Piecewise Linear Function Of Time Research Articles

Reliably Assessing Duration of Protection for Coronavirus Disease 2019 Vaccines.

Decision making about vaccination and boosting schedules for coronavirus disease 2019 (COVID-19) hinges on reliable methods for evaluating the longevity of vaccine protection. We show that modeling of protection as a piecewise linear function of time since vaccination for the log hazard ratio of the vaccine effect provides more reliable estimates of vaccine effectiveness at the end of an observation period and also detects plateaus in protective effectiveness more reliably than the standard method of estimating a constant vaccine effect over each time period. This approach will be useful for analyzing data pertaining to COVID-19 vaccines and other vaccines for which rapid and reliable understanding of vaccine effectiveness over time is desired.

Fast reconstruction of time-dependent market volatility for European options

This paper presents a robust and fast numerical algorithm to reconstruct the implied volatility as a piecewise linear function of time. This is done from a set of market observations in the Black–Scholes world. We use a fully implicit finite difference scheme to solve the partial differential equations. To find the time-dependent volatility function, we minimize the cost function defined as the sum of the squared errors between the theoretical prices and the prices observed on the market. On the last time step, right before each maturity, we apply a decomposition of the numerical option value with respect to the volatility which increases the stability and the solvability of the problem considered. We employ a predictor–corrector technique due to the non-uniqueness of the volatility function minimizer. The paper is concluded with profound numerical experiments with synthetic and real market data.

Understanding and monitoring the evolution of the Covid-19 epidemic from medical emergency calls: the example of the Paris area

We portray the evolution of the Covid-19 epidemic during the crisis of March–April 2020 in the Paris area, by analyzing the medical emergency calls received by the EMS of the four central departments of this area (Centre 15 of SAMU 75, 92, 93 and 94). Our study reveals strong dissimilarities between these departments. We show that the logarithm of each epidemic observable can be approximated by a piecewise linear function of time. This allows us to distinguish the different phases of the epidemic, and to identify the delay between sanitary measures and their influence on the load of EMS. This also leads to an algorithm, allowing one to detect epidemic resurgences. We rely on a transport PDE epidemiological model, and we use methods from Perron–Frobenius theory and tropical geometry.

Labor Productivity in France: Is the Slowdown of its Growth Inevitable or are there Levers to fight it?

Labour productivity in most advanced countries has slowed in successive stages since the 1970’s and, after the 2008 crisis, it has reached its lowest level since World War II. Here we analyze the trend evolution of the aggregate labor productivity per hour worked in France, over the last four decades, with two main questions. 1) Is the slowdown of labor productivity growth a process which began far before the Great Recession and should it continue with just temporary yet persistent effects of the 2008 crisis and 2) are there levers to reverse the current trend? We proceed in two steps. First, the trend (log) productivity is described as a deterministic piecewise linear function of time, involving so-called structural breaks, and, second, without breaks, as a linear function of fundamentals derived from an augmented growth model including Human capital. We propose a thorough econometric investigation with multiple robustness analyses involving sector-specific analyses. The structural specifica-tion we retain is able to explain the evolution of labor productivity over the last four decades. The accumulation of human capital has been the main driver of productivity growth over the period of interest while France is close to the technological frontier.

Bayesian Detection of Piecewise Linear Trends in Replicated Time-Series with Application to Growth Data Modelling.

We consider the situation where a temporal process is composed of contiguous segments with differing slopes and replicated noise-corrupted time series measurements are observed. The unknown mean of the data generating process is modelled as a piecewise linear function of time with an unknown number of change-points. We develop a Bayesian approach to infer the joint posterior distribution of the number and position of change-points as well as the unknown mean parameters. A-priori, the proposed model uses an overfitting number of mean parameters but, conditionally on a set of change-points, only a subset of them influences the likelihood. An exponentially decreasing prior distribution on the number of change-points gives rise to a posterior distribution concentrating on sparse representations of the underlying sequence. A Metropolis-Hastings Markov chain Monte Carlo (MCMC) sampler is constructed for approximating the posterior distribution. Our method is benchmarked using simulated data and is applied to uncover differences in the dynamics of fungal growth from imaging time course data collected from different strains. The source code is available on CRAN.

Which part of a chain breaks?

This work investigates the dynamics of a one-dimensional homogeneous harmonic chain on a horizontal table. One end is anchored to a wall; the other (free) end is pulled by an external force. A Green's function is derived to calculate the response to a generic pulling force. As an example, I assume that the magnitude of the pulling force increases with time at a uniform rate β. If the number of beads and springs used to model the chain is large, the extension of each spring takes a simple closed form, which is a piecewise-linear function of time. Under an additional assumption that a spring breaks when its extension exceeds a certain threshold, results show that for large β the spring breaks near the pulling end, whereas the breaking point can be located close to the wall by choosing small β. More precisely, the breaking point moves back and forth along the chain as β decreases, which has been called “anomalous” breaking in the context of the pull-or-jerk experiment. Although the experiment has been explained in terms of inertia, its meaning can be fully captured by discussing the competition between intrinsic and extrinsic time scales of forced oscillation.

Fermi’s golden rule: its derivation and breakdown by an ideal model

Fermi’s golden rule is of great importance in quantum dynamics. However, in many textbooks on quantum mechanics, its contents and limitations are obscured by the approximations and arguments in the derivation, which are inevitable because of the generic setting considered. Here we propose to introduce it by an ideal model, in which the quasi-continuum band consists of equaldistant levels extending from to , and each of them couples to the discrete level with the same strength. For this model, the transition probability in the first order perturbation approximation can be calculated analytically by invoking the Poisson summation formula. It turns out to be a piecewise linear function of time, demonstrating on the one hand the key features of Fermi’s golden rule, and on the other hand that the rule breaks down beyond the Heisenberg time, even when the first order perturbation approximation itself is still valid.

Propagation of Longitudinal Deformation Wave Along a Hoisting Rope Carrying an Intermediate Concentrated Load

Motion of the hoisting rope carrying an intermediate concentrated load is described by the one-dimensional wave equation in the region consisting of two sections separated by a moving boundary condition. The system is moved by the driving force acting at the upper cross section of the rope. Position of the intermediate load and consequently the lengths of the rope sections vary in the time depending on the magnitude of driving force. Solution of the wave equation is represented as a sum of integrals with variable limits of integration. The problem is reduced to solving the sequence of ordinary differential equations which describe a motion of the load in the fixed coordinate system and the paths of the rope ends in the moving coordinate system connected with the load. The argument of functions involved in the right-hand side of these equations lag behind the argument of the derivatives in the left-hand side of equations by a short time interval. A description of the unknown functions in a parametric form makes possible to eliminate retarded arguments from the equations. The problem is solved by using a technique of the sequential continuation of solution for time intervals corresponding to propagation of the deformation wave in the opposite directions. A computer program has been developed for solving the problem. Results of the numerical solution are presented in the case that the driving force is a piecewise linear function of time and is discontinuous at the peak point.

Registration of terrestrial mobile laser data on 2D or 3D geographic database by use of a non-rigid ICP approach.

Abstract. This article presents a generic and efficient method to register terrestrial mobile data with imperfect location on a geographic database with better overall accuracy but less details. The registration method proposed in this paper is based on a semi-rigid point to plane ICP ("Iterative Closest Point"). The main applications of such registration is to improve existing geographic databases, particularly in terms of accuracy, level of detail and diversity of represented objects. Other applications include fine geometric modelling and fine façade texturing, object extraction such as trees, poles, road signs marks, facilities, vehicles, etc. The geopositionning system of mobile mapping systems is affected by GPS masks that are only partially corrected by an Inertial Navigation System (INS) which can cause an important drift. As this drift varies non-linearly, but slowly in time, it will be modelled by a translation defined as a piecewise linear function of time which variation over time will be minimized (rigidity term). For each iteration of the ICP, the drift is estimated in order to minimise the distance between laser points and planar model primitives (data attachment term). The method has been tested on real data (a scan of the city of Paris of 3.6 million laser points registered on a 3D model of approximately 71,400 triangles).

An accurate numerical solution for the transient heating of an evaporating spherical droplet

A recently derived numerical algorithm for one-dimensional time-dependent Stefan problems is extended for the purposes of solving a moving boundary problem for the transient heating of an evaporating spherical droplet. The Keller box finite-difference scheme is used, in tandem with the so-called boundary immobilization method. An important component of the work is the careful use of variable transformations that must be built into the numerical algorithm in order to preserve second-order accuracy in both time and space – an issue not previously discussed in relation to this widely-used scheme. In addition, we demonstrate that our solution is in close agreement with the solution obtained using an alternative numerical scheme that employs an analytic solution of the heat conduction equation inside the droplet, for which the droplet radius was assumed to be a piecewise linear function of time. The advantages of the new method are discussed.

A THEORETICAL STUDY OF ELECTROMAGNETIC TRANSIENTS IN A LARGE PLATE DUE TO VOLTAGE IMPACT EXCITATION

Maxwell's equations are solved to determine transient electromagnetic fields inside as well as outside of a large conducting plate of an arbitrary thickness. The plate is carrying a uniformly distributed excitation winding on its surfaces. Transient fields are produced due to sudden application of a d.c. voltage at the terminals of the excitation winding. On the basis of a linear treatment of this initial value problem it is concluded that the transient fields may decay at a faster rate for conducting plates with smaller values of relaxation time. It is also shown that the growth of flux in a perfectly nonconducting plate is a piecewise linear function of time and the current in its excitation winding is a series of step function of time.

Stochastic progression of new states in psychotherapy.

This paper presents a study of the stochastic behaviour of the entry of a 5-dimensional, 2400 state, system into new states. Each state consists of five measurements on variables that are sensitive measures of the degree of activated unconscious communication. The data come from six psychotherapy consultation sessions. The study sought to investigate the manner in which patient and therapist progress into new configurations (new states) of the five variables, and the way tension accumulates to the point of overcoming the inertia in the patient/therapist system. We show that the time series that consists of the time intervals between successive new states is a generalized Poisson process. That is, the process is inherently governed by a Poisson parameter that is a continuous piecewise linear function of time. It increases rapidly in the opening moments of an interview and decreases slowly thereafter. This non-linear, non-stationary Markov model fits the data well according to the Kolmogorov-Smirnoff criterion. We discuss the model's implications for the nature of psychotherapy and its participants.

Maximum a p o s t e r i o r i estimation of narrow-band signal parameters

Maximum a posteriori (MAP) techniques are applied to the problem of estimating the unknown parameters of a frequency modulated narrow-band signal in additive white Gaussian noise. Signals are assumed to have unknown amplitude, initial phase, and frequency versus time history, although a priori information concerning the frequency trajectory may be available. In addition, it is assumed that the signal frequency trajectory can be adequately modeled by a continuous piecewise linear function of time. It is shown that the MAP estimator maximizes a linear combination of (1) coherent match of signal induced outputs to received observations, and (2) a term dependent only upon a priori knowledge. This second term reduces to a measure of trajectory smoothness in the special case of a Gauss–Markov a priori frequency model. Lastly, a MAP detection scheme is suggested for cases in which the presence of such signals is an issue.