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An introduction to discrete-valued time series
A much-needed introduction to the field of discrete-valued time series, with a focus on count-data time series Time series analysis is an essential tool in a wide array of fields, including business, economics, computer science, epidemiology, finance, manufacturing and meteorology, to name just a few. Despite growing interest in discrete-valued time series—especially those arising from counting specific objects or events at specified times—most books on time series give short shrift to that increasingly important subject area. This book seeks to rectify that state of affairs by providing a much needed introduction to discrete-valued time series, with particular focus on count-data time series. The main focus of this book is on modeling. Throughout numerous examples are provided illustrating models currently used in discrete-valued time series applications. Statistical process control, including various control charts (such as cumulative sum control charts), and performance evaluation are treated at length. Classic approaches like ARMA models and the Box-Jenkins program are also featured with the basics of these approaches summarized in an Appendix. In addition, data examples, with all relevant R code, are available on a companion website. • Provides a balanced presentation of theory and practice, exploring both categorical and integer-valued series • Covers common models for time series of counts as well as for categorical time series, and works out their most important stochastic properties • Addresses statistical approaches for analyzing discrete-valued time series and illustrates their implementation with numerous data examples • Covers classical approaches such as ARMA models, Box-Jenkins program and how to generate functions • Includes dataset examples with all necessary R code provided on a companion website An Introduction to Discrete-Valued Time Series is a valuable working resource for researchers and practitioners in a broad range of fields, including statistics, data science, machine learning, and engineering. It will also be of interest to postgraduate students in statistics, mathematics and economics.
eBook
Ergodicity of Markov Processes via Nonstandard Analysis
The Markov chain ergodic theorem is well-understood if either the time-line or the state space is discrete. However, there does not
exist a very clear result for general state space continuous-time Markov processes. Using methods from mathematical logic and
nonstandard analysis, we introduce a class of hyperfinite Markov processes-namely, general Markov processes which behave like finite
state space discrete-time Markov processes. We show that, under moderate conditions, the transition probability of hyperfinite Markov
processes align with the transition probability of standard Markov processes. The Markov chain ergodic theorem for hyperfinite Markov
processes will then imply the Markov chain ergodic theorem for general state space continuous-time Markov processes.
eBook
Data-driven consensus control of fully distributed event-triggered multi-agent systems
This study investigates the consensus control issue in discrete-time linear multi-agent systems (MASs) using data-driven control under undirected communication networks. To alleviate the communication burden, an adaptive event-triggered control strategy involving only local information is proposed and a model-based stability condition is derived that guarantees the asymptotic consensus of MASs. Furthermore, a data-based consensus condition for unknown MASs is established by combining a data-based system representation with the model-based stability condition, using only pre-collected noisy input-state data instead of the accurate system information a priori. Specifically, both model-based and data-driven event-triggered controllers can be utilized without requiring any global information. The validity and correctness of the controllers and associated theoretical results are demonstrated via numerical simulations.
Journal Article
Long-time properties of generic Floquet systems are approximately periodic with the driving period
A Floquet quantum system is governed by a Hamiltonian that is periodic in time. Consider the space of piecewise time-independent Floquet systems with (geometrically) local interactions. We prove that for all but a measure zero set of systems in this space, starting from a random product state, many properties (including expectation values of observables and the entanglement entropy of a macroscopically large subsystem) at long times are approximately periodic with the same period as the Hamiltonian. Thus, in almost every Floquet system of arbitrarily large but finite size, discrete time-crystalline behavior does not persist to strictly infinite time.
Journal Article
Phase diagram and optimal control for n-tupling discrete time crystal
A remarkable consequence of spontaneously breaking the time translational symmetry in a system, is the emergence of time crystals. In periodically driven systems, discrete time crystals (DTC) can be realized which have a periodicity that is n times the driving period. However, all of the experimental observations have been performed for period-doubling and period-tripling DTC. Novel physics can arise by simulating many-body physics in the time domain, which would require a genuine realisation of the n-tupling DTC. A system of ultra-cold bosonic atoms bouncing resonantly on an oscillating mirror is one of the models that can realise large period DTC. The preparation of DTC demands control in creating the initial distribution of the ultra-cold bosonic atoms along with the mirror frequency. In this work, we demonstrate that such DTC is robust against perturbations to the initial distribution of atoms. We show how Bayesian methods can be used to enhance control in the preparation of the initial state as well as to efficiently calculate the phase diagram for such a model. Moreover, we examine the stability of DTCs by analyzing quantum many-body fluctuations and show that they do not reveal signatures of heating.
Journal Article
Many-body effects and quantum fluctuations for discrete time crystals in Bose–Einstein condensates
We present a fully comprehensive multi-mode quantum treatment based on the truncated Wigner approximation (TWA) to study many-body effects and effects of quantum fluctuations on the formation of a discrete time crystal (DTC) in a Bose–Einstein condensate (BEC) bouncing resonantly on a periodically driven atom mirror. Zero-range contact interactions between the bosonic atoms are assumed. Our theoretical approach avoids the restrictions both of mean-field theory, where all bosons are assumed to remain in a single mode, and of time-dependent Bogoliubov theory, which assumes boson depletion from the condensate mode is small. We show that the mean-field and time-dependent Bogoliubov approaches can be derived as approximations to the TWA treatment. Differing initial conditions, such as a finite temperature BEC, can also be treated. For realistic initial conditions corresponding to a harmonic trap condensate mode function, our TWA calculations performed for period-doubling agree broadly with recent mean-field calculations for times out to at least 2000 mirror oscillations, except at interaction strengths very close to the threshold value for DTC formation where the position probability density differs significantly from that determined from mean-field theory. For typical attractive interaction strengths above the threshold value for DTC formation and for the chosen trap and driving parameters, the TWA calculations indicate a quantum depletion due to quantum many-body fluctuations of less than about two atoms out of a total of 600 atoms at times corresponding to 2000 mirror oscillations, in agreement with time-dependent Bogoliubov theory calculations. On the other hand, for interaction strengths very close to the threshold value for DTC formation, the TWA calculations predict a large quantum depletion—as high as about 260 atoms out of 600. We also show that the mean energy per particle of the DTC does not increase significantly for times out to at least 2000 mirror oscillations and typically oscillates around an average value close to its initial value; so TWA theory predicts the absence of thermalisation. Finally, we find that the dynamical behaviour of our system is largely independent of whether the boson–boson interaction is attractive or repulsive, and that it is possible to create a stable DTC based on repulsive interactions.
Journal Article
Discrete-time delay systems: part 1. Global fully actuated case
A basic introduction to the fully actuated system (FAS) approaches for discrete-time systems with delays is given. Firstly, general dynamical discrete-time FAS models with time-varying state delays and constant input delays are proposed. The FAS models are classified into affine ones and non-affine ones, and also ones with and without interconnections. Secondly, controllers for such FASs are designed, which result in constant linear closed-loop systems with arbitrarily assignable eigenstructure. Different from the case of FAS with state delays only, the controller for a discrete-time FAS with an input delay involves a prediction scheme which is constructed based on the open-loop system. The contribution of this paper has laid a fundamental basis for FAS approaches to discrete-time delay systems, and further specific analysis and design problems can be established similar to the continuous-time system case.
Journal Article
H∞ Filtering for Nonlinear Discrete-time Singular Systems in Encrypted State
This paper studies the
H
∞
filtering problem of discrete-time singular nonlinear systems in encrypted state which are represented by Takagi-Sugeno (T-S) fuzzy model, meantime, quantization, signal missing and filter failure are considered. This paper selects the measurement output and the filter output for quantization, the sensor failure of the systems, the loss of the estimated signal and filter output signals are considered. Then, the admissible condition of the filtering error system is calculated and verified, and the condition meets the specific
H
∞
performance index. By quoting a new Lyapunov function, the design conditions of the filter and the adjustment parameters of the quantizers are obtained. Finally, the feasibility of this method is verified by a circuit example.
Journal Article