Kriging approach for the experimental cross-section covariances estimation

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1 EPJ Web of Conferences, () DOI:./ epjconf/ C Owned by te autors, publised by EDP Sciences, Kriging approac for te experimental cross-section covariances estimation S. Varet,a, P. Dossantos-Uzarralde, N. Vayatis, A. Garlaud, and E. Bauge CEA-DAM-DIF, 9 Arpajon Cedex ENS Cacan ENSIIE Abstract. In te classical use of a generalized χ to determine te evaluated cross section uncertainty, we need te covariance matrix of te experimental cross sections. Te usual propagation error metod to estimate te covariances is ardly usable and te lack of data prevents from using te direct empirical estimator. We propose in tis paper to apply te kriging metod wic allows to estimate te covariances via te distances between te points and wit some assumptions on te covariance matrix structure. All te results are illustrated wit te Mn nucleus measurements. Introduction Te use of a generalized χ for te evaluated cross section uncertainty determination needs to take into account te correlations and te uncertainties of te experimental measurements. As te covariances are not provided, we need to estimate te covariance matrix Σ F of te experimental cross sections wit its inverse. As in many cases only one measure per energy is available te direct empirical estimator is not available and te classical metod of error propagation [] is rarely usable due to te lack of information on te experimental parameters and due to te linearity assumption tat is not fullfilled. Te kriging is a valuable tool for covariances evaluation under te stationnarity assumption (constant mean, constant variance and covariances unvariant by translation) ([],[] and []). Here tis metod is introduced in te field of experimental cross-section covariances determination. Most of te time and until now, we dispose of only one measurement per energy. In tis case kriging is appropriate since it is a linear interpolation metod. Te aim of tis work is to estimate te experimental covariances troug te kriging. Te particularity of tis metod is tat te coefficients of te linear decomposition are determined wit wat we call te variogram. Te variogram is a continuous function wic depends on te distance (on te x axis in dimension ) of te data to interpolate. Under some assumptions, like te second order stationarity of te data, we can rely on te covariance function and te variogram. Tus, te resulting kriging model depends on te data covariance matrix and in particular it depends on te inverse covariance matrix. Te covariance matrix is estimated troug te estimation of te variogram. As te inverse covariance matrix is needed, te covariance matrix estimation must be invertible. In te litterature some functions for te variogram are known to lead to an invertible covariance matrix. Terefore in te kriging metod te covariance matrix is estimated in adjusting a parametrical function to te empirical variogram. a suzanne.varet@cea.fr Tis is an Open Access article distributed under te terms of te Creative Commons Attribution License., wic permits unrestricted use, distribution, and reproduction in any medium, provided te original work is properly cited. Article available at ttp:// or ttp://dx.doi.org/./epjconf/

2 EPJ Web of Conferences Here, we illustrate tis approac wit real data measurement of Mn nucleus. In te next section we introduce te needed notations. We present te kriging for te covariance evaluation in te tird section. Te entire approac is illustrated in te case of te Mn nucleus in te fourt section. Notations We assume tat te response (in our case te experimental cross-section) is a stocastic process, writen F (E), wic depends on te s-dimensional variable E (in our case s = and E is te energy) and tat we observe only at a finite number N of points E j, j =,..., N. We write F = (F,..., F N ) t te random vector of te measured cross section at te N distinct energies. Te covariance matrix of F is denoted by Σ F. We assume tat we can observe one realisation of te F vector tat we write F mes,..., F mes N. Te kriging Te kriging is a linear interpolation metod. More precisely te kriging metod supposes tat we can write te response (in our case te experimental cross section) at a given point E E (in our case at a fixed energy) as a linear combination of te observed responses F mes,..., F mes F (E) = i= λ i F mes i. In order to find te coefficients λ i, i =,..., N, we suppose tat we can write te experimental cross section F (E) = µ(e) + δ(e) were µ(e) is a deterministic component wic represents te trend function and δ(e) is a centered stocastic process wic represents te residuals. In te simple kriging µ(e) is assumed to be a known constant, in te ordinary kriging µ(e) is assumed to be an unknown constant and in te universal kriging µ(e) is assumed to be a linear combination of functions wic depend on te position E tat must be estimated. As a first approximation we suppose in tis paper tat we are in te ordinary kriging case. Te coefficients λ i, i =,..., N are estimated in minimising te variance of te prediction wit a non bias constraint. To deal wit te unicity of te observation at a given point E i, te kriging imposes a stationarity assumption on F (E) (or equivalently in te ordinary kriging case on δ(e)). In kriging, tere is two types of stationnarity assumptions: te second order stationnarity (constant mean and covariance invariant by translation) and te intrinsic stationarity (te increment process as a constant mean and a variance invariant by translation). If F (E) is assumed to be a second order stationary process ten it is a intrinsic stationary process but te inverse is not true in general. Te variogram γ is define as te variance of te increment process: N : = Var(F (E + ) F (E)). () We assume now tat F (E) is a second order stationary process. Tat is { IE(F (E)) = m E E Cov(F (E + ), F (E)) = C() E, E + E. Te C function is te covariance function also called te covariogram. Under te second order stationarity assumption, te covariogram can be relied on te variogram by te following equation = (C() C()). () () t denotes te transpose of a vector. () -p.

3 WONDER- Tis relation is a convenient way for te covariances estimation. Tus, for te experimental cross section covariances estimation, we assume tat F (E) is a second order stationary process. Now we must estimate te variogram γ. Te empirical estimation ˆγ of γ is ˆ = D() (i,j) D() (F mes i F mes j ) () were D() = {(i, j) N suc tat E i E j = } and D() is te cardinal of D(). Te value C() is te variance of F (E) (assumed constant by te second order stationarity ypotesis) estimated by were F = N i= F mes i ˆσ = N i= (F mes i F ) () is te empirical mean. Ten we can estimate te covariances by Ĉ() = ˆσ ˆ. However te obtained matrix can be non invertible. Moreover we need a continuous covariance function in order to interpolate at energies were no measurement ave been done. In te litterature some continuous functions for te covariogram are known to lead to an invertible covariance matrix. Terefore we must find te most adapted covariogram function to te empirical variogram. In tis paper we ave used te exponential, gaussian and sperical functions: ( exponential: C θ () = σ exp gaussian: C θ () = σ exp θ ( ) θ ) sperical: C θ () = if > θ and C θ () = σ ( θ + ( θ )) oterwise. Te covariance matrix estimation Σ F is ten given by: C θ () C θ ( )... C θ ( N ) Σ F = () C θ ( N ) C θ ( N )... C θ () Te kriging model F is ten given, in minimising te mean square error, by: F mes m F (E) = m + (C θ ( E E ),..., C θ ( E E N )) Σ F.. m F mes N Application to te M n nucleus We present in tis section te results of te variogram study for te M n nucleus measurements. Te measurements of te total and te (n,γ) cross sections are extracted from te EXFOR database ([]) and te (n,n) cross sections measurements are extracted from []. Please note tat in order to prevent from te effect of ig variations in te energy range and in te cross section range, we center and reduce te data. Ten we calibrate te variogram model to te empirical variogram of te reduced data. Note also tat wen te number of measurements is ig enoug, in order to find a compromise between te D() cardinal and te ˆγ sample size, we introduce a tolerance ε (a positive real) on te distance in te D() definition. Tat is D ε () = {(i, j) N suc tat ε E i E j + ε}. Te variogram -p.

4 EPJ Web of Conferences model parameters are calibrated via a cross validation wit te estimation of te mean square error as criterion: mse(ˆγ, γ) = H (ˆγ( k ) γ( k )) H k= were H is te ˆγ sample size. For te M n (n,n), total and (n,γ) cross sections, te empirical variogram and te tree variogram models (exponential, gaussian and sperical) are illustrated on figures (a) (a) and (a) respectively. For eac of tese cross sections, te covariance matrix associated to te lowest mean square error model is represented ((b) (b) and (b) respectively). For te Mn (n,n) cross section, te covariance matrix obtained wit te variogram calibration seems to indicate tat te correlations are iger at ig energies. In te contrary, te Mn (n,γ) covariance matrix seems to indicate tat te correlations are iger at low energies. Finally, in te case of te total cross section, te correlations seems similar for all energies. Empirical variogram (tolerance = ) Exponential model (θ = ) Gaussian model (θ =.) Sperical model (θ = )... (a) Empirical variogram for te Mn (n,n) (b) Covariance matrix for te Mn (n,n) cross section wit gaussian model.. Fig.. Variogram and covariance matrix for te Mn (n,n)... empirical variogram exponential model gaussian model sperical model (a) Empirical variogram for te Mn total (b) Covariance matrix for te Mn total cross section wit exponential model. Fig.. Variogram and covariance matrix for te Mn total -p.

5 WONDER-.. Empirical variogram (tolerance =.8) Exponential model (θ =.) Gaussian model (θ =.) Sperical model (θ =.) x (a) Empirical variogram for te Mn (n,γ) (b) Covariance matrix for te Mn (n,γ) cross section wit sperical model. Fig.. Variogram and covariance matrix for te Mn (n,γ) Conclusion In tis paper an ordinary kriging metod for experimental cross section covariance matrix estimation as been presented. Te aim of tis work is to apply tis metod to observed experimental cross section data. We focus on te situation were te lack of measurements prevents from using te direct empirical estimator and te informations on te experimental protocol are not available. To deal wit te problem of one measurement per energy, te kriging approac makes strong assumptions on te covariance matrix structure. In tis work we ave assumed te variance stationarity and te decreasing correlations as te distance between measurements increases. In tis context te covariance matrix can be easily approximated via te variogram study. Te results could be compared wit tose obtained wit anoter optimisation metod []. Finally te consideration of less restrictive assumptions, like a non stationary mean, lead to a universal kriging approac wic still remains a point to explore. References. M. Ionescu-Bujor and D. G. Cacuci, A comparative review of sensitivity and uncertainty analysis of large-scale systems I: Deterministic metods, Nuclear Science and Engineering (), 89. O. Roustant, D. Ginsbourger and Y. Deville, DiceKriging, DiceOptim: Two R packages for te analysis of computer experiments by kriging-based metamodelling and optimization, (re-sent after minor revision in June ), Journal of Statistical Software,.. M. Stein, Interpolation of spatial data, some teory for kriging, 999, Springer.. E. Vasquez and E. Walter, Coix d une covariance pour la prédiction par krigeage de séries cronologiques écantillonnées irrégulièrement I-Revue, GRETSI, Groupe d Etudes du Traitement du Signal et des Images,.. ttp:// A. Milocco, A. Trkov, and R. Capote Noy Nuclear data evaluation of Mn by te EMPIRE code wit empasis on te capture cross-section, Nuclear Engineering and Design (), -.. S. Varet, P. Dossantos-Uzarralde, N. Vayatis and E. Bauge, Pseudo-measurement simulations and bootstrap for te experimental cross-section covariances estimation wit quality quantification, WONDER. -p.