Poisson processes

Nature In monitoring a moving threadhne, one criterion of quality would be the frequency of broken filaments. These can be identified as they occur through the threadhne by a broken-filament detector mounted adjacent to the threadhne. In this context, the random occurrences of broken filaments can be modeled by the Poisson distribution. This is called a Poisson process and corresponds to a probabilistic description of the frequency of defects or, in general, what are called arrivals at points on a continuous line or in time. Other examples include ... 

Figure 8.19. Random one-dimensional fragmentation—a Poisson process.

Our next example concerns the Poisson process, which plays a central role in a variety of problems such as waiting lines, inventory control, electrical noise, the firing of neurons, and radioactive decay. We will discuss the application of the Poisson process to the study of certain kinds of electrical noise in a later section. 

The time functions N(t) underlying the Poisson process are all of the general form shown in Fig. 3-9, where it is assumed that N (t) is finite except perhaps at t = oo. Such functions iV(t) are sometimes referred to as emitting functions because of the relations... 

The Poisson process is now defined by specifying a particular set of probability density functions that enable us to calculate all possible... 

The physical significance of the parameter n entering into the definition of the Poisson process can be established by noting that, for 8 > ,... 

The Poisson process represents only one possible way of assigning joint distribution functions to the increments of counting functions however, in many problems, one can argue that the Poisson process is the most reasonable choice that can be made. For example, let us consider the stream of electrons flowing from cathode to plate in a vacuum tube, and let us further assume that the plate current is low enough so that the electrons do not interact with one another in the... 

There are many ways we could assign probability distribution functions to the increments N(t + sk) — N(t + tk) and simultaneously satisfy the independent increment requirement expressed by Eq. (3-237) however, if we require a few additional properties, it is possible to show that the only possible probability density assignment is the Poisson process assignment defined by Eq. (3-231). One example of such additional requirements is the following50... 

The shot noise process is defined in terms of the Poisson process by means of the formula... 

Poisson process is often a reasonable model for the arrival times of... 

Exercise Show that the density function of the sum of ft independent, identical random variables with Hie common density function Ae-A is given by A(Ax)n 1e-Ju/(ft — 1) . Note that the time intervals between events that occur by a Poisson process are exponentially distributed. 

The Monte Carlo method as described so far is useful to evaluate equilibrium properties but says nothing about the time evolution of the system. However, it is in some cases possible to construct a Monte Carlo algorithm that allows the simulated system to evolve like a physical system. This is the case when the dynamics can be described as thermally activated processes, such as adsorption, desorption, and diffusion. Since these processes are particularly well defined in the case of lattice models, these are particularly well suited for this approach. The foundations of dynamical Monte Carlo (DMC) or kinetic Monte Carlo (KMC) simulations have been discussed by Eichthom and Weinberg (1991) in terms of the theory of Poisson processes. The main idea is that the rate of each process that may eventually occur on the surface can be described by an equation of the Arrhenius type ... 

The "one-hit" hypothesis states that the tumor initiation is a Poisson process, in that each additional molecule of a carcinogen produces an equal increment in the probability of a response, and that all such "hits" are independent. Consequently,... 

Nonsolvent bath, polymer precipitation by immersion in, 15 808-811 Nonspecific elution, in affinity chromatography, 6 398, 399 Nonstationary Poisson process, in reliability modeling, 26 989 Non-steady-state conduction, 9 105 Nonsteroidal antiinflamatory agents/drugs (NSAIDs) 21 231 for Alzheimer s disease, 2 820 for cancer prevention, 2 826 Nonsulfide collectors, 16 649 Nonsulfide flotation, 16 649-650 Nonsulfide mineral flotation collectors used in, 16 648-649t modifiers used in, 16 650, 651t Nonsulfide ores, 16 598, 624... 

The outflow of a CSTR is a Poisson process, i.e., fluid elements are randomly selected regardless of theirposition in the reactor. The waiting time before selection for a Poisson process has an exponential probability distribution. 

Examples include birth—death processes, die Poisson process, and die random telegraph process. 

Figure 4.3 Distribution of events on the time axis in a Poisson process over the time interval [0—0] the probability of occurrence for one event in a time interval depends on its duration only (see text).

The exponential distribution with parameter X is the distribution of waiting times ( distance in time) between events which take place at a mean rate of X. It is also the distribution of distances between features which have a uniform probability of occurrence (Poisson process), such as the simplest model of faults on a map. The gamma distribution with parameter n and X l, where n is an integer is the distribution of the waiting time between the first and the nth successive events in a Poisson process. Alternatively, the distribution /(t), such as... 

We can model this as a Poisson process and use the simple formula ... 

This new terminology is proposed jointly in the present publication and in Pitard Pierre Gy s Theory of Sampling and C.O. IngamelTs Poisson Process Approach — Pathways to Representative SampUng and Appropriate Industrial Standards . Doctoral thesis, Aalborg University, campus Esbjerg (2009). ISBN 978-87-7606-032-9. 

Hence shot noise is fully determined by a single parameter, viz., its density. The alternative name Poisson process indicates that it may be regarded as a stochastic process, as will be seen in IV.2. 

It is understood that P1(1 = 0 for n2 < n. Thus each sample function y(t) is a succession of steps of unit height and at random moments. It is uniquely determined by the time points at which the steps take place. These time points constitute a random set of dots on the time axis. Their number between any two times tl912 is distributed according to the Poisson distribution (2.6). Hence Y(t) is called Poisson process and describes the same situation as (II.2.6). 

Exercise. Find for the Poisson process the same quantities as in (2.7). 

Exercise. Find the jump moments and the macroscopic equation for the decay process and for the Poisson process. 

An important example of a one-step process with constant transition probabilities is the Poisson process, defined by... 

It is a random walk over the integers n = 0,1,2,... with steps to the right alone, but at random times. The relation to chapter II becomes more clear by the following alternative definition. Every random set of events can be treated in terms of a stochastic process Y by defining Y(t) to be the number of events between some initial time t = 0 and t. Each sample function consists of unit steps and takes only integral values n = 0,1, 2,... (fig. 5). In general this Y is not Markovian, but if the events are independent (in the sense of II.2) there is a probability q(t) dt for a step to occur between t and t + dt, regardless of what happened before. If, moreover, q does not depend on time, Y is a Poisson process. 

Exercise. In the discrete-time random walk the steps were taken at fixed time points. Suppose now that the times of the steps are randomly distributed as given by the Poisson process. Show that this is identical to the situation described by (2.1). )... 

Exercise. The Poisson process (IV.2.6) has independent increments. Show that it gives rise to a white noise with Tm = 1 for all ra. 

A process with independent increments can be generated by compounding Poisson processes in the following way. Take a random set of dots on the time axis forming shot noise as in (II.3.14) the density fx will now be called p. Define a process Z(t) by stipulating that, at each dot, Z jumps by an amount z (positive or negative), which is random with probability density w(z). Clearly the increment of Z between t and t + T is independent of previous history and its probability distribution has the form (IV.4.7). It is easy to compute. 

On comparing this expression with (6.4) one sees that the compound Poisson process defined here is the same as Z(t) in (6.3) if... 

Exercise. Argue that our compound Poisson process may be regarded as a superposition of simple Poisson processes as defined in IV.2. 

Exercise. Show that the random walk is a compound Poisson process. 

---