Integral Geometry of Linearly Combined Gaussian and Student t, and Skew t Random Fields

28/08/2013
OAI : oai:www.see.asso.fr:2552:4893
DOI :

Résumé

no preview

Média

Voir la vidéo

Métriques

136
0
2.11 Mo
 application/pdf
bitcache://1499aff0c4e8d14dacc555fcb4161f0086a2d445

Licence

Creative Commons Aucune (Tous droits réservés)

Sponsors

Sponsors scientifique

logo_smf_cmjn.gif

Sponsors financier

gdrmia_logo.png
logo_inria.png
image010.png
logothales.jpg

Sponsors logistique

logo-minesparistech.jpg
logo-universite-paris-sud.jpg
logo_supelec.png
Séminaire Léon Brillouin Logo
sonycsl.jpg
logo_cnrs_2.jpg
logo_ircam.png
logo_imb.png
<resource  xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance"
                xmlns="http://datacite.org/schema/kernel-4"
                xsi:schemaLocation="http://datacite.org/schema/kernel-4 http://schema.datacite.org/meta/kernel-4/metadata.xsd">
        <identifier identifierType="DOI">10.23723/2552/4893</identifier><creators><creator><creatorName>Yann Gavet</creatorName></creator><creator><creatorName>Ola Ahmad</creatorName></creator><creator><creatorName>Jean-Charles Pinoli</creatorName></creator></creators><titles>
            <title>Integral Geometry of Linearly Combined Gaussian and Student t, and Skew t Random Fields</title></titles>
        <publisher>SEE</publisher>
        <publicationYear>2013</publicationYear>
        <resourceType resourceTypeGeneral="Text">Text</resourceType><dates>
	    <date dateType="Created">Mon 16 Sep 2013</date>
	    <date dateType="Updated">Sun 25 Dec 2016</date>
            <date dateType="Submitted">Mon 17 Dec 2018</date>
	</dates>
        <alternateIdentifiers>
	    <alternateIdentifier alternateIdentifierType="bitstream">1499aff0c4e8d14dacc555fcb4161f0086a2d445</alternateIdentifier>
	</alternateIdentifiers>
        <formats>
	    <format>application/pdf</format>
	</formats>
	<version>27099</version>
        <descriptions>
            <description descriptionType="Abstract"></description>
        </descriptions>
    </resource>
.

Integral Geometry of Linearly Combined Gaussian and Student t, and Skew t Random Fields Yann Gavet, Ola Ahmad and Jean-Charles Pinoli École Nationale Supérieure des Mines de Saint-Etienne, LGF 5307, France ahmad@emse.fr, gavet@emse.fr, pinoli@emse.fr GSI2013 - Geometric Science of Information, Paris August 28-30 2013 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work General stochastic problem: Y = Data + error e.g., Data is a matrix of unknown random variables of N dimensions represented on a group of voxels, (2D or 3D images). How can they be approximated or represented? Non-parametric methods "often numeric" have no reference of probability. Parametric methods: e.g.; Random fields theory priori knowledge of Y based on the measurements. few significant parameters & geometric information that control and interpret some physical problems. 4 / 32 General stochastic problem: Y = Data + error e.g., Data is a matrix of unknown random variables of N dimensions represented on a group of voxels, (2D or 3D images). How can they be approximated or represented? Non-parametric methods "often numeric" have no reference of probability. Parametric methods: e.g.; Random fields theory priori knowledge of Y based on the measurements. few significant parameters & geometric information that control and interpret some physical problems. 4 / 32 Application example: Total hip implant 5 / 32 Statistical analysis via stochastic modelling Real phenomenon: biology, physics, mechanics, ... Stochastic represen- tation of problem Experimental observa- tions & measurements Geometric features (MF or LKCs,...) cal- culated from the model Emprical features Parameters estimation. Validity testning of model. Decision & analysis of phenomena. 6 / 32 Why Gaussian random fields? Completely characterized by their first and second order moments, mean and covariance function. Smooth and twice-differentiable Why not Gaussian random field ? Real observations are often not Gaussian 7 / 32 Why Gaussian random fields? Completely characterized by their first and second order moments, mean and covariance function. Smooth and twice-differentiable Why not Gaussian random field ? Real observations are often not Gaussian 7 / 32 Example: Worn engineered surface Rough Skewed Heavy-tailed distributions Need to go beyond the Gaussian 8 / 32 Beyond Gaussian random fields Related Gaussian RFs F : Rk R, f(x) = F(g(x)) g1, ..., gk are i.i.d Gaussian RFs. Examples: χ2 , F, t RFs Mixed random fields : High flexibility f(x) = β1Z(x) + β2G(x), G, Z are independent random fields. 9 / 32 Beyond Gaussian random fields Related Gaussian RFs F : Rk R, f(x) = F(g(x)) g1, ..., gk are i.i.d Gaussian RFs. Examples: χ2 , F, t RFs Mixed random fields : High flexibility f(x) = β1Z(x) + β2G(x), G, Z are independent random fields. 9 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 10 / 32 Random Fields A random field Y(x) : x S indexed by some space S, (e.g., S RN ), satisfies that any arbitrary p collection, (Y(x1), ..., Y(xp)) follows a multivariate probability density function with (p p) covariance matrix ΩY 11 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 12 / 32 Excursion sets Excursion sets A random set over a level h of Y: Eh = x S : Y(x) h Example: thresholding the surface at some height level 13 / 32 Excursion sets Excursion sets A random set over a level h of Y: Eh = x S : Y(x) h Example: thresholding the surface at some height level 13 / 32 Integral geometry Estimation of intrinsic volumes of Eh E[ k (Eh(Y, S))] = N−k j=0 j + k j j+k (S)ρj(h) 0(.) = χ(.) : Euler-Poincaré characteristic j(.) : j th dimensional volume ρj(.) : EC densities How to get j and ρj? 14 / 32 Integral geometry Estimation of intrinsic volumes of Eh E[ k (Eh(Y, S))] = N−k j=0 j + k j j+k (S)ρj(h) 0(.) = χ(.) : Euler-Poincaré characteristic j(.) : j th dimensional volume ρj(.) : EC densities How to get j and ρj? 14 / 32 Integral geometry Estimation of intrinsic volumes of Eh E[ k (Eh(Y, S))] = N−k j=0 j + k j j+k (S)ρj(h) 0(.) = χ(.) : Euler-Poincaré characteristic j(.) : j th dimensional volume ρj(.) : EC densities How to get j and ρj? 14 / 32 Integral geometry Estimation of intrinsic volumes of Eh E[ k (Eh(Y, S))] = N−k j=0 j + k j j+k (S)ρj(h) 0(.) = χ(.) : Euler-Poincaré characteristic j(.) : j th dimensional volume ρj(.) : EC densities How to get j and ρj? 14 / 32 Integral geometry Estimation of intrinsic volumes of Eh E[ k (Eh(Y, S))] = N−k j=0 j + k j j+k (S)ρj(h) 0(.) = χ(.) : Euler-Poincaré characteristic j(.) : j th dimensional volume ρj(.) : EC densities How to get j and ρj? 14 / 32 Integral geometry j(S) : N(S) = σ−N S det(Λ(x)) 1=2 dx N−1(S) = 1 2 σ−(N−1) @S det(Λ@S(x)) 1=2 N−1(dx) ρj(.) : Morse theory: ρj(h) = E ˙Y+ (j)det( ¨Y|j−1) ˙Y|j−1 = 0, Y = h p ˙Y|j−1 (0; h) 15 / 32 Integral geometry j(S) : N(S) = σ−N S det(Λ(x)) 1=2 dx N−1(S) = 1 2 σ−(N−1) @S det(Λ@S(x)) 1=2 N−1(dx) ρj(.) : Morse theory: ρj(h) = E ˙Y+ (j)det( ¨Y|j−1) ˙Y|j−1 = 0, Y = h p ˙Y|j−1 (0; h) 15 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 16 / 32 Gaussian t random field Definition Y(x) = G(x) + βT (x), β > 0, x S G: is a stationary Gaussian random field. T : is a homogeneous student’s t random field with ν degrees of freedom. Linear transformed pdf at each fixed point x of both normal and t pdfs: pY (y) = Γ +1 2 2πβΓ 2 2 1=2 ∞ −∞ 1 + (y u)2 β2ν −ν+1 2 e−u2 2 du 17 / 32 EC densities of Gaussian t random field Theorem [Ahmad and Pinoli(2013a)] The EC densities, ρj(.) of a two-dimensional real-valued Gaussian t random field with ν 2 degrees of freedom, and β > 0 are given, at level h, by: where ΛG = λGI2, and Λ = λI2 is the second spectral moments matrix of G, and Λ = λI2 is associated with T 18 / 32 Simulation example Simulated and analytical Minkowski functionals for the Gaussian−t random field of 5 degrees of freedom and β = 0.2. 19 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 20 / 32 Application to surface characterization Machined surface observed from Polyethylene material [Ahmad and Pinoli(2012)]: Fitting the empirical and analytical intrinsic volumes of the real surface and the Gaussian−t random field of 5 degrees of freedom and β = 1.2 21 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 22 / 32 Skew t Random field Definition G0(x), G1(x), ..., Gk (x), (x S), i.i.d stationary centred Gaussian random fields, with (N N) spectral moment matrix Λ. Z Normal(0, 1) is independent of G0, G1, ..., Gk : Y(x) = δ Z + 1 δ2G0(x) k i=1 G2 i (x)/k 1=2 , , δ2 < 1 (1) defines a skew t RF with k degrees of freedom, and skewness index α = δ/ 1 δ2 23 / 32 Example: two-dimensional skew t RFs 24 / 32 EC densities Theorem [Ahmad and Pinoli(2013c)] The EC densities, ρj(.) of a two-dimensional real-valued skew t random field with k degrees of freedom and skewness parameter α R, are given by: 25 / 32 Simulation example Simulated and analytical Minkowski functionals for the skew−t random field of 5 degrees of freedom and skewness index α = 0.7. 26 / 32 Outline Introduction Background & Motivation Preliminaries Random Fields Integral geometry of random fields Linear mixtures random fields Gaussian t random field Application Skew random fields Skew t random field Application Conclusions and Future work 27 / 32 Application to worn engineered surfaces Worn engineered surface observed from Polyethylene material [Ahmad and Pinoli(2013b)]: Fitting the empirical and analytical intrinsic volumes of the real surface and the skew−t random field of 6 degrees of freedom and δ = 0.5 28 / 32 Application to worn engineered surfaces Worn engineered surface observed from Polyethylene material [Ahmad and Pinoli(2013b)]: 28 / 32 Conclusion Random fields are computationally feasible, voxel-based, probabilistic models that can be used to approximate and represent some physical problems. Integral geometry provides interesting geometric information of the excursion sets, called intrinsic volumes. These geometric characteristics can be calculated analytically to fit the real measurements with some probabilistic model, and to estimate its parameters. Skew t random field is an appropriate model for statistical representation of worn engineered surfaces. 29 / 32 Future Work Using skew t random field for statistical analysis of surface roughness evolution. Space-scale random fields for multi-scale characterization. Space-time random fields for prediction of future behaviour, and for estimation of roughness evolution of rough engineered surfaces. Opened question: Intrinsic volumes of probabilistic models of non explicit or closed analytical form. 30 / 32 References: Ola Ahmad and Jean-Charles Pinoli. On the linear combination of the gaussian and student’s t random field and the integral geometry of its excursion sets. Statistics & Probability Letters, 83(2):559 – 567, 2013a. ISSN 0167-7152. doi: 10.1016/j.spl.2012.10.022. Ola Suleiman Ahmad and Jean-Charles Pinoli. On the linear combination of the gaussian and student-t random fields and the geometry of its excursion sets. In Lecture Notes in Engineering and Computer Science: Proceedings of the World Congress on Engineering and Computer Science 2012, WCECS 2012, 24-26 October, San Francisco, USA, pages 1–5, 2012. Ola Suleiman Ahmad and Jean-Charles Pinoli. Lipschitz-killing curvatures of the excursion sets of skew student-t random fields. In 2nd Annual International Conference on Computational Mathematics, Computational Geometry & Statistics, volume 1, Feb 2013b. doi: 10.5176/2251-1911_CMCGS13.05. Ola Suleiman Ahmad and Jean-Charles Pinoli. Lipschitz-killing curvatures of the excursion sets of skew student’ s t random fields. Stochastic Models, 29(2):273–289, 2013c. ISSN 1532-6349. doi: 10.1080/15326349.2013.783290. 31 / 32 Thank You for Your Attention ahmad@emse.fr, gavet@emse.fr, pinoli@emse.fr