Extreme value statistics of smooth random Gaussian fields
We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of p_G which is expected to hol...
| Main Authors: | , , , , |
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| Format: | Journal article |
| Language: | English |
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2011
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| _version_ | 1826272656960258048 |
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| author | Colombi, S Davis, O Devriendt, J Prunet, S Silk, J |
| author_facet | Colombi, S Davis, O Devriendt, J Prunet, S Silk, J |
| author_sort | Colombi, S |
| collection | OXFORD |
| description | We consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of p_G which is expected to hold in a regime where local maxima of the field are moderately high and weakly clustered. When the patch size becomes sufficiently large, the negative of the logarithm of the cumulative extreme value distribution is simply equal to the average of the Euler Characteristic of the field in the excursion x > x_max inside the patches. The Gumbel statistics therefore represents an interesting alternative probe of the genus as a test of non Gaussianity, e.g. in cosmic microwave background temperature maps or in three-dimensional galaxy catalogs. It can be approximated, except in the remote positive tail, by a negative Weibull type form, converging slowly to the expected Gumbel type form for infinitely large patch size. Convergence is facilitated when large scale correlations are weaker. We compare the analytic predictions to numerical experiments for the case of a scale-free Gaussian field in two dimensions, achieving impressive agreement between approximate theory and measurements. We also discuss the generalization of our formalism to non-Gaussian fields. |
| first_indexed | 2024-03-06T22:16:02Z |
| format | Journal article |
| id | oxford-uuid:536c7d23-7171-4ff8-a7b0-2773c0a9d9b2 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T22:16:02Z |
| publishDate | 2011 |
| record_format | dspace |
| spelling | oxford-uuid:536c7d23-7171-4ff8-a7b0-2773c0a9d9b22022-03-26T16:31:33ZExtreme value statistics of smooth random Gaussian fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:536c7d23-7171-4ff8-a7b0-2773c0a9d9b2EnglishSymplectic Elements at Oxford2011Colombi, SDavis, ODevriendt, JPrunet, SSilk, JWe consider the Gumbel or extreme value statistics describing the distribution function p_G(x_max) of the maximum values of a random field x within patches of fixed size. We present, for smooth Gaussian random fields in two and three dimensions, an analytical estimate of p_G which is expected to hold in a regime where local maxima of the field are moderately high and weakly clustered. When the patch size becomes sufficiently large, the negative of the logarithm of the cumulative extreme value distribution is simply equal to the average of the Euler Characteristic of the field in the excursion x > x_max inside the patches. The Gumbel statistics therefore represents an interesting alternative probe of the genus as a test of non Gaussianity, e.g. in cosmic microwave background temperature maps or in three-dimensional galaxy catalogs. It can be approximated, except in the remote positive tail, by a negative Weibull type form, converging slowly to the expected Gumbel type form for infinitely large patch size. Convergence is facilitated when large scale correlations are weaker. We compare the analytic predictions to numerical experiments for the case of a scale-free Gaussian field in two dimensions, achieving impressive agreement between approximate theory and measurements. We also discuss the generalization of our formalism to non-Gaussian fields. |
| spellingShingle | Colombi, S Davis, O Devriendt, J Prunet, S Silk, J Extreme value statistics of smooth random Gaussian fields |
| title | Extreme value statistics of smooth random Gaussian fields |
| title_full | Extreme value statistics of smooth random Gaussian fields |
| title_fullStr | Extreme value statistics of smooth random Gaussian fields |
| title_full_unstemmed | Extreme value statistics of smooth random Gaussian fields |
| title_short | Extreme value statistics of smooth random Gaussian fields |
| title_sort | extreme value statistics of smooth random gaussian fields |
| work_keys_str_mv | AT colombis extremevaluestatisticsofsmoothrandomgaussianfields AT daviso extremevaluestatisticsofsmoothrandomgaussianfields AT devriendtj extremevaluestatisticsofsmoothrandomgaussianfields AT prunets extremevaluestatisticsofsmoothrandomgaussianfields AT silkj extremevaluestatisticsofsmoothrandomgaussianfields |