Discrete-time Value Research Articles

Boundedness of the discrete Hilbert transform in discrete Hölder spaces

The Hilbert transform plays an important role in the theory and practice of signal processing operations in continuous system theory. The Hilbert transform was the motivation for the development of modern harmonic analysis. Its discrete version is also widely used in many areas of science and technology and plays an important role in digital signal processing. The essential motivation behind thinking about discrete transforms is that experimental data are most frequently not taken in a continuous manner but sampled at discrete time values. Since much of the data collected in both the physical sciences and engineering are discrete, the discrete Hilbert transform is a rather useful tool in these areas for the general analysis of this type of data. The Hilbert transform has been well studied on classical function spaces such as H¨older, Lebesgue, Morrey, etc. But its discrete version, which also has numerous applications, has not been fully studied in discrete analogues of these spaces. In this paper, we discuss the discrete Hilbert transform in discrete H¨older spaces and obtain its boundedness in these spaces.

Assessing paediatric safeguarding in rural Australian health services.

Establish the incidence, burden and characteristics of paediatric safeguarding concerns in rural Australian emergency department practice. Retrospective cohort study of burns, injury and poisoning presentations across 16 months involving 1472 paediatric cases. Five per cent of presentations had confirmed safeguarding concern. These were highest during the 2200-0600 staffing period. Mean age was 7.7 years, 43.8% were female. Multivariable regression models show age 2-6 years (odds ratio (OR), 3.27; 95% confidence interval (CI), 1.35-7.93); delayed presentation (OR, 2.3; 95% CI, 1.47-3.59); and police accompaniment (OR, 9.46; 95% CI, 2.61-34.26) are associated with increased safeguarding concerns. Most concerns (91.8%) related to injuries, largely musculoskeletal, wounds and head injuries. Thermal burns were more common than chemical and electrical. Children aged 2-6 are at higher risk for harm than previously recognised and children aged 0-2 years were over-represented in staff-suspected concerns. Those accompanied by police had significant association with confirmed safeguarding concerns which were under-suspected by staff or assumed to have been already reported. In rural practice, 'unreasonable delay' was found to be a better measure of concern than a discrete time value. Transient family arrangements, unsecured accommodation, geographical isolation, cultural safety and unique home environments must be taken into when completing injury assessments. For regional health services to successfully identify children at risk, interagency collaboration, staff education and local patterns of concern should be targeted. Rostering changes should increase after-hours assessment capacity by specialty paediatric staff.

Note on the (non-)smoothness of discrete time value functions in optimal stopping

We consider the discrete time stopping problem \[ V(t,x) = \sup _{\tau }\mathbb {E}_{(t,x)}g(\tau , X_{\tau }), \] where $X$ is a random walk. It is well known that the value function $V$ is in general not smooth on the boundary of the continuation set $\partial C$. We show that under some conditions $V$ is not smooth in the interior of $C$ either. Even more, under some additional conditions we show that $V$ is not differentiable on a dense subset of $C$. As a guiding example we consider the Chow-Robbins game. We give evidence that $\partial C$ is not smooth and that $C$ is not convex, in the Chow-Robbins game and other examples.

Oblique Survival Trees in Discrete Event Time Analysis.

One of the main objectives of survival analysis is to predict the failure time that is usually considered as a continuous variable. In longitudinal studies, the data are often collected at every certain period of time, for example, monthly, quarterly, or yearly. Such data require appropriate techniques to handle the discrete time values that often have incomplete information about the failure occurrence-so-called "censored cases." Tree-based models are common, assumption-free methods of survival prediction. In this paper, the author proposes three recursive partitioning techniques able to cope with discrete-time censored survival data, which, in contrast to already-existing models limited to univariate trees, allow splits to have a form of any hyperplane. The performance of proposed methods, expressed as a mean absolute error, was examined on the basis of both synthetic and real data sets available in the literature and compared with existing tree-based models. To demonstrate the applicability of the methods in identifying subgroups of patients with a similar survival experience and to assess the influence of covariates on the risk of failure, a Veteran's Administration lung cancer data set was used. The results confirm proposed models to be good prediction tools for discrete-time survival data.

A new methodology for Functional Principal Component Analysis from scarce data. Application to stroke rehabilitation.

Functional Principal Component Analysis (FPCA) is an increasingly used methodology for analysis of biomedical data. This methodology aims to obtain Functional Principal Components (FPCs) from Functional Data (time dependent functions). However, in biomedical data, the most common scenario of this analysis is from discrete time values. Standard procedures for FPCA require obtaining the functional data from these discrete values before extracting the FPCs. The problem appears when there are missing values in a non-negligible sample of subjects, especially at the beginning or the end of the study, because this approach can compromise the analysis due to the need to extrapolate or dismiss subjects with missing values. In this paper, we present an alternative methodology extracting the FPCs directly from the sampled data, avoiding the need to have functional data before extracting them. We demonstrate the feasibility of our approach from real data obtained from the analysis of balance recovery after stroke. Finally, we demonstrate that FPCA can obtain differences between groups when these differences are more related to the dynamics of the process than data values at given points.

Finite element solution of nonlinear eddy current problems with periodic excitation and its industrial applications

An efficient finite element method to take account of the nonlinearity of the magnetic materials when analyzing three-dimensional eddy current problems is presented in this paper. The problem is formulated in terms of vector and scalar potentials approximated by edge and node based finite element basis functions. The application of Galerkin techniques leads to a large, nonlinear system of ordinary differential equations in the time domain.The excitations are assumed to be time-periodic and the steady-state periodic solution is of interest only. This is represented either in the frequency domain as a finite Fourier series or in the time domain as a set of discrete time values within one period for each finite element degree of freedom. The former approach is the (continuous) harmonic balance method and, in the latter one, discrete Fourier transformation will be shown to lead to a discrete harmonic balance method. Due to the nonlinearity, all harmonics, both continuous and discrete, are coupled to each other.The harmonics would be decoupled if the problem were linear, therefore, a special nonlinear iteration technique, the fixed-point method is used to linearize the equations by selecting a time-independent permeability distribution, the so-called fixed-point permeability in each nonlinear iteration step. This leads to uncoupled harmonics within these steps.As industrial applications, analyses of large power transformers are presented. The first example is the computation of the electromagnetic field of a single-phase transformer in the time domain with the results compared to those obtained by traditional time-stepping techniques. In the second application, an advanced model of the same transformer is analyzed in the frequency domain by the harmonic balance method with the effect of the presence of higher harmonics on the losses investigated. Finally a third example tackles the case of direct current (DC) bias in the coils of a single-phase transformer.

Optimal Execution for Uncertain Market Impact: Derivation and Characterization of a Continuous-Time Value Function

We study an optimal execution problem with uncertain market impact to derive a more realistic market model. We construct a discrete-time model as a value function for optimal execution. Market impact is formulated as the product of a deterministic part increasing with execution volume and a positive stochastic noise part. Then, we derive a continuous-time model as a limit of a discrete-time value function. We find that the continuous-time value function is characterized by a stochastic control problem with a Levy process.

Time Factor in Consolidation: Critical Review

The magnitude of consolidation settlement is often calculated using Terzaghi's expression for average degree of consolidation (U) with respect to time. Developed during a time of limited computing capabilities, Terzaghi's series solution to the one-dimensional consolidation equation was generalized using dimensionless time factor (T), where a single U-T curve is used to describe the consolidation behaviour of both singly and doubly drained strata. As a result, any comparisons between one- and two-way drainage are indirect and confined to discrete values of time. By introducing a modified time factor T* in terms of layer thickness (D) instead of the maximum drainage pat length (Hdr), it is now possible to observe the effect of drainage conditions over a continuous range of time for a variety of asymmetric initial excess pore pressure distributions. although two separate U-T plots are required (for singly and doubly drained cases), the time factor at specific times remains the same for both cases, enabling a direct visual comparison. The importance of a revised time factor is evident when observing the endpoint of consolidation, which occurs as U approaches 100%. This occurs at T*≅0.5 for two-way drainage and at T*≅2 for one-way drainage, an observation not possible using the traditional expression for time factor.

The Use of Statistical Tests to Calibrate the Black-Scholes Asset Dynamics Model Applied to Pricing Options with Uncertain Volatility

A new method for calibrating the Black-Scholes asset price dynamics model is proposed. The data used to test the calibration problem included observations of asset prices over a finite set of (known) equispaced discrete time values. Statistical tests were used to estimate the statistical significance of the two parameters of the Black-Scholes model: the volatility and the drift. The effects of these estimates on the option pricing problem were investigated. In particular, the pricing of an option with uncertain volatility in the Black-Scholes framework was revisited, and a statistical significance was associated with the price intervals determined using the Black-Scholes-Barenblatt equations. Numerical experiments involving synthetic and real data were presented. The real data considered were the daily closing values of the S&P500 index and the associated European call and put option prices in the year 2005. The method proposed here for calibrating the Black-Scholes dynamics model could be extended to other science and engineering models that may be expressed in terms of stochastic dynamical systems.

Analysis and synthesis of discrete-time systems

This paper presents an introduction to the analysis and synthesis of sampled-data (discrete-time) systems, i.e. systems in which some or all signals can change values only at discrete values of time. The description of these systems is presented using state-space concepts. Following an introduction to linear discrete-time systems including systems where two or more samplers operate at different frequencies, here is a brief introduction to non-linear systems including the study of stability using the Second Method of Lyapunov. The latter part of the paper describes pulse-width modulated discrete systems. The final section considers the synthesis of systems designed to reach equilibrium states in the minimum number of sampling periods. The concepts discussed in the paper are illustrated with a large number of examples.

Wavelet analyses and applications

It is shown how a modern extension of Fourier analysis known as wavelet analysis is applied to signals containing multiscale information. First, a continuous wavelet transform is used to analyse the spectrum of a nonstationary signal (one whose form changes in time). The spectral analysis of such a signal gives the strength of the signal in each frequency as a function of time. Next, the theory is specialized to discrete values of time and frequency, and the resulting discrete wavelet transform is shown to be useful for data compression. This paper is addressed to a broad community, from undergraduate to graduate students to general physicists and to specialists in other fields than wavelets.

OPTIMAL CONTROL IN HYBRID SYSTEMS

Necessary conditions for optimality have been established for hybrid systems. Unfortunately, these conditions lead to multi-point boundary value problems and they do not prevent from the combinatorial explosion when no constraints are given on the transitions. Different approaches have been proposed to approximate the solution, such as relaxed dynamic programming, non linear programming and sensitivity functions.In hybrid systems context, the necessary conditions for optimal control are now well known. These conditions mix discrete and continuous classical necessary conditions on the optimal control. The discrete dynamic involves dynamic programming methods whereas between the a priori unknown discrete values of time, optimization of the continuous dynamic is performed using the maximum principle (MP) or Hamilton Jacobi Bellmann equations(HJB). At the switching instants, a set of boundary tranversality necessary conditions ensure a global optimization of the hybrid system.These theoretical conditions were applied to minimum time problem and to linear quadratic optimization.But it is practically very hard to perform such an optimization. The major raison is that discrete dynamic requires evaluating the optimal cost along all branches of the tree of all possible discrete trajectories.Dynamic programming is then used, but the duration between two switchings and the continuous optimization procedure make the task really hard. This makes the complexity increasing and only problems with a poor coupling between continuous and discrete parts can be reasonably solved. Nowadays, it seems obvious that only approximated solutions can be found. Various schemes have been imagined.Recent works have proposed to solve optimal switching problems by using a fixed switching schedule. By switched systems we mean a class of hybrid dynamical systems consisting of a family of continuous (or discrete) time subsystems and a rule (to be determined) that governs the switching between them.The optimization consists then in determining the optimal switching instants and the optimal continuous control assuming the number of switchings and the order of active subsystems already given. Then a nonlinear search method is used to determine the optimal solution.after the calculus of the derivatives of the value function with respect to the switching instants.Relaxed Dynamic programming : a relaxed procedure based on upper and lower bounds of the optimal cost was recently introduced. It proved to give good results for piece-wise affine systems and to obtain a suboptimal state feedback solution in the case of a quadratic criteria Algorithms based on the maximum principle for both multiple controlled and autonomous switchings with fixed schedule have been proposed. The algorithms use the transversality conditions at switching instants. Then, the authors develop a combinational search in order to determine the optimal switching schedule Interesting results on state or output feedback have been given with the regions of the state space where an optimal mode switch should occur.In complement of all the methods resulting from the resolution of the necessary conditions of optimality, we propose to use a multiple-phase multiple-shooting formulation which enables the use of standard constraint nonlinear programming methods. This formulation is applied to hybrid systems with autonomous and controlled switchings and seems to be of interest in practice due to the simplicity of implementation.Sensitivity analysis is the key point of all the methods based on non linear programming. It can also be used to determine limit cycles and the optimal strategy to reach them. All these items are discussed in the plenary session.

Numerical method for the analysis of time-varying singular systems

A new method that generates approximate values of the solution of the time-varying singular problem is presented in the paper. The method of regularisation is used to overcome the ill-posedness of this problem and a relation between the unknown state variable and its derivative is generated. A pseudospectral method is used to discretise this relationship. The discrete relationship between the state variable and its derivative is reduced to a set of algebraic equations, which are solved to produce values of the state variable at discrete values of time. Performance of this method is evaluated through its application to the solution of an example of a time-varying singular system. Results from this example show that this method generates excellent approximate values for the state variable of the singular system.

Hydrodynamical Quantum State Reconstruction

The density matrix of a nonrelativistic wave-packet in an arbitrary, one-dimensional and time-dependent potential can be reconstructed by measuring hydrodynamical moments of the Wigner distribution. An n-th order Taylor polynomial in the off-diagonal variable is obtained by measuring the probability distribution at n+1 discrete time values.

A new numeric technique for high-speed evaluation of power system frequency

A new numerical technique for evaluating power system frequency from either a voltage or current signal is presented. The technique uses discrete time values of the input signal, taken at a fixed sampling rate, to provide an estimate of power system frequency accurate to within typically 0.001 Hz. The technique is shown to be capable of tracking frequency under dynamic power systems conditions and is immune to the effect of harmonics. It is further shown that the algorithm can be easily adapted to process three-phase signals such that the positive phase sequence component can be utilised thus increasing the reliability of the measurement under fault conditions.

Analysis of local uniform grid refinement

Numerical methods for time-dependent PDEs usually integrate on a fixed grid, a priori chosen for the whole time interval. Similar to a fixed stepsize, a fixed grid may be inefficient when solutions possess large local gradients. While most schemes can easily adapt the stepsize, as in genuine ODE and method-of-lines schemes, the question of how to automatically adapt the grid to rapid spatial transitions is much more involved. The subject of this paper is local uniform grid refinement (LUGR) for finite different methods. The idea of LUGR is to cover the spatial domain with nested, finer-and-finer, locally uniform subgrids. LUGR is applicable both to stationary and time-dependent problems. For time-dependent problems the local subgrids are adapted at discrete values of time to follow moving transitions. The aim of this paper is to discuss, for the class of finite difference methods under consideration, a general error analysis that shows the interplay between local truncation and interpolation errors. This analysis points the way to a theoretically optimal strategy for the local refinement, optimal in the sense that this strategy controls accumulation of interpolation errors and simultaneously strives for the spatial accuracy that would be obtained on the finest grid when used without adaptation. Attention is paid to both the stationary and time-dependent case, while for time-dependent problems the emphasis lies on combining LUGR with Runge-Kutta time stepping.

The Choice of Time for Zeroing a Disturbance in a Minimum-Fuel Consumption Control Problem

The choice of time T for zeroing an initial disturbance with minimum-fuel consumption influences the dependence of the switching instances on the phase variables. For a set of discrete values of time T this relation can be found easily and mechanized in a simple manner. The restriction of the free choice of time T is so mild that it does not impair the usefulness of the scheme.

The analysis of sampled data servomechanisms performed on the IBM Type 650

Because of the need for more accurate and versatile control capabilities than can be obtained economically with continuous systems, discrete filters (digital computers) are being substituted as control elements in many applications. The exact response of linear systems which combine continuous and discrete filters can be obtained for discrete values of time by a series of well-defined mathematical processes. Since these processes require many computations of a repetitive nature to obtain complete response data, manual methods of computing are often not practicable and general-purpose digital computers such as the IBM (International Business Machines Corporation) type 650 are used to assist in the analysis of these systems.