A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models
This paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by...
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| Format: | Article |
| Language: | English |
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MDPI AG
2020-03-01
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| Series: | Entropy |
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| Online Access: | https://www.mdpi.com/1099-4300/22/3/306 |
| _version_ | 1811305090051473408 |
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| author | Mingming Li Huafei Sun Didong Li |
| author_facet | Mingming Li Huafei Sun Didong Li |
| author_sort | Mingming Li |
| collection | DOAJ |
| description | This paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by providing various useful pre-derived expressions of some fundamental geometric quantities, such as Fisher-Riemannian metrics, <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-connections and <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-curvatures. Then, we proceed to consider some average methods based on these geometric structures. Specifically, we study the formulation and computation of the midpoint of two points and the Karcher mean of multiple points. In conclusion, we find some parallel results for the average problems on these two complementary models. |
| first_indexed | 2024-04-13T08:18:57Z |
| format | Article |
| id | doaj.art-6e26787e9d114870869b82232f6ab9d5 |
| institution | Directory Open Access Journal |
| issn | 1099-4300 |
| language | English |
| last_indexed | 2024-04-13T08:18:57Z |
| publishDate | 2020-03-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Entropy |
| spelling | doaj.art-6e26787e9d114870869b82232f6ab9d52022-12-22T02:54:42ZengMDPI AGEntropy1099-43002020-03-0122330610.3390/e22030306e22030306A Geometric Approach to Average Problems on Multinomial and Negative Multinomial ModelsMingming Li0Huafei Sun1Didong Li2School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, ChinaSchool of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, ChinaDepartment of Mathematics, Duke University, Durham, NC 27708, USAThis paper is concerned with the formulation and computation of average problems on the multinomial and negative multinomial models. It can be deduced that the multinomial and negative multinomial models admit complementary geometric structures. Firstly, we investigate these geometric structures by providing various useful pre-derived expressions of some fundamental geometric quantities, such as Fisher-Riemannian metrics, <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-connections and <inline-formula> <math display="inline"> <semantics> <mi>α</mi> </semantics> </math> </inline-formula>-curvatures. Then, we proceed to consider some average methods based on these geometric structures. Specifically, we study the formulation and computation of the midpoint of two points and the Karcher mean of multiple points. In conclusion, we find some parallel results for the average problems on these two complementary models.https://www.mdpi.com/1099-4300/22/3/306structure characterizationaverage problemgeometric midpointskarcher mean |
| spellingShingle | Mingming Li Huafei Sun Didong Li A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models Entropy structure characterization average problem geometric midpoints karcher mean |
| title | A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models |
| title_full | A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models |
| title_fullStr | A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models |
| title_full_unstemmed | A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models |
| title_short | A Geometric Approach to Average Problems on Multinomial and Negative Multinomial Models |
| title_sort | geometric approach to average problems on multinomial and negative multinomial models |
| topic | structure characterization average problem geometric midpoints karcher mean |
| url | https://www.mdpi.com/1099-4300/22/3/306 |
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