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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., Comment: 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.

We perform stochastic microsimulations of the dynamics of England and Wales population after the British referendum on EU membership, considering different possible outcomes. Employing available survey data, we model the demographics of the region over the next generation, as shaped by births, deaths and international migration. The migration patterns between England and Wales and the remaining EU countries are modified according to the possible scenarios of their future relations. We find that Brexit will accelerate the overall population ageing and the deepening imbalance between workers and retirees but reduce the population growth and the fraction of women of reproductive age. In the alternative scenarios of remaining in the EU these effects will be partially forestalled by the influx of immigrants from current and prospective EU countries and their children. In all considered scenarios the native British population declines. Our study demonstrates that microsimulations can be a useful tool for designing and evaluating the country's policies in the advent of fundamental transformations., Comment: 13 pages, to appear in the International Journal of Microsimulations

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

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., Comment: 9 pages, 8 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., Comment: 21 pages, 10 figures

We present the Reduced Operator Approximation: a simple, physically transparent and computationally efficient method of modelling open quantum systems. It employs the Heisenberg picture of the quantum dynamics, which allows us to focus on the system degrees of freedom in a natural and easy way. We describe different variants of the method, low- and high-order in the system-bath interaction operators, defining them for either general quantum harmonic oscillator baths or specialising them for independent baths with Lorentzian spectral densities. Its wide applicability is demonstrated on the examples of systems coupled to different baths (with varying system-bath interaction strength and bath memory length), and compared with the exact pseudomode and the popular quantum state diffusion approach. The method captures the decoherence of the system interacting with the bath, while conserving the total energy. Our results suggest that quantum coherence effects persist in open quantum systems for much longer times than previously thought., Comment: 22+4 pages, 7 figures

In their Letter [Phys. Rev. Lett. 105, 137402 (2010)], Eisfeld et al. predicted the exchange broadening and blue-shift of the absorption band as well as a nonuniversal disorder scaling of the localization length of excitons in J-aggregates with Levy disorder, which they contrasted with the previously analyzed Gaussian and Lorentzian case [Phys. Rev. B, 79, 1 (2009)]. The observations were explained by chain segmentation due to outliers, and its interplay with localization of states in effective potential wells created by typical random site energies. We argue that the previously known theory does not break down and thus anticipates the properties of absorption band investigated in the commented work., Comment: Phys. Rev. Lett. 109, 259701 (2012) (subm on 29 Nov 2011)

We present exact expressions for the sum of states of noninteracting classical and quantum particles occupying a finite number of modes with arbitrary spacings. Exploiting a probabilistic analogy, we derive an analytic fourth-order approximation to the density of states, which captures its variance and kurtosis, and is superior to the previous, commonly used methods for all three particle statistics. Our approach employs a simple exact method of calculating the moments of the microcanonical density of states for quantum particles, which requires less computational effort than the commonly used saddle-point approximation. We test our methods numerically and discuss their applicability to various physical systems., Comment: 8 pages, 6 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)

The collective behavior of spins in a dilute magnetic semiconductor is determined by their mutual interactions and influenced by the underlying crystal structure. Hence, we begin with the atomic quantum-mechanical description of this system using the proposed variational-perturbation calculus, and then turn to the emerging macroscopic picture employing phenomenological constants. Within this framework we study spin waves and exchange stiffness in the p-d Zener model of (Ga,Mn)As, its thin layers and bulk crystals described by the spds* tight-binding approximation. Analyzing the anisotropic part of exchange, we find that the Dzyaloshinskii-Moriya interaction may lead to the cycloidal spin arrangement and uniaxial in-plane anisotropy in thin layers, resulting in a surface-like anisotropy in thicker films. We also derive and discuss the spin-wave contribution to magnetization and Curie temperature. Our theory reconstructs the values of stiffness determined from the temperature dependence of magnetization, but reproduces only partly those obtained from analyzing precession modes in (Ga,Mn)As thin films., Comment: 17 pages, 12 figures (added Eq. 36, replaced Fig. 10)

The relativistic nature of the electron motion underlies the intrinsic part of the anomalous Hall effect, believed to dominate in ferromagnetic (Ga,Mn)As. In this paper, we concentrate on the crystal band structure as an important facet to the description of this phenomenon. Using different k.p and tight-binding computational schemes, we capture the strong effect of the bulk inversion asymmetry on the Berry curvature and the anomalous Hall conductivity. At the same time, we find it not to affect other important characteristics of (Ga,Mn)As, namely the Curie temperature and uniaxial anisotropy fields. Our results extend the established theories of the anomalous Hall effect in ferromagnetic semiconductors and shed new light on its puzzling nature.