This is a multi-part message in MIME format.
------=_NextPart_000_000C_01C183E6.195A0050
Content-Type: text/plain;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
Dear all,
I have a question on finding steady-state probabilities in a Markov =
process.
Suppose there are J alternatives, I have a J-by-J transition matrix, =
which is called A, between period t and period t+1.
Each element in this J-by-J transition matrix takes the following form:
a_{ij}=3D{rho}* S_{ij}+(1-{rho})* {theta _{j}}
I want to find the steady state probabilities, for these J alternatives, =
i.e.,
I want to find a row vector X, which is 1-by-J, which satisfies,
X=3DX*A.=20
However, the above system is redundant. I need one more restriction:=20
The sum of all the elements in X will sum up to 1. To satisfy this =
restriction, I need to find X which satisfies,
X*A'=3De
A'=3DA-I, i.e., A minus a J-by-J identity matrix, THEN(!!), we'll =
replace the last column of A-I by a column of 1's.
e=3D(0,....0,1), i.e., the first J-1 elements are 0, but the last =
element is 1.
When X*A'=3De, X would be the steady state probabilities and satisfy =
the restriction: all elements in X sum up to 1.
I could not solve X, because I found I couldn't figure out the algebra =
when calculating the inverse of A'.
I wonder anyone could help me with this problem, or tell me where I =
could find a way to solve it using some statistics theorem.
Thank you very much for your helps.
------=_NextPart_000_000C_01C183E6.195A0050
Content-Type: text/html;
charset="iso-8859-1"
Content-Transfer-Encoding: quoted-printable
<!DOCTYPE HTML PUBLIC "-//W3C//DTD HTML 4.0 Transitional//EN">
<HTML><HEAD>
<META http-equiv=3DContent-Type content=3D"text/html; =
charset=3Diso-8859-1">
<META content=3D"MSHTML 5.50.4807.2300" name=3DGENERATOR>
<STYLE></STYLE>
</HEAD>
<BODY>
<DIV><FONT face=3DArial size=3D2>Dear all,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I have a question on finding =
steady-state=20
probabilities in a Markov process.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Suppose there are J alternatives, I =
have a J-by-J=20
transition matrix, which is called A, between period t and period=20
t+1.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Each element in this J-by-J transition =
matrix takes=20
the following form:</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>a_{ij}=3D{rho}* S_{ij}+(1-{rho})* =
{theta=20
_{j}}</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I want to find the steady state =
probabilities, for=20
these J alternatives, i.e.,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I want to find a row vector X, which is =
1-by-J,=20
which satisfies,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>X=3DX*A. </FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>However, the above system is =
redundant. =20
I need one more restriction: </FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>The sum of all the elements in X will =
sum up to 1.=20
To satisfy this restriction, I need to find X which =
satisfies,</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>X*A'=3De</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>A'=3DA-I, i.e., A minus a J-by-J =
identity matrix,=20
THEN(!!), we'll replace the last column of A-I by a column of =
1's.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>e=3D(0,....0,1), i.e., the first J-1 =
elements are 0,=20
but the last element is 1.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>When <FONT face=3DArial =
size=3D2>X*A'=3De, X=20
would be the steady state probabilities and satisfy the restriction: all =
elements in X sum up to 1.</FONT></FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I could not solve X, because I found I =
couldn't=20
figure out the algebra when calculating the inverse of A'.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>I wonder anyone could help me with this =
problem, or=20
tell me where I could find a way to solve it using some statistics=20
theorem.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2>Thank you very much for your =
helps.</FONT></DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV>
<DIV><FONT face=3DArial size=3D2></FONT> </DIV></BODY></HTML>
------=_NextPart_000_000C_01C183E6.195A0050--
=================================================================
Instructions for joining and leaving this list and remarks about
the problem of INAPPROPRIATE MESSAGES are available at
http://jse.stat.ncsu.edu/
=================================================================
