Transition-Matrix Estimators

We begin with the microstate probability i(i — j) of making a move from configuration i to j, each characterized by a volume, number of particles, and set of coordinates q. This probability and its reverse satisfy the detailed balance condition  

Now consider the macrostates I and J to which the microstates belong. These macrostates may be characterized by a combination of N, V, U or any other well-defined order parameter. If we sum all instances of (3.58) for the pairs of microstates i e / and j e J, we can write 

Note that we have employed discrete notation, although rigorously this sum is an integral in configuration space. The choice is for clarity in the following derivations, and our conclusions will be unaffected by the notation. With minor manipulation, (3.59) gives this important result 

Fortunately for us, measurement of the macroscopic transition probabilities is straightforward. We could accomplish this, for example, by counting the number of times moves are made between every I and J macrostate in our simulation. The estimate for 7 ( / — J) would then be the number of times a move from I to J occurred, divided by the total number of attempted moves from I. The latter is simply given by the sum of counts for transitions from I to any state. A more precise procedure that retains more information than simple counts is to record the acceptance probabilities themselves, regardless of the actual acceptance of the moves [46, 47]. In this case, one adds a fractional probability to the running tallies, rather than a count (the number one). This data is stored in a matrix, which we will notate C(I, J) and which initially contains all zeros. With each move, we then update C as 

Of practical importance, we must consider the situation in which the proposed move J lies outside our macrostate boundaries. For such moves, we only need to update the diagonal term as 

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Transition-matrix estimators are typically more accurate than their histogram counterparts [25,26,46], and they offer greater flexibility in accumulating simulation data from multiple state conditions. This statistical improvement over histograms is likely due to the local nature of transition probabilities, which are more readily equilibrated than global measures such as histograms [25], Fenwick and Escobedo... 

Transition matrix estimators have received less attention than the multicanonical and Wang-Landau methods, but have been applied to a small collection of informative examples. Smith and Bruce [111, 112] applied the transition probability approach to the determination of solid-solid phase coexistence in a square-well model of colloids. Erring ton and coworkers [113, 114] have also used the method to determine liquid-vapor and solid-liquid [115] equilibria in the Lennard-Jones system. Transition matrices have also been used to generate high-quality data for the evaluation of surface tension [114, 116] and for the estimation of order parameter weights in phase-switch simulations [117]. 

The elements of the transition matrix from state j to state i can be estimated in the transition state theory approximation... 

Dynamical Entropy In order to capture the dynamics of a CML pattern, Kaneko has constructed what amounts to it mutual information between two successive patterns at a given time interval [kaneko93]. It is defined by first obtaining an estimate, through spatio-temporal samplings, of the probability transition matrix Td,d = transition horn domain of size D to a domain of size D. The dynamical entropy, Sd, is then given by... 

Two formal approaches have been established to solve isotopomer balances for biochemical networks in a generally applicable way (i) the transition matrix approach by Wiechert [22] and (ii) the isotopomer mapping matrix (IMM) approach by Schmidt et al. [14]. The matrix transition approach is based on a transformation of isotopomer balances into cumomer balances exhibiting a much greater simplicity. As shown, non-linear isotopomer balances can always be analytically solved by this approach [16]. The matrix transition approach was applied for experimental design of tracer experiments and for parameter estimation from labeling data [16,23]. 

The combination of all the local densities of states, g(E L,xj), represents a lumped picture of the pocket. This description is intermediate between the overall density of states for the whole pocket g(E Lmax) and the exhaustive description of every microscopic detail of the energy landscape within the pocket one can achieve in the discrete case. With this information, it is possible21 to construct a transition matrix M(T) in the lumped configuration space that allows the simulation of the evolution of the system for temperature T, and thus yields estimates for Teq(R) and tcsc(R)-... 

Figure 6.21 shows a cyclic three-state Markov chain, /r = S , S 2, S 3. Given an initial state and the matrix of transition probabilities, one can not only estimate the state of the chain at any future instant but can also determine the probability of observing a certain sequence, using the state transition matrix. The examples below demonstrate these cases. 

Example An analysis is presented where the consequences of brand switching between three different brands of laundry detergent, X, Y and Z are explored. A market survey is conducted to estimate the following transition matrix for the probability of moving between brands each month ... 

While there have been a number of papers written on the basic structure problem since Ref. [40] appeared, [55], none of them go qualitatively further than the calculations of that work. Curiously, an experimental paper, [5], claims to have reduced the theoretical error through comparison with new measurements of transition matrix elements. It is the author s opinion, however, that this essentially semiempirical approach is dangerous, and prefers to leave the 1 percent error estimate unchanged at present. However, there are three places in which considerable activity has taken place that we address in turn, the vector polarizability / , the Breit interaction, and radiative corrections. 

To model the state of the plant, a discrete Markov process is used. To calculate the transition matrix Q of a discrete Markov process, the transition probabilities between both states have to be estimated. All transitions of the recorded inflow data is used. The time series of plant states LOt are calculated by... 

In this section we shall consider how to choose a transition matrix P consistent with (16) that is (in some sense) the optimum matrix. We first note that we have a good deal of freedom in our choice of P since (16) is not very restrictive. To decide among these matrices we require a suitable criterion and an obvious choice would be to choose P to minimize the resulting variance, var (M ),of the estimate [Eq. (19)] over successive states of a Markov chain... 

The first passage time D(/) and the stationary probabilities vector rr (and consequently 4) can be estimated from Xt = (X, i > 0) as follows. A Markov process Xj (initial state / and transition matrix M) is observed with the two time sequences /j and L so that ... 

In the calculations of the Rb2 PA, we estimate the -X E+ — electronic-dipole moment to be p, = 3 au. This value corresponds to die 55 i/2(m = 1/2) — 5P3/2(m = 3/2) transition in Rb in a circularly polarized field, and is consistent with all the other 5S - 5P matrix elements for the Rb atom in a polarized laser field. The exact values of the atomic reduced dipole matrix elements can be found in Ref. [92]. The electronic-dipole transition matrix elements in KRb are taken as 3.5 eao fortheXiE+ o- transition, and 0.5 eao for the -o- l n transition, the order-of-magnitude estimate based on the data from Refs. [93,94],... 

A typical state space model for stand-alone GPS would have 8 states, the spatial coordinates and their velocities, and the clock offset and frequency. The individual pseudo-range measurements can be processed sequentially, which means that the Kalman gains can be calculated as scalars without the need for matrix inversions. There is no minimum number of measurements required to obtain an updated position estimate. The measurements are processed in an optimum fashion and if not enough for good geometry, the estimate of state error variance [P (fc)] will grow. If two sateUites are available, the clock bias terms are just propagated forward via the state transition matrix. 

In this case there is no exponential growth and one is left with estimating the mass renewal function the notation is the same as for > 1, just keep in mind that now b = 0, so A° (n) = Ma, (n), Ta,f n) = Ma p n) fi/ and P( ) is the Markov transition matrix of J, with invariant measure V, Va = Ca a- We Will uot give the details of the proof, that can be found in [Caravenna et al. (2005)], but we stress that the proof is a matter of deahng with a return distribution that is a random superposition of return laws with 0=1/2 and trivial L( ), so the N dependence in Theorem 3.4(2) does not come as a surprise once we consider the corresponding result (2.15)... 

In order to estimate the computer memory requirement it is important to know the dimensions of the matrices involved in the calculation. If fVrank(0 and Mrank(0 e the maximum expansion order and the number of azimuthal modes for the /th particle, then the dimension of the transition matrix T is dim(T ) = 2Af max(/) X 2fVi ax(0> where... 

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