Processes
Class SFSMarkovChain

java.lang.Object
  Processes.StochasticProcess
      Processes.MarkovChain
          Processes.SFSMarkovChain

Direct Known Subclasses:

SFSMarkovChainImpl, SFSStoppableMarkovChain

public abstract class SFSMarkovChain

extends MarkovChain

A stationary finite state (SFS) Markov chain. * *

A Markov chain with finitely many states 0,1,...,N-1. * and transition probabilities q(t,i.j)=q(i,j) independent of time * and given as abstract class methods. These methods are to be defined * in subclasses or in the body of a constructor call instantiating an * SFSMarkovChain. * Recall from MarkovChain how we sample the transition * probabilities.

* *

The methods a(int), b(int) provide information about the * possible movements of the chain: the chain can move * from state i only to the states j with a(i)<=j<=b(i). * This information speeds up all computations.

* *

We override the very inefficient sampling and timeStep methods * of the general MarkovChain with a faster algorithm by precomputing * the partition of the unit interval conditional on the state X(t)=i * according to the probabilities q(t,i,j)=q(i,j), a(i)<=j<=b(i):

* *

partition * [0,1) = union { [I(i,j),I(i,j+1)) : a(i)<=j<=b(i) } * of [0,1) into subintervals [I(i,j),I(i,j+1)) of length q(i,j):

* *

I(i,j)=sum_{k=a(i)}^{j-1}q(i,j), j=a(i),...,b(i) * (with I(i,a(i))=0).

* *

WARNING: methods do not check if parameter values are in their intended * ranges. Parameters outside these ranges lead to undefined and meaningless * behaviour.

* * @author Michael J. Meyer

Constructor Summary

SFSMarkovChain(int T, int j_0)
Constructor leaving all initializations to subclasses.

SFSMarkovChain(int T, int j_0, int N)
Constructor performing all initializations using the abstract * methods a(i), b(i).

SFSMarkovChain(int T, int j_0, int N, int[] a, int[] b)
Constructor performing all initializations.

Method Summary

abstract int	a(int i) Lower bound for states.
abstract int	b(int i) Upper bound for states.
double	I(int i, int j) Points I(i,j) of the partition of *[0,1) used to sample the transition probabilities q(i,j).
abstract double	q(int i, int j) Time independent transition probabilities.
double	q(int t, int i, int j) Definition of super.q(t,i,j).
void	timeStep(int t) Evolves the path of the chain from time t to time t+1.

Methods inherited from class Processes.StochasticProcess

get_dt, get_path, get_T, get_X_0, newPath, newPathBranch, pathSegment, pathSegment, pathSegment, sampledAt, simulationInit, timeStep

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail

SFSMarkovChain

public SFSMarkovChain(int T,
                      int j_0)

Constructor leaving all initializations to subclasses.

* * This constructor is to be called from a subclass if initializations * in the subclas have to be carried out before SFSMarkovChain can be * initialized. In this case all initializations need to be carried out in * the subclass constructor. * * @param T time steps to horizon * @param j_0 initial state

SFSMarkovChain

public SFSMarkovChain(int T,
                      int j_0,
                      int N)

Constructor performing all initializations using the abstract * methods a(i), b(i).

* *

This constructor performs some work but does not define the methods * a(i), b(i), q(i,j). You can instantiate the class by calling this * constructor and defining these methods in the body of the constructor * call.

* * @param T time steps to horizon * @param j_0 initial state * @param N number of states

SFSMarkovChain

public SFSMarkovChain(int T,
                      int j_0,
                      int N,
                      int[] a,
                      int[] b)

Constructor performing all initializations.

* * The methods a(i), b(i) are defined from the precopmputed arrays * a[], b[] as * public int a(int i){ return a[i]; } * public int b(int i){ return b[i]; } * * @param T time steps to horizon * @param j_0 initial state * @param N number of states * @param a array defining the a(int) values * @param b array defining the b(int) values

Method Detail

q

public abstract double q(int i,
                         int j)

Time independent transition probabilities.

* * q(i,j) is the probability that the chain transitions from state i * to state j in the next step.

a

public abstract int a(int i)

Lower bound for states.

* * From state i only the states j=a(i),...,b(i) can be reached * in the next step.

b

public abstract int b(int i)

Upper bound for states.

* * From state i only the states j=a(i),...,b(i) can be reached * in the next step.

q

public double q(int t,
                int i,
                int j)

Definition of super.q(t,i,j).

* * This method is replaced with q(int, int) in path computations.

Specified by:

q in class MarkovChain

I

public double I(int i,
                int j)

Points I(i,j) of the partition of *[0,1) used to sample the transition probabilities q(i,j). * Here i=X(t) is the current state of the chain.

timeStep

public void timeStep(int t)

Evolves the path of the chain from time t to time t+1.

* * Overrides MarkovChain.timeStep replacing j(t,i,u) * with the much faster j(i,u).

Overrides:

timeStep in class MarkovChain