Pub Date : 2020-09-27DOI: 10.2140/astat.2021.12.139
William Q. Erickson
The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on the set $[n]={1,dots,n}$. We present the details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in the special case of three distributions, the EMD is half the sum of the pairwise EMD's. Extending the methods of Bourn and Willenbring (arXiv:1903.03673), we compute the expected value of this generalized EMD on random $d$-tuples of distributions, using a generating function which coincides with the Hilbert series of the Segre embedding. We then use the EMD to analyze a real-world data set of grade distributions.
{"title":"A generalization for the expected value of the\u0000earth mover’s distance","authors":"William Q. Erickson","doi":"10.2140/astat.2021.12.139","DOIUrl":"https://doi.org/10.2140/astat.2021.12.139","url":null,"abstract":"The earth mover's distance (EMD), also called the first Wasserstein distance, can be naturally extended to compare arbitrarily many probability distributions, rather than only two, on the set $[n]={1,dots,n}$. We present the details for this generalization, along with a highly efficient algorithm inspired by combinatorics; it turns out that in the special case of three distributions, the EMD is half the sum of the pairwise EMD's. Extending the methods of Bourn and Willenbring (arXiv:1903.03673), we compute the expected value of this generalized EMD on random $d$-tuples of distributions, using a generating function which coincides with the Hilbert series of the Segre embedding. We then use the EMD to analyze a real-world data set of grade distributions.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-09-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77389319","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-07-02DOI: 10.2140/astat.2021.12.125
Elina Robeva, M. Sun
Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2times n$ grid configurations in the plane. Furthermore, we connect the former to the large Schroder numbers. We also show that the numbers of bimonotone and total subdivisions of a $2times n$ grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any $m times n$ grid. Finally, we prove that all bimonotone triangulations of an $m times n$ grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an $mtimes n$ grid.
{"title":"Bimonotone subdivisions of point configurations in the plane","authors":"Elina Robeva, M. Sun","doi":"10.2140/astat.2021.12.125","DOIUrl":"https://doi.org/10.2140/astat.2021.12.125","url":null,"abstract":"Bimonotone subdivisions in two dimensions are subdivisions all of whose sides are either vertical or have nonnegative slope. They correspond to statistical estimates of probability distributions of strongly positively dependent random variables. The number of bimonotone subdivisions compared to the total number of subdivisions of a point configuration provides insight into how often the random variables are positively dependent. We give recursions as well as formulas for the numbers of bimonotone and total subdivisions of $2times n$ grid configurations in the plane. Furthermore, we connect the former to the large Schroder numbers. We also show that the numbers of bimonotone and total subdivisions of a $2times n$ grid are asymptotically equal. We then provide algorithms for counting bimonotone subdivisions for any $m times n$ grid. Finally, we prove that all bimonotone triangulations of an $m times n$ grid are connected by flips. This gives rise to an algorithm for counting the number of bimonotone (and total) triangulations of an $mtimes n$ grid.","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77060549","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Comparative Study of Deep Learning Models for Natural Language Processing (NLP)","authors":"","doi":"10.52783/jas.v11i1.1432","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1432","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70995800","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"An Intelligent System for Detection of Mental Stress Using Machine Learning","authors":"","doi":"10.52783/jas.v11i1.1438","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1438","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"AI-based Natural Language Processing (NLP) Systems","authors":"","doi":"10.52783/jas.v11i1.1431","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1431","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70995912","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Deep Learning-based Approach for Medical Image Analysis and Diagnosis","authors":"","doi":"10.52783/jas.v11i1.1436","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1436","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Automated Process of Fruit Recognition Using Computer Vision","authors":"","doi":"10.52783/jas.v11i1.1439","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1439","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996373","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Cloud-based IIoT System for Quality Control in Manufacturing: Techniques and Applications","authors":"","doi":"10.52783/jas.v11i1.1433","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1433","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70996361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"A Hybrid IIoT-based System for Real-time Monitoring and Control of Industrial Processes","authors":"","doi":"10.52783/jas.v11i1.1434","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1434","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70995943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Anomaly Detection and Diagnosis in IIoT Systems: A Review of Techniques and Applications","authors":"","doi":"10.52783/jas.v11i1.1435","DOIUrl":"https://doi.org/10.52783/jas.v11i1.1435","url":null,"abstract":"","PeriodicalId":41066,"journal":{"name":"Journal of Algebraic Statistics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70995993","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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