Markovmodeller

Markovmodeller, a term used in Danish and Norwegian for Markov models, are a class of stochastic models used to describe systems that move between a finite or countable set of states over time. The key feature is the Markov property: the future state depends only on the present state and not on past history. A Markov model is specified by a state space and rules that govern transitions between states.

In discrete time, the model is a discrete-time Markov chain (DTMC) with a transition matrix P, where Pij is the probability of moving from state i to state j in one time step. In continuous time, the process is a continuous-time Markov chain (CTMC) and is described by a rate matrix (generator) Q, with exponentially distributed waiting times in each state. The distribution of states over time can be computed from the initial distribution and P (or Q), and many models focus on stationary distributions and ergodicity.

Extensions include hidden Markov models (HMMs), where the observed data are emitted by an unobserved Markov chain, and Markov decision processes (MDPs), which integrate decision-making and optimization into the dynamics. Related concepts include non-homogeneous Markov chains, which have time-dependent transition structures, and reinforcement learning, which often solves MDP-like problems.

Parameter estimation and inference for Markovmodeller frameworks often use maximum likelihood or Bayesian methods. For HMMs, the Baum–Welch algorithm (an expectation–maximization approach) is commonly employed. Applications span queueing theory, finance, reliability engineering, genetics, speech recognition, and natural language processing, among others. The Markov modelling approach remains valued for its mathematical tractability and its ability to capture systems with memoryless dynamics.