DiscreteMarkovProcess

DiscreteMarkovProcess[i0,m]

represents a discrete-time, finite-state Markov process with transition matrix m and initial state i0.

DiscreteMarkovProcess[p0,m]

represents a Markov process with initial state probability vector p0.

represents a Markov process with transition matrix from the graph g.

Details

• DiscreteMarkovProcess is also known as a discrete-time Markov chain.
• DiscreteMarkovProcess is a discrete-time and discrete-state random process.
• The states of DiscreteMarkovProcess are integers between 1 and , where is the length of transition matrix m.
• The transition matrix m specifies conditional transition probabilities mi,jProbability[x[k+1]jx[k]i], where x[k] is the state of the process at time k. »
• A discrete Markov process can be seen as a random walk on a graph, where the probability of transitioning from state to state is specified by mi,j.
• EstimatedProcess[data,DiscreteMarkovProcess[n]] indicates that a process with n states should be estimated.
• The transition matrix in the case of a graph g is constructed to give equal probability of transitioning to each incident vertex.
• DiscreteMarkovProcess allows m to be an × matrix with non-negative elements and rows that sum to 1, i0 is an integer between 1 and , and p0 is a vector of length of non-negative elements that sum to 1.
• DiscreteMarkovProcess can be used with such functions as MarkovProcessProperties, PDF, Probability, and RandomFunction.

Examples

open allclose all

Basic Examples(2)

Define a discrete Markov process:

 In[1]:=

Simulate it:

 In[2]:=
 Out[2]=
 In[3]:=
 Out[3]=

Find the PDF for the state at time :

 In[1]:=
 In[2]:=
 Out[2]=

Find the long-run proportion of time the process is in state 2:

 In[3]:=
 Out[3]=