Authors: Andrej Gajdoš, Jozef Hanč, Martina Hančová
Faculty of Science, P. J. Šafárik University in Košice, Slovakia
email: [email protected], [email protected]

Supplementary materials for EBLUP-NE

Estimating variances in FDSLRMs via EBLUPs and convex optimization

Index

Electricity consumption - toy model 1

• PY-estimation-electricity1-SciPyCVXPY.ipynb, EBLUP-NE in SciPy, CVXPY

• PY-estimation-electricity1-SageMath.ipynb, EBLUP-NE in SageMath

• R-estimation-electricity1-CVXR.ipynb, EBLUP-NE in CVXR

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

Electricity consumption - toy model 2

• PY-estimation-electricity2-SciPyCVXPY.ipynb, EBLUP-NE in SciPy, CVXPY

• PY-estimation-electricity2-SageMath.ipynb, EBLUP-NE in SageMath

• R-estimation-electricity2-CVXR.ipynb, EBLUP-NE in CVXR

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

Tourism

• tourism.ipynb, FDSLRM modeling in R

• PY-estimation-tourism-SciPyCVXPY.ipynb, EBLUP-NE in SciPy, CVXPY

• R-estimation-tourism-CVXR.ipynb, EBLUP-NE in CVXR

• R-estimation-tourism-standardRtools.ipynb, EBLUP-NE in nlme, MMEinR, sommer, fdslrm

Cyber attacks

• cyberattacks.ipynb, FDSLRM modeling in R

• PY-estimation-cyberattacks-SciPyCVXPY.ipynb, EBLUP-NE in SciPy, CVXPY

• R-estimation-cyberattacks-CVXR.ipynb, EBLUP-NE in CVXR

• R-estimation-cyberattacks-standardRtools.ipynb, EBLUP-NE in nlme, MMEinR, sommer, fdslrm

References

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.

Abstract of the paper

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