**Authors:** Andrej Gajdoš, Jozef Hanč, Martina Hančová

*Faculty of Science, P. J. Šafárik University in Košice, Slovakia*

email: [email protected], [email protected]

** Binder for EBLUP-NE using standard R tools and CVXR **

Interactive execution of Jupyter Notebooks. Use SHIFT-Enter or menu for executing cells in open notebooks.

- R-estimation-electricity1-standardRtools.ipynb, EBLUP-NE in
*nlme, MMEinR, sommer, fdslrm* - R-estimation-electricity1-CVXR.ipynb, EBLUP-NE in
*CVXR*

- R-estimation-electricity2-standardRtools.ipynb, EBLUP-NE in
*nlme, MMEinR, sommer, fdslrm* - R-estimation-electricity2-CVXR.ipynb, EBLUP-NE in
*CVXR*

- tourism.ipynb, FDSLRM modeling in
*R* - R-estimation-tourism-standardRtools.ipynb, EBLUP-NE in
*nlme, MMEinR, sommer, fdslrm* - R-estimation-tourism-CVXR.ipynb, EBLUP-NE in
*CVXR*

- cyberattacks.ipynb, FDSLRM modeling in
*R* - R-estimation-cyberattacks-standardRtools.ipynb, EBLUP-NE in
*nlme, MMEinR, sommer, fdslrm* - R-estimation-cyberattacks-CVXR.ipynb, EBLUP-NE in
*CVXR*

This notebook belongs to suplementary materials of the paper submitted to Statistical Papers and available at https://arxiv.org/abs/1905.07771.

- Hančová, M., Vozáriková, G., Gajdoš, A., Hanč, J. (2019). Estimating variance components in time series linear regression models using empirical BLUPs and convex optimization, https://arxiv.org/, 2019.

We propose a two-stage estimation method of variance components in time series models known as FDSLRMs, whose observations can be described by a linear mixed model (LMM). We based estimating variances, fundamental quantities in a time series forecasting approach called kriging, on the empirical (plug-in) best linear unbiased predictions of unobservable random components in FDSLRM.

The method, providing invariant non-negative quadratic estimators, can be used for any absolutely continuous probability distribution of time series data. As a result of applying the convex optimization and the LMM methodology, we resolved two problems $-$ theoretical existence and equivalence between least squares estimators, non-negative (M)DOOLSE, and maximum likelihood estimators, (RE)MLE, as possible starting points of our method and a practical lack of computational implementation for FDSLRM. As for computing (RE)MLE in the case of $ n $ observed time series values, we also discovered a new algorithm of order $\mathcal{O}(n)$, which at the default precision is $10^7$ times more accurate and $n^2$ times faster than the best current Python(or R)-based computational packages, namely CVXPY, CVXR, nlme, sommer and mixed.

We illustrate our results on three real data sets $-$ electricity consumption, tourism and cyber security $-$ which are easily available, reproducible, sharable and modifiable in the form of interactive Jupyter notebooks.

- Gajdoš A., Hanč J., and Hančová M. (2019).
*fdslrm EBLUP-NE*. GitHub repository, P.J. Šafárik University in Košice, Slovakia. https://github.com/fdslrm/EBLUP-NE