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Is this a non-homogeneous Poisson process?
Clash Royale CLAN TAG#URR8PPP
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Consider the population growth model where $P'(t) = rP(t)$, where $P(t)$ is the population at time $t$ with the constant growth rate $r$. Modeling this as a homogeneous Poisson process, at time $t$, $P(t+1)$ can be obtained by simulating from a Poisson distribution with mean $lambda = rP(t)$. So far so good.
Now, if $r$ is a function of $P(t)$ with constants $alpha$ and $K$, such that
$$
r(t) = alpha log(K/P(t)),
$$
is it still a non-homogeneous Poisson process? If not what process this might follow?
If the process is not known, suppose we know $P(0)$, choosing initial $alpha$ and $K$, can we simulate this as a homogeneous Poisson process starting with $r(0)$ and iterating $r$ and $P$ as the process evolves?
Thanks!
stochastic-processes poisson-process
asked Aug 6 at 0:22
user2167741
1059
add a comment |Â
up vote
0
down vote
favorite
Consider the population growth model where $P'(t) = rP(t)$, where $P(t)$ is the population at time $t$ with the constant growth rate $r$. Modeling this as a homogeneous Poisson process, at time $t$, $P(t+1)$ can be obtained by simulating from a Poisson distribution with mean $lambda = rP(t)$. So far so good.
Now, if $r$ is a function of $P(t)$ with constants $alpha$ and $K$, such that
$$
r(t) = alpha log(K/P(t)),
$$
is it still a non-homogeneous Poisson process? If not what process this might follow?
If the process is not known, suppose we know $P(0)$, choosing initial $alpha$ and $K$, can we simulate this as a homogeneous Poisson process starting with $r(0)$ and iterating $r$ and $P$ as the process evolves?
Thanks!
stochastic-processes poisson-process
asked Aug 6 at 0:22
user2167741
1059
randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00
add a comment |Â
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Consider the population growth model where $P'(t) = rP(t)$, where $P(t)$ is the population at time $t$ with the constant growth rate $r$. Modeling this as a homogeneous Poisson process, at time $t$, $P(t+1)$ can be obtained by simulating from a Poisson distribution with mean $lambda = rP(t)$. So far so good.
Now, if $r$ is a function of $P(t)$ with constants $alpha$ and $K$, such that
$$
r(t) = alpha log(K/P(t)),
$$
is it still a non-homogeneous Poisson process? If not what process this might follow?
If the process is not known, suppose we know $P(0)$, choosing initial $alpha$ and $K$, can we simulate this as a homogeneous Poisson process starting with $r(0)$ and iterating $r$ and $P$ as the process evolves?
Thanks!
stochastic-processes poisson-process
asked Aug 6 at 0:22
user2167741
1059
Consider the population growth model where $P'(t) = rP(t)$, where $P(t)$ is the population at time $t$ with the constant growth rate $r$. Modeling this as a homogeneous Poisson process, at time $t$, $P(t+1)$ can be obtained by simulating from a Poisson distribution with mean $lambda = rP(t)$. So far so good.
Now, if $r$ is a function of $P(t)$ with constants $alpha$ and $K$, such that
$$
r(t) = alpha log(K/P(t)),
$$
is it still a non-homogeneous Poisson process? If not what process this might follow?
If the process is not known, suppose we know $P(0)$, choosing initial $alpha$ and $K$, can we simulate this as a homogeneous Poisson process starting with $r(0)$ and iterating $r$ and $P$ as the process evolves?
Thanks!
stochastic-processes poisson-process
asked Aug 6 at 0:22
user2167741
1059
asked Aug 6 at 0:22
user2167741
1059
asked Aug 6 at 0:22
user2167741
1059
asked Aug 6 at 0:22
user2167741
1059
1059
randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00
add a comment |Â
randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00
randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00
add a comment |Â
1 Answer
1
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votes
up vote
1
down vote
accepted
Inhomogeneous Poisson processes are typically understood to have the following features:
- the number of points in each finite interval has a Poisson distribution,
- the number of points in disjoint intervals are independent random variables,
- the distributions depend on the time-varying (deterministic) rate $lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process
Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.
Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.
For more background on theory and the simulation algorithm, this is an excellent reference:
Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.
answered Aug 6 at 11:23
mloning
556
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Inhomogeneous Poisson processes are typically understood to have the following features:
- the number of points in each finite interval has a Poisson distribution,
- the number of points in disjoint intervals are independent random variables,
- the distributions depend on the time-varying (deterministic) rate $lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process
Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.
Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.
For more background on theory and the simulation algorithm, this is an excellent reference:
Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.
answered Aug 6 at 11:23
mloning
556
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
add a comment |Â
up vote
1
down vote
accepted
Inhomogeneous Poisson processes are typically understood to have the following features:
- the number of points in each finite interval has a Poisson distribution,
- the number of points in disjoint intervals are independent random variables,
- the distributions depend on the time-varying (deterministic) rate $lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process
Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.
Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.
For more background on theory and the simulation algorithm, this is an excellent reference:
Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.
answered Aug 6 at 11:23
mloning
556
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Inhomogeneous Poisson processes are typically understood to have the following features:
- the number of points in each finite interval has a Poisson distribution,
- the number of points in disjoint intervals are independent random variables,
- the distributions depend on the time-varying (deterministic) rate $lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process
Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.
Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.
For more background on theory and the simulation algorithm, this is an excellent reference:
Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.
answered Aug 6 at 11:23
mloning
556
Inhomogeneous Poisson processes are typically understood to have the following features:
- the number of points in each finite interval has a Poisson distribution,
- the number of points in disjoint intervals are independent random variables,
- the distributions depend on the time-varying (deterministic) rate $lambda(t)$, instead of the stationary assumptions for the homogeneous Poisson process
Your specification seems to violate (2), as the number of points depend on the history of the process. You may want to have a look at cluster processes, where the offspring in your context would represent the points in a cluster. Hawkes processes are another class that is widely used which depend on the process history.
Regarding simulation, you should be able to simulate the process by using Ogata's modified thinning algorithm, as you can compute the intensity of the process at each point.
For more background on theory and the simulation algorithm, this is an excellent reference:
Daley, D. J.; Vere-Jones, D., An introduction to the theory of point processes. Vol. I: Elementary theory and methods., Probability and Its Applications. New York, NY: Springer. xxi, 469 p. (2003). ZBL1026.60061.
answered Aug 6 at 11:23
mloning
556
answered Aug 6 at 11:23
mloning
556
answered Aug 6 at 11:23
mloning
556
answered Aug 6 at 11:23
mloning
556
556
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
add a comment |Â
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Terrific! Thank you for the explanation. I was looking for a good book to learn these elementary things and there you pointed it out. Many thanks. Off these discussions, I found this stack exchange much more people friendly and it is because of people like you who take their time to explain for non-experts.
– user2167741
Aug 6 at 12:01
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Good! Another good and more introductory source are the Lecture notes on temporal point processes by Rassmussen. Also please accept the answer if this is what you were looking for!
– mloning
Aug 6 at 12:06
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
Sorry - accepted it. And thanks again. Will check this out too.
– user2167741
Aug 6 at 12:14
add a comment |Â
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randomservices.org/random/poisson/Nonhomogeneous.html This might help.
– herb steinberg
Aug 6 at 0:28
Thanks! Honestly, this is terse for me to understand. I moved from mathematics to biology a long time back and now my questions are centered around biology research. It seems enormously difficult to translate intra-tumor heterogeneity into measure theoretic terms :-(
– user2167741
Aug 6 at 0:38
Frankly, I never heard of the term "homogeneous" used in connection with Poisson processes. Try Google - you might find references that you like better.
– herb steinberg
Aug 6 at 4:00