Classification of states and chains in Markov Chain
$begingroup$
Let $S={{1,2,3,4,5}}$. Find out which states are: persistent, transient, null, non-null, periodic, aperiodic, ergodic and absorbing. Find closed and irreducible sets of a states. For closed sets that are irreducible find mean recurrence times and stationary distribution, when
$
P=
begin{bmatrix}
1 & 0 & 0 & 0 & 0 \
1/4 & 1/4 & 1/8 & 1/4 & 1/8 \
0 & 0 & 1/4 & 1/4 & 1/2 \
0 & 0 & 1/2 & 1/2 & 0 \
0 & 0 & 1/4 & 3/4 & 0
end{bmatrix}
$
Closed sets: ${{3,4}}, {{3,5}}$
Irreducible sets: ${{3,4}}, {{3,5}}$
Sets ${{3,4}}, {{3,5}}$ consists of states: persistent and non-null
Transient set: ${{2}}$
I'm not sure how to find out if a state is periodic or aperiodic. If $p_{ii}>0$, then the period equals $1$, and so all the states are periodic?
Please check the above, correct me if I'm wrong anywhere, and help me with the other part of the exercise. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
$endgroup$
add a comment |
$begingroup$
Let $S={{1,2,3,4,5}}$. Find out which states are: persistent, transient, null, non-null, periodic, aperiodic, ergodic and absorbing. Find closed and irreducible sets of a states. For closed sets that are irreducible find mean recurrence times and stationary distribution, when
$
P=
begin{bmatrix}
1 & 0 & 0 & 0 & 0 \
1/4 & 1/4 & 1/8 & 1/4 & 1/8 \
0 & 0 & 1/4 & 1/4 & 1/2 \
0 & 0 & 1/2 & 1/2 & 0 \
0 & 0 & 1/4 & 3/4 & 0
end{bmatrix}
$
Closed sets: ${{3,4}}, {{3,5}}$
Irreducible sets: ${{3,4}}, {{3,5}}$
Sets ${{3,4}}, {{3,5}}$ consists of states: persistent and non-null
Transient set: ${{2}}$
I'm not sure how to find out if a state is periodic or aperiodic. If $p_{ii}>0$, then the period equals $1$, and so all the states are periodic?
Please check the above, correct me if I'm wrong anywhere, and help me with the other part of the exercise. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
$endgroup$
add a comment |
$begingroup$
Let $S={{1,2,3,4,5}}$. Find out which states are: persistent, transient, null, non-null, periodic, aperiodic, ergodic and absorbing. Find closed and irreducible sets of a states. For closed sets that are irreducible find mean recurrence times and stationary distribution, when
$
P=
begin{bmatrix}
1 & 0 & 0 & 0 & 0 \
1/4 & 1/4 & 1/8 & 1/4 & 1/8 \
0 & 0 & 1/4 & 1/4 & 1/2 \
0 & 0 & 1/2 & 1/2 & 0 \
0 & 0 & 1/4 & 3/4 & 0
end{bmatrix}
$
Closed sets: ${{3,4}}, {{3,5}}$
Irreducible sets: ${{3,4}}, {{3,5}}$
Sets ${{3,4}}, {{3,5}}$ consists of states: persistent and non-null
Transient set: ${{2}}$
I'm not sure how to find out if a state is periodic or aperiodic. If $p_{ii}>0$, then the period equals $1$, and so all the states are periodic?
Please check the above, correct me if I'm wrong anywhere, and help me with the other part of the exercise. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
$endgroup$
Let $S={{1,2,3,4,5}}$. Find out which states are: persistent, transient, null, non-null, periodic, aperiodic, ergodic and absorbing. Find closed and irreducible sets of a states. For closed sets that are irreducible find mean recurrence times and stationary distribution, when
$
P=
begin{bmatrix}
1 & 0 & 0 & 0 & 0 \
1/4 & 1/4 & 1/8 & 1/4 & 1/8 \
0 & 0 & 1/4 & 1/4 & 1/2 \
0 & 0 & 1/2 & 1/2 & 0 \
0 & 0 & 1/4 & 3/4 & 0
end{bmatrix}
$
Closed sets: ${{3,4}}, {{3,5}}$
Irreducible sets: ${{3,4}}, {{3,5}}$
Sets ${{3,4}}, {{3,5}}$ consists of states: persistent and non-null
Transient set: ${{2}}$
I'm not sure how to find out if a state is periodic or aperiodic. If $p_{ii}>0$, then the period equals $1$, and so all the states are periodic?
Please check the above, correct me if I'm wrong anywhere, and help me with the other part of the exercise. Any will be much appreciated.
probability-theory stochastic-processes markov-chains
probability-theory stochastic-processes markov-chains
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
asked Dec 15 '18 at 9:16
MacAbraMacAbra
269210
269210
add a comment |
add a comment |
1 Answer
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${1}$ and ${3,4,5}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ ${3,4}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
$endgroup$
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
add a comment |
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$begingroup$
${1}$ and ${3,4,5}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ ${3,4}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
$endgroup$
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
add a comment |
$begingroup$
${1}$ and ${3,4,5}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ ${3,4}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
$endgroup$
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
add a comment |
$begingroup$
${1}$ and ${3,4,5}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ ${3,4}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
$endgroup$
${1}$ and ${3,4,5}$ are closed sets, $2$ is transient and all recurrent states are aperiodic. [ ${3,4}$ is not a closed set because you can go from $3$ to $5$]. Of course, $1$ is an absorbing state.
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
answered Dec 15 '18 at 12:22
Kavi Rama MurthyKavi Rama Murthy
74.1k53270
74.1k53270
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
add a comment |
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
1
1
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
$begingroup$
Thank you for correcting my closed sets. But what about irreducible sets? I think these two are also irreducible, can I find any other ones? Can you explain me a little more why all recurrent states are aperiodic, and generally how do I find if a state is periodic or not?
$endgroup$
– MacAbra
Dec 15 '18 at 15:03
1
1
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
$begingroup$
From $3$ or $4$ you cannot go to $5$ so ${3,4,5}$ is not irreducible. Period of state $i$ is the gcd of ${n:p_{ii}^{(n)} >0$. If $p_{ii} >0$ the the state is aperiodoc (i.e. has period $1$) because the gcd of any set of integers containing $1$ is $1$.
$endgroup$
– Kavi Rama Murthy
Dec 15 '18 at 23:44
add a comment |
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