Probabilities time-invariant

Finally, the prediction error model assumes that the parameter values do not change with respect to time, that is, they are time invariant. A quick and simple test of the invariance of the model is to split the data into two parts and cross validate the models using the other data set. If both models perform successfully, then the parameters are probably time invariant, at least over the time interval considered. 

Two consequences of this simple analysis are far-reaching. First, the common perception that normal or log-normal functions may be used as catch-all probability density functions is physically untenable since these functions are not time-invariant relative to most geological processes (mixing, differentiation,. ..). Second, there is more information on the physics of geological processes contained in the density function of concentrations, ratios, and other geochemical parameters than what is reflected by their mean or variance. Obviously, this information is deeply buried and convoluted, but deserves attention anyway. 

The Markov model is a special case of the semi-Markov model in which all the retention-time variables are exponentially distributed, Ai Exp(/v(), and Ki is the parameter of the exponential. In this case, the semi-Markov model parameters are k, = ha and ujtl = h,l j//q for j / i and i,j = 1,..., m. This results from the assumption of the Markov model given in (9.1), which implies that the conditional transfer probability from i to j in a time increment At is time-invariant, or in other words is independent of the age of the particle in the compartment. Particles with such a constant flow rate, or hazard rate, are said to lack memory of their past retention time in the compartment. 

When discussing diffusion, one inevitably needs to solve diffusion equations. The Laplace transform has proven to be the most effective solution for these differential equations, as it converts them to polynomial equations. The Laplace transform is also a powerful technique for both steady-state and transient analysis of linear time-invariant systems such as electric circuits. It dramatically reduces the complexity of the mathematical calculations required to solve integral and differential equations. Furthermore, it has many other important applications in areas such as physics, control engineering, signal processing, and probability theory. 

With the dimension of multivariable MFC systems ever increasing, the probability of dealing with a MIMO process that contains an integrator or an unstable unit also increases. For such units FIR models, as used by certain traditional commercial algorithms such as dynamic matrix control (DMC), is not feasible. Integrators or unstable units raise no problems if state-space or DARMAX model MFC formulations are used. As we will discuss later, theory developed for MFC with state-space or DARMAX models encompasses all linear, time-invariant, lumped-parameter systems and consequently has broader applicability. 

The time Invariant load-strain curves appear to be an inherent property of the tissue. The values were independent of the porosity of the platens and the strain rates that the tissue was compressed at. Up to about 30% compression the data could be fitted, by linear regression, to a straight line. As such, the curve represents the equilibrium load-strain relationship of the swollen matrix. It should be pointed out, however, that while the strains calculated for this and the other curves are probably close to the actual strains, they nevertheless are approximations, due to the assumptions of shape of the penetrating tissue and the possible slight compression of the tissue by the rod. 

The convolution derivation assumes a time-invariant system. This follows from the fact that the probability function Pr(t) in the derivation (see the appendix) does not depend on T, the time the drug molecules enter at the input point, P. The convolution derivation also assumes an instantaneous sampling procedure. 

In conclusion, we have introduced a neutral type of linear response experiment for nonlinear kinetics involving multiple reaction intermediates. We have shown that the susceptibility functions from the response equations are given by the probability densities of the transit time in the system. We have shown that a transit time is a sum of different lifetimes corresponding to different reaction pathways, and that in the particular case of a time-invariant system our definition of the transit time is consistent with Easterby s definition [23]. 

The qualitative and quantitative analysis of the probability densities of the transit time may lead to interesting hints concerning the reaction mechanism and the kinetics of the process. This is an important problem that needs further development. Here we present only a simple illustration based on the study of the response of a time-invariant (stationary) system. 

The method proposed in (Andrieu (2002), Sudret (2004)) provides an approach for calculating the outcrossing rate using the binormal law. This outcrossing rate is time integrated, making it possible thus to calculate cumulative failure probability using classical (time invariant) tools. 

The probability distribution of the maximum value (i.e. the largest extreme) is often approximated by one of the asymptotic extreme value distributions. Hence for structures subjected to a single time-varying action, a random process model is replaced by a random variable model and the principles and methods for time invariant models may he apphed. 

The chain is time invariant if and only if these conditional probabilities are the same for all i it is then indecomposable if and only if there is an output value u such that for all output values we have Pr(M,+y = m ,= )> 0 (where i is arbitrary) for some j 1,2,.... ... 

Numerical evaluation of the numerator of Eq. (7) reduces to a time-invariant two-component parallel system reliability analysis. It is clear that the first part of Eq. (5) represents the building block for the solution of both time-invariant and time-variant reliability problems (Der Kiureghian 1996). Using Eq. (7), Poisson approximation to the failure probability, Pf oisson(T), is obtained as (under the hypothesis that... 

A currently under development hybrid time-invariant reliability method, referred to herein as DP-RS-Sim method and able to enhance the FORM/SORM estimates of time-invariant and time-variant failure probabilities for structural and/or geotechnical systems, is briefly presented and illustrated below. The DP-RS-Sim method combines (1) the DP search (used in FORM and SORM), (2) the Response Surface (RS) method to approximate in analytical (polynomial) form the LSF near the DP, and (3) a simulation technique (Sim), to be applied on the response surface representation of the actual LSF. 

Estimate of the time-invariant failure probability using crude MCS or any other more advanced simulation technique (e.g., IS) applied on the analytical response surface approximation of the actual LSF. 

We are concerned with the determination of the functions L(/), h(/), and p l l ), from suitably designed population data to be determined from experiments. We assume that the cells are growing in a batch culture under balanced exponential growth conditions. Under these circumstances we have fid t) = where /(/) is the time-invariant probability dis-... 

Eor time-invariant structural reliability, the problem is generally described by a performance function G(X), in which X is a vector of random variables of system parameters, and the probability of failure (limit state being reached) can be formulated by... 

For dynamic structural reliability, the problem is generally described as a structural system with parameters Xj under the action of time-varying loadings S(t, Xj), where S is a vector random process of time, and the process is specified by a set of parameters Xj. Let X denote all parameters in the structure and loadings, where X is a vector of quantities that are uncertain but time invariant. The performance function must include a vector of random response variables, R, because R is dependent not only on X but also on the random process, S. The probability of failure can be similarly formulated by... 

For a general structural system and limit state, similar to the case of barrier crossing at a constant level of a single-degree-of-freedom oscillator, Gueri and Rackwitz (1986) included R as basic variables, together with X, and reduced the dynamic problem to a time-invariant one with n + m basic variables therefore, the first-order reliability method can be used, where m is the number of response variables. The required information includes the explicit form of the performance function, G the probability distributions not only of system parameters, X, but also of response variables, R and even the joint PDF,, R(x, r). The method is very complicated because of the number of response variables and the difficulty of determining the distribution of response variables of interest. For some limit states, it may not be possible to identify R and determine the performance function in the explicit form. 

The time invariance in the spatial distribution of p is another manifestation of the self-similarity of the structures generated by time- and spatially periodic chaotic flows. Such invariant statistical properties can be demonstrated by computing the probability density H(log p). If the probability density function of the scaled variable (p/(p is computed according to eq. (3-13), the distributions are scaled automatically by the mean density. 

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. 

We will restrict ourselves to time invariant Markov chains where the transition probabilities only depend on the states, not the time n. These are also called homogeneous Markov chains. In this case, we can leave out the time index and the transition probability matrix of the Markov chain is given by... 

If the probability of transition from state i to state j is the same for all times, the Markov chain is called time-invariant or homogeneous. 

The one-step transition probabilities for a time-invariant Markov chain with a finite state space can be put in a matrix P, where the pij is the probability of a transition from state i to stated in one-step. This is the conditional probability... 

The reference peak ground acceleration corresponds to the reference return period, Tncr, of the design seismic action for structures of ordinary importance ( Importance Class II ). It is reminded that under the Poisson assumption of earthquake occurrence (i.e., that the number of earthquakes in an interval of time depends only on the length of the interval in a time-invariant way), the return period, Tr, of seismic events exceeding a certain threshold is related to the probability, P, that the threshold will be exceeded in Ti years as... 

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