Re: simulating a 2-state Markov process
- To: mathgroup at smc.vnet.net
- Subject: [mg41387] Re: [mg41369] simulating a 2-state Markov process
- From: Bobby Treat <drmajorbob at mailblocks.com>
- Date: Sat, 17 May 2003 05:49:53 -0400 (EDT)
- Sender: owner-wri-mathgroup at wolfram.com
Here's the initial conditions as a probability vector, where a is the
probability of starting with Heads. (In your post, a=1)
initial := {a, 1 - a};
Here's the transition probability matrix:
tpm := {{1 - p, q}, {p, 1 - q}};
After one transition, the probabilities add up to one as expected:
tpm.initial
Simplify@(Plus @@ %)
{a (1-p)+(1-a) q,a p+(1-a) (1-q)}
1
After n transitions, the probability vector is computed by
state[n_Integer] /; n >= 0 := Nest[tpm.# &, initial, n]
or this may work better, if you're not specifying the parameters:
state[n_Integer] /; n >= 0 := Nest[Simplify[tpm.#] &, initial, n]
a, p, and q can be set as desired and state[] is automatically defined
accordingly, or you can compute state[n]/.{a->1, p->0.1, q->0.3}, for
instance.
The latter will not work after you have given values to the parameters.
The former is much faster, but you may want to see parametric results.
Clear[a, p, q] gives you the choice again.
If speed is an issue, you will want to specify p and q, and define the
state vector after n transitions this way:
p = 0.1; q = 0.3;
Clear[n]
{values, vectors} = Eigensystem[tpm];
fastState[n_Integer] /; n >= 0 =
Simplify[Transpose[vectors].DiagonalMatrix[values^n].
Inverse[Transpose[vectors]].initial]
{-0.75*0.6^n + 0.75*1.^n + 1.*0.6^n*a,
0.7499999999999998*0.6^n +
0.24999999999999994*1.^n -
0.9999999999999998*0.6^n*a}
Bobby
-----Original Message-----
From: David E. Burmaster <deb at alceon.com>
To: mathgroup at smc.vnet.net
Subject: [mg41387] [mg41369] simulating a 2-state Markov process
Dear MathGroup,
Can someone please suggest an efficient way to use Mathematica to
draw realizations from a 2-state Markov process as outlined below?
i want to specify the transition probabilities and the number of
transitions to simulate.
=-=
for notation....
1. let's denote the two states as H and T (for Heads and Tails, even
though i am thinking about a process more general than flipping a
coin)
2. let's denote the four conditional transition probabilities as follows
prob(T | H) = p
prob(H | H) = 1-p
and
prob(H | T) = q
prob(T | T) = 1-q
3. for initial conditions, let the process begin in state H
=-=
within this framework, p and q can take any values in the unit square
(0, 1] x (0, 1]
=-=-=
thank you for your help
dave
--
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Alceon Corporation
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