What is the natural extension of Poisson Process to two macroscopic variables

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AFAIK, Poisson Process is an object that takes in an event rate $R$, and produces a time-series of events, such that the expected number of events within any time interval $Delta t$ is $R Delta t$. The rate $R(t)$ may also depend on time, and can be estimated by averaging events within a time window if the changes are not too rapid.



Now, Poisson process assumes that there is no structure in the events, other than their rate. Assume I have noisy random event data from an experiment, and I would like to explore the data for potential additional structure. What is the next simplest process to try, that has, say, two input parameters?



If I turn on my imagination, I am thinking similarities with Taylor series. In a truncated series expansion, every next term increases the accuracy, but also requires better knowledge of the function (experimentally - more data) to be estimated. Is there something similar to series expansion for random processes? For example, in terms of moments - mean, variance, skew, etc.







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    AFAIK, Poisson Process is an object that takes in an event rate $R$, and produces a time-series of events, such that the expected number of events within any time interval $Delta t$ is $R Delta t$. The rate $R(t)$ may also depend on time, and can be estimated by averaging events within a time window if the changes are not too rapid.



    Now, Poisson process assumes that there is no structure in the events, other than their rate. Assume I have noisy random event data from an experiment, and I would like to explore the data for potential additional structure. What is the next simplest process to try, that has, say, two input parameters?



    If I turn on my imagination, I am thinking similarities with Taylor series. In a truncated series expansion, every next term increases the accuracy, but also requires better knowledge of the function (experimentally - more data) to be estimated. Is there something similar to series expansion for random processes? For example, in terms of moments - mean, variance, skew, etc.







    share|cite|improve this question
























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      AFAIK, Poisson Process is an object that takes in an event rate $R$, and produces a time-series of events, such that the expected number of events within any time interval $Delta t$ is $R Delta t$. The rate $R(t)$ may also depend on time, and can be estimated by averaging events within a time window if the changes are not too rapid.



      Now, Poisson process assumes that there is no structure in the events, other than their rate. Assume I have noisy random event data from an experiment, and I would like to explore the data for potential additional structure. What is the next simplest process to try, that has, say, two input parameters?



      If I turn on my imagination, I am thinking similarities with Taylor series. In a truncated series expansion, every next term increases the accuracy, but also requires better knowledge of the function (experimentally - more data) to be estimated. Is there something similar to series expansion for random processes? For example, in terms of moments - mean, variance, skew, etc.







      share|cite|improve this question














      AFAIK, Poisson Process is an object that takes in an event rate $R$, and produces a time-series of events, such that the expected number of events within any time interval $Delta t$ is $R Delta t$. The rate $R(t)$ may also depend on time, and can be estimated by averaging events within a time window if the changes are not too rapid.



      Now, Poisson process assumes that there is no structure in the events, other than their rate. Assume I have noisy random event data from an experiment, and I would like to explore the data for potential additional structure. What is the next simplest process to try, that has, say, two input parameters?



      If I turn on my imagination, I am thinking similarities with Taylor series. In a truncated series expansion, every next term increases the accuracy, but also requires better knowledge of the function (experimentally - more data) to be estimated. Is there something similar to series expansion for random processes? For example, in terms of moments - mean, variance, skew, etc.









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      edited Aug 28 at 7:54

























      asked Aug 27 at 16:23









      Aleksejs Fomins

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          I'd say that you are looking for a Poisson Process with branches.



          Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
          $$N(t)=sum_j=1^nN_j(t).$$
          Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.



          If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.






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            up vote
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            I'd say that you are looking for a Poisson Process with branches.



            Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
            $$N(t)=sum_j=1^nN_j(t).$$
            Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.



            If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.






            share|cite|improve this answer
























              up vote
              0
              down vote













              I'd say that you are looking for a Poisson Process with branches.



              Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
              $$N(t)=sum_j=1^nN_j(t).$$
              Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.



              If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.






              share|cite|improve this answer






















                up vote
                0
                down vote










                up vote
                0
                down vote









                I'd say that you are looking for a Poisson Process with branches.



                Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
                $$N(t)=sum_j=1^nN_j(t).$$
                Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.



                If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.






                share|cite|improve this answer












                I'd say that you are looking for a Poisson Process with branches.



                Consider for example a Poisson Process $N(t)_t$ having rate $lambda$, and suppose that each event is classified into either one of $n$ types. If you assume that the classification of an event is independent to the classification of any other event, an event is classified as a type $j$ with probability $p_j$ and $N_j(t)_t$ is the process that counts the number of type $j$ events, then
                $$N(t)=sum_j=1^nN_j(t).$$
                Moreover, it can be proved that $N_j(t)_t$ is also a Poisson Process whose rate is $lambda p_j$ and each $N_j(t)_t$ is independent of every $N_k(t)_t$ with $kneq j$.



                If you need not too rigurous book to read stuff like this you can check "Introduction to Probability Models" by Sheldon M. Ross.







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Aug 30 at 1:24









                Esteban Gutiérrez

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