To get started, we must load the TikZ package: For our Markov chain diagrams we will use the automata and positioning libraries within TikZ, these are loaded using the \usetikzlibrary command: The tikzpicture environment is where our diagram will be built: Drawing the Markov chain is broken into two steps: Within the tikzpicture environment, states can be added using the \node command: For each node, you must specify a unique id. At each step, links are provided to the LaTeX source on Overleaf where you can view and edit the examples. The state option controls the appearance of the node. CS1 maint: multiple names: authors list (, Markov chains on a measurable state space, Learn how and when to remove this template message, https://en.wikipedia.org/w/index.php?title=Examples_of_Markov_chains&oldid=959568458, Articles needing additional references from June 2016, All articles needing additional references, Creative Commons Attribution-ShareAlike License, This page was last edited on 29 May 2020, at 12:17. The final LaTeX source for this example can be found on Overleaf. Let’s add some color and adjust the styling of a little bit to make this perfect for presentations. {\displaystyle X_{0}=10} The \draw command takes the form, where the start and end ids reference nodes previously defined. To start, we will change the colors of the nodes. It doesn't depend on how things got to their current state. Some additional options are used with the \node command. after the draw command. Our states will now take the form: Clearly we need to correct the positioning of the labels. Using the transition matrix it is possible to calculate, for example, the long-term fraction of weeks during which the market is stagnant, or the average number of weeks it will take to go from a stagnant to a bull market. This will draw arrows that bend to the right (from the perspective of the node at the starting end of the arrow) between the nodes. And suppose that at a given observation period, say period, the probability of the system being in a particular state depends on its status at the n-1 period, such a system is called Markov Chain or Markov process. The next thing we may want to add to our diagram is labels for the transition probabilities between states. We will arrange the nodes in an equilateral triangle. On a given day, the probability of it raining or being sunny the following day is determined by the current days weather. To busy to read the explanation? 6 X Suppose that you start with $10, and you wager $1 on an unending, fair, coin toss indefinitely, or until you lose all of your money. The process described here is an approximation of a Poisson point process – Poisson processes are also Markov processes. In the above-mentioned dice games, the only thing that matters is the current state of the board. , 5 1.1 An example and some interesting questions Example 1.1. Example 1: A Simple Weather Model. The nodes can be moved further apart to avoid appearing crowded by changing right=of s to right=2cm of s. This will place the nodes 2cm way from each other. It is not necessary to know when they popped, so knowing Next, the color, width and arrow head of the arrows will be adjusted. [1][2], The probabilities of weather conditions (modeled as either rainy or sunny), given the weather on the preceding day, 6 P(Dry) = 0.3 x 0.2 x 0.8 x 0.6 = 0.0288 "rainy", and the rows can be labelled in the same order. North American Actuarial Journal, 11(4):92–109, 2007. The weather on day 2 (the day after tomorrow) can be predicted in the same way: In this example, predictions for the weather on more distant days are increasingly It will be calculatedas: P({Dry, Dry, Rain, Rain}) = P(Rain|Rain) .P(Rain|Dry) . 1 t In a game such as blackjack, a player can gain an advantage by remembering which cards have already been shown (and hence which cards are no longer in the deck), so the next state (or hand) of the game is not independent of the past states. n } This page contains examples of Markov chains and Markov processes in action. Adding text to a node is optional, although if you chose not to you must use empty curly brackets {} . The states represent whether a hypothetical stock market is exhibiting a bull market, bear market, or stagnant market trend during a given week. A finite-state machine can be used as a representation of a Markov chain. In the above-mentioned dice games, the only thing that matters is the current state of the board. Note that no options have been used here for either arrow. A game of snakes and ladders or any other game whose moves are determined entirely by dice is a Markov chain, indeed, an absorbing Markov chain. Labeling the state space {1 = bull, 2 = bear, 3 = stagnant} the transition matrix for this example is, The distribution over states can be written as a stochastic row vector x with the relation x(n + 1) = x(n)P. So if at time n the system is in state x(n), then three time periods later, at time n + 3 the distribution is, In particular, if at time n the system is in state 2 (bear), then at time n + 3 the distribution is. This can be done by adding arrows that connect nodes to themselves. The loop above option draws the arrow above the state. In the previous example, the rainy node was positioned using right=of s. In this example we will use absolute position where we specify where we want the node to be. If you need a nice visual for a slideshow, video or website, you may consider some additional styling. Considerthe given probabilities for the two given states: Rain and Dry. If In this example we will create an illustration of this Markov chain.

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