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Background Analysis R Shiny
COM-Poisson (CMP): Introduction Multivariate CMP Discussion
Multivariate COM-Poisson: A Flexible Count Distribution that - - PowerPoint PPT Presentation
Multivariate COM-Poisson: A Flexible Count Distribution that - - PowerPoint PPT Presentation
Multivariate COM-Poisson: A Flexible Count Distribution that Accommodates Data Dispersion Yixuan (Sherry) Wu Georgetown University Mentor: Prof. Kimberly Sellers COM-Poisson Multivariate (CMP): Discussion CMP Introduction Background
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Background Analysis R Shiny
Available Count Models:
- Poisson Distribution
- PMF for random variable Y:
P(Y = y) = ππ§πβπ
π§!
, π§ = 0, 1, 2, β―
- Assumption:
- Ξ» = mean = variance β equi-dispersion
- Multivariate analog of this distribution exists
https://en.wikipedia.org/wiki/Poisson_d istribution#/media/File:Poisson_pmf.svg
Background & Motivation
y
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Background & Motivation
Available Count Models:
- Negative Binomial Distribution
- Accounts for over-dispersion, but not under-dispersion
- Generalized Poisson Distribution
- Not good when huge amount of under-dispersion
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Background & Motivation
Available Count Models:
- Negative Binomial Distribution
- Accounts for over-dispersion, but not under-dispersion
- Generalized Poisson Distribution
- Not good when huge amount of under-dispersion
Challenges:
- 1. Accounting for under-dispersion
- 2. Uncertain Dispersion
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Background Analysis R Shiny
COM-Poisson Distribution
(Conway & Maxwell, 1962; Shmueli et al., 2005) COM-Poisson distribution PMF for random variable Y:
π π = π§ =
ππ§ π§! ππ(π,π) , π§ = 0, 1, 2, β―
where π π, π = ΰ·
π=0 β ππ (π!)π ; π β₯ 0; π = πΉ(ππ)
Captures both over-dispersion and under-dispersion
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Background Analysis R Shiny
COM-Poisson Distribution
(Conway & Maxwell, 1962; Shmueli et al., 2005) COM-Poisson distribution PMF for random variable Y:
π π = π§ =
ππ§ π§! ππ(π,π) , π§ = 0, 1, 2, β―
where π π, π = ΰ·
π=0 β ππ (π!)π ; π β₯ 0; π = πΉ(ππ)
Special Cases: π = 1 β π π, π = ππ β π π = π§ =
ππ§ πβπ π§!
βΌ ππππ‘π‘ππ π π = 0, π < 1 β π π, π =
1 1βπ
β π π = π§ = ππ§(1 β π) βΌ π»πππππ’π ππ (π = 1 β π) π β β β π π, π = 1 + π β π π = π§ =
π 1+π βΌ πΆππ πππ£πππ (π = π 1+π)
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Background Analysis R Shiny
COM-Poisson Distribution Properties
Probability Generating Function (PGF):
Ξ π’β = πΉ π’π =
π ππ’, π π π, π
Moment Generating Function (MGF):
ππ π’β = πΉ etY =
π πππ’, π π(π, π)
Expected Value:
E π = π
π log π(π,π) ππ
Variance: Var π =
ππΉ(π) π log π
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Let π βΌ CMP(π, π) and (π1, β― , ππ|π) have conditional PGF
Multivariate COM-Poisson Distribution
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Let π βΌ CMP(π, π) and (π1, β― , ππ|π) have conditional PGF Using compounding technique, unconditional PGF:
Multivariate COM-Poisson Distribution
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Tri-variate Case
- Probability Generating Function:
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- The current challenge with computing PMF:
- Tri-variate PMF
βComputational Challenge to estimate parameters βQuestions?
Next Step & Challenges
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Trivariate Case - PGF
d_1 d_3 d_2
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Compounding Method:
For Derivation of Multivariate Poisson Distribution
Let π βΌ Poisson(π) and (π1, β― , ππ|π) have conditional PGF
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Compounding Method:
For Derivation of Multivariate Poisson Distribution
Let π βΌ Poisson(π) and (π1, β― , ππ|π) have conditional PGF
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Background Analysis R Shiny