Stochastic Process Simulation

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I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation,
$$
P(t+1) = R P(t)
$$
In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.




stochastic-processes




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    up vote
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    I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation,
    $$
    P(t+1) = R P(t)
    $$
    In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.




    stochastic-processes




    share|cite|improve this question





















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation,
      $$
      P(t+1) = R P(t)
      $$
      In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.




      stochastic-processes




      share|cite|improve this question











      I am trying to understand some basic stochastic process simulation for population growth. Let us say, a population grows exponentially at a constant rate $R$, given by the equation,
      $$
      P(t+1) = R P(t)
      $$
      In the book "A Biologist's Guide to Mathematical Modeling in Ecology and Evolution" the authors generate various realizations of this model by taking a value of the rate parameter (like $R = 1.2$) and sampling from a Poisson distribution with mean $RP(t)$ to get $P(t+1)$. In reality, in some cases, we may not know the rate parameter $R$. Is there a way to estimate $R$ if we know the mean value of $P(t)$ and $P(t+1)$ at some point $t$, by such simulations? Thanks.





      stochastic-processes





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      asked Aug 2 at 3:48









      user2167741

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          Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.






          share|cite|improve this answer





















          • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
            – user2167741
            Aug 2 at 11:35











          • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
            – user2167741
            Aug 2 at 14:26










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          1 Answer
          1






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          active

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          up vote
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          accepted










          Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.






          share|cite|improve this answer





















          • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
            – user2167741
            Aug 2 at 11:35











          • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
            – user2167741
            Aug 2 at 14:26














          up vote
          1
          down vote



          accepted










          Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.






          share|cite|improve this answer





















          • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
            – user2167741
            Aug 2 at 11:35











          • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
            – user2167741
            Aug 2 at 14:26












          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.






          share|cite|improve this answer













          Let $p(t) = logP(t), r = log R$. Then the equation becomes $p(t+1)=r+p(t)$, and you can turn that into something like $p(t+t_0)-p(t_0)=rt$. You can run a standard linear regression on the latter.







          share|cite|improve this answer













          share|cite|improve this answer



          share|cite|improve this answer











          answered Aug 2 at 6:56









          ertl

          427110




          427110











          • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
            – user2167741
            Aug 2 at 11:35











          • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
            – user2167741
            Aug 2 at 14:26
















          • Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
            – user2167741
            Aug 2 at 11:35











          • Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
            – user2167741
            Aug 2 at 14:26















          Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
          – user2167741
          Aug 2 at 11:35





          Brilliant! So you mean to say we can simulate from a Poisson for $p(t+t_0)$ and $p(t_0)$ and do a regression. But the $log$ of a Poisson will not be a Poisson right? I suppose we can still sample from Poisson and take a log.
          – user2167741
          Aug 2 at 11:35













          Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
          – user2167741
          Aug 2 at 14:26




          Also, the above equation should be $p(t+t_0)-p(t) = rt_0$.
          – user2167741
          Aug 2 at 14:26












           

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