Is this a non-homogeneous Poisson process?

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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




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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














up vote
0
down vote

favorite
1












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




share|cite|improve this question



















  • 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












up vote
0
down vote

favorite
1









up vote
0
down vote

favorite
1






1





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




share|cite|improve this question











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





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Aug 6 at 0:22









user2167741

1059




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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
















  • 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










1 Answer
1






active

oldest

votes

















up vote
1
down vote



accepted










Inhomogeneous Poisson processes are typically understood to have the following features:



  1. the number of points in each finite interval has a Poisson distribution,

  2. the number of points in disjoint intervals are independent random variables,

  3. 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.





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  • 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










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1 Answer
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active

oldest

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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:



  1. the number of points in each finite interval has a Poisson distribution,

  2. the number of points in disjoint intervals are independent random variables,

  3. 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.





share|cite|improve this answer





















  • 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














up vote
1
down vote



accepted










Inhomogeneous Poisson processes are typically understood to have the following features:



  1. the number of points in each finite interval has a Poisson distribution,

  2. the number of points in disjoint intervals are independent random variables,

  3. 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.





share|cite|improve this answer





















  • 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












up vote
1
down vote



accepted







up vote
1
down vote



accepted






Inhomogeneous Poisson processes are typically understood to have the following features:



  1. the number of points in each finite interval has a Poisson distribution,

  2. the number of points in disjoint intervals are independent random variables,

  3. 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.





share|cite|improve this answer













Inhomogeneous Poisson processes are typically understood to have the following features:



  1. the number of points in each finite interval has a Poisson distribution,

  2. the number of points in disjoint intervals are independent random variables,

  3. 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.






share|cite|improve this answer













share|cite|improve this answer



share|cite|improve this answer











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
















  • 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












 

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