Electronic Journal of Statistics

Regularised forecasting via smooth-rough partitioning of the regression coefficients

Hyeyoung Maeng and Piotr Fryzlewicz

Full-text: Open access

Abstract

We introduce a way of modelling temporal dependence in random functions $X(t)$ in the framework of linear regression. Based on discretised curves $\{X_{i}(t_{0}),X_{i}(t_{1}),\ldots ,X_{i}(t_{T})\}$, the final point $X_{i}(t_{T})$ is predicted from $\{X_{i}(t_{0}),X_{i}(t_{1}),\ldots ,X_{i}(t_{T-1})\}$. The proposed model flexibly reflects the relative importance of predictors by partitioning the regression parameters into a smooth and a rough regime. Specifically, unconstrained (rough) regression parameters are used for observations located close to $X_{i}(t_{T})$, while the set of regression coefficients for the predictors positioned far from $X_{i}(t_{T})$ are assumed to be sampled from a smooth function. This both regularises the prediction problem and reflects the ‘decaying memory’ structure of the time series. The point at which the change in smoothness occurs is estimated from the data via a technique akin to change-point detection. The joint estimation procedure for the smoothness change-point and the regression parameters is presented, and the asymptotic behaviour of the estimated change-point is analysed. The usefulness of the new model is demonstrated through simulations and four real data examples, involving country fertility data, pollution data, stock volatility series and sunspot number data. Our methodology is implemented in the R package srp, available from CRAN.

Article information

Source
Electron. J. Statist., Volume 13, Number 1 (2019), 2093-2120.

Dates
Received: October 2018
First available in Project Euclid: 22 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejs/1561168839

Digital Object Identifier
doi:10.1214/19-EJS1573

Mathematical Reviews number (MathSciNet)
MR3973133

Zentralblatt MATH identifier
07080069

Keywords
Change-point detection prediction penalised spline functional linear regression

Rights
Creative Commons Attribution 4.0 International License.

Citation

Maeng, Hyeyoung; Fryzlewicz, Piotr. Regularised forecasting via smooth-rough partitioning of the regression coefficients. Electron. J. Statist. 13 (2019), no. 1, 2093--2120. doi:10.1214/19-EJS1573. https://projecteuclid.org/euclid.ejs/1561168839


Export citation

References

  • Aneiros-Pérez, G. and Vieu, P. (2008). Nonparametric time series prediction: A semi-functional partial linear modeling., Journal of Multivariate Analysis, 99:834–857.
  • Antoniadis, A., Paparoditis, E., and Sapatinas, T. (2006). A functional wavelet–kernel approach for time series prediction., Journal of the Royal Statistical Society, Series B, 68:837–857.
  • Aue, A., Norinho, D. D., and Hörmann, S. (2015). On the prediction of stationary functional time series., Journal of the American Statistical Association, 110:378–392.
  • Bandi, F. M. and Phillips, P. C. (2003). Fully nonparametric estimation of scalar diffusion models., Econometrica, 71:241–283.
  • Bosq, D. (2000)., Linear Processes in Function Spaces. New York: Springer-Verlag.
  • Cai, T. T. and Hall, P. (2006). Prediction in functional linear regression., The Annals of Statistics, 34:2159–2179.
  • Cardot, H., Crambes, C., Kneip, A., and Sarda, P. (2007). Smoothing splines estimators in functional linear regression with errors-in-variables., Computational Statistics and Data Analysis, 51:4832–4848.
  • Cardot, H., Ferraty, F., and Sarda, P. (2003). Spline estimators for the functional linear model., Statistica Sinica, 13:571–591.
  • Chen, K., Delicado, P., and Müller, H.-G. (2017). Modelling function-valued stochastic processes, with applications to fertility dynamics., Journal of the Royal Statistical Society, Series B, 79:177–196.
  • Crambes, C., Kneip, A., and Sarda, P. (2009). Smoothing splines estimators for functional linear regression., The Annals of Statistics, 37:35–72.
  • Ferraty, F., Hall, P., and Vieu, P. (2010). Most-predictive design points for functional data predictors., Biometrika, 97:807–824.
  • Ferraty, F. and Vieu, P. (2002). The functional nonparametric model and application to spectrometric data., Computational Statistics, 17:545–564.
  • Gabrys, R., Horváth, L., and Kokoszka, P. (2010). Tests for error correlation in the functional linear model., Journal of the American Statistical Association, 105:1113–1125.
  • Goia, A. (2012). A functional linear model for time series prediction with exogenous variables., Statistics and Probability Letters, 82:1005–1011.
  • Goia, A. and Vieu, P. (2015). A partitioned single functional index model., Computational Statistics, 30:673–692.
  • Hall, P. and Hooker, G. (2016). Truncated linear models for functional data., Journal of the Royal Statistical Society, Series B, 78:637–653.
  • Han, K., Müller, H.-G., and Park, B. U. (2017). Smooth backfitting for additive modeling with small errors-in-variables, with an application to additive functional regression for multiple predictor functions., Preprint.
  • Horváth, L., Kokoszka, P., and Reeder, R. (2013). Estimation of the mean of functional time series and a two-sample problem., Journal of the Royal Statistical Society, Series B, 75:103–122.
  • Horváth, L., Kokoszka, P., and Rice, G. (2014). Testing stationarity of functional time series., Journal of Econometrics, 179:66–82.
  • James, G. M., Wang, J., and Zhu, J. (2009). Functional linear regression that’s interpretable., The Annals of Statistics, 37:2083–2108.
  • Ji, H. and Müller, H.-G. (2017). Optimal designs for longitudinal and functional data., Journal of the Royal Statistical Society, Series B, 79:859–876.
  • Kneip, A., Poß, D., Sarda, P., et al. (2016). Functional linear regression with points of impact., The Annals of Statistics, 44:1–30.
  • Kong, D., Xue, K., Yao, F., and Zhang, H. H. (2016). Partially functional linear regression in high dimensions., Biometrika, 103:147–159.
  • Kristensen, D. (2010). Nonparametric filtering of the realized spot volatility: A kernel-based approach., Econometric Theory, 26:60–93.
  • Lin, Z., Cao, J., Wang, L., and Wang, H. (2015). A smooth and locally sparse estimator for functional linear regression via functional scad penalty., Preprint.
  • McKeague, I. W. and Sen, B. (2010). Fractals with point impact in functional linear regression., The Annals of Statistics, 38:2559–2586.
  • Müller, H.-G., Sen, R., and Stadtmüller, U. (2011). Functional data analysis for volatility., Journal of Econometrics, 165:233–245.
  • Ramsay, J. O. and Silverman, B. W. (2005)., Functional Data Analysis. New York: Springer-Verlag.
  • Reiss, P. T., Goldsmith, J., Shang, H. L., and Ogden, R. T. (2017). Methods for scalar-on-function regression., International Statistical Review, 85:228–249.
  • Reno, R. (2008). Nonparametric estimation of the diffusion coefficient of stochastic volatility models., Econometric Theory, 24:1174–1206.
  • Ruppert, D. (2002). Selecting the number of knots for penalized splines., Journal of Computational and Graphical Statistics, 11:735–757.
  • Schwarz, G. (1978). Estimating the dimension of a model., The Annals of Statistics, 6:461–464.
  • Shin, H. (2009). Partial functional linear regression., Journal of Statistical Planning and Inference, 139:3405–3418.
  • Shin, H. and Lee, M. H. (2012). On prediction rate in partial functional linear regression., Journal of Multivariate Analysis, 103:93–106.
  • Wood, S. N. (2006)., Generalized additive models: an introduction with R. CRC press.
  • Zhou, J. and Chen, M. (2012). Spline estimators for semi-functional linear model., Statistics and Probability Letters, 82:505–513.
  • Zhou, J., Chen, Z., and Peng, Q. (2016). Polynomial spline estimation for partial functional linear regression models., Computational Statistics, 31:1107–1129.
  • Zhou, J., Wang, N.-Y., and Wang, N. (2013). Functional linear model with zero-value coefficient function at sub-regions., Statistica Sinica, 23:25–50.
  • Zhu, H., Yao, F., and Zhang, H. H. (2014). Structured functional additive regression in reproducing kernel hilbert spaces., Journal of the Royal Statistical Society, Series B, 76:581–603.