Markov model

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In probability theory, a Markov model is a stochastic model used to model pseudo-randomly changing systems. It is assumed that future states depend only on the current state, not on the events that occurred before it (that is, it assumes the Markov property). Generally, this assumption enables reasoning and computation with the model that would otherwise be intractable. For this reason, in the fields of predictive modelling and probabilistic forecasting, it is desirable for a given model to exhibit the Markov property.

Contents

Introduction

There are four common Markov models used in different situations, depending on whether every sequential state is observable or not, and whether the system is to be adjusted on the basis of observations made:

Markov models
System state is fully observableSystem state is partially observable
System is autonomous Markov chain Hidden Markov model
System is controlled Markov decision process Partially observable Markov decision process

Markov chain

The simplest Markov model is the Markov chain. It models the state of a system with a random variable that changes through time. In this context, the Markov property indicates that the distribution for this variable depends only on the distribution of a previous state. An example use of a Markov chain is Markov chain Monte Carlo, which uses the Markov property to prove that a particular method for performing a random walk will sample from the joint distribution.

Hidden Markov model

A hidden Markov model is a Markov chain for which the state is only partially observable or noisily observable. In other words, observations are related to the state of the system, but they are typically insufficient to precisely determine the state. Several well-known algorithms for hidden Markov models exist. For example, given a sequence of observations, the Viterbi algorithm will compute the most-likely corresponding sequence of states, the forward algorithm will compute the probability of the sequence of observations, and the Baum–Welch algorithm will estimate the starting probabilities, the transition function, and the observation function of a hidden Markov model.

One common use is for speech recognition, where the observed data is the speech audio waveform and the hidden state is the spoken text. In this example, the Viterbi algorithm finds the most likely sequence of spoken words given the speech audio.

Markov decision process

A Markov decision process is a Markov chain in which state transitions depend on the current state and an action vector that is applied to the system. Typically, a Markov decision process is used to compute a policy of actions that will maximize some utility with respect to expected rewards.

Partially observable Markov decision process

A partially observable Markov decision process (POMDP) is a Markov decision process in which the state of the system is only partially observed. POMDPs are known to be NP complete, but recent approximation techniques have made them useful for a variety of applications, such as controlling simple agents or robots. [1]

Markov random field

A Markov random field, or Markov network, may be considered to be a generalization of a Markov chain in multiple dimensions. In a Markov chain, state depends only on the previous state in time, whereas in a Markov random field, each state depends on its neighbors in any of multiple directions. A Markov random field may be visualized as a field or graph of random variables, where the distribution of each random variable depends on the neighboring variables with which it is connected. More specifically, the joint distribution for any random variable in the graph can be computed as the product of the "clique potentials" of all the cliques in the graph that contain that random variable. Modeling a problem as a Markov random field is useful because it implies that the joint distributions at each vertex in the graph may be computed in this manner.

Hierarchical Markov models

Hierarchical Markov models can be applied to categorize human behavior at various levels of abstraction. For example, a series of simple observations, such as a person's location in a room, can be interpreted to determine more complex information, such as in what task or activity the person is performing. Two kinds of Hierarchical Markov Models are the Hierarchical hidden Markov model [2] and the Abstract Hidden Markov Model. [3] Both have been used for behavior recognition [4] and certain conditional independence properties between different levels of abstraction in the model allow for faster learning and inference. [3] [5]

Tolerant Markov model

A Tolerant Markov model (TMM) is a probabilistic-algorithmic Markov chain model. [6] It assigns the probabilities according to a conditioning context that considers the last symbol, from the sequence to occur, as the most probable instead of the true occurring symbol. A TMM can model three different natures: substitutions, additions or deletions. Successful applications have been efficiently implemented in DNA sequences compression. [6] [7]

Markov-chain forecasting models

Markov-chains have been used as a forecasting methods for several topics, for example price trends, [8] wind power [9] and solar irradiance. [10] The Markov-chain forecasting models utilize a variety of different settings, from discretizing the time-series [9] to hidden Markov-models combined with wavelets [8] and the Markov-chain mixture distribution model (MCM). [10]

See also

Related Research Articles

<span class="mw-page-title-main">Markov chain</span> Random process independent of past history

A Markov chain or Markov process is a stochastic model describing a sequence of possible events in which the probability of each event depends only on the state attained in the previous event. Informally, this may be thought of as, "What happens next depends only on the state of affairs now." A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC). A continuous-time process is called a continuous-time Markov chain (CTMC). Markov processes are named in honor of the Russian mathematician Andrey Markov.

A hidden Markov model (HMM) is a Markov model in which the observations are dependent on a latent Markov process. An HMM requires that there be an observable process whose outcomes depend on the outcomes of in a known way. Since cannot be observed directly, the goal is to learn about state of by observing By definition of being a Markov model, an HMM has an additional requirement that the outcome of at time must be "influenced" exclusively by the outcome of at and that the outcomes of and at must be conditionally independent of at given at time Estimation of the parameters in an HMM can be performed using maximum likelihood. For linear chain HMMs, the Baum–Welch algorithm can be used to estimate the parameters.

A Bayesian network is a probabilistic graphical model that represents a set of variables and their conditional dependencies via a directed acyclic graph (DAG). While it is one of several forms of causal notation, causal networks are special cases of Bayesian networks. Bayesian networks are ideal for taking an event that occurred and predicting the likelihood that any one of several possible known causes was the contributing factor. For example, a Bayesian network could represent the probabilistic relationships between diseases and symptoms. Given symptoms, the network can be used to compute the probabilities of the presence of various diseases.

The Viterbi algorithm is a dynamic programming algorithm for obtaining the maximum a posteriori probability estimate of the most likely sequence of hidden states—called the Viterbi path—that results in a sequence of observed events. This is done especially in the context of Markov information sources and hidden Markov models (HMM).

In statistics, Markov chain Monte Carlo (MCMC) is a class of algorithms used to draw samples from a probability distribution. Given a probability distribution, one can construct a Markov chain whose elements' distribution approximates it – that is, the Markov chain's equilibrium distribution matches the target distribution. The more steps that are included, the more closely the distribution of the sample matches the actual desired distribution.

<span class="mw-page-title-main">Markov property</span> Memoryless property of a stochastic process

In probability theory and statistics, the term Markov property refers to the memoryless property of a stochastic process, which means that its future evolution is independent of its history. It is named after the Russian mathematician Andrey Markov. The term strong Markov property is similar to the Markov property, except that the meaning of "present" is defined in terms of a random variable known as a stopping time.

A graphical model or probabilistic graphical model (PGM) or structured probabilistic model is a probabilistic model for which a graph expresses the conditional dependence structure between random variables. They are commonly used in probability theory, statistics—particularly Bayesian statistics—and machine learning.

In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for sampling from a specified multivariate probability distribution when direct sampling from the joint distribution is difficult, but sampling from the conditional distribution is more practical. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.

In electrical engineering, statistical computing and bioinformatics, the Baum–Welch algorithm is a special case of the expectation–maximization algorithm used to find the unknown parameters of a hidden Markov model (HMM). It makes use of the forward-backward algorithm to compute the statistics for the expectation step.

<span class="mw-page-title-main">Markov random field</span> Set of random variables

In the domain of physics and probability, a Markov random field (MRF), Markov network or undirected graphical model is a set of random variables having a Markov property described by an undirected graph. In other words, a random field is said to be a Markov random field if it satisfies Markov properties. The concept originates from the Sherrington–Kirkpatrick model.

A partially observable Markov decision process (POMDP) is a generalization of a Markov decision process (MDP). A POMDP models an agent decision process in which it is assumed that the system dynamics are determined by an MDP, but the agent cannot directly observe the underlying state. Instead, it must maintain a sensor model and the underlying MDP. Unlike the policy function in MDP which maps the underlying states to the actions, POMDP's policy is a mapping from the history of observations to the actions.

Conditional random fields (CRFs) are a class of statistical modeling methods often applied in pattern recognition and machine learning and used for structured prediction. Whereas a classifier predicts a label for a single sample without considering "neighbouring" samples, a CRF can take context into account. To do so, the predictions are modelled as a graphical model, which represents the presence of dependencies between the predictions. What kind of graph is used depends on the application. For example, in natural language processing, "linear chain" CRFs are popular, for which each prediction is dependent only on its immediate neighbours. In image processing, the graph typically connects locations to nearby and/or similar locations to enforce that they receive similar predictions.

In the mathematical theory of stochastic processes, variable-order Markov (VOM) models are an important class of models that extend the well known Markov chain models. In contrast to the Markov chain models, where each random variable in a sequence with a Markov property depends on a fixed number of random variables, in VOM models this number of conditioning random variables may vary based on the specific observed realization.

In the mathematical theory of probability, the entropy rate or source information rate is a function assigning an entropy to a stochastic process.

In computer science, a predictive state representation (PSR) is a way to model a state of controlled dynamical system from a history of actions taken and resulting observations. PSR captures the state of a system as a vector of predictions for future tests (experiments) that can be done on the system. A test is a sequence of action-observation pairs and its prediction is the probability of the test's observation-sequence happening if the test's action-sequence were to be executed on the system. One of the advantage of using PSR is that the predictions are directly related to observable quantities. This is in contrast to other models of dynamical systems, such as partially observable Markov decision processes (POMDPs) where the state of the system is represented as a probability distribution over unobserved nominal states.

This page lists articles related to probability theory. In particular, it lists many articles corresponding to specific probability distributions. Such articles are marked here by a code of the form (X:Y), which refers to number of random variables involved and the type of the distribution. For example (2:DC) indicates a distribution with two random variables, discrete or continuous. Other codes are just abbreviations for topics. The list of codes can be found in the table of contents.

The following outline is provided as an overview of and topical guide to machine learning:

References

  1. ↑ Kaelbling, L. P.; Littman, M. L.; Cassandra, A. R. (1998). "Planning and acting in partially observable stochastic domains". Artificial Intelligence. 101 (1–2): 99–134. CiteSeerX   10.1.1.390.8474 . doi: 10.1016/S0004-3702(98)00023-X . ISSN   0004-3702.
  2. ↑ Fine, S.; Singer, Y. (1998). "The hierarchical hidden markov model: Analysis and applications". Machine Learning. 32 (1): 41–62. doi: 10.1023/A:1007469218079 .
  3. 1 2 Bui, H. H.; Venkatesh, S.; West, G. (2002). "Policy recognition in the abstract hidden markov model". Journal of Artificial Intelligence Research. 17: 451–499. doi: 10.1613/jair.839 . hdl: 10536/DRO/DU:30044252 .
  4. ↑ Theocharous, G. (2002). Hierarchical Learning and Planning in Partially Observable Markov Decision Processes (PhD). Michigan State University.
  5. ↑ Luhr, S.; Bui, H. H.; Venkatesh, S.; West, G. A. W. (2003). "Recognition of Human Activity through Hierarchical Stochastic Learning". PERCOM '03 Proceedings of the First IEEE International Conference on Pervasive Computing and Communications. pp. 416–422. CiteSeerX   10.1.1.323.928 . doi:10.1109/PERCOM.2003.1192766. ISBN   978-0-7695-1893-0. S2CID   13938580.
  6. 1 2 Pratas, D.; Hosseini, M.; Pinho, A. J. (2017). "Substitutional tolerant Markov models for relative compression of DNA sequences". PACBB 2017 – 11th International Conference on Practical Applications of Computational Biology & Bioinformatics, Porto, Portugal. pp. 265–272. doi:10.1007/978-3-319-60816-7_32. ISBN   978-3-319-60815-0.
  7. ↑ Pratas, D.; Pinho, A. J.; Ferreira, P. J. S. G. (2016). "Efficient compression of genomic sequences". Data Compression Conference (DCC), 2016. IEEE. pp. 231–240. doi:10.1109/DCC.2016.60. ISBN   978-1-5090-1853-6. S2CID   14230416.
  8. 1 2 de Souza e Silva, E.G.; Legey, L.F.L.; de Souza e Silva, E.A. (2010). "Forecasting oil price trends using wavelets and hidden Markov models". Energy Economics. 32.
  9. 1 2 Carpinone, A; Giorgio, M; Langella, R.; Testa, A. (2015). "Markov chain modeling for very-short-term wind power forecasting". Electric Power Systems Research. 122: 152–158. doi: 10.1016/j.epsr.2014.12.025 .
  10. 1 2 Munkhammar, J.; van der Meer, D.W.; Widén, J. (2019). "Probabilistic forecasting of high-resolution clear-sky index time-series using a Markov-chain mixture distribution model". Solar Energy. 184: 688–695. doi:10.1016/j.solener.2019.04.014. S2CID   146076100.
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