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We propose a simple and efficient approach to generate a prediction intervals (PI) for approximated and forecasted trends. Our method leverages a weighted asymmetric loss function to estimate the lower and upper bounds of the PI, with the weights determined by its coverage probability. We provide a concise mathematical proof of the method, show how it can be extended to derive PIs for parametrised functions and discuss its effectiveness when training deep neural networks. The presented tests of the method on a real-world forecasting task using a neural network-based model show that it can produce reliable PIs in complex machine learning scenarios., 14 pages, 3 figures

We employ stochastic dynamic microsimulations to analyse and forecast the pension cost dependency ratio for England and Wales from 1991 to 2061, evaluating the impact of the ongoing state pension reforms and changes in international migration patterns under different Brexit scenarios. To fully account for the recently observed volatility in life expectancies, we propose mortality rate model based on deep learning techniques, which discovers complex patterns in data and extrapolated trends. Our results show that the recent reforms can effectively stave off the "pension crisis" and bring back the system on a sounder fiscal footing. At the same time, increasingly more workers can expect to spend greater share of their lifespan in retirement, despite the eligibility age rises. The population ageing due to the observed postponement of death until senectitude often occurs with the compression of morbidity, and thus will not, perforce, intrinsically strain healthcare costs. To a lesser degree, the future pension cost dependency ratio will depend on the post-Brexit relations between the UK and the EU, with "soft" alignment on the free movement lowering the relative cost of the pension system compared to the "hard" one. In the long term, however, the ratio has a rising tendency.

In their Letter "Excitons in Molecular Aggregates with Levy Disorder: Anomalous Localization and Exchange Broadening of Optical Spectra", Eisfeld et al. predicted the existence of exchange broadening and blue-shift of the absorption band and a non-universal disorder scaling of the localisation length of absorption band excitons in J-aggregates with heavy-tailed disorder, which they contrasted with the previously analysed Gaussian and Lorentzian case. The observations were explained by another localisation mechanism, the chain segmentation by outliers in heavy tails of the disorder distribution. We argue that the previously known theory does not break down and consequently anticipates the properties of the absorption band investigated in the commented work., 9 pages, 8 figures

The Comment to the Letter "Excitons in Molecular Aggregates with Levy Disorder: Anomalous Localization and Exchange Broadening of Optical Spectra" appeared in Phys. Rev. Lett. 109, 259701 (submitted on November 28, 2011). I prepared the below response to the rebuttal received from the Letter's Authors in the review process. It contains useful comments, which further address the errors in the Letter and other flaws in Authors' understanding of the topic revealed in discussion., Comment: 9 pages, 7 figures

We present a stochastic model of population dynamics exploiting cross-sectional data in trend analysis and forecasts for groups and cohorts of a population. While sharing the convenient features of classic Markov models, it alleviates the practical problems experienced in longitudinal studies. Based on statistical and information-theoretical analysis, we adopt maximum likelihood estimation to determine model parameters, facilitating the use of a range of model selection methods. Their application to several synthetic and empirical datasets shows that the proposed approach is robust, stable and superior to a regression-based one. We extend the basic framework to simulate ageing cohorts, processes with finite memory, distinguishing their short and long-term trends, introduce regularisation to avoid the ecological fallacy, and generalise it to mixtures of cross-sectional and (possibly incomplete) longitudinal data. The presented model illustrations yield new and interesting results, such as an implied common driving factor in obesity for all generations of the English population and "yo-yo" dieting in the U.S. data., 21 pages, 10 figures

This appendix to the paper by Werpachowska and Dietl "Theory of spin waves in ferromagnetic (Ga,Mn)As" (DOI: 10.1103/PhysRevB.82.085204) is a response to inquiries about the derivation of the Loewdin calculus and its numerical implementation. It presents the detailed version of the Loewdin calculus for the multiband Hamiltonian taking into account both the p-d and s-d exchange couplings. This should explain how to implement the sp-d exchange interaction in the tight-binding computational scheme. I also include the complete procedure of the Bogoliubov transform for systems with spatial inversion asymmetry., Comment: 8 + epsilon pages, no figures, one column (corrected)