In this paper we develop a data-driven mixture of vector autoregressive models with exogenous components. The process is assumed to change regimes according to an underlying Markov process. In contrast to the hidden Markov setup, we allow the transition probabilities of the underlying Markov process to depend on past time series values and exogenous variables. Such processes have potential applications to modeling brain signals. For example, brain activity at time t (measured by electroencephalograms) will can be modeled as a function of both its past values as well as exogenous variables (such as visual or somatosensory stimuli). Furthermore, we establish stationarity, geometric ergodicity and the existence of moments for these processes under suitable conditions on the parameters of the model. Such properties are important for understanding the stability properties of the model as well as deriving the asymptotic behavior of various statistics and model parameter estimators.
Given a directed graph G = (N,A), a tension is a function from A to R which satisfies Kirchhoff\\\'s law for voltages. There are two well-known tension problems on graphs. In the minimum cost tension problem (MCT), a cost vector is given and a tension satisfying lower and upper bounds is seeked such that the total cost is minimum. In the maximum tension problem (MaxT), the graph contains 2 special nodes and an arc between them. The aim is to find the maximum tension on this arc. In this study we assume that both problems are feasible and have finite optimal solutions and analyze their inverse versions under rectilinear and Chebyshev distances. In the inverse minimum cost tension problem we adjust the cost parameter to make a given feasible solution the optimum, whereas in inverse maximum tension problem the bounds of the arcs are modified. We show, by extending the results of Ahuja and Orlin (2002), that these inverse tension problems are in a way \\\"dual\\\" to the inverse network flows. We prove that the inverse minimum cost tension problem under rectilinear norm is equivalent to solving a minimum cost tension problem, while under unit weight Chebyshev norm it can be solved by finding a minimum mean cost residual cut. Moreover, inverse maximum tension problem under rectilinear norm can be solved as a maximum tension problem on the same graph with new arc bounds. Finally, we provide a generalization of the inverse problems to monotropic programming problems with linear costs.
Finding a delivery plan for cancer radiation treatment using multileaf collimators operating in ''step-and-shoot mode'' can be formulated mathematically as a problem of decomposing an integer matrix into a weighted sum of binary matrices having the consecutive-ones property - and sometimes other properties related to the collimator technology. The efficiency of the delivery plan is measured by both the sum of weights in the decomposition, known as the total beam-on time, and the number of different binary matrices appearing in it, referred to as the cardinality, the latter being closely related to the set-up time of the treatment. In practice, the total beam-on time is usually restricted to its minimum possible value, (which is easy to find), and a decomposition that minimises cardinality (subject to this restriction) is sought.