Time stochastic processes

Discrete-time Markov chains are discrete-time stochastic processes with a discrete state space. Let the state of the random variable at time t be represented by y, then the stochastic process can be represented by (yi, y2, y, ...). 

HMMs are statistical models for systems that behave like a Markov chain, which a discrete-time stochastic process formed on the basis of the Markov property. 

A multi-factor model of the whole yield curve has been presented by Heath et al. (1992). This is a seminal work and a ground-breaking piece of research. The approach models the forward curve as a process arising from the entire initial yield curve, rather than the short-rate only. The spot rate is a stochastic process and the derived yield curve is a function of a number of stochastic factors. The HJM model uses the current yield curve and forward rate curve, and then specifies a continuous time stochastic process to describe the evolution of the yield curve over a specified time period. 

Recall that a Levy process X (t) is a continuous-time stochastic process that has independent and stationary increments. It represents a natural generalization of a simple random walk defined as a sum of independent identically distributed random variables. The independence of increments ensures that Levy processes are Markov processes. The main feature of a Levy process is that it is infinitely divisible for... 

Finally, we observe that by calculating cr t ) and generating random increments Rn J (0, y(tn)) we have a discrete representation of the continuous time stochastic process Y(t), effectively an exact stroboscopic representation. Also it is natural to view the stochastic integral as the solution of the stochastic differential equation (SDE) with zero initial condition ... 

Furthermore, let a(f) be a non-decreasing, right-continuous, real valued function for i > 0 with a(0) = 0. The gamma process with shape function a t) > 0 and scale parameter /S > 0 is a continuous-time stochastic process X(f), f > 0 with the following properties... 

M. F. Schlesinger [1984] Williams-Watts Dielectric Relaxation A Fractal Time Stochastic Process, J. Statist. Phys. 36, 639-648. 

The cybernetic description of systems of different types is characterized by concepts and definitions like feedback, delay time, stochastic processes, and stability. These aspects will, for the time being, be demonstrated on the control loop as an example of a simple cybernetic system, but one that contains all typical properties. Thereby the importance of the probability calculus and communication in cybernetics can be clearly explained. This is then followed by a general representation of cybernetic systems. 

We were interested in measuring the effect of cyber-attacks on the system under study. We chose to compare the behavior of a base-line model, i.e. a model without cyber-attacks, with the behavior of the model in which cyber-attacks are enabled ( system under attack ). For the comparison we chose as a reward (utility function) the deviation of the supplied power, in the presence of failures and attacks, from the known maximum power supplied of 10,940 MW. This reward has been used in the analysis of power systems by others [9]. Other suitable candidates would be the size of cascades as we have done in the past [3]. We compute the reward at any state-machine event in the model and log these values during the simulations. Clearly, for every simulation run, the value of the supplied power varies over time to form a continuous-time stochastic process. We study the following two statistics of this process ... 

The first characteristic of a stochastic process is the time points when the process is allowed to change values. If the process can only change values at discrete values such as t = 1,2,..., it is called a discrete time stochastic process. If the process can change values at any time point t > 0 it is called a continuous time stochastic process. We only consider discrete time stochastic processes, where changes are only allowed at times n = 1,2, ... 

Diffusion is a stochastic process associated with the Brownian motion of atoms. For simplicity we assume a one-dimensional Brownian motion where a particle moves a lattice unit <2 in a short time period Td in a direction either forward or backward. After N timesteps the displacement of the particle from the starting point is... 

To see the connection between this stochastic process and a chemically reacting system, consider the first step of Scheme IX. Each (real) molecule of A has an equal and constant probability of reacting in time t. In the simulation, each position in the grid has an equal and constant probability (p) of being selected. For this first-order reaction, the chemical system is described by... 

A stochastic process is a family of random variables X( depending on a parameter t (e.g., time). This definition may be extended to include... 

The statistic models consider surface roughness as a stochastic process, and concern the averaged or statistic behavior of lubrication and contact. For instance, the average flow model, proposed by Patir and Cheng [2], combined with the Greenwood and Williamsons statistic model of asperity contact [3] has been one of widely accepted models for mixed lubrication in early times. 

A homogeneity index or significance coefficienf has been proposed to describe area or spatial homogeneity characteristics of solids based on data evaluation using chemometrical tools, such as analysis of variance, regression models, statistics of stochastic processes (time series analysis) and multivariate data analysis (Singer and... 

In the framework of this ultimate model [33] there are m2 constants of the rate of the chain propagation kap describing the addition of monomer to the radical Ra whose reactivity is controlled solely by the type a of its terminal unit. Elementary reactions of chain termination due to chemical interaction of radicals Ra and R is characterized by m2 kinetic parameters k f . The stochastic process describing macromolecules, formed at any moment in time t, is a Markov chain with transition matrix whose elements are expressed through the concentrations Ra and Ma of radicals and monomers at this particular moment in the following way [1,34] ... 

An exhaustive statistical description of living copolymers is provided in the literature [25]. There, proceeding from kinetic equations of the ideal model, the type of stochastic process which describes the probability measure on the set of macromolecules has been rigorously established. To the state Sa(x) of this process monomeric unit Ma corresponds formed at the instant r by addition of monomer Ma to the macroradical. To the statistical ensemble of macromolecules marked by the label x there corresponds a Markovian stochastic process with discrete time but with the set of transient states Sa(x) constituting continuum. Here the fundamental distinction from the Markov chain (where the number of states is discrete) is quite evident. The role of the probability transition matrix in characterizing this chain is now played by the integral operator kernel ... 

Radioactive decay is a stochastic process that occurs at random in a large number of atoms of an isotope (see Textbox 13). The exact time when any particular atom decayed or will decay can be neither established nor predicted. The average rate of decay of any radioactive isotope is, however, constant and predictable. It is usually expressed in terms of a half-life, the amount of time it takes for half of the atoms in a sample of a radioactive isotope to decay to a stable form. 

Autocorrelation and time series analysis have been successfully applied in testing spatial inhomogeneities (Ehrlich and Kluge [1989], Do-erffel et al. [1990]). This techniques are generalized in the theory of stochastic processes (Bohacek [1977a, b]) which is widely used in chemical process analysis and about them. 

The main goal of time-series analysis (Box and Jenkins [1976], Chat-field [1989], Metzler and Nickel [1986]) apart from process analysis is time-dependent sampling. In both cases fluctuations in time x(t) matter and can be considered as a simple stochastic process or as time series. 

Fig. 2.8. Autocorrelation function (ACF) of a stochastic process. rc is the correlation time (time constant)...

Both time- and position-dependent concentration functions can be dealt with by the theory of stochastic processes (Bohacek [1977]). Time functions playing a role in process analysis can be assessed not only by means of information amount M(n)t but also - sometimes in a more effective way -by means of the information flow, J, which is generally given by... 

The aim of this chapter is to describe approaches of obtaining exact time characteristics of diffusion stochastic processes (Markov processes) that are in fact a generalization of FPT approach and are based on the definition of characteristic timescale of evolution of an observable as integral relaxation time [5,6,30—41]. These approaches allow us to express the required timescales and to obtain almost exactly the evolution of probability and averages of stochastic processes in really wide range of parameters. We will not present the comparison of these methods because all of them lead to the same result due to the utilization of the same basic definition of the characteristic timescales, but we will describe these approaches in detail and outline their advantages in comparison with the FPT approach. 

Thus, the problem on the growth of a block copolymer chain in the course of the interphase radical copolymerization may be formulated in terms of a stochastic process with two regular states corresponding to two types of terminal units (i.e. active centers) of a macroradical. The fact of independent formation of its blocks means in terms of a stochastic process the independence of times ta of the uninterrupted residence in every a-th stay of any realization of this process. Stochastic processes possessing such a property have been scrutinized in the Renewal Theory [75]. On the basis of the main ideas of this theory, the set of kinetic equations describing the interphase copolymerization have been derived [74],... 

But in real radar applications many different noise and clutter background signal situations can occur. The target echo signal practically always appears before a background signal, which is filled with point, area or even extended clutter and additional superimposed noise. Furthermore the location of this background clutter varies in time, position and intensity. Clutter is, in real applications, a complicated time and space variant stochastic process. 

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