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A recursive algorithm for computing inferences in imprecise Markov chains Universiteit Gent
We present an algorithm that can efficiently compute a broad class of inferences for discrete-time imprecise Markov chains, a generalised type of Markov chains that allows one to take into account partially specified probabilities and other types of model uncertainty. The class of inferences that we consider contains, as special cases, tight lower and upper bounds on expected hitting times, on hitting probabilities and on expectations of ...
Elucidating Transition State Behaviour from Mobility Data by Cascades of Markov Chains Vrije Universiteit Brussel
ith the ongoing trend towards digitisation, vast amounts of, often very fine-grained, data are being collected. The ultimate goal is to capture and understand the behaviour of a system, such as the traffic in a city. However, making sense of such data is not straightforward due to its high level of detail and complex dependencies in time and space. Exploring heuristic approaches is essential to arrive at data representations that enable better ...
Imprecise Markov chains and their limit behavior Universiteit Gent
On the embedding problem for discrete-time Markov chains. Vrije Universiteit Brussel
When a discrete-time homogenous Markov chain is observed at time intervals that correspond to its time unit, then the transition probabilities of the chain can be estimated using known maximum likelihood estimators. In this paper we consider a situation when a Markov chain is observed on time intervals with length equal to twice the time unit of the Markov chain. The question then arises of characterizing probability matrices whose square ...
On the embedding problem for three-state Markov chains. Vrije Universiteit Brussel
The present paper investigates the embedding problem for time-homogeneous Markov chains. A discrete-time Markov chain with time unit 1 is embeddable in case there exists a compatible Markov chain regarding time unit 1/m (with m > 1 integer number). An embeddable Markov chain has a transition matrix for which there exists an m-th root that is a probability matrix. The present paper examines the embedding problem for discrete-time Markov chains ...
Necessary embedding conditions for state-wise monotone Markov chains. Vrije Universiteit Brussel
In previous work, the embedding problem is examined within the entire set of discrete-time Markov chains. However, for several phenomena, the states of a Markov model are ordered categories and the transition matrix is state-wise monotone. The present paper investigates the embedding problem for the specific subset of state-wise monotone Markov chains. We prove necessary conditions on the transition matrix of a discrete-time Markov chain with ...
On monotone Markov chains and properties of monotone matrix roots Vrije Universiteit Brussel
Monotone matrices are stochastic matrices that satisfy the monotonicity conditions as introduced by Daley in 1968. Monotone Markov chains are useful in modeling phenomena in several areas. Most previous work examines the embedding problem for Markov chains within the entire set of stochastic transition matrices, and only a few studies focuses on the embeddability within a specific subset of stochastic matrices. The present paper examines for ...
Average behaviour of imprecise Markov chains : a single pointwise ergodic theorem for six different models Universiteit Gent
We study the average behaviour of imprecise Markov chains; a generalised type of Markov chain where local probabilities are partially specified, and where structural assumptions such as Markovianity are weakened. In particular, we prove a pointwise ergodic theorem that provides (strictly) almost sure bounds on the long term average of any real function of the state of such an imprecise Markov chain. Compared to an earlier ergodic theorem by De ...
Matrix roots and embedding conditions for three-state discrete-time Markov chains with complex eigenvalues Vrije Universiteit Brussel
The present paper examines matrix root properties and embedding conditions for discrete-time Markov chains with three states and a transition matrix having complex eigenvalues. Necessary as well as sufficient conditions for the existence of an m-th stochastic root of the transition matrix, are investigated. Matrix roots are expressed in analytical form based on the spectral decomposition of the transition matrix and properties of these matrix ...