Testing Self-Similarity Through Lamperti Transformations

Handle URI:
http://hdl.handle.net/10754/621509
Title:
Testing Self-Similarity Through Lamperti Transformations
Authors:
Lee, Myoungji; Genton, Marc G. ( 0000-0001-6467-2998 ) ; Jun, Mikyoung
Abstract:
Self-similar processes have been widely used in modeling real-world phenomena occurring in environmetrics, network traffic, image processing, and stock pricing, to name but a few. The estimation of the degree of self-similarity has been studied extensively, while statistical tests for self-similarity are scarce and limited to processes indexed in one dimension. This paper proposes a statistical hypothesis test procedure for self-similarity of a stochastic process indexed in one dimension and multi-self-similarity for a random field indexed in higher dimensions. If self-similarity is not rejected, our test provides a set of estimated self-similarity indexes. The key is to test stationarity of the inverse Lamperti transformations of the process. The inverse Lamperti transformation of a self-similar process is a strongly stationary process, revealing a theoretical connection between the two processes. To demonstrate the capability of our test, we test self-similarity of fractional Brownian motions and sheets, their time deformations and mixtures with Gaussian white noise, and the generalized Cauchy family. We also apply the self-similarity test to real data: annual minimum water levels of the Nile River, network traffic records, and surface heights of food wrappings. © 2016, International Biometric Society.
KAUST Department:
Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division
Citation:
Lee M, Genton MG, Jun M (2016) Testing Self-Similarity Through Lamperti Transformations. JABES 21: 426–447. Available: http://dx.doi.org/10.1007/s13253-016-0258-1.
Publisher:
Springer Nature
Journal:
Journal of Agricultural, Biological, and Environmental Statistics
Issue Date:
14-Jul-2016
DOI:
10.1007/s13253-016-0258-1
Type:
Article
ISSN:
1085-7117; 1537-2693
Sponsors:
This work was partially supported by NSF Grant DMS-1208421 and Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).
Appears in Collections:
Articles; Computer, Electrical and Mathematical Sciences and Engineering (CEMSE) Division

Full metadata record

DC FieldValue Language
dc.contributor.authorLee, Myoungjien
dc.contributor.authorGenton, Marc G.en
dc.contributor.authorJun, Mikyoungen
dc.date.accessioned2016-11-03T08:31:03Z-
dc.date.available2016-11-03T08:31:03Z-
dc.date.issued2016-07-14en
dc.identifier.citationLee M, Genton MG, Jun M (2016) Testing Self-Similarity Through Lamperti Transformations. JABES 21: 426–447. Available: http://dx.doi.org/10.1007/s13253-016-0258-1.en
dc.identifier.issn1085-7117en
dc.identifier.issn1537-2693en
dc.identifier.doi10.1007/s13253-016-0258-1en
dc.identifier.urihttp://hdl.handle.net/10754/621509-
dc.description.abstractSelf-similar processes have been widely used in modeling real-world phenomena occurring in environmetrics, network traffic, image processing, and stock pricing, to name but a few. The estimation of the degree of self-similarity has been studied extensively, while statistical tests for self-similarity are scarce and limited to processes indexed in one dimension. This paper proposes a statistical hypothesis test procedure for self-similarity of a stochastic process indexed in one dimension and multi-self-similarity for a random field indexed in higher dimensions. If self-similarity is not rejected, our test provides a set of estimated self-similarity indexes. The key is to test stationarity of the inverse Lamperti transformations of the process. The inverse Lamperti transformation of a self-similar process is a strongly stationary process, revealing a theoretical connection between the two processes. To demonstrate the capability of our test, we test self-similarity of fractional Brownian motions and sheets, their time deformations and mixtures with Gaussian white noise, and the generalized Cauchy family. We also apply the self-similarity test to real data: annual minimum water levels of the Nile River, network traffic records, and surface heights of food wrappings. © 2016, International Biometric Society.en
dc.description.sponsorshipThis work was partially supported by NSF Grant DMS-1208421 and Award No. KUS-C1-016-04 made by King Abdullah University of Science and Technology (KAUST).en
dc.publisherSpringer Natureen
dc.subjectFractional Brownian sheeten
dc.titleTesting Self-Similarity Through Lamperti Transformationsen
dc.typeArticleen
dc.contributor.departmentComputer, Electrical and Mathematical Sciences and Engineering (CEMSE) Divisionen
dc.identifier.journalJournal of Agricultural, Biological, and Environmental Statisticsen
dc.contributor.institutionInstitute for Applied Mathematics and Computational Science, Texas A&M University, College Station, TX, United Statesen
dc.contributor.institutionDepartment of Statistics, Texas A&M University, College Station, TX, United Statesen
kaust.authorGenton, Marc G.en
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