Positive definite functions: from Schoenberg to space-time challenges
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| Main Author: | |
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| Format: | Book |
| Language: | English |
| Published: |
Castelló de la Plana
Departament of Mathematics, University Jaume I
[2008]
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 198 Seiten 21 cm |
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Record in the Search Index
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| adam_text | Contents 1 Introduction to Spatial Statistics 1.1 Space-time geostatistics: a brief theoretical setup .... I The Language of Positive Definite Functions 1 7 13 2 Stieltjes-Pick-Bernstein-Schoenberg and their Connection to Complete Monotonicity 15 2.1 Positive definite and completely monotonic functions . . 16 2.2 Conditionally negative definite functions........................ 20 2.3 Stieltjes functions................................................................ 23 2.4 Pick functions...................................................................... 26 2.5 Bernstein functions............................................................ 32 2.6 Logarithmically completely monotonic functions........... 35 2.7 Some completely monotonic functions related to the Gamma function................................................................................ 38 3 On the Extension Problem for Covariance Functions 3.1 Covariance functions and variograms.............................. 3.2 Positive definiteness and integral representations........... 3.3 Growth ............................................................................... 3.4 The extension problem...................................................... 3.5 Sufficient conditions for admissibility.............................. 3.6 Description of all extensions............................................. 47 48 49 51 53 54 58 4 Problems Related to Positive Definite Functions 4.1 General properties of positive definite functions........... 63 64 i
4.2 4.3 4.4 4.5 4.6 4.7 4.8 II Schoenberg’s problem and the positive definiteness of the function (1 — ρλ(τ))+.................................................. 72 The structure of solutions of Problem 1............................ 76 A criterion of positive definiteness in terms of completely monotone functions..................................................... 79 Sufficient conditions for ƒ Є Φ(Rn, p)................................ 84 Positive definiteness of the function cos βρ{χ)............................................... 85 Radial functions with compactsupport.............................. 89 Some unsolved problems..................................................... 106 Some Topics in Space-Time Statistics 5 Spatiotemporal Statistics and Geostatistics 5.1 Introduction........................................................................... 5.2 The spatiotemporal continuum............................................ 5.2.1 Separable metric structures .................................. 5.2.2 Composite metric structures.................................. 5.2.3 Fractal metric structures........................................ 5.3 Spatiotemporal random field theory................................... 5.3.1 Spatiotemporal dependence functions................... 5.3.2 Model permissibility............................................... 5.3.3 Generalized random fields...................................... 5.4 Techniques............................................................................. 5.4.1 Knowledge bases.....................................................
5.4.2 Fundamental equations............................................ 5.4.3 Implementation of spatiotemporal statistics: SEKDGUI software library.............................................. 5.5 Applications.......................................................................... 115 117 118 119 120 121 122 123 124 129 130 131 133 133 137 138 6 Quasi Arithmetic Covariance Functions and their Possi ble Reducibility through Functional Equations 155 6.1 Introduction......................................................................... 156 6.2 The Perrin-Senoussi problem............................................. 158 6.3 Quasi-arithmetic covariance functions.............................. 159 6.3.1 Quasi-arithmetic and increasing means................ 159 ii
6.4 6.5 6.6 6.7 6.3.2 Quasi-arithmetic compositions............................... 161 6.3.3 Permissibility and quasi-arithmeticity................... 162 A new reducibility problem connected to quasi-arithmeticity 166 Theoretical results related to increasing mean-type func tional equations........................................................... 168 6.5.1 Л/ is weighted quasi-arithmetic............................ 170 6.5.2 Μ is the logarithmic mean..................................... 173 Consequences for irreducible correlation functions .... 174 Conclusions................................................................ 175 7 Covariance Tapering in Spatial Statistics 7.1 7.2 7.3 7.4 7.5 181 Introduction................................................................ 182 Equivalent probability measures under covariance tapering 183 Tail properties of tapering spectral density............ 186 Asymptotic properties of estimation under tapering . . . 191 Covariance tapering for multivariate processes..... 195 iii
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| adam_txt |
Contents 1 Introduction to Spatial Statistics 1.1 Space-time geostatistics: a brief theoretical setup . I The Language of Positive Definite Functions 1 7 13 2 Stieltjes-Pick-Bernstein-Schoenberg and their Connection to Complete Monotonicity 15 2.1 Positive definite and completely monotonic functions . . 16 2.2 Conditionally negative definite functions. 20 2.3 Stieltjes functions. 23 2.4 Pick functions. 26 2.5 Bernstein functions. 32 2.6 Logarithmically completely monotonic functions. 35 2.7 Some completely monotonic functions related to the Gamma function. 38 3 On the Extension Problem for Covariance Functions 3.1 Covariance functions and variograms. 3.2 Positive definiteness and integral representations. 3.3 Growth . 3.4 The extension problem. 3.5 Sufficient conditions for admissibility. 3.6 Description of all extensions. 47 48 49 51 53 54 58 4 Problems Related to Positive Definite Functions 4.1 General properties of positive definite functions. 63 64 i
4.2 4.3 4.4 4.5 4.6 4.7 4.8 II Schoenberg’s problem and the positive definiteness of the function (1 — ρλ(τ))+. 72 The structure of solutions of Problem 1. 76 A criterion of positive definiteness in terms of completely monotone functions. 79 Sufficient conditions for ƒ Є Φ(Rn, p). 84 Positive definiteness of the function cos βρ{χ). 85 Radial functions with compactsupport. 89 Some unsolved problems. 106 Some Topics in Space-Time Statistics 5 Spatiotemporal Statistics and Geostatistics 5.1 Introduction. 5.2 The spatiotemporal continuum. 5.2.1 Separable metric structures . 5.2.2 Composite metric structures. 5.2.3 Fractal metric structures. 5.3 Spatiotemporal random field theory. 5.3.1 Spatiotemporal dependence functions. 5.3.2 Model permissibility. 5.3.3 Generalized random fields. 5.4 Techniques. 5.4.1 Knowledge bases.
5.4.2 Fundamental equations. 5.4.3 Implementation of spatiotemporal statistics: SEKDGUI software library. 5.5 Applications. 115 117 118 119 120 121 122 123 124 129 130 131 133 133 137 138 6 Quasi Arithmetic Covariance Functions and their Possi ble Reducibility through Functional Equations 155 6.1 Introduction. 156 6.2 The Perrin-Senoussi problem. 158 6.3 Quasi-arithmetic covariance functions. 159 6.3.1 Quasi-arithmetic and increasing means. 159 ii
6.4 6.5 6.6 6.7 6.3.2 Quasi-arithmetic compositions. 161 6.3.3 Permissibility and quasi-arithmeticity. 162 A new reducibility problem connected to quasi-arithmeticity 166 Theoretical results related to increasing mean-type func tional equations. 168 6.5.1 Л/ is weighted quasi-arithmetic. 170 6.5.2 Μ is the logarithmic mean. 173 Consequences for irreducible correlation functions . 174 Conclusions. 175 7 Covariance Tapering in Spatial Statistics 7.1 7.2 7.3 7.4 7.5 181 Introduction. 182 Equivalent probability measures under covariance tapering 183 Tail properties of tapering spectral density. 186 Asymptotic properties of estimation under tapering . . . 191 Covariance tapering for multivariate processes. 195 iii |
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| index_date | 2024-07-03T21:00:30Z |
| indexdate | 2024-07-10T09:41:34Z |
| institution | BVB |
| language | English |
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| physical | XIV, 198 Seiten 21 cm |
| publishDate | 2008 |
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| publisher | Departament of Mathematics, University Jaume I |
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| spelling | Mateu, Jorge Verfasser (DE-588)111636865X aut Positive definite functions from Schoenberg to space-time challenges Jorge Mateu, Emilio Porcu Castelló de la Plana Departament of Mathematics, University Jaume I [2008] XIV, 198 Seiten 21 cm txt rdacontent n rdamedia nc rdacarrier Funciones definidas positivas embne Análisis espacial (Estadística) embne Positiv-definite Funktion (DE-588)4175422-0 gnd rswk-swf Räumliche Statistik (DE-588)4386767-4 gnd rswk-swf Positiv-definite Funktion (DE-588)4175422-0 s Räumliche Statistik (DE-588)4386767-4 s DE-604 Porcu, Emilio ca. 20./21. Jh. Sonstige (DE-588)1274384788 oth Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=033939134&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Mateu, Jorge Positive definite functions from Schoenberg to space-time challenges Funciones definidas positivas embne Análisis espacial (Estadística) embne Positiv-definite Funktion (DE-588)4175422-0 gnd Räumliche Statistik (DE-588)4386767-4 gnd |
| subject_GND | (DE-588)4175422-0 (DE-588)4386767-4 |
| title | Positive definite functions from Schoenberg to space-time challenges |
| title_auth | Positive definite functions from Schoenberg to space-time challenges |
| title_exact_search | Positive definite functions from Schoenberg to space-time challenges |
| title_exact_search_txtP | Positive definite functions from Schoenberg to space-time challenges |
| title_full | Positive definite functions from Schoenberg to space-time challenges Jorge Mateu, Emilio Porcu |
| title_fullStr | Positive definite functions from Schoenberg to space-time challenges Jorge Mateu, Emilio Porcu |
| title_full_unstemmed | Positive definite functions from Schoenberg to space-time challenges Jorge Mateu, Emilio Porcu |
| title_short | Positive definite functions |
| title_sort | positive definite functions from schoenberg to space time challenges |
| title_sub | from Schoenberg to space-time challenges |
| topic | Funciones definidas positivas embne Análisis espacial (Estadística) embne Positiv-definite Funktion (DE-588)4175422-0 gnd Räumliche Statistik (DE-588)4386767-4 gnd |
| topic_facet | Funciones definidas positivas Análisis espacial (Estadística) Positiv-definite Funktion Räumliche Statistik |
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