마르코프 체인

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Pythagoras0 (토론 | 기여)님의 2020년 12월 21일 (월) 00:34 판 (→‎매스매티카 파일)
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  • 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

마르코프 체인1.png

  • 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


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  1. Markov chains, named after Andrey Markov, are mathematical systems that hop from one "state" (a situation or set of values) to another.[1]
  2. Of course, real modelers don't always draw out Markov chain diagrams.[1]
  3. This means the number of cells grows quadratically as we add states to our Markov chain.[1]
  4. One use of Markov chains is to include real-world phenomena in computer simulations.[1]
  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.[2]
  6. Thus Markov chain analysis is ideal for providing insights on life history or anything related to timing.[2]
  7. Life expectancy – The fundamental matrix in Markov chain analysis provides a measure of expected time in each state before being absorbed.[2]
  8. Markov chains are a fairly common, and relatively simple, way to statistically model random processes.[3]
  9. A popular example is r/SubredditSimulator, which uses Markov chains to automate the creation of content for an entire subreddit.[3]
  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.[3]
  11. This example illustrates many of the key concepts of a Markov chain.[3]
  12. A (time-homogeneous) Markov chain built on states A and B is depicted in the diagram below.[4]
  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.[5]
  14. Each number represents the probability of the Markov process changing from one state to another state, with the direction indicated by the arrow.[6]
  15. A countably infinite sequence, in which the chain moves state at discrete time steps, gives a discrete-time Markov chain (DTMC).[6]
  16. A continuous-time process is called a continuous-time Markov chain (CTMC).[6]
  17. However, many applications of Markov chains employ finite or countably infinite state spaces, which have a more straightforward statistical analysis.[6]
  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.[7]
  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.[7]
  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.[7]
  21. As Markov chains have become commonplace tools, the story of their origin has largely faded from memory.[7]
  22. A Markov chain describes the transitions between a given set of states using transition probabilities.[8]
  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.[8]
  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, ….[8]
  25. Markov chains are usually analyzed in matrix notation.[8]
  26. But the concept of modeling sequences of random events using states and transitions between states became known as a Markov chain.[9]
  27. One of the first and most famous applications of Markov chains was published by Claude Shannon.[9]
  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.[10]
  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.[10]
  30. In the improved multivariate Markov chain model, Ching et al. incorporated positive and negative association parts.[11]
  31. With the developments of Markov chain models and their applications, the number of the sequences may be larger.[11]
  32. It is inevitable that a large categorical data sequence group will cause high computational cost in multivariate Markov chain model.[11]
  33. In Section 2, we review two lemmas and several Markov chain models.[11]
  34. The Season 1 episode "Man Hunt" (2005) of the television crime drama NUMB3RS features Markov chains.[12]
  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.[13]
  36. A Markov chain is a mathematical model that describes movement (transitions) from one state to another.[13]
  37. In a first-order Markov chain, this probability only depends on the current state, and not any further transition history.[13]
  38. We can build more "memory" into our model by using a higher-order Markov chain.[13]
  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.[14]
  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.[15]
  41. Consider the problem of modelling memory effects in discrete-state random walks using higher-order Markov chains.[16]
  42. As an illustration of model selection for multistep Markov chains, this manuscript re-examines the hot hand phenomenon from a different analytical philosophy.[16]
  43. The memoryless property of Markov chains refers to h = 1.[16]
  44. This manuscript addressed general methods of degree selection for multistep Markov chain models.[16]
  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.[17]
  46. The algorithm known as PageRank, which was originally proposed for the internet search engine Google, is based on a Markov process.[17]
  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.[17]
  48. A Markov chain is represented using a probabilistic automaton (It only sounds complicated!).[17]
  49. This unique guide to Markov chains approaches the subject along the four convergent lines of mathematics, implementation, simulation, and experimentation.[18]
  50. An introduction to simple stochastic matrices and transition probabilities is followed by a simulation of a two-state Markov chain.[18]
  51. The notion of steady state is explored in connection with the long-run distribution behavior of the Markov chain.[18]
  52. Predictions based on Markov chains with more than two states are examined, followed by a discussion of the notion of absorbing Markov chains.[18]
  53. We next give examples that illustrate various properties of quantum Markov chains.[19]
  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.[20]
  55. The size and stationarity of the reforecast ensemble dataset permits straightforward estimation of the parameters of certain Markov chain models.[20]
  56. Finally, opportunities to apply the Markov chain model to decision support are highlighted.[20]
  57. In this same framework, the importance of recognizing inhomogeneity in the Markov chain parameters when specifying decision intervals is illustrated.[20]
  58. The extra questions are interesting and off the well-beaten path of questions that are typical for an introductory Markov Chains course.[21]
  59. CUP 1997 (Chapter 1, Discrete Markov Chains is freely available to download.[21]
  60. (Each of these books contains a readable chapter on Markov chains and many nice examples.[21]
  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.[21]
  62. Markov chains are used to model probabilities using information that can be encoded in the current state.[22]
  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.[22]
  64. Representing a Markov chain as a matrix allows for calculations to be performed in a convenient manner.[22]
  65. A Markov chain is one example of a Markov model, but other examples exist.[22]
  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.[23]
  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.[24]
  68. Unfortunately the Markov chain Monte Carlo literature provides limited guidance for practical risk management.[24]
  69. Before introducing Markov chain Monte Carlo we will begin with a short review of the Monte Carlo method.[24]
  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.[24]

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