Hierarchical time series is a set of data sequences organized by aggregation constraints to represent many real-world applications in research and the industry. Forecasting of hierarchical time series is a challenging and time-consuming problem owing to ensuring the forecasting consistency among the hierarchy levels based on their dimensional features. The excellent empirical performance of our Deep Long Short-Term Memory (DLSTM) approach on various forecasting tasks motivated us to extend it to solve the forecasting problem through hierarchical architectures. Toward this target, we develop the DLSTM model in auto-encoder (AE) fashion and take full advantage of the hierarchical architecture for better time series forecasting. DLSTM-AE works as an alternative approach to traditional and machine learning approaches that have been used to manipulate hierarchical forecasting. However, training a DLSTM in hierarchical architectures requires updating the weight vectors for each LSTM cell, which is time-consuming and requires a large amount of data through several dimensions. Transfer learning can mitigate this problem by training first the time series at the bottom level of the hierarchy using the proposed DLSTM-AE approach. Then, we transfer the learned features to perform synchronous training for the time series of the upper levels of the hierarchy. To demonstrate the efficiency of the proposed approach, we compare its performance with existing approaches using two case studies related to the energy and tourism domains. An evaluation of all approaches was based on two criteria, namely, the forecasting accuracy and the ability to produce coherent forecasts through through the hierarchy. In both case studies, the proposed approach attained the highest accuracy results among all counterparts and produced more coherent forecasts.

Machine learning methods have been adopted in the literature as contenders to conventional methods to solve the energy time series forecasting (TSF) problems. Recently, deep learning methods have been emerged in the artificial intelligence field attaining astonishing performance in a wide range of applications. Yet, the evidence about their performance in to solve the energy TSF problems, in terms of accuracy and computational requirements, is scanty. Most of the review articles that handle the energy TSF problem are systematic reviews, however, a qualitative and quantitative study for the energy TSF problem is not yet available in the literature. The purpose of this paper is twofold, first it provides a comprehensive analytical assessment for conventional, machine learning, and deep learning methods that can be utilized to solve various energy TSF problems. Second, the paper carries out an empirical assessment for many selected methods through three real-world datasets. These datasets related to electrical energy consumption problem, natural gas problem, and electric power consumption of an individual household problem. The first two problems are univariate TSF and the third problem is a multivariate TSF. Compared to both conventional and machine learning contenders, the deep learning methods attain a significant improvement in terms of accuracy and forecasting horizons examined. In the meantime, their computational requirements are notably greater than other contenders. Eventually, the paper identifies a number of challenges, potential research directions, and recommendations to the research community may serve as a basis for further research in the energy forecasting domain.

In this paper, some properties of multi-qubit states traveling in non-inertial frames are investigated, where we assume that all particles are accelerated. These properties are including fidelities, capacities and entanglement of the accelerated channels for three different states, namely, Greeberger-Horne-Zeilinger (GHZ) state, GHZ-like state and W-state. It is shown here that all these properties are decreased as the accelerations of the moving particles are increased. The obtained results show that the GHZ-state is the most robust state comparing to the others, where the degradation rate is less than that for the other states particularly in the second Rindler region. Also, it is shown here that the entangled property doesn't change in the accelerated frames. Additionally, the paper shows that the degree of entanglement decreases as the accelerations of the particles increase in the first Rindler region. However in the second region, where all subsystems are disconnected at zero acceleration, entangled states are generated as the acceleration increases.