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 SageMath

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References

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

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.