메뉴 건너뛰기




Volumn 2019, Issue , 2011, Pages 1-136

Topological complexity of smooth random functions: École d'Été de Probabilités de Saint-Flour XXXIX - 2009

Author keywords

[No Author keywords available]

Indexed keywords


EID: 79957745768     PISSN: 00758434     EISSN: None     Source Type: Book Series    
DOI: 10.1007/978-3-642-19580-8_1     Document Type: Article
Times cited : (11)

References (94)
  • 1
    • 23044534453 scopus 로고    scopus 로고
    • Finding topology in a factory: Configuration spaces
    • A. Abrams and R. Ghrist. Finding topology in a factory: configuration spaces. Amer. Math. Monthly, 109(2):140-150, 2002.
    • (2002) Amer. Math. Monthly , vol.109 , Issue.2 , pp. 140-150
    • Abrams, A.1    Ghrist, R.2
  • 2
    • 0003529366 scopus 로고
    • John Wiley & Sons Ltd., Chichester Reprinted in 2010 by SIAM, Classics of Applied Mathematics Series
    • R.J. Adler. The Geometry of Random Fields. John Wiley & Sons Ltd., Chichester, 1981. Reprinted in 2010 by SIAM, Classics of Applied Mathematics Series.
    • (1981) The Geometry of Random Fields
    • Adler, R.J.1
  • 4
    • 0034392041 scopus 로고    scopus 로고
    • On excursion sets tube formulae and maxima of random fields
    • R.J. Adler. On excursion sets, tube formulae, and maxima of random fields. Annals of Applied Prob., 10:1-74, 2000.
    • (2000) Annals of Applied Prob. , vol.10 , pp. 1-74
    • Adler, R.J.1
  • 12
    • 38349022394 scopus 로고    scopus 로고
    • False discovery rates for spatial signals
    • R. Benjamini, Y. Heller. False discovery rates for spatial signals. J. Amer. Statist. Assoc., 102(480):1272-1281, 2007.
    • (2007) J. Amer. Statist. Assoc. , vol.102 , Issue.480 , pp. 1272-1281
    • Benjamini, R.1    Heller, Y.2
  • 13
    • 0036433746 scopus 로고    scopus 로고
    • Lipschitz-Killing invariants
    • A. Bernig and L. Br̈ocker. Lipschitz-Killing invariants. Math. Nachr., 245:5-25, 2002.
    • (2002) Math. Nachr. , vol.245 , pp. 5-25
    • Bernig, A.1    Br̈ocker, L.2
  • 16
    • 18244425953 scopus 로고    scopus 로고
    • Integral geometry of tame sets
    • L. Br̈ocker and M. Kuppe. Integral geometry of tame sets. Geom. Dedicata, 82(1-3):285-323, 2000.
    • (2000) Geom. Dedicata , vol.82 , Issue.1-3 , pp. 285-323
    • Br̈ocker, L.1    Kuppe, M.2
  • 17
    • 37849007580 scopus 로고    scopus 로고
    • A statistical approach to persistent homology
    • P. Bubenik and P.T. Kim. A statistical approach to persistent homology. Homology, Homotopy and Applications, 9(2):337-362, 2007.
    • (2007) Homology Homotopy and Applications , vol.9 , Issue.2 , pp. 337-362
    • Bubenik, P.1    Kim, P.T.2
  • 21
    • 67349232288 scopus 로고    scopus 로고
    • The theory of multidimensional persistence
    • G. Carlsson and A. Zomorodian. The theory of multidimensional persistence. Discrete Comput. Geom., 42(1):71-93, 2009.
    • (2009) Discrete Comput. Geom. , vol.42 , Issue.1 , pp. 71-93
    • Carlsson, G.1    Zomorodian, A.2
  • 22
    • 0007288995 scopus 로고
    • On the kinematic formula in integral geometry
    • S.S. Chern. On the kinematic formula in integral geometry. J. Math. Mech., 16:101-118, 1966.
    • (1966) J. Math. Mech. , vol.16 , pp. 101-118
    • Chern, S.S.1
  • 24
    • 55149102004 scopus 로고    scopus 로고
    • False discovery rate revisited: Fdr and topological inference using gaussian random fields
    • Justin R. Chumbley and Karl J. Friston. False discovery rate revisited: Fdr and topological inference using gaussian random fields. NeuroImage, 44(1):62-70, 2009
    • (2009) NeuroImage , vol.44 , Issue.1 , pp. 62-70
    • Chumbley, J.R.1    Friston, K.J.2
  • 25
    • 85009406857 scopus 로고    scopus 로고
    • DARPA
    • DARPA. Topological data analysis. 2010. www.darpa.mil/dso/thrusts/math/ funmath/tda/index.htm.
    • (2010) Topological Data Analysis
  • 26
    • 41849094721 scopus 로고    scopus 로고
    • Coverage in sensor networks via persistent homology
    • V. de Silva and R. Ghrist. Coverage in sensor networks via persistent homology. Algebr. Geom. Topol., 7:339-358, 2007.
    • (2007) Algebr. Geom. Topol. , vol.7 , pp. 339-358
    • De Silva, V.1    Ghrist, R.2
  • 28
    • 34447096791 scopus 로고    scopus 로고
    • Nodal densities of planar Gaussian random waves
    • M.R. Dennis. Nodal densities of planar Gaussian random waves. Eur. Phys. J., 145:191-210, 2007.
    • (2007) Eur. Phys. J. , vol.145 , pp. 191-210
    • Dennis, M.R.1
  • 29
    • 0000599192 scopus 로고
    • A dozen de Finetti-style results in search of a theory
    • P. Diaconis and D. Freedman. A dozen de Finetti-style results in search of a theory. Ann. Inst. H. Poincaŕe Probab. Statist., 23(2, suppl.):397-423, 1987.
    • (1987) Ann. Inst. H. Poincaŕe Probab. Statist. , vol.23 , Issue.2 SUPPL. , pp. 397-423
    • Diaconis, P.1    Freedman, D.2
  • 30
    • 0000436287 scopus 로고
    • Finite de Finetti theorems in linear models and multivariate analysis
    • P.W. Diaconis, M.L. Eaton, and S.L. Lauritzen. Finite de Finetti theorems in linear models and multivariate analysis. Scand. J. Statist., 19(4):289-315, 1992.
    • (1992) Scand. J. Statist. , vol.19 , Issue.4 , pp. 289-315
    • Diaconis, P.W.1    Eaton, M.L.2    Lauritzen, S.L.3
  • 31
    • 0000660999 scopus 로고
    • The sizes of compact subsets of hilbert space and continuity of Gaussian processes
    • R.M. Dudley. The sizes of compact subsets of hilbert space and continuity of Gaussian processes. J. Funct. Anal., 1:290-330, 1967.
    • (1967) J. Funct. Anal. , vol.1 , pp. 290-330
    • Dudley, R.M.1
  • 32
    • 0002485879 scopus 로고
    • Sample functions of the Gaussian process
    • R.M. Dudley. Sample functions of the Gaussian process. Ann. Probab., 1:66-103, 1973.
    • (1973) Ann. Probab. , vol.1 , pp. 66-103
    • Dudley, R.M.1
  • 34
    • 70349220130 scopus 로고    scopus 로고
    • Persistent homology\a survey Surveys on discrete and computational geometry
    • volume 453 of Contemp. Math. Providence, RI
    • H. Edelsbrunner and J. Harer. Persistent homology\a survey. In Surveys on discrete and computational geometry, volume 453 of Contemp. Math., pages 257-282. Amer. Math. Soc., Providence, RI, 2008.
    • (2008) Amer. Math. Soc. , pp. 257-282
    • Edelsbrunner, H.1    Harer, J.2
  • 40
    • 41849100389 scopus 로고    scopus 로고
    • Barcodes: The persistent topology of data
    • electronic
    • R. Ghrist. Barcodes: the persistent topology of data. Bull. Amer. Math. Soc. (N.S.), 45(1):61-75 (electronic), 2008.
    • (2008) Bull. Amer. Math. Soc. (N.S.) , vol.45 , Issue.1 , pp. 61-75
    • Ghrist, R.1
  • 45
    • 0002357581 scopus 로고
    • The sponge-like topology of large-scale structure in the universe
    • J.R. Gott III, M. Dickinson, and AL Melott. The sponge-like topology of large-scale structure in the universe. The Astrophysical Journal, 306:341-357, 1986.
    • (1986) The Astrophysical Journal , vol.306 , pp. 341-357
    • Gott Iii, J.R.1    Dickinson, M.2    Melott, A.L.3
  • 50
    • 0001070336 scopus 로고
    • Tubes and spheres in n-spaces and a class of statistical problems
    • H. Hotelling. Tubes and spheres in n-spaces and a class of statistical problems. Amer. J. Math., 61:440-460, 1939.
    • (1939) Amer. J. Math. , vol.61 , pp. 440-460
    • Hotelling, H.1
  • 51
    • 0142196254 scopus 로고    scopus 로고
    • Kinematic and Crofton formulae of integral geometry: Recent variants and extensions
    • In C. Barceĺo i Vidal, editor Universitat de Girona
    • D. Hug and R. Schneider. Kinematic and Crofton formulae of integral geometry: recent variants and extensions. In C. Barceĺo i Vidal, editor, Homenatge al Professor Lus Santaĺo i Sors, pages 51-80. Universitat de Girona, 2002.
    • (2002) Homenatge Al Professor Lus Santaĺo i Sors , pp. 51-80
    • Hug, D.1    Schneider, R.2
  • 52
    • 0003876050 scopus 로고    scopus 로고
    • Cambridge University Press, Cambridge
    • S. Janson. Gaussian Hilbert Spaces. Cambridge University Press, Cambridge, 1997.
    • (1997) Gaussian Hilbert Spaces
    • Janson, S.1
  • 53
    • 0001318038 scopus 로고
    • Hotelling's theorem on the volume of tubes: Some illustrations in simultaneous inference and data analysis
    • Søren Johansen and Iain M. Johnstone. Hotelling's theorem on the volume of tubes: some illustrations in simultaneous inference and data analysis. Ann. Statist., 18(2):652-684, 1990.
    • (1990) Ann. Statist. , vol.18 , Issue.2 , pp. 652-684
    • Johansen, S.1    Johnstone, I.M.2
  • 54
    • 0000817308 scopus 로고
    • On the average number of real roots of a random algebraic equation
    • M. Kac. On the average number of real roots of a random algebraic equation. Bull. Amer. Math. Soc., 43:314-320, 1943.
    • (1943) Bull. Amer. Math. Soc. , vol.43 , pp. 314-320
    • Kac, M.1
  • 55
    • 0000892128 scopus 로고
    • Large excursions of Gaussian processes
    • M. Kac and D. Slepian. Large excursions of Gaussian processes. Ann. Math. Statist., 30:1215-1228, 1959.
    • (1959) Ann. Math. Statist. , vol.30 , pp. 1215-1228
    • Kac, M.1    Slepian, D.2
  • 56
    • 79551707495 scopus 로고    scopus 로고
    • Topological inference for EEG and MEG data
    • J.M Kilner and K.J. Friston. Topological inference for EEG and MEG data. Ann. Appl. Stat., 4(3):1272-1290, 2010.
    • (2010) Ann. Appl. Stat. , vol.4 , Issue.3 , pp. 1272-1290
    • Kilner, J.M.1    Friston, K.J.2
  • 58
    • 0001000930 scopus 로고
    • On Hotelling's approach to testing for a nonlinear parameter in a regression
    • M. Knowles and D Siegmund. On Hotelling's approach to testing for a nonlinear parameter in a regression. Int. Statist. Rev, 57:205-220, 1989.
    • (1989) Int. Statist. Rev , vol.57 , pp. 205-220
    • Knowles, M.1    Siegmund, D.2
  • 59
    • 0026739891 scopus 로고
    • Language localization with activation positron emission tomography scanning
    • R. Leblanc, E. Meyer, D. Bub, R. J. Zatorre, and A. C. Evans. Language localization with activation positron emission tomography scanning. Neurosurgery, 31(2):369-372, 1992.
    • (1992) Neurosurgery , vol.31 , Issue.2 , pp. 369-372
    • Leblanc, R.1    Meyer, E.2    Bub, D.3    Zatorre, R.J.4    Evans, A.C.5
  • 62
    • 0000868825 scopus 로고
    • Local maxima of Gaussian fields
    • G. Lindgren. Local maxima of Gaussian fields. Ark. Mat., 10:195-218, 1972.
    • (1972) Ark. Mat. , vol.10 , pp. 195-218
    • Lindgren, G.1
  • 63
    • 0001323815 scopus 로고
    • On the statistical distribution of the heights of sea waves
    • M.S. Longuet-Higgins. On the statistical distribution of the heights of sea waves. J. Marine Res., 11:245-266, 1952.
    • (1952) J. Marine Res. , vol.11 , pp. 245-266
    • Longuet-Higgins, M.S.1
  • 64
    • 0000109756 scopus 로고
    • The statistical analysis of a random moving surface
    • M.S. Longuet-Higgins. The statistical analysis of a random moving surface. Phil. Trans. Roy. Soc., A249:321-387, 1957.
    • (1957) Phil. Trans. Roy. Soc. A , vol.249 , pp. 321-387
    • Longuet-Higgins, M.S.1
  • 68
    • 0003655789 scopus 로고    scopus 로고
    • American Mathematical Society, Providence, RI Translated from the Russian by V. V. Piterbarg, Revised by the author
    • V.I. Piterbarg. Asymptotic methods in the theory of Gaussian processes and fields. American Mathematical Society, Providence, RI, 1996. Translated from the Russian by V. V. Piterbarg, Revised by the author.
    • (1996) Asymptotic Methods in the Theory of Gaussian Processes and Fields.
    • Piterbarg, V.I.1
  • 69
    • 0040406290 scopus 로고
    • The Distribution of the Maxima of a Random Curve
    • S. O. Rice. The Distribution of the Maxima of a Random Curve. Amer. J. Math., 61(2):409-416, 1939.
    • (1939) Amer. J. Math. , vol.61 , Issue.2 , pp. 409-416
    • Rice, S.O.1
  • 70
    • 84944485718 scopus 로고
    • Mathematical analysis of random noise
    • S.O. Rice. Mathematical analysis of random noise. Bell System Tech. J., 24:46-156, 1945.
    • (1945) Bell System Tech. J. , vol.24 , pp. 46-156
    • Rice, S.O.1
  • 73
    • 1542629810 scopus 로고    scopus 로고
    • Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps
    • J. Schmalzing and K.M. Gorski. Minkowski functionals used in the morphological analysis of cosmic microwave background anisotropy maps. Monthly Notices of the Royal Astronomical Society, 297(2):355-365, 1998 (Pubitemid 128502232)
    • (1998) Monthly Notices of the Royal Astronomical Society , vol.297 , Issue.2 , pp. 355-365
    • Schmalzing, J.1    Gorski, K.M.2
  • 75
    • 0004454158 scopus 로고
    • Testing for a signal with unknown location and scale in a stationary Gaussian random field
    • D.O. Siegmund and K.J. Worsley. Testing for a signal with unknown location and scale in a stationary Gaussian random field. Ann. Statist, 23:608-639, 1995.
    • (1995) Ann. Statist , vol.23 , pp. 608-639
    • Siegmund, D.O.1    Worsley, K.J.2
  • 78
    • 0002077465 scopus 로고
    • Tail probabilities of the maxima of Gaussian random fields
    • Jiayang Sun. Tail probabilities of the maxima of Gaussian random fields. Ann. Probab., 21(1):34-71, 1993.
    • (1993) Ann. Probab. , vol.21 , Issue.1 , pp. 34-71
    • Sun, J.1
  • 79
    • 0036331007 scopus 로고    scopus 로고
    • On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains
    • DOI 10.1214/aoap/1026915624
    • A. Takemura and S. Kuriki. On the equivalence of the tube and Euler characteristic methods for the distribution of the maximum of Gaussian fields over piecewise smooth domains. Ann. of Appl. Prob., 12(2):768-796, 2002. (Pubitemid 37023758)
    • (2002) Annals of Applied Probability , vol.12 , Issue.2 , pp. 768-796
    • Takemura, A.1    Kuriki, S.2
  • 81
    • 33644944828 scopus 로고    scopus 로고
    • A Gaussian kinematic formula
    • DOI 10.1214/009117905000000594
    • J.E. Taylor. A Gaussian kinematic formula. Ann. Probab., 34(1): 122-158, 2006. (Pubitemid 43409921)
    • (2006) Annals of Probability , vol.34 , Issue.1 , pp. 122-158
    • Taylor, J.E.1
  • 82
    • 23244455084 scopus 로고    scopus 로고
    • Validity of the expected Euler characteristic heuristic
    • DOI 10.1214/009117905000000099
    • J.E. Taylor, A. Takemura, and R.J. Adler. Validity of the expected Euler characteristic heuristic. Ann. Probab., 33(4):1362-1396, 2005. (Pubitemid 41095879)
    • (2005) Annals of Probability , vol.33 , Issue.4 , pp. 1362-1396
    • Taylor, J.1    Takemura, A.2    Adler, R.J.3
  • 84
    • 35348839230 scopus 로고    scopus 로고
    • Detecting sparse signals in random fields with an application to brain mapping
    • J.E. Taylor and K.J. Worsley. Detecting sparse signals in random fields, with an application to brain mapping. J. Amer. Statist. Assoc, 102(479):913-928, 2007.
    • (2007) J. Amer. Statist. Assoc , vol.102 , Issue.479 , pp. 913-928
    • Taylor, J.E.1    Worsley, K.J.2
  • 87
    • 0000045032 scopus 로고
    • On the volume of tubes
    • H. Weyl. On the volume of tubes. Amer. J. Math., 61:461-472, 1939.
    • (1939) Amer. J. Math. , vol.61 , pp. 461-472
    • Weyl, H.1
  • 88
    • 0002018283 scopus 로고
    • Local maxima and the expected Euler characteristic of excursion sets of χ, F and t fields
    • K. J. Worsley. Local maxima and the expected Euler characteristic of excursion sets of χ, F and t fields. Adv. in Appl. Probab., 26(1):13-42, 1994.
    • (1994) Adv. in Appl. Probab. , vol.26 , Issue.1 , pp. 13-42
    • Worsley, K.J.1
  • 89
    • 0002087618 scopus 로고    scopus 로고
    • The geometry of random images
    • K. J. Worsley. The geometry of random images. Chance, 9(1):27-40, 1997.
    • (1997) Chance , vol.9 , Issue.1 , pp. 27-40
    • Worsley, K.J.1
  • 90
    • 0001464776 scopus 로고
    • Boundary corrections for the expected Euler characteristic of excursion sets of random fields with an application to astrophysics
    • K.J. Worsley. Boundary corrections for the expected Euler characteristic of excursion sets of random fields, with an application to astrophysics. Adv. Appl. Probab., 27:943-959, 1995.
    • (1995) Adv. Appl. Probab. , vol.27 , pp. 943-959
    • Worsley, K.J.1
  • 91
    • 21844486695 scopus 로고
    • Estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets with applications to medical images
    • K.J. Worsley. Estimating the number of peaks in a random field using the hadwiger characteristic of excursion sets, with applications to medical images. Ann. Statist., 23:640-669, 1995.
    • (1995) Ann. Statist. , vol.23 , pp. 640-669
    • Worsley, K.J.1
  • 92
    • 0035713340 scopus 로고    scopus 로고
    • 2 random field, with an application to fMRI
    • DOI 10.1239/aap/1011994029
    • K.J. Worsley. Testing for signals with unknown location and scale in a χ random field, with an application to fMRI. Adv. in Appl. Probab., 33(4):773-793, 2001. (Pubitemid 34149103)
    • (2001) Advances in Applied Probability , vol.33 , Issue.4 , pp. 773-793
    • Worsley, K.J.1
  • 94
    • 79957751394 scopus 로고
    • Springer-Verlag, Berlin Mesure ǵeoḿetrique des ensembles de niveau. [The geometric measure of level sets]
    • M. Wschebor. Surfaces aĺeatoires, volume 1147 of Lecture Notes in Mathematics. Springer-Verlag, Berlin, 1985. Mesure ǵeoḿetrique des ensembles de niveau. [The geometric measure of level sets].
    • (1985) Surfaces Aĺeatoires Volume 1147 of Lecture Notes in Mathematics
    • Wschebor, M.1


* 이 정보는 Elsevier사의 SCOPUS DB에서 KISTI가 분석하여 추출한 것입니다.