- Markov Processes
- Stochastic Comparisons for Non-Markov Processes
- Bayesian Inference of Markov Processes
- Markov Processes - CRC Press Book
Figure 3. Figure 4.
There was no ground truth data available for this study to assess image accuracy. However, the classification methodology used for image of year was replicated for classifying image. Both images were collected in the rainy season at the same time of the day using the same satellite sensor. For these reasons, we assumed that the overall accuracy of classification should be identical to the classification. Overall accuracies obtained for and images were, respectively, The Kappa indices for years and were, respectively, Tables Tablas de contingencia fuera del PNSC.
The value obtained to measure the association between the contingency table inside PNSC Table 5 and the Chapman-Kolmogrov equation is 0. This value is clearly below the critical value of the distribution for a significance level of 0. For the remaining area, where the LUCC has been more dynamic, the chi-square calculated to measure the association between the Chapman-Kolmogrov matrix and the contingency table is now clearly above the critical value of the distribution for 0.
However, this property can only be defined for recurrent Markov chains . A Markov chain is said to be recurrent if it is certain that the chain will return to the same state, but uncertain when that will happen aperiodic. We have no reason to assume that LUCC is a recurrent aperiodic chain; therefore we must ignore the stationary equation. On the other hand, it is reasonable to assume that LUCC inside PNSC will continue to be this Markov chain in the near future because all the factors affecting this process will continue to be regulated by the park administration.
Table 9. Contingency table inside PNSC. This prediction is not spatial because Markov chains assume spatial independence of the area units. Estimates for what happens outside the PNSC are not presented because there was no significant statistical evidence to state that it was a Markov chain. It may follow any other probabilistic law but not a Markovian one. This paper describes an integrated approach of remote sensing and stochastic modeling techniques in explaining LUCC in Sintra-Cascais area.
These findings reinforce the existence of the PNSC as an important factor in the stability of this highly dynamic area. Although Markov chains constitute a good tool for describing and projecting LUCC quantities, they are insufficient for spatial explicit LUCC predictions, because they assume statistical independence of spatial units. The methodology here presented can be employed to investigate if it is correct or not to use Markov transition probabilities in their modeling processes. Future research includes the experimentation of spatially explicit models to better understand the LUCC dynamics of this area.
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Stochastic Comparisons for Non-Markov Processes
Faculty of Geographical Sciences. High-resolution integrated modelling of the spatial dynamics of urban and regional systems. Computers, environment and urban systems, Vol. Cities and complexity: Understanding cities with cellular automata, agent-based models and fractals.
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Vol Markov processes for stochastic modeling. Definition of Hitting times; Stochastic identities and Recurrent systems of linear equations for moments of hitting times; Test functions and upper bounds for moments of hitting times; Return times; Examples; Problems. Law of Large Numbers and stationary distributions for Markov chains and semi-Markov processes; System of linear equations for stationary distributions; Stationary distributions and expectations of return times; Quasi-stationary distributions; Examples; Problems.
Regenerative processes; Regenerative properties of Markov chains and semi-Markov processes; Renewal Equation; Renewal function; Regenerative properties of stochastic processes with semi-Markov modulation; Examples; Problems. Renewal theorem in discrete and continuous time; Individual ergodic theorems for Markov chains and semi-Markov processes; Individual ergodic theorems for stochastic processes with semi-Markov modulation;.
Markov Processes - CRC Press Book
Perron-Frobenious Theorem and Eigenvalues Decomposition. Coupling for Markov chains; Coupling for semi-Markov processes; Explicit upper bounds in ergodic theorems; Examples: Problems. Additive-type functionals for Markov chains and semi-Markov processes; Stoachatic identities and Equations for moments and distributions. Limit theorems for random sums and randomly stopped stochastic processes. Perturbation conditions; Markov Chains and Semi-Markov processes with absorption; Singularly perturbed Markov Chains and semi-Markov Processes; Laurent asymptotic expansions; Laurent asymptotic expansions for hitting times; Asymptotic expansions for stationary and quasi-stationary distributions; Limit theorems for hitting times; Examples; Problems.
Kendal classification of queuing systems. Queuing systems; Biological processes; Reduction of phase space algorithms; Explicit formulas for hitting functionals, stationary and quasi stationary distributions; Example; Problems.
Risk processes; Ruin probabilities; Cramer-Lundberg approximations; Simulation of risk processes; Modulated risk processes; Examples; Problems. Atomic Markov chains; Markov-type price processes; Time skeleton approximations for Markov-type price processes; Time-space approximations for Markov-type price processes; Stochastic approximations for European and American type options; Examples; Problems. Introduction to Probability Models.