Generalized Theorems for Nonlinear State Space Reconstruction
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{"title"=>"Generalized theorems for nonlinear state space reconstruction", "type"=>"journal", "authors"=>[{"first_name"=>"Ethan R.", "last_name"=>"Deyle", "scopus_author_id"=>"37070429000"}, {"first_name"=>"George", "last_name"=>"Sugihara", "scopus_author_id"=>"7005912152"}], "year"=>2011, "source"=>"PLoS ONE", "identifiers"=>{"doi"=>"10.1371/journal.pone.0018295", "sgr"=>"79953660291", "issn"=>"19326203", "pui"=>"361550142", "isbn"=>"10.1371/journal.pone.0018295", "pmid"=>"21483839", "scopus"=>"2-s2.0-79953660291"}, "id"=>"91da0ae5-62af-3095-9776-e964a19005d4", "abstract"=>"Takens' theorem (1981) shows how lagged variables of a single time series can be used as proxy variables to reconstruct an attractor for an underlying dynamic process. State space reconstruction (SSR) from single time series has been a powerful approach for the analysis of the complex, non-linear systems that appear ubiquitous in the natural and human world. The main shortcoming of these methods is the phenomenological nature of attractor reconstructions. Moreover, applied studies show that these single time series reconstructions can often be improved ad hoc by including multiple dynamically coupled time series in the reconstructions, to provide a more mechanistic model. Here we provide three analytical proofs that add to the growing literature to generalize Takens' work and that demonstrate how multiple time series can be used in attractor reconstructions. These expanded results (Takens' theorem is a special case) apply to a wide variety of natural systems having parallel time series observations for variables believed to be related to the same dynamic manifold. The potential information leverage provided by multiple embeddings created from different combinations of variables (and their lags) can pave the way for new applied techniques to exploit the time-limited, but parallel observations of natural systems, such as coupled ecological systems, geophysical systems, and financial systems. This paper aims to justify and help open this potential growth area for SSR applications in the natural sciences.", "link"=>"http://www.mendeley.com/research/generalized-theorems-nonlinear-state-space-reconstruction-3", "reader_count"=>158, "reader_count_by_academic_status"=>{"Unspecified"=>3, "Professor > Associate Professor"=>11, "Researcher"=>33, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>59, "Student > Postgraduate"=>1, "Student > Master"=>19, "Other"=>6, "Student > Bachelor"=>11, "Lecturer"=>4, "Professor"=>5}, "reader_count_by_user_role"=>{"Unspecified"=>3, "Professor > Associate Professor"=>11, "Researcher"=>33, "Student > Doctoral Student"=>6, "Student > Ph. D. Student"=>59, "Student > Postgraduate"=>1, "Student > Master"=>19, "Other"=>6, "Student > Bachelor"=>11, "Lecturer"=>4, "Professor"=>5}, "reader_count_by_subject_area"=>{"Unspecified"=>10, "Agricultural and Biological Sciences"=>27, "Business, Management and Accounting"=>2, "Veterinary Science and Veterinary Medicine"=>1, "Computer Science"=>10, "Earth and Planetary Sciences"=>12, "Economics, Econometrics and Finance"=>2, "Engineering"=>24, "Environmental Science"=>26, "Biochemistry, Genetics and Molecular Biology"=>1, "Mathematics"=>12, "Medicine and Dentistry"=>7, "Neuroscience"=>9, "Physics and Astronomy"=>12, "Psychology"=>2, "Social Sciences"=>1}, "reader_count_by_subdiscipline"=>{"Medicine and Dentistry"=>{"Medicine and Dentistry"=>7}, "Social Sciences"=>{"Social Sciences"=>1}, "Physics and Astronomy"=>{"Physics and Astronomy"=>12}, "Psychology"=>{"Psychology"=>2}, "Mathematics"=>{"Mathematics"=>12}, "Unspecified"=>{"Unspecified"=>10}, "Environmental Science"=>{"Environmental Science"=>26}, "Engineering"=>{"Engineering"=>24}, "Neuroscience"=>{"Neuroscience"=>9}, "Earth and Planetary Sciences"=>{"Earth and Planetary Sciences"=>12}, "Economics, Econometrics and Finance"=>{"Economics, Econometrics and Finance"=>2}, "Agricultural and Biological Sciences"=>{"Agricultural and Biological Sciences"=>27}, "Computer Science"=>{"Computer Science"=>10}, "Business, Management and Accounting"=>{"Business, Management and Accounting"=>2}, "Biochemistry, Genetics and Molecular Biology"=>{"Biochemistry, Genetics and Molecular Biology"=>1}, "Veterinary Science and Veterinary Medicine"=>{"Veterinary Science and Veterinary Medicine"=>1}}, "reader_count_by_country"=>{"Colombia"=>2, "Argentina"=>1, "United States"=>13, "United Kingdom"=>3, "Switzerland"=>2, "Cuba"=>1, "Canada"=>2, "Czech Republic"=>1, "Netherlands"=>2, "Belgium"=>1, "Luxembourg"=>1, "Taiwan"=>2, "Poland"=>1, "Brazil"=>1, "Germany"=>2}, "group_count"=>3}

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Figshare

  • {"files"=>["https://ndownloader.figshare.com/files/787281"], "description"=>"<p>The Lorenz attractor <a href=\"http://www.plosone.org/article/info:doi/10.1371/journal.pone.0018295#pone.0018295-Lorenz1\" target=\"_blank\">[37]</a> is shown with three shadow manifolds created from lag-coordinate transformations. The typical parameters were used: , , and , giving the three coupled equations as , , and . The solution was computed using a fourth order Runge-Kutta method with a time step of , and the time lag used to create the shadow manifolds was . (A) The trajectory shown in the , , and coordinates of the original system reveals a two-lobed manifold. (B) A univariate transformation using time lags of the -coordinate, , preserves this two-lobed structure (and other topological properties), verifying Takens' theorem. (C) A univariate transformation using time lags of the -coordinate, , does not preserve the two-lobed structure. Local neighborhoods of the original attractor are, however, preserved. Thus, though this mapping violates a genericity assumption of the original theorem and is not an embedding, it is an immersion of the original manifold. (D) A multivariate transformation using both the - and -coordinates, , fulfills the assumptions of Theorems 2 and 7. As predicted, it also preserves the two-lobed structure of the Lorenz and is a valid embedding.</p>", "links"=>[], "tags"=>["attractor"], "article_id"=>457657, "categories"=>["Biological Sciences", "Mathematics", "Ecology", "Genetics"], "users"=>["Ethan R. Deyle", "George Sugihara"], "doi"=>"https://dx.doi.org/10.1371/journal.pone.0018295.g001", "stats"=>{"downloads"=>6, "page_views"=>40, "likes"=>0}, "figshare_url"=>"https://figshare.com/articles/_Lorenz_attractor_with_three_shadow_manifolds_/457657", "title"=>"Lorenz attractor with three shadow manifolds.", "pos_in_sequence"=>0, "defined_type"=>1, "published_date"=>"2013-02-20 19:32:15"}

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Relative Metric

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