scholarly journals Markov Chains for Computer Music Generation

2021 ◽  
Vol 11 (2) ◽  
pp. 167-195
Author(s):  
Ilana Shapiro ◽  
Mark Huber
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Computer Music ◽  
New Music ◽  
Random Generation ◽  
Markov Chain Models ◽  
Music Generation

Random generation of music goes back at least to the 1700s with the introduction of Musical Dice Games. More recently, Markov chain models have been used as a way of extracting information from a piece of music and generating new music. We explain this approach and give Python code for using it to first draw out a model of the music and then create new music with that model.

The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities

1975 ◽  
Vol 12 (04) ◽  
pp. 744-752 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Transition Probabilities ◽  
General State ◽  
Transition Matrices ◽  
Positive Recurrence ◽  
Markov Chain Models ◽  
Countable Space ◽  
General State Space

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

Classified Discrete-Time Markov Chains

2015 ◽  
pp. 87-115
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Discrete Time ◽  
Formal Definition ◽  
Markov Chain Models ◽  
Definition Of ◽  
Markovian Systems

The main focus of this chapter is on the formalization of classified DTMCs. The chapter begins by presenting the formalization of some foundational notions of classified states, which are categorized based on reachability, periodicity, or absorbing features. Then, these results along with the formal definition of a DTMC, presented in the previous chapter, are used to formalize classified Markov chains, such as aperiodic and irreducible DTMCs. Based on these concepts, some long-term properties are verified for the purpose of formally checking the correctness of the functions of Markovian systems or analyzing the performance of Markov chain models.

scholarly journals Markov chain models of refugee migration data

2020 ◽  
Vol 85 (6) ◽  
pp. 892-912
Author(s):  
Vincent Huang ◽  
James Unwin
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
United Nations ◽  
State Of The Art ◽  
Refugee Crisis ◽  
Agent Based Model ◽  
Agent Based ◽  
Markov Chain Models ◽  
Migration Data ◽  
High Commissioner

Abstract The application of Markov chains to modelling refugee crises is explored, focusing on local migration of individuals at the level of cities and days. As an explicit example, we apply the Markov chains migration model developed here to United Nations High Commissioner for Refugees data on the Burundi refugee crisis. We compare our method to a state-of-the-art ‘agent-based’ model of Burundi refugee movements, and highlight that Markov chain approaches presented here can improve the match to data while simultaneously being more algorithmically efficient.

The robustness of positive recurrence and recurrence of Markov chains under perturbations of the transition probabilities

1975 ◽  
Vol 12 (4) ◽  
pp. 744-752 ◽  
Author(s):  
Richard L. Tweedie
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Finite Number ◽  
Transition Probabilities ◽  
General State ◽  
Transition Matrices ◽  
Positive Recurrence ◽  
Markov Chain Models ◽  
Countable Space ◽  
General State Space

In many Markov chain models, the immediate characteristic of importance is the positive recurrence of the chain. In this note we investigate whether positivity, and also recurrence, are robust properties of Markov chains when the transition laws are perturbed. The chains we consider are on a fairly general state space : when specialised to a countable space, our results are essentially that, if the transition matrices of two irreducible chains coincide on all but a finite number of columns, then positivity of one implies positivity of both; whilst if they coincide on all but a finite number of rows and columns, recurrence of one implies recurrence of both. Examples are given to show that these results (and their general analogues) cannot in general be strengthened.

scholarly journals Sampling Random Chordal Graphs by MCMC (Student Abstract)

2020 ◽  
Vol 34 (10) ◽  
pp. 13929-13930
Author(s):  
Wenbo Sun ◽  
Ivona Bezáková
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chains ◽  
Chordal Graph ◽  
Chordal Graphs ◽  
Random Generation ◽  
Mcmc Method ◽  
Bounded Treewidth ◽  
Graph Class ◽  
General Graphs

Chordal graphs are a widely studied graph class, with applications in several areas of computer science, including structural learning of Bayesian networks. Many problems that are hard on general graphs become solvable on chordal graphs. The random generation of instances of chordal graphs for testing these algorithms is often required. Nevertheless, there are only few known algorithms that generate random chordal graphs, and, as far as we know, none of them generate chordal graphs uniformly at random (where each chordal graph appears with equal probability). In this paper we propose a Markov chain Monte Carlo (MCMC) method to sample connected chordal graphs uniformly at random. Additionally, we propose a Markov chain that generates connected chordal graphs with a bounded treewidth uniformly at random. Bounding the treewidth parameter (which bounds the largest clique) has direct implications on the running time of various algorithms on chordal graphs. For each of the proposed Markov chains we prove that they are ergodic and therefore converge to the uniform distribution. Finally, as initial evidence that the Markov chains have the potential to mix rapidly, we prove that the chain on graphs with bounded treewidth mixes rapidly for trees (chordal graphs with treewidth bound of one).

Asymptotic behaviour of sample weighted circuits representing recurrent Markov chains

1990 ◽  
Vol 27 (03) ◽  
pp. 545-556 ◽  
Author(s):  
S. Kalpazidou
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Asymptotic Behaviour ◽  
Probabilistic Interpretation ◽  
Stationary Markov Chain

The asymptotic behaviour of the sequence (𝒞 n (ω), wc,n (ω)/n), is studied where 𝒞 n (ω) is the class of all cycles c occurring along the trajectory ωof a recurrent strictly stationary Markov chain (ξ n ) until time n and wc,n (ω) is the number of occurrences of the cycle c until time n. The previous sequence of sample weighted classes converges almost surely to a class of directed weighted cycles (𝒞∞, ω c ) which represents uniquely the chain (ξ n ) as a circuit chain, and ω c is given a probabilistic interpretation.

On the absorption probabilities and mean time for absorption for discrete Markov chains

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Nikolaos Halidias
Keyword(s):  
Markov Chain ◽  
Random Walk ◽  
Markov Chains ◽  
Generating Function ◽  
Discrete Time ◽  
Probability Generating Function ◽  
The Mean ◽  
Mean Time ◽  
Absorption Probabilities

Abstract In this note we study the probability and the mean time for absorption for discrete time Markov chains. In particular, we are interested in estimating the mean time for absorption when absorption is not certain and connect it with some other known results. Computing a suitable probability generating function, we are able to estimate the mean time for absorption when absorption is not certain giving some applications concerning the random walk. Furthermore, we investigate the probability for a Markov chain to reach a set A before reach B generalizing this result for a sequence of sets A 1 , A 2 , … , A k {A_{1},A_{2},\dots,A_{k}} .

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