The topic of Markov Random Fields (MRFs) has been well studied in the past, and has found practical use in various image processing, and machine learning applications. Where coding is concerned, MRF specific schemes have been largely unexplored. In this thesis, an overview is given of recent developments and challenges in the lossless coding of MRFs. Specifically, we concentrate on difficulties caused by computational intractability due to the partition function of the MRF. One proposed solution to this problem is to segment the MRF with a cutset, and encode the components separately. Using this method, arithmetic coding is possible via the Belief Propagation (BP) algorithm. We consider two cases of the BP algorithm: MRFs with only simple cliques, and MRFs with complex cliques. In the latter case, we study a minimum radius condition requirement for ensuring that all cliques are accounted for during coding. This condition also simplifies the process of conditioning on observed sites. Finally, using these results, we develop a systematic procedure of clustering and choosing cutsets. / Thesis (Master, Mathematics & Statistics) -- Queen's University, 2013-08-12 14:50:00.596

The issue of angle instability has been widely discussed in the power engineering literature. Many control techniques have been proposed to provide the complementary synchronizing and damping torques through generators and/or network connected power apparatus such as FACTs, braking resistors and DC links. The synchronizing torque component keeps all generators in synchronism while damping torque reduces oscillations and returns the power system to its pre-fault operating condition. One of the main factors limiting the transfer capacity of the electrical transmission network is the separation of the power system at weak links which can be understood by analogy with a large spring-mass system. However, this weak-links related problem is not dealt with in existing control designs because it is non-trivial during transient period to determine credible weak links in a large power system which may consist of hundreds of strong and weak links. The difficulty of identifying weak links has limited the performance of existing controls when it comes to the synchronization of generators and damping of oscillations. Such circumstances also restrict the operation of power systems close to its transient stability limits. These considerations have led to the primary research question in this thesis, "To what extent can the synchronization of generators and damping of oscillations be maximized to fully extend the transient stability limits of power systems and to improve the transfer capacity of the network?" With the recent advances in power electronics technology, the extension of transfer capacity is becoming more readily achievable. Complementary to the use of power electronics technology to improve transfer capacity, this research develops an improved control strategy by examining the dynamics of the modes of separation associated with the strong and weak links of the reduced transmission network. The theoretical framework of the control strategy is based on Energy Decomposition and Unstable Equilibrium Points. This thesis recognizes that under extreme loadings of the transmission network containing strong and weak links, weak-links are most likely to dictate the transient stability limits of the power system. We conclude that in order to fully extend the transient stability limits of power system while maximizing the value of control resources, it is crucial for the control strategy to aim its control effort at the energy component that is most likely to cause a separation. The improvement in the synchronization amongst generators remains the most important step in the improvement of the transfer capacity of the power system network.

This thesis is concerned with the efficient computation of Jacobian matrices of nonlinear vector maps using automatic differentiation (AD). Specifically, we propose the use of two directed edge separator methods, the weighted minimum separator and natural order separator methods, to exploit the structure of the computational graph of the nonlinear system.This allows for the efficient determination of the Jacobian matrix using AD software. We will illustrate the promise of this approach with computational experiments.