What is the natural extension of Poisson Process to two macroscopic variables
Clash Royale CLAN TAG#URR8PPP
up vote
2
down vote
favorite
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.
stochastic-processes poisson-process
add a comment |Â
up vote
2
down vote
favorite
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.
stochastic-processes poisson-process
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
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.
stochastic-processes poisson-process
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.
stochastic-processes poisson-process
edited Aug 28 at 7:54
asked Aug 27 at 16:23
Aleksejs Fomins
373111
373111
add a comment |Â
add a comment |Â
1 Answer
1
active
oldest
votes
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.
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
add a comment |Â
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.
add a comment |Â
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.
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.
answered Aug 30 at 1:24
Esteban Gutiérrez
13810
13810
add a comment |Â
add a comment |Â
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2896372%2fwhat-is-the-natural-extension-of-poisson-process-to-two-macroscopic-variables%23new-answer', 'question_page');
);
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Sign up or log in
StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password