Markov process: The population distribution of the system after $n$-transitions
$begingroup$
I am familiar with discrete markov processes and idea of their convergence towards a stable distribution after some number of steps. But my question is much simpler--more like a validation of some intuition.
Say I am looking at the transition of population between 3 cities. I have a $3times3$ transition matrix $M$ with entries $m_{ij}$ for the transition probability for a person moving from city $i$ to city $j$. I also have a vector $v_0$ of initial values--in this case the initial populations of the 3 cities.
My question is how to determine the population of the 3 cities after $k$ transitions? That is, we would like to know $v_k$.
Seems like the simplest results is akin to what we would see in a simple differential equation. Something like:
$$
text{population after first transition} = v_1 = v_0M
$$
The population after a second transition would be:
$$
v_2 = v_1M = (v_0M)M
$$
If we iterate this forward, then we would get:
$$
v_k = v_0M^k
$$
This seems simple enough, but just wanted to make sure I did not make some careless error or leave something out. Of course I can do an eigenvector decomposition of $M^k$ to quickly determine the matrix power for something larger than a $3times3$ matrix.
markov-process
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
$endgroup$
add a comment |
$begingroup$
I am familiar with discrete markov processes and idea of their convergence towards a stable distribution after some number of steps. But my question is much simpler--more like a validation of some intuition.
Say I am looking at the transition of population between 3 cities. I have a $3times3$ transition matrix $M$ with entries $m_{ij}$ for the transition probability for a person moving from city $i$ to city $j$. I also have a vector $v_0$ of initial values--in this case the initial populations of the 3 cities.
My question is how to determine the population of the 3 cities after $k$ transitions? That is, we would like to know $v_k$.
Seems like the simplest results is akin to what we would see in a simple differential equation. Something like:
$$
text{population after first transition} = v_1 = v_0M
$$
The population after a second transition would be:
$$
v_2 = v_1M = (v_0M)M
$$
If we iterate this forward, then we would get:
$$
v_k = v_0M^k
$$
This seems simple enough, but just wanted to make sure I did not make some careless error or leave something out. Of course I can do an eigenvector decomposition of $M^k$ to quickly determine the matrix power for something larger than a $3times3$ matrix.
markov-process
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
$endgroup$
add a comment |
$begingroup$
I am familiar with discrete markov processes and idea of their convergence towards a stable distribution after some number of steps. But my question is much simpler--more like a validation of some intuition.
Say I am looking at the transition of population between 3 cities. I have a $3times3$ transition matrix $M$ with entries $m_{ij}$ for the transition probability for a person moving from city $i$ to city $j$. I also have a vector $v_0$ of initial values--in this case the initial populations of the 3 cities.
My question is how to determine the population of the 3 cities after $k$ transitions? That is, we would like to know $v_k$.
Seems like the simplest results is akin to what we would see in a simple differential equation. Something like:
$$
text{population after first transition} = v_1 = v_0M
$$
The population after a second transition would be:
$$
v_2 = v_1M = (v_0M)M
$$
If we iterate this forward, then we would get:
$$
v_k = v_0M^k
$$
This seems simple enough, but just wanted to make sure I did not make some careless error or leave something out. Of course I can do an eigenvector decomposition of $M^k$ to quickly determine the matrix power for something larger than a $3times3$ matrix.
markov-process
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
$endgroup$
I am familiar with discrete markov processes and idea of their convergence towards a stable distribution after some number of steps. But my question is much simpler--more like a validation of some intuition.
Say I am looking at the transition of population between 3 cities. I have a $3times3$ transition matrix $M$ with entries $m_{ij}$ for the transition probability for a person moving from city $i$ to city $j$. I also have a vector $v_0$ of initial values--in this case the initial populations of the 3 cities.
My question is how to determine the population of the 3 cities after $k$ transitions? That is, we would like to know $v_k$.
Seems like the simplest results is akin to what we would see in a simple differential equation. Something like:
$$
text{population after first transition} = v_1 = v_0M
$$
The population after a second transition would be:
$$
v_2 = v_1M = (v_0M)M
$$
If we iterate this forward, then we would get:
$$
v_k = v_0M^k
$$
This seems simple enough, but just wanted to make sure I did not make some careless error or leave something out. Of course I can do an eigenvector decomposition of $M^k$ to quickly determine the matrix power for something larger than a $3times3$ matrix.
markov-process
markov-process
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
edited Dec 17 '18 at 4:34
krishnab
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
asked Dec 17 '18 at 2:03
krishnabkrishnab
454415
454415
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
You appear to be confusing two different, but related processes. The $3times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3times3$. Now, if you want to examine the expected population distributions, you can use your individual $3times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.
answered Dec 17 '18 at 3:08
amdamd
31k21051
$endgroup$
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
$endgroup$
– krishnab
Dec 17 '18 at 3:50
2
$begingroup$
@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
$endgroup$
– krishnab
Dec 17 '18 at 17:02
1
$begingroup$
@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
$endgroup$
– amd
Dec 17 '18 at 21:23
|
show 5 more comments
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$begingroup$
You appear to be confusing two different, but related processes. The $3times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3times3$. Now, if you want to examine the expected population distributions, you can use your individual $3times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.
answered Dec 17 '18 at 3:08
amdamd
31k21051
$endgroup$
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
$endgroup$
– krishnab
Dec 17 '18 at 3:50
2
$begingroup$
@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
$endgroup$
– krishnab
Dec 17 '18 at 17:02
1
$begingroup$
@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
$endgroup$
– amd
Dec 17 '18 at 21:23
|
show 5 more comments
$begingroup$
You appear to be confusing two different, but related processes. The $3times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3times3$. Now, if you want to examine the expected population distributions, you can use your individual $3times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.
answered Dec 17 '18 at 3:08
amdamd
31k21051
$endgroup$
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
$endgroup$
– krishnab
Dec 17 '18 at 3:50
2
$begingroup$
@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
$endgroup$
– krishnab
Dec 17 '18 at 17:02
1
$begingroup$
@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
$endgroup$
– amd
Dec 17 '18 at 21:23
|
show 5 more comments
$begingroup$
You appear to be confusing two different, but related processes. The $3times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3times3$. Now, if you want to examine the expected population distributions, you can use your individual $3times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.
answered Dec 17 '18 at 3:08
amdamd
31k21051
$endgroup$
You appear to be confusing two different, but related processes. The $3times3$ matrix $M$ gives the transition probabilities for an individual person moving from city to city. As you follow this one person around, the matrix $M^k$ then gives the $k$-step transition probabilities, that is, the probability that that particular person ends up in city $j$ after $k$ steps if she started in city $i$.
For the aggregate populations, on the other hand, the system is much larger. Depending on how you model it, you might have $N^3$ states, where $N$ is the total population (in this model you track the location of each individual), or $binom{N+2}{2}$ states (in this model you track the population of each city, with a total population of $N$). Either way, the transition matrix is a lot larger than $3times3$. Now, if you want to examine the expected population distributions, you can use your individual $3times3$ matrix: by linearity of expectation, if you have a three-element vector with the city populations, then multiplying it by $M^k$ gives you the expected population distribution, not the actual one, after $k$ steps.
answered Dec 17 '18 at 3:08
amdamd
31k21051
answered Dec 17 '18 at 3:08
amdamd
31k21051
answered Dec 17 '18 at 3:08
amdamd
31k21051
answered Dec 17 '18 at 3:08
amdamd
31k21051
31k21051
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
$endgroup$
– krishnab
Dec 17 '18 at 3:50
2
$begingroup$
@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
$endgroup$
– krishnab
Dec 17 '18 at 17:02
1
$begingroup$
@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
$endgroup$
– amd
Dec 17 '18 at 21:23
|
show 5 more comments
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
$endgroup$
– krishnab
Dec 17 '18 at 3:50
2
$begingroup$
@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
$endgroup$
– krishnab
Dec 17 '18 at 17:02
1
$begingroup$
@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
$endgroup$
– amd
Dec 17 '18 at 21:23
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
thanks for catching my confusion. Right on. I get what you are saying by having $N^3$ states where each element of the transition matrix represents the probability of an individual $n_h$ moving from city $i$ to city $j$. So if I have 4 individuals in the population, then I would have to look at combinations like v = [1 1 3 2], and [2 1 3 2], ( where cities = 1,2.3) and $v_h$ for individual $h in 1...4$. So there are $3^4$ possible states and the transition matrix $M$ has $3^4$ elements. Is that right so far?
$endgroup$
– krishnab
Dec 17 '18 at 3:30
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
@amd I'm a little confused by this answer. I can't see what difference it would make to track individuals rather than just populations unless you were actually concerned with the likelihood that some particular individual was in some city after k steps. To me, the OPs interpretation of the problem in terms of population vectors seems much more natural and useful. Maybe I've misunderstood something.
$endgroup$
– Tyberius
Dec 17 '18 at 3:41
$begingroup$
I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
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– krishnab
Dec 17 '18 at 3:50
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I think I still not clear about getting the expected population distribution. If I have a vector $s$ of total populations in each of the 3 cities. Lets use new letters for clarity. Say $s_0$ = [ 200 300 150]. I want to know what $s_k$ is. So you are saying that I could multiply $s_0$ by the $3x3$ matrix $M^k$ and get the expected population distribution? Is there a way to estimate the variance of such an estimate? I suppose if I used the larger individual population models of $N$ I could sample a set of rows, and get bootstrap estimates for the variance. But can't do that with the $3x3$ matrix
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– krishnab
Dec 17 '18 at 3:50
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2
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@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
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– krishnab
Dec 17 '18 at 17:02
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@amd and Tyberius Just to close out the conversation. Key point is that the population process is stochastic. So if I was to simulate the movement of individuals $N$ between cities one at a time, I would get a different "actual" population in each city after $k$ iterations. That is just the nature of a stochastic process. But, we often characterize stochastic processes by their expected values of the variable. So the expected value of the population vector--the notation $E(s_k)$ is equal to the initial value $s_0$ times the $3x3$ city transition matrix.
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– krishnab
Dec 17 '18 at 17:02
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1
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@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
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– amd
Dec 17 '18 at 21:23
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@Tyberius The one thing that I would add at this point is that, although you have an underlying Markov process that governs the population movement, the way that the expected (average) population distribution evolves is a stochastic process in only the most trivial sense: you know exactly what the next state will be.
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– amd
Dec 17 '18 at 21:23
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