a research paper about FDSLRM modeling with supplementary materials - software, notebooks
Estimating variances in time series linear regression models using empirical BLUPs and convex optimization
a research paper and supplementary materials - software, notebooks
by Martina Hančová, Gabriela Vozáriková, Andrej Gajdoš, Jozef Hanč [email protected]
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 O(n), which at the default precision is 107 times more accurate and n2 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.
Research paper
The research paper has been submitted for publishing in Statistical Papers.
A preprint version is available at https://arxiv.org/abs/1905.07771.
Software 
The notebooks folders (Modeling, PYnotebooks, Rnotebooks) contain Python based and R based Jupyter notebooks which are detailed records of our computing with explaining narratives ilustrating explored concepts and methods.
Notebooks can be studied and viewed statically in Jupyter nbviewer with full visualisation. If there is a need, they can be also viewed directly on Github index.ipynb, also as a raw code.
For interactive executing notebooks as live documents without any need to install or compile the software, use the following links
There is also another very comfort alternative, fully tested by us, called CoCalc providing interactive computing with our Jupyter notebooks.
All source code is distributed under the MIT license.
Related materials
The misc folder contains our previously published papers related to the fdslrm project.
Acknowledgements
This work was supported by the Slovak Research and Development Agency under the contract No. APVV-17-0568, the Scientific Grant Agency of the Slovak Republic (VEGA), VEGA grant No.1/0311/18 and the Internal Research Grant System of Faculty of Science, P.J. Šafárik University in Košice (VVGS PF UPJŠ) — project VVGS-PF-2018-792.