마르코프 체인

예

• 다음의 문장이 주어져 있다
  • 모든 국민은 학문과 예술의 자유를 가진다.
  • 모든 국민은 사생활의 비밀과 자유를 침해받지 아니한다.
• 두 문장에 나오는 단어들은 이 확률과정의 상태공간 S를 이룬다
  • S={"모든", "국민은", "비밀과", "예술의", "자유를", "학문과", "가진다.", "사생활의", "침해받지", "아니한다."}
• "모든" 이라는 단어(상태)에서 출발하여, 연결된 선을 따라 다음 단어로 이동하며 (전이), 마침표가 있는 단어에 이르면 이 과정을 종료한다
• 한 단어에서 다음 단어로 넘어갈 확률은 두 단어가 연결된 빈도로부터 얻어진다

• 이러한 확률과정을 통하여, 새로운 문장을 생성할 수 있게 된다
  • 예: "모든 국민은 학문과 예술의 자유를 침해받지 아니한다"
• 이는 현재의 한 단어만이 다음 단어에 영향을 주는 '바이그램' 모형이다

관련된 항목들

• 랜덤워크(random walk)

매스매티카 파일

• https://drive.google.com/file/d/0B8XXo8Tve1cxTGczRjYxemZueTg/view

노트

위키데이터

• ID : Q176645

말뭉치

1. Markov chains, named after Andrey Markov, are mathematical systems that hop from one "state" (a situation or set of values) to another.
2. Of course, real modelers don't always draw out Markov chain diagrams.
3. This means the number of cells grows quadratically as we add states to our Markov chain.
4. One use of Markov chains is to include real-world phenomena in computer simulations.
5. Markov chain analysis is concerned in general with how long individual entities spend, on average, in different states before being absorbed and on first passage times to absorbing states.
6. Thus Markov chain analysis is ideal for providing insights on life history or anything related to timing.
7. Life expectancy – The fundamental matrix in Markov chain analysis provides a measure of expected time in each state before being absorbed.
8. Markov chains are a fairly common, and relatively simple, way to statistically model random processes.
9. A popular example is r/SubredditSimulator, which uses Markov chains to automate the creation of content for an entire subreddit.
10. Overall, Markov Chains are conceptually quite intuitive, and are very accessible in that they can be implemented without the use of any advanced statistical or mathematical concepts.
11. This example illustrates many of the key concepts of a Markov chain.
12. A (time-homogeneous) Markov chain built on states A and B is depicted in the diagram below.
13. If the Markov chain in Figure 21.3 is used to model time-varying propagation loss, then each state in the chain corresponds to a different loss.
14. Each number represents the probability of the Markov process changing from one state to another state, with the direction indicated by the arrow.
15. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC).
16. A continuous-time process is called a continuous-time Markov chain (CTMC).
17. However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis.
18. His analysis did not alter the understanding or appreciation of Pushkin’s poem, but the technique he developed—now known as a Markov chain—extended the theory of probability in a new direction.
19. In physics the Markov chain simulates the collective behavior of systems made up of many interacting particles, such as the electrons in a solid.
20. And Markov chains themselves have become a lively area of inquiry in recent decades, with efforts to understand why some of them work so efficiently—and some don’t.
21. As Markov chains have become commonplace tools, the story of their origin has largely faded from memory.
22. A Markov chain describes the transitions between a given set of states using transition probabilities.
23. A Markov chain evolves in discrete time and moves step by step from state to state; the step size can be chosen arbitrarily, and depending on the application, it could be 1 day, or 1 month, or 1 year.
24. Transition probabilities only depend on the current state the Markov chain is in at time t, and not on any previous states at t−1,t−2, ….
25. Markov chains are usually analyzed in matrix notation.
26. But the concept of modeling sequences of random events using states and transitions between states became known as a Markov chain.
27. One of the first and most famous applications of Markov chains was published by Claude Shannon.
28. For a Markov chain to be ergodic, two technical conditions are required of its states and the non-zero transition probabilities; these conditions are known as irreducibility and aperiodicity.
29. It follows from Theorem 21.2.1 that the random walk with teleporting results in a unique distribution of steady-state probabilities over the states of the induced Markov chain.
30. In the improved multivariate Markov chain model, Ching et al. incorporated positive and negative association parts.
31. With the developments of Markov chain models and their applications, the number of the sequences may be larger.
32. It is inevitable that a large categorical data sequence group will cause high computational cost in multivariate Markov chain model.
33. In Section 2, we review two lemmas and several Markov chain models.
34. The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.
35. Along the way, you are welcome to talk with other students (don't forget we have a Blackboard Q&A discussion forum!) and explore how others have used Markov chains.
36. A Markov chain is a mathematical model that describes movement (transitions) from one state to another.
37. In a first-order Markov chain, this probability only depends on the current state, and not any further transition history.
38. We can build more "memory" into our model by using a higher-order Markov chain.
39. The paper also investigates the definition of stochastic monotonicity on a more general state space, and the properties of integer-valued stochastically monotone Markov Chains.
40. Markov chain Monte Carlo using the Metropolis-Hastings algorithm is a general method for the simulation of stochastic processes having probability densities known up to a constant of proportionality.
41. Consider the problem of modelling memory effects in discrete-state random walks using higher-order Markov chains.
42. As an illustration of model selection for multistep Markov chains, this manuscript re-examines the hot hand phenomenon from a different analytical philosophy.
43. The memoryless property of Markov chains refers to h = 1.
44. This manuscript addressed general methods of degree selection for multistep Markov chain models.
45. A Markov chain is a mathematical system usually defined as a collection of random variables, that transition from one state to another according to certain probabilistic rules.
46. The algorithm known as PageRank, which was originally proposed for the internet search engine Google, is based on a Markov process.
47. A discrete-time Markov chain involves a system which is in a certain state at each step, with the state changing randomly between steps.
48. A Markov chain is represented using a probabilistic automaton (It only sounds complicated!).
49. This unique guide to Markov chains approaches the subject along the four convergent lines of mathematics, implementation, simulation, and experimentation.
50. An introduction to simple stochastic matrices and transition probabilities is followed by a simulation of a two-state Markov chain.
51. The notion of steady state is explored in connection with the long-run distribution behavior of the Markov chain.
52. Predictions based on Markov chains with more than two states are examined, followed by a discussion of the notion of absorbing Markov chains.
53. We next give examples that illustrate various properties of quantum Markov chains.
54. First, the probability sequence is treated as consisting of successive realizations of a stochastic process, and the stochastic process is modeled using Markov chain theory.
55. The size and stationarity of the reforecast ensemble dataset permits straightforward estimation of the parameters of certain Markov chain models.
56. Finally, opportunities to apply the Markov chain model to decision support are highlighted.
57. In this same framework, the importance of recognizing inhomogeneity in the Markov chain parameters when specifying decision intervals is illustrated.
58. The extra questions are interesting and off the well-beaten path of questions that are typical for an introductory Markov Chains course.
59. CUP 1997 (Chapter 1, Discrete Markov Chains is freely available to download.
60. (Each of these books contains a readable chapter on Markov chains and many nice examples.
61. See also, Sheldon Ross and Erol Pekoz, A Second Course in Probability, 2007 (Chapter 5 gives a readable treatment of Markov chains and covers many of the topics in our course.
62. Markov chains are used to model probabilities using information that can be encoded in the current state.
63. More technically, information is put into a matrix and a vector - also called a column matrix - and with many iterations, a collection of probability vectors makes up Markov chains.
64. Representing a Markov chain as a matrix allows for calculations to be performed in a convenient manner.
65. A Markov chain is one example of a Markov model, but other examples exist.
66. So far, we have discussed discrete-time Markov chains in which the chain jumps from the current state to the next state after one unit time.
67. Markov chain Monte Carlo is one of our best tools in the desperate struggle against high-dimensional probabilistic computation, but its fragility makes it dangerous to wield without adequate training.
68. Unfortunately the Markov chain Monte Carlo literature provides limited guidance for practical risk management.
69. Before introducing Markov chain Monte Carlo we will begin with a short review of the Monte Carlo method.
70. Finally we will discuss how these theoretical concepts manifest in practice and carefully study the behavior of an explicit implementation of Markov chain Monte Carlo.

관련 링크

• 수학노트 부설 마르코프 체인 연구소