
[4] Aneiros, G.; Vilar, J. M.; Cao, R.; Muñoz, A., Functional prediction for the residual demand in electricity spot markets, IEEE Trans. Power Syst., 28, 4, 4201-4208 (2013)
[5] Aneiros, G.; Vilar, J. M.; Raña, P., Short-term forecast of daily curves of electricity demand and price, Int. J. Electr. Power Energy Syst., 80, 96-108 (2016)
[6] Antoch, J.; Prchal, L.; De Rosa, M. R.; Sarda, P., Electricity consumption prediction with functional linear regression using spline estimators, J. Appl. Stat., 37, 12, 2027-2041 (2010) · Zbl 1511.62405
[7] Antoniadis, A.; Paparoditis, E.; Sapatinas, T., A functional wavelet-kernel approach for time series prediction, J. R. Stat. Soc. Ser. B Stat. Methodol., 68, 5, 837-857 (2006) · Zbl 1110.62122
[8] Arribas-Gil, A.; Romo, J., Shape outlier detection and visualization for functional data: the outliergram, Biostatistics, 15, 4, 603-619 (2014)
[9] Aue, A.; Norinho, D. D.; Hörmann, S., On the prediction of stationary functional time series, J. Am. Stat. Assoc.: Theory Methods, 110, 509, 378-392 (2015) · Zbl 1373.62462
[10] Biau, G.; Cerou, F.; Guyader, A., Rates of convergence of the functional \(k\)-nearest neighbor estimate, IEEE Trans. Inform. Theory, 56, 4, 2034-2040 (2010) · Zbl 1366.62080
[11] Bosq, D., Linear Processes in Function Spaces: Theory and Applications (2000), Springer: Springer New York · Zbl 0962.60004
[12] Burba, F.; Ferraty, F.; Vieu, P., K-nearest neighbour method in functional nonparametric regression, J. Nonparametr. Stat., 21, 4, 453-469 (2009) · Zbl 1161.62017
[13] Cérou, F.; Guyader, A., Nearest neighbor classification in infinite dimension, ESAIM: PS, 10, 340-355 (2006) · Zbl 1187.62115
[14] Cho, H.; Goude, Y.; Brossat, X.; Yao, Q., Modeling and forecasting daily electricity load curves: A hybrid approach, J. Am. Stat. Assoc.: Appl. Case Stud., 108, 501, 7-21 (2013) · Zbl 1379.62091
[15] Claeskens, G.; Hubert, M.; Slaets, L.; Vakili, K., Multivariate functional halfspace depth, J. Amer. Statist. Assoc., 109, 505, 411-423 (2014) · Zbl 1367.62162
[16] Cuesta-Albertos, J. A.; Febrero-Bande, M.; Oviedo de la Fuente, M., The \(D D^G\)-classifier in the functional setting, TEST, 26, 1, 119-142 (2017) · Zbl 1422.62216
[17] Cuesta-Albertos, J.; Nieto-Reyes, A., The random Tukey depth, Comput. Statist. Data Anal., 52, 11, 4979-4988 (2008) · Zbl 1452.62344
[18] Cuevas, A., A partial overview of the theory of statistics with functional data, J. Statist. Plann. Inference, 147, 1-23 (2014) · Zbl 1278.62012
[19] Cuevas, A.; Febrero-Bande, M.; Fraiman, R., Robust estimation and classification for functional data via projection-based depth notions, Comput. Statist., 22, 3, 481-496 (2007) · Zbl 1195.62032
[20] Dai, W.; Genton, M. G., Multivariate functional data visualization and outlier detection, J. Comput. Graph. Statist., 27, 4, 923-934 (2018) · Zbl 07499002
[21] Febrero-Bande, M.; Galeano, P.; González-Manteiga, W., Outlier detection in functional data by depth measures, with application to identify abnormal NOx levels, Environmetrics, 19, 4, 331-345 (2008)
[22] Ferraty, F.; Nagy, S., Scalar-on-function local linear regression and beyond, Biometrika (2021)
[23] Ferraty, F.; Vieu, P., Nonparametric Functional Data Analysis: Theory and Practice (2006), Springer-Verlag New York · Zbl 1119.62046
[24] Fraiman, R.; Muniz, G., Trimmed means for functional data, Test, 10, 2, 419-440 (2001) · Zbl 1016.62026
[25] Gijbels, I.; Nagy, S., On a general definition of depth for functional data, Statist. Sci., 32, 4, 630-639 (2017) · Zbl 1381.62098
[26] Gneiting, T.; Raftery, A., Strictly proper scoring rules, prediction, and estimation, J. Am. Stat. Assoc.: Rev. Article, 102, 477, 359-378 (2007) · Zbl 1284.62093
[27] Goia, A.; Vieu, P., An introduction to recent advances in high/infinite dimensional statistics, J. Multivariate Anal., 146, 1-6 (2016), Special Issue on Statistical Models and Methods for High or Infinite Dimensional Spaces · Zbl 1384.00073
[28] Hörmann, S.; Kokoszka, P. P., Functional time series, (Time Series Analysis. Time Series Analysis, Handbook of Statistics, 30 (2012), Elsevier B.V.: Elsevier B.V. Netherlands), 157-186
[29] Hubert, M.; Rousseeuw, P.; Segaert, P., Multivariate and functional classification using depth and distance, Adv. Data Anal. Classif., 11, 3, 445-466 (2017) · Zbl 1414.62247
[30] Hyndman, R. J.; Booth, H., Stochastic population forecasts using functional data models for mortality, fertility and migration, Int. J. Forecast., 24, 3, 323-342 (2008)
[31] Hyndman, R. J.; Shang, H. L., Forecasting functional time series, J. Korean Stat. Soc., 38, 3, 199-211 (2009) · Zbl 1293.62267
[32] Hyndman, R. J.; Shang, H. L., Rainbow plots, bagplots, and boxplots for functional data, J. Comput. Graph. Stat., 19, 1, 29-45 (2010)
[33] Hyndman, R. J.; Shang, H. L., ftsa: Functional time series analysis (2020), URL https://CRAN.R-project.org/package=ftsa R package version 6.0
[34] Hyndman, R. J.; Ullah, M., Robust forecasting of mortality and fertility rates: A functional data approach, J. Comput. Graph. Stat., 51, 10, 4942-4956 (2007) · Zbl 1162.62434
[35] Ieva, F.; Paganoni, A. M., Depth measures for multivariate functional data, Comm. Statist. Theory Methods, 42, 7, 1265-1276 (2013) · Zbl 1347.62093
[36] Kara, L.-Z.; Laksaci, A.; Rachdi, M.; Vieu, P., Data-driven kNN estimation in nonparametric functional data analysis, J. Multivariate Anal., 153, 176-188 (2017) · Zbl 1351.62084
[37] Klepsch, J.; Klüppelberg, C., An innovations algorithm for the prediction of functional linear processes, J. Multivariate Anal., 155, 252-271 (2017) · Zbl 1397.62346
[38] Kokoszka, P.; Reimherr, M., Introduction To Functional Data Analysis (2017), Chapman and Hall/CRC · Zbl 1411.62004
[39] Kraus, D., Components and completion of partially observed functional data, J. R. Stat. Soc. B, 77, 4, 777-801 (2015) · Zbl 1414.62212
[40] Kudraszow, N. L.; Vieu, P., Uniform consistency of kNN regressors for functional variables, Statist. Probab. Lett., 83, 8, 1863-1870 (2013) · Zbl 1277.62113
[41] Li, D.; Robinson, P. M.; Shang, H. L., Local whittle estimation of long-range dependence for functional time series, J. Time Series Anal. (2020)
[42] Li, D.; Robinson, P. M.; Shang, H. L., Long-range dependent curve time series, J. Am. Stat. Assoc.: Theory Methods, 115, 530, 957-971 (2020) · Zbl 1445.62231
[43] Lian, H., Convergence of functional k-nearest neighbor regression estimate with functional responses, Electron. J. Stat., 5, 31-40 (2011) · Zbl 1274.62291
[44] Ling, N.; Vieu, P., Nonparametric modelling for functional data: selected survey and tracks for future, Statistics, 52, 4, 934-949 (2018) · Zbl 1411.62084
[45] Liu, N., Ambient particulate air pollution and daily mortality in 652 cities, N. Engl. J. Med., 381, 8, 705-715 (2019)
[46] López-Pintado, S.; Romo, J., Depth-based classification for functional data, (Regina Y. Liu, R. S.; Souvaine, D. L., Data Depth: Robust Multivariate Analysis, Computational Geometry and Applications, Vol. 72 (2006), AMS and DIMACS), 103
[47] López-Pintado, S.; Romo, J., On the concept of depth for functional data, J. Am. Stat. Assoc.: Theory Methods, 104, 486, 718-734 (2009) · Zbl 1388.62139
[48] López-Pintado, S.; Romo, J.; Torrente, A., Robust depth-based tools for the analysis of gene expression data, Biostatistics, 11, 2, 254-264 (2010) · Zbl 1437.62542
[49] López-Pintado, S.; Sun, Y.; Lin, J. K.; Genton, M. G., Simplicial band depth for multivariate functional data, Adv. Data Anal. Classif., 8, 3, 321-338 (2014) · Zbl 1414.62066
[50] Lütkepohl, H., New Introduction to Multiple Time Series Analysis (2006), Sprinter-Verlag: Sprinter-Verlag New York · Zbl 1141.62071
[51] Makridakis, S.; Spiliotis, E.; Assimakopoulos, V., Statistical and machine learning forecasting methods: Concerns and ways forward, PLOS ONE, 13, 3, 1-26 (2018)
[52] Martínez, F.; Frías, M. P.; Charte, F.; Rivera, A. J., A methodology for applying k-nearest neighbor to time series forecasting, Artif. Intell. Rev., 52, 3, 2019-2037 (2019)
[53] Martínez, F.; Frías, M. P.; Charte, F.; Rivera, A. J., Time series forecasting with KNN in r: the tsfknn package, R J., 11, 2, 229-242 (2019)
[54] Nagy, S.; Gijbels, I.; Hlubinka, D., Depth-based recognition of shape outlying functions, J. Comput. Graph. Statist., 26, 4, 883-893 (2017)
[55] Narisetty, N. N.; Nair, V. N., Extremal depth for functional data and applications, J. Am. Stat. Assoc.: Theory Methods, 111, 516, 1705-1714 (2015)
[56] Nieto-Reyes, A.; Battey, H., A topologically valid definition of depth for functional data, Statist. Sci., 31, 1, 61-79 (2016) · Zbl 1436.62720
[57] Paparoditis, E.; Sapatinas, T., Short-term load forecasting: The similar shape functional time-series predictor, IEEE Trans. Power Syst., 28, 4, 3818-3825 (2013)
[58] Perretti, C. T.; Munch, S. B.; Sugihara, G., Model-free forecasting outperforms the correct mechanistic model for simulated and experimental data, Proc. Natl. Acad. Sci., 110, 13, 5253-5257 (2013)
[59] Ramsay, J.; Silverman, B., Functional Data Analysis (2005), Springer: Springer New York · Zbl 1079.62006
[60] Raña, P.; Vilar, J.; Aneiros, G., On the use of functional additive models for electricity demand and price prediction, IEEE Access, 6, 9603-9613 (2018)
[61] Rasmussen, C. E.; Williams, C. K.I., Gaussian Processes for Machine Learning (2005), The MIT Press: The MIT Press Cambridge
[62] Serfling, R.; Wijesuriya, U., Depth-based nonparametric description of functional data, with emphasis on use of spatial depth, Comput. Statist. Data Anal., 105, 24-45 (2017) · Zbl 1466.62192
[63] Shang, H. L., Functional time series approach for forecasting very short-term electricity demand, J. Appl. Stat., 40, 1, 152-168 (2013) · Zbl 1514.62861
[64] Shang, H. L., Functional time series forecasting with dynamic updating: An application to intraday particulate matter concentration, Econometrics Stat., 1, 184-200 (2017)
[65] Shang, H. L., Visualizing rate of change: an application to age-specific fertility rates, J. R. Stat. Soc. Ser. A, 182, 1, 249-262 (2019)
[66] Shang, H. L.; Hyndman, R. J., Nonparametric time series forecasting with dynamic updating, Math. Comput. Simulation, 81, 7, 1310-1324 (2011) · Zbl 1215.62098
[67] Shang, H. L.; Yang, Y.; Kearney, F., Intraday forecasts of a volatility index: functional time series methods with dynamic updating, Ann. Oper. Res., 282, 331-354 (2018) · Zbl 1434.62196
[68] Singh, S. K.; McMillan, H.; Bádossy, A.; Fateh, C., Nonparametric catchment clustering using the data depth function, Hydrol. Sci. J., 61, 15, 2649-2667 (2016)
[69] Sonmez, O.; Aue, A.; Rice, G., fChange: Change point analysis in functional data (2019), URL https://cran.r-project.org/package=fChange R package version 0.2.1, removed from CRAN
[70] Sugihara, G.; Grenfell, B. T.; May, R. M.; Tong, H., Nonlinear forecasting for the classification of natural time series, Philos. Trans. R. Soc. Lond. Ser. A Phys. Eng. Sci., 348, 1688, 477-495 (1994) · Zbl 0864.92001
[71] Sugihara, G.; May, R. M., Nonlinear forecasting as a way of distinguishing chaos from measurement error in time series, Nature, 344, 6268, 734-741 (1990)
[72] Sun, Y.; Genton, M. G., Functional boxplots, J. Comput. Graph. Stat., 20, 2, 316-334 (2011)
[73] Sun, Y.; Genton, M. G.; Nychka, D. C., Exact fast computation of band depth for large functional datasets: How quickly can one million curves be ranked?, Stat, 1, 68-74 (2012)
[74] Tarabelloni, N.; Ieva, F.; Biasi, R.; Paganoni, A. M., Use of depth measure for multivariate functional data in disease prediction: An application to electrocardiograph signals, Int. J. Biostat., 11, 189-201 (2015)
[75] Tupper, L. L.; Matteson, D. S.; Anderson, C. L.; Zephyr, L., Band depth clustering for nonstationary time series and wind speed behavior, Technometrics, 60, 2, 245-254 (2017)
[76] Vilar, J.; Aneiros, G.; Raña, P., Prediction intervals for electricity demand and price using functional data, Int. J. Electr. Power Energy Syst., 96, 457-472 (2018)
[77] Vilar, J. M.; Cao, R.; Aneiros, G., Forecasting next-day electricity demand and price using nonparametric functional methods, Int. J. Electr. Power Energy Syst., 39, 1, 48-55 (2012)
[78] Wang, J.-L.; Chiou, J.-M.; Müller, H.-G., Functional data analysis, Annu. Rev. Stat. Appl., 3, 1, 257-295 (2016)
[79] Wang, E.; Cook, D.; Hyndman, R. J., Calendar-based graphics for visualizing people’s daily schedules, J. Comput. Graph. Statist., 29, 3, 1-13 (2020) · Zbl 07499291
[80] Winkler, R. L., A decision-theoretic approach to interval estimation, J. Am. Stat. Assoc.: Theory Methods, 67, 337, 187-191 (1972) · Zbl 0231.62012
[81] Yao, F.; Müller, H. G.; Wang, J. L., Functional data analysis for sparse longitudinal data, J. Am. Stat. Assoc.: Theory Methods, 100, 470, 577-590 (2005) · Zbl 1117.62451
[82] Zhang, G.; Eddy Patuwo, B.; Y. Hu, M., Forecasting with artificial neural networks:: The state of the art, Int. J. Forecast., 14, 1, 35-62 (1998)
[83] Zhang, S.; Jank, W.; Shmueli, G., Real-time forecasting of online auctions via functional K-nearest neighbors, Int. J. Forecast., 26, 4, 666-683 (2010)
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