Continuous state-space

The block diagram of the system is shown in Figure 9.10. Continuous state-space model From equations (9.77)-(9.81)... 

The idea that a universal principle should be able to provide the a priori probability in each case and thereby establish the link between the mathematical probability and the actual world, lost ground during the nineteenth century. An important argument was that in the case of a continuous state space the prescription is not even well-defined since it depends on the choice of variable. A uniform distribution in velocity space is not the same as a uniform distribution in the energy scale. This difficulty has been beautifully demonstrated by Bertrand ). Take a fixed circle of... 

A fundamental property of the master equation is As t -> oo all solutions tend to the stationary solution or - in the case of decomposable or splitting W - to one of the stationary solutions. Again this statement is strictly true only for a finite number of discrete states. For an infinite number of states, and a fortiori for a continuous state space, there are exceptions, e.g., the random walk (2.11). Yet it is a useful rule of thumb for a physicist who knows that many systems tend to equilibrium. We shall therefore not attempt to give a general proof covering all possible cases, but restrict ourselves to a finite state space. There exist several ways of proving the theorem. Of course, they all rely on the property (2.5), which defines the class of W-matrices. 

In the t3rpical case the d3mamical process under investigation lives on a continuous state space such that the transfer operator does not have the form of a nice stochastic matrix. Therefore, discretization of the transfer operator is needed to 3deld a stochastic matrix with which one can proceed as in the example above. 

State and input coefficient matrices in continuous state-space systems... 

In the case of a discrete time Markov chain with a continuous state space, we may simply suppress the variable t in the above formulas, and write... 

The state-space (or site space according to the terminology soon to be introduced) can be chosen either continuous (A <= U ) or discrete (A c= Z ). To emphasise the existence of elementary particles of a population (as, for example, sometimes in reaction kinetics and in population dynamics), a discrete state-space formalism can be used. Continuum mechanics is an illustration for preferring a continuous state-space, since the mass points can arbitrarily occupy space. 

Noncontinuous processes in continuous state-space occur when the condition (5.5) is not fulfilled. In this case we need more general equations than the Kolmogorov equations. The main point is that analogously to (5.6) and (5.7) the yth velocity of conditional moments Dj can be defined ... 

The terminology is nonstandard, and in physical literature the Kramers-Mpyal expansion is given as a (nonsystematic) procedure to approximate discrete state-space processes by continuous processes. The point that we want to emphasise here is the clear fact that, even in the case of a continuous state-space, the process itself can be noncontinuous, when the Lindeberg condition is not fulfilled. The functions for the higher coefficients do not necessarily have to vanish. 

A great amount of stochastic physics investigates the approximation of jump processes by diffusion processes, i.e. of the master equation by a Fokker-Planck equation, since the latter is easier to solve. The rationale behind this procedure is the fact that the usual deterministic (CCD) and stochastic (CDS) models differ from each other in two aspects. The CDS model offers a stochastic description with a discrete state space. In most applications, where the number of particles is large and may approach Avogadro s number, the discreteness should be of minor importance. Since the CCD model adopts a continuous state-space, it is quite natural to adopt CCS model as an approximation for fluctuations. 

Other kinds of Fokker-Planck equations can be also derived. The continuous state-space stochastic model of a chemical reaction, which considers the reaction as a diffusion process , neglects the essential discreteness of the mesoscopic events. However, some shortcomings of (5.65) have been eliminated by using a direct Fokker-Planck equation obtained by means of nonlinear transport theory (Grabert et al., 1983). 

Continuous time, continuous state-space deterministic models... 

Zhou, Y, Ma, L., Mathew, J., Sun, Y. Wolff, R. 2009. Asset life prediction using multiple degradation indicators and failure events a continuous state space model approach Operation and Reliability No. 4. 

In Section 5.1 we introduce the stochastic processes. In Section 5.2 we will introduce Markov chains and define some terms associated with them. In Section 5.3 we find the n-step transition probability matrix in terms of one-step transition probability matrix for time invariant Markov chains with a finite state space. Then we investigate when a Markov ehain has a long-run distribution and discover the relationship between the long-run distribution of the Markov chain and the steady state equation. In Section 5.4 we classify the states of a Markov chain with a discrete state space, and find that all states in an irreducible Markov chain are of the same type. In Section 5.5 we investigate sampling from a Markov chain. In Section 5.6 we look at time-reversible Markov chains and discover the detailed balance conditions, which are needed to find a Markov chain with a given steady state distribution. In Section 5.7 we look at Markov chains with a continuous state space to determine the features analogous to those for discrete space Markov chains. 

MARKOV CHAINS WITH CONTINUOUS STATE SPACE... 

Sometimes the state space of a Markov chain consists of all possible values in an interval. In that case, we would say the Markov chain has a one-dimensional continuous state space. In other Markov chains, the state space consists of all possible values in a rectangular region of dimension p and we would say the Markov chain has a p-dimensional continuous state space. In both of these cases, there are an uncountably infinite number of possible values in the state space. This is far too many to have a transition probability function associated with each pair of values. The probability of a transition between all but a countable number of pairs of possible values must be zero. Otherwise the sum of the transition probabilities would be infinite. A state to state transition probability function won t work for all pairs of states. Instead we define the transition probabilities from each possible state X to each possible measurable set of states A We call... 

The main results that hold for discrete Markov chains continue to hold for Markov chains with a continuous state space with some modifications. In the single dimensional case, the probability of a measurable set A can be found from the probability of the transition CDF which is... 

This is analogous to Equation 5.11. When the continuous state space has dimension p, the integrals are multiple integrals over p dimensions, and the joint density function is found by taking p partial derivatives. 

In general, the probability of passing to state j from state i must be zero at all but at most a countably infinite number of states, or else they would not have a finite sum. This means, that we can t classify states by their first return probabilities and return probabilities as given in Tables 5.1 and 5.2, because these would equal 0 for almost all states. Redefining the definitions for recurrence is beyond the scope of this book, and Gamerman (1997) outlines the changes required. What is important to us is that under the required modifications, the main results we found for discrete Markov chains continue to hold for Markov chains with continuous state space. These are ... 

For Markov chains with a continuous state space, there are too many states for us to use a transition probability function. Instead we define a transition kernel which measures the probability of going from each individual state to every measurable set of states. 

For a Markov chain with continuous state space, if the transition kernel is absolutely continuous, the Chapman-Kolmogorov and the steady state equations are written as integral equations involving the transition density function. 

This says the long-run probability of a state equals the weighted sum of one-step probabilities of entering that state from all states each weighted by its long-run probability. The comparable steady state equation that ir 0), the long-run distribution of a Markov chain with a continuous state space, satisfies is given by... 

Within a structural health monitoring fi amework, the target commonly is the tracking of dynamic response, described by the general continuous state-space formulation... 

Markov process A stochastic process is a random process in which the evolution from a state X(t ) to X(t +i) is indeterminate (i.e. governed by the laws of probability) and can be expressed by a probability distribution function. Diffusion can be classified as a stochastic process in a continuous state space (r) possessing the Markov property as... 

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