### Table 3 The outer loop iteration numbers and the corresponding Markov chain length and l values Inner valve values The lengths of Markov chain

2004

"... In PAGE 4: ...90. Table3 gives the outer loop iteration numbers needed for various Markov chain length and l values. In these cases 3in was set at 0.... ..."

### Table 2 The outer loop iteration numbers and the corresponding Markov chain length and 3in values

2004

"... In PAGE 4: ... The use of three strategies mentioned before would dramatically speedup the rate of convergence. Table2 lists the numbers of outer loop iteration needed for convergence corresponding to the various choices of Markov chain length and 3in values. In these cases l was set at 0.... ..."

### Table 2: Parameter estimates for diagonal skew normal on four securities Parameter estimates for the diagonal model of Sahu et al. (2002) used to t the daily stock returns of General Electric, Lucent Technologies, Cisco Systems, and Sun Microsystems from April 1996 to March 2002. These estimates are the result of a Bayesian Markov Chain Monte Carlo iterative sampling routine. These parameters combine to give the mean ( + (2= )1=2 1), variance ( + (1 2= ) 0), and skewness (see Appendix for formula).

2004

Cited by 7

### Table 1: Comparison of Markov Chains and Random Fields on four di erent collections of polyphonic music. For Markov Chains we show the N-gram model that gave best performance on the testing set. For Random Fields we specify the number of iterations of the induction algorithm. For every collection, Random Fields result in lower testing perplexity and higher area under the ROC curve

"... In PAGE 9: ... We observe that Random Fields noticeably outperform Markov Chains. The lower portion of Table1 summarizes the quantitativedifferencebetweentheROCcurveson the fourdatasets. We use area under the ROC curve as a single-number measure of relative performance.... ..."

### Table 3: Parameter estimates for full skew normal on global asset allocation benchmark Parameter estimates for full model of Sahu et al. (2002) used to t the weekly bench- mark indices Lehman Brothers government bonds, LB corporate bonds, and LB mortgage bonds, MSCI EAFE (non-U.S. developed market equity), MSCI EMF (emerging market free investments), Russell 1000 (large cap), and Russell 2000 (small cap) from January 1989 to June 2002. These estimates are the result of a Bayesian Markov Chain Monte Carlo iterative sampling routine. These parameters combine to give the mean ( + (2= )1=2 1), variance ( + (1 2= ) 0), and skewness (see Appendix for formula).

2004

Cited by 7

### Table 1 gives the sizes of their reachable state space and the set of recurrent states. The back-annotated version achieves over three times the state compression ratio of the estimated version which is much higher than that of the FIFOs. This is largely because the modules in the DIFFEQs are loosely coupled and rather sequential. Due to the state compression, the number of power iterations to convergence is dramatically reduced. The curves in Figure 11 illustrate convergence of the distance of the probability vectors from two consecutive iterations. They clearly suggest that the Markov chains in both estimated and back-annotated versions possess 22

1998

"... In PAGE 23: ...9 gt; 4,287 35 gt;122 8,915 10,040 0.89 Table1 : State compression and iteration number reduction in the DIFFEQ and PCI analyses. Examples CPU time without compression (sec) CPU time with state compression (sec) Speedup Power iteration Compression Power iteration Expansion Total DiffeqEst.... In PAGE 24: ...eceiver side is also geometrically distributed with a parameter of 0.9. The mutual exclusion element mutex has a unit delay and is assumed to be fair with simultaneously arriving requests. Table1 lists the sizes of the reachable state space, the recurrent state set and the state space after compression. The model achieves a state compression ratio close to 6.... ..."

Cited by 6

### Table 4 Iterations and execution times for k-step SOR preconditioning ap- plied to GMRES(10) for four Markov chain matrices As can be observed, for the Marc1 and Marc2 matrices, the GMRES iteration preconditioned with SOR(1) and SOR(2) did not converge in the maximum of 500 steps allowed. However, SOR(k) converges for k 4 and the convergence keeps improving. As k increases the cost of each preconditioning step increases and in the end the overall cost increases again. The optimum number of steps seems to be about k = 10 for the rst three matrices.

1995

Cited by 6

### Table 2: Results for Gauss-Seidel Iteration. It appears that the order imposed by MARCA is practically as e cient as the periodic ordering. This may be explained by the fact that the ordering given by MARCA is well suited to Gauss-Seidel iterations. Indeed, this ordering also takes the ow of the Markov chain into account and thus leads to long \high stepping transitions quot; [11]. These results should therefore be interpreted, not as a poor performance by the periodic ordering, but rather as a desirable characteristic of the ordering provided by MARCA. Nevertheless, the periodic ordering guarantees the convergence of Gauss-Seidel, and is by itself, a good reason to choose it.

in On the Use of Periodicity Properties for the Efficient Numerical Solution of Certain Markov Chains

### Table 1 Markov chain of five states

"... In PAGE 2: ... Thus, we have two parallel sequences of Bernoulli trials, which are intermittently shifted forward. The shifting process can be described as a finite Markov chain apos; with the five states shown in Table1 .... ..."

### Table 1: Markov chain based methods for PMS

1999

"... In PAGE 3: ...1 Literature survey In this section we present and compare the Markov chain-based approaches [2, 3, 5, 10, 16, 21, 22] and the one based on SAN in [4] for the dependability model- ing and analysis of PMS. The most relevant aspects of the comparison are summa- rized in Table1 .A key point that impacts most of the other aspects is represented by the sin- gle/separate modeling of the phases: it affects the reusability/flexibility of previously built models, the modeling of dependencies among phases and the complexity of the solution steps.... ..."

Cited by 11