Poisson point process application and terminology
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I am trying to understand Poisson process that is used in a derivation of maximum likelihood estimation of intrinsic dimension given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=maximum_likelihood_estimation_of_intrinsic_dimension
In this work, the authors consider a Poisson point process distribution $f(x)$ and assume observations as a homogeneous Poisson process in a small sphere of radius $R$. I am unable to follow a few terms and will be grateful if somebody could help explaining these concerns:
Question 1: Is $theta$ (used in the first wiki link) or $lambda$ (used in the second wiki link) the intensity of the point process?
Question 2: The authors assume that the density $f(x)$ is constant and $theta = log f(x)$ as constant. Why is there a log and a constant assumption?
Is this a property of a homogeneous Poisson process? I am unsure if constant assumption means that it is a homogeneous Poisson process in the derivation.
What does $lambda = log_e f(x)$ = constant assumption imply and is it a property of a homogeneous point process?
statistics probability math
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SKMSKM
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I am trying to understand Poisson process that is used in a derivation of maximum likelihood estimation of intrinsic dimension given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=maximum_likelihood_estimation_of_intrinsic_dimension
In this work, the authors consider a Poisson point process distribution $f(x)$ and assume observations as a homogeneous Poisson process in a small sphere of radius $R$. I am unable to follow a few terms and will be grateful if somebody could help explaining these concerns:
Question 1: Is $theta$ (used in the first wiki link) or $lambda$ (used in the second wiki link) the intensity of the point process?
Question 2: The authors assume that the density $f(x)$ is constant and $theta = log f(x)$ as constant. Why is there a log and a constant assumption?
Is this a property of a homogeneous Poisson process? I am unsure if constant assumption means that it is a homogeneous Poisson process in the derivation.
What does $lambda = log_e f(x)$ = constant assumption imply and is it a property of a homogeneous point process?
statistics probability math
asked 2 days ago
SKMSKM
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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add a comment |
$begingroup$
I am trying to understand Poisson process that is used in a derivation of maximum likelihood estimation of intrinsic dimension given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=maximum_likelihood_estimation_of_intrinsic_dimension
In this work, the authors consider a Poisson point process distribution $f(x)$ and assume observations as a homogeneous Poisson process in a small sphere of radius $R$. I am unable to follow a few terms and will be grateful if somebody could help explaining these concerns:
Question 1: Is $theta$ (used in the first wiki link) or $lambda$ (used in the second wiki link) the intensity of the point process?
Question 2: The authors assume that the density $f(x)$ is constant and $theta = log f(x)$ as constant. Why is there a log and a constant assumption?
Is this a property of a homogeneous Poisson process? I am unsure if constant assumption means that it is a homogeneous Poisson process in the derivation.
What does $lambda = log_e f(x)$ = constant assumption imply and is it a property of a homogeneous point process?
statistics probability math
asked 2 days ago
SKMSKM
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
$endgroup$
I am trying to understand Poisson process that is used in a derivation of maximum likelihood estimation of intrinsic dimension given here https://wiki.math.uwaterloo.ca/statwiki/index.php?title=maximum_likelihood_estimation_of_intrinsic_dimension
In this work, the authors consider a Poisson point process distribution $f(x)$ and assume observations as a homogeneous Poisson process in a small sphere of radius $R$. I am unable to follow a few terms and will be grateful if somebody could help explaining these concerns:
Question 1: Is $theta$ (used in the first wiki link) or $lambda$ (used in the second wiki link) the intensity of the point process?
Question 2: The authors assume that the density $f(x)$ is constant and $theta = log f(x)$ as constant. Why is there a log and a constant assumption?
Is this a property of a homogeneous Poisson process? I am unsure if constant assumption means that it is a homogeneous Poisson process in the derivation.
What does $lambda = log_e f(x)$ = constant assumption imply and is it a property of a homogeneous point process?
statistics probability math
statistics probability math
asked 2 days ago
SKMSKM
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
SKMSKM
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
SKMSKM
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 2 days ago
SKMSKM
11
asked 2 days ago
SKMSKM
11
11
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
SKM is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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