Local maxima of Gaussian fields View Full Text


Ontology type: schema:ScholarlyArticle      Open Access: True


Article Info

DATE

1972-12

AUTHORS

Georg Lindgren

ABSTRACT

The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]–[7], which include results for moderateu as well as foru→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rn}, with mean zero and the covariance functionr(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$ \end{document}WhereA(t) andb(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with heightu fort=0; see [4] for terminology.In Section 2 we examine the process ξu(t) asu→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial int1…,tn with random coefficients. This result is quite analogous with the one-dimensional case.In Section 3 we study the locations of the local minima of ξu(t) asu → ∞. In the non-isotropic caser(t) may have a local minimum at some pointt0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu neart0, and that τu-t0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case. More... »

PAGES

195-218

References to SciGraph publications

  • 1971-03. Extreme values of stationary normal processes in PROBABILITY THEORY AND RELATED FIELDS
  • 1972-12. Wave-length and amplitude for a stationary Gaussian process after a high maximum in PROBABILITY THEORY AND RELATED FIELDS
  • Identifiers

    URI

    http://scigraph.springernature.com/pub.10.1007/bf02384809

    DOI

    http://dx.doi.org/10.1007/bf02384809

    DIMENSIONS

    https://app.dimensions.ai/details/publication/pub.1035563631


    Indexing Status Check whether this publication has been indexed by Scopus and Web Of Science using the SN Indexing Status Tool
    Incoming Citations Browse incoming citations for this publication using opencitations.net

    JSON-LD is the canonical representation for SciGraph data.

    TIP: You can open this SciGraph record using an external JSON-LD service: JSON-LD Playground Google SDTT

    [
      {
        "@context": "https://springernature.github.io/scigraph/jsonld/sgcontext.json", 
        "about": [
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/01", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Mathematical Sciences", 
            "type": "DefinedTerm"
          }, 
          {
            "id": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/0101", 
            "inDefinedTermSet": "http://purl.org/au-research/vocabulary/anzsrc-for/2008/", 
            "name": "Pure Mathematics", 
            "type": "DefinedTerm"
          }
        ], 
        "author": [
          {
            "affiliation": {
              "alternateName": "Dept. of Mathematical Statistics, University of Lund, Box 725, S-22007, Lund 7, Sweden", 
              "id": "http://www.grid.ac/institutes/grid.4514.4", 
              "name": [
                "Dept. of Mathematical Statistics, University of Lund, Box 725, S-22007, Lund 7, Sweden"
              ], 
              "type": "Organization"
            }, 
            "familyName": "Lindgren", 
            "givenName": "Georg", 
            "id": "sg:person.012274436413.00", 
            "sameAs": [
              "https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012274436413.00"
            ], 
            "type": "Person"
          }
        ], 
        "citation": [
          {
            "id": "sg:pub.10.1007/bf00538473", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1039466392", 
              "https://doi.org/10.1007/bf00538473"
            ], 
            "type": "CreativeWork"
          }, 
          {
            "id": "sg:pub.10.1007/bf00532515", 
            "sameAs": [
              "https://app.dimensions.ai/details/publication/pub.1024636452", 
              "https://doi.org/10.1007/bf00532515"
            ], 
            "type": "CreativeWork"
          }
        ], 
        "datePublished": "1972-12", 
        "datePublishedReg": "1972-12-01", 
        "description": "The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]\u2013[7], which include results for moderateu as well as foru\u2192\u00b1\u221e. In this paper we generalize these results to a homogeneous Gaussian field {\u03be(t) t \u2208 Rn}, with mean zero and the covariance functionr(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.In Section 1 it is shown that if \u03be has a local maximum with heightu at0 then \u03be(t) can be expressed as\\documentclass[12pt]{minimal}\n\t\t\t\t\\usepackage{amsmath}\n\t\t\t\t\\usepackage{wasysym}\n\t\t\t\t\\usepackage{amsfonts}\n\t\t\t\t\\usepackage{amssymb}\n\t\t\t\t\\usepackage{amsbsy}\n\t\t\t\t\\usepackage{mathrsfs}\n\t\t\t\t\\usepackage{upgreek}\n\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\n\t\t\t\t\\begin{document}\n$$\\xi _u (t) = uA(t) - \\xi _u^\\prime  b(t) + \\Delta (t),$$\n\\end{document}WhereA(t) andb(t) are certain functions, \u03b8u is a random vector, and \u0394(t) is a non-homogeneous Gaussian field. Actually \u03beu(t) is the old process \u03be(t) conditioned in the horizontal window sense to have a local maximum with heightu fort=0; see [4] for terminology.In Section 2 we examine the process \u03beu(t) asu\u2192\u2212\u221e, and show that, after suitable normalizations, it tends to a fourth degree polynomial int1\u2026,tn with random coefficients. This result is quite analogous with the one-dimensional case.In Section 3 we study the locations of the local minima of \u03beu(t) asu \u2192 \u221e. In the non-isotropic caser(t) may have a local minimum at some pointt0. Then it is shown in 3.2 that \u03beu(t) will have a local minimum at some point \u03c4u neart0, and that \u03c4u-t0 after a normalization is asymptoticallyn-variate normal asu\u2192\u221e. This is in accordance with the one-dimensional case.", 
        "genre": "article", 
        "id": "sg:pub.10.1007/bf02384809", 
        "inLanguage": "en", 
        "isAccessibleForFree": true, 
        "isPartOf": [
          {
            "id": "sg:journal.1136676", 
            "issn": [
              "0004-2080", 
              "1871-2487"
            ], 
            "name": "Arkiv f\u00f6r matematik", 
            "publisher": "International Press of Boston", 
            "type": "Periodical"
          }, 
          {
            "issueNumber": "1-2", 
            "type": "PublicationIssue"
          }, 
          {
            "type": "PublicationVolume", 
            "volumeNumber": "10"
          }
        ], 
        "keywords": [
          "normalization", 
          "cases", 
          "results", 
          "different types", 
          "INT1", 
          "authors", 
          "higher maximum", 
          "types", 
          "function", 
          "minimum", 
          "process", 
          "maximum", 
          "Rn", 
          "covariance", 
          "certain functions", 
          "terminology", 
          "location", 
          "ASU", 
          "accordance", 
          "present authors", 
          "vector", 
          "structure", 
          "field", 
          "AT0", 
          "sense", 
          "coefficient", 
          "paper", 
          "Section 1", 
          "Section 2", 
          "Section 3", 
          "stationary Gaussian process", 
          "Gaussian process", 
          "local maxima", 
          "homogeneous Gaussian field", 
          "Gaussian field", 
          "random vectors", 
          "old process", 
          "suitable normalization", 
          "random coefficients", 
          "one-dimensional case", 
          "local minima", 
          "local structure", 
          "\u03b8u", 
          "prescribed heightu", 
          "heightu", 
          "moderateu", 
          "Nosko", 
          "heightu at0", 
          "non-homogeneous Gaussian field", 
          "horizontal window sense", 
          "window sense", 
          "fourth degree polynomial int1", 
          "degree polynomial int1", 
          "polynomial int1", 
          "pointt0", 
          "point \u03c4u neart0", 
          "\u03c4u neart0", 
          "neart0", 
          "\u03c4u-t0", 
          "asymptoticallyn-variate"
        ], 
        "name": "Local maxima of Gaussian fields", 
        "pagination": "195-218", 
        "productId": [
          {
            "name": "dimensions_id", 
            "type": "PropertyValue", 
            "value": [
              "pub.1035563631"
            ]
          }, 
          {
            "name": "doi", 
            "type": "PropertyValue", 
            "value": [
              "10.1007/bf02384809"
            ]
          }
        ], 
        "sameAs": [
          "https://doi.org/10.1007/bf02384809", 
          "https://app.dimensions.ai/details/publication/pub.1035563631"
        ], 
        "sdDataset": "articles", 
        "sdDatePublished": "2022-01-01T18:01", 
        "sdLicense": "https://scigraph.springernature.com/explorer/license/", 
        "sdPublisher": {
          "name": "Springer Nature - SN SciGraph project", 
          "type": "Organization"
        }, 
        "sdSource": "s3://com-springernature-scigraph/baseset/20220101/entities/gbq_results/article/article_145.jsonl", 
        "type": "ScholarlyArticle", 
        "url": "https://doi.org/10.1007/bf02384809"
      }
    ]
     

    Download the RDF metadata as:  json-ld nt turtle xml License info

    HOW TO GET THIS DATA PROGRAMMATICALLY:

    JSON-LD is a popular format for linked data which is fully compatible with JSON.

    curl -H 'Accept: application/ld+json' 'https://scigraph.springernature.com/pub.10.1007/bf02384809'

    N-Triples is a line-based linked data format ideal for batch operations.

    curl -H 'Accept: application/n-triples' 'https://scigraph.springernature.com/pub.10.1007/bf02384809'

    Turtle is a human-readable linked data format.

    curl -H 'Accept: text/turtle' 'https://scigraph.springernature.com/pub.10.1007/bf02384809'

    RDF/XML is a standard XML format for linked data.

    curl -H 'Accept: application/rdf+xml' 'https://scigraph.springernature.com/pub.10.1007/bf02384809'


     

    This table displays all metadata directly associated to this object as RDF triples.

    126 TRIPLES      22 PREDICATES      88 URIs      78 LITERALS      6 BLANK NODES

    Subject Predicate Object
    1 sg:pub.10.1007/bf02384809 schema:about anzsrc-for:01
    2 anzsrc-for:0101
    3 schema:author N0362c78ed1b045ad96ce33fd36640cf9
    4 schema:citation sg:pub.10.1007/bf00532515
    5 sg:pub.10.1007/bf00538473
    6 schema:datePublished 1972-12
    7 schema:datePublishedReg 1972-12-01
    8 schema:description The structure of a stationary Gaussian process near a local maximum with a prescribed heightu has been explored in several papers by the present author, see [5]–[7], which include results for moderateu as well as foru→±∞. In this paper we generalize these results to a homogeneous Gaussian field {ξ(t) t ∈ Rn}, with mean zero and the covariance functionr(t). The local structure of a Gaussian field near a high maximum has also been studied by Nosko, [8], [9], who obtains results of a slightly different type.In Section 1 it is shown that if ξ has a local maximum with heightu at0 then ξ(t) can be expressed as\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\xi _u (t) = uA(t) - \xi _u^\prime b(t) + \Delta (t),$$ \end{document}WhereA(t) andb(t) are certain functions, θu is a random vector, and Δ(t) is a non-homogeneous Gaussian field. Actually ξu(t) is the old process ξ(t) conditioned in the horizontal window sense to have a local maximum with heightu fort=0; see [4] for terminology.In Section 2 we examine the process ξu(t) asu→−∞, and show that, after suitable normalizations, it tends to a fourth degree polynomial int1…,tn with random coefficients. This result is quite analogous with the one-dimensional case.In Section 3 we study the locations of the local minima of ξu(t) asu → ∞. In the non-isotropic caser(t) may have a local minimum at some pointt0. Then it is shown in 3.2 that ξu(t) will have a local minimum at some point τu neart0, and that τu-t0 after a normalization is asymptoticallyn-variate normal asu→∞. This is in accordance with the one-dimensional case.
    9 schema:genre article
    10 schema:inLanguage en
    11 schema:isAccessibleForFree true
    12 schema:isPartOf N8fbb7432d8d64cd38c084f3bb0af1a67
    13 Nfd56cee092a44833a9f8b39b8ae355c4
    14 sg:journal.1136676
    15 schema:keywords ASU
    16 AT0
    17 Gaussian field
    18 Gaussian process
    19 INT1
    20 Nosko
    21 Rn
    22 Section 1
    23 Section 2
    24 Section 3
    25 accordance
    26 asymptoticallyn-variate
    27 authors
    28 cases
    29 certain functions
    30 coefficient
    31 covariance
    32 degree polynomial int1
    33 different types
    34 field
    35 fourth degree polynomial int1
    36 function
    37 heightu
    38 heightu at0
    39 higher maximum
    40 homogeneous Gaussian field
    41 horizontal window sense
    42 local maxima
    43 local minima
    44 local structure
    45 location
    46 maximum
    47 minimum
    48 moderateu
    49 neart0
    50 non-homogeneous Gaussian field
    51 normalization
    52 old process
    53 one-dimensional case
    54 paper
    55 point τu neart0
    56 pointt0
    57 polynomial int1
    58 prescribed heightu
    59 present authors
    60 process
    61 random coefficients
    62 random vectors
    63 results
    64 sense
    65 stationary Gaussian process
    66 structure
    67 suitable normalization
    68 terminology
    69 types
    70 vector
    71 window sense
    72 θu
    73 τu neart0
    74 τu-t0
    75 schema:name Local maxima of Gaussian fields
    76 schema:pagination 195-218
    77 schema:productId N503ed5fa68324253bf5b72fe352ccb32
    78 Na8339fa341d8465ab819686329d8354b
    79 schema:sameAs https://app.dimensions.ai/details/publication/pub.1035563631
    80 https://doi.org/10.1007/bf02384809
    81 schema:sdDatePublished 2022-01-01T18:01
    82 schema:sdLicense https://scigraph.springernature.com/explorer/license/
    83 schema:sdPublisher Na3e465b402c04e52b5f8f91afeadaa60
    84 schema:url https://doi.org/10.1007/bf02384809
    85 sgo:license sg:explorer/license/
    86 sgo:sdDataset articles
    87 rdf:type schema:ScholarlyArticle
    88 N0362c78ed1b045ad96ce33fd36640cf9 rdf:first sg:person.012274436413.00
    89 rdf:rest rdf:nil
    90 N503ed5fa68324253bf5b72fe352ccb32 schema:name dimensions_id
    91 schema:value pub.1035563631
    92 rdf:type schema:PropertyValue
    93 N8fbb7432d8d64cd38c084f3bb0af1a67 schema:issueNumber 1-2
    94 rdf:type schema:PublicationIssue
    95 Na3e465b402c04e52b5f8f91afeadaa60 schema:name Springer Nature - SN SciGraph project
    96 rdf:type schema:Organization
    97 Na8339fa341d8465ab819686329d8354b schema:name doi
    98 schema:value 10.1007/bf02384809
    99 rdf:type schema:PropertyValue
    100 Nfd56cee092a44833a9f8b39b8ae355c4 schema:volumeNumber 10
    101 rdf:type schema:PublicationVolume
    102 anzsrc-for:01 schema:inDefinedTermSet anzsrc-for:
    103 schema:name Mathematical Sciences
    104 rdf:type schema:DefinedTerm
    105 anzsrc-for:0101 schema:inDefinedTermSet anzsrc-for:
    106 schema:name Pure Mathematics
    107 rdf:type schema:DefinedTerm
    108 sg:journal.1136676 schema:issn 0004-2080
    109 1871-2487
    110 schema:name Arkiv för matematik
    111 schema:publisher International Press of Boston
    112 rdf:type schema:Periodical
    113 sg:person.012274436413.00 schema:affiliation grid-institutes:grid.4514.4
    114 schema:familyName Lindgren
    115 schema:givenName Georg
    116 schema:sameAs https://app.dimensions.ai/discover/publication?and_facet_researcher=ur.012274436413.00
    117 rdf:type schema:Person
    118 sg:pub.10.1007/bf00532515 schema:sameAs https://app.dimensions.ai/details/publication/pub.1024636452
    119 https://doi.org/10.1007/bf00532515
    120 rdf:type schema:CreativeWork
    121 sg:pub.10.1007/bf00538473 schema:sameAs https://app.dimensions.ai/details/publication/pub.1039466392
    122 https://doi.org/10.1007/bf00538473
    123 rdf:type schema:CreativeWork
    124 grid-institutes:grid.4514.4 schema:alternateName Dept. of Mathematical Statistics, University of Lund, Box 725, S-22007, Lund 7, Sweden
    125 schema:name Dept. of Mathematical Statistics, University of Lund, Box 725, S-22007, Lund 7, Sweden
    126 rdf:type schema:Organization
     




    Preview window. Press ESC to close (or click here)


    ...