## Electronic Journal of Statistics

### Regularised forecasting via smooth-rough partitioning of the regression coefficients

#### 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
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

#### 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

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