A Correlated Sampling Method for Multivariate Normal and Log-normal Distributions

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1 A Correlated Sampling Method for Multivariate Normal and Log-normal Distributions Gašper Žerovni, Andrej Trov, Ivan A. Kodeli Jožef Stefan Institute Jamova cesta 39, SI-000 Ljubljana, Slovenia ABSTRACT A correlated random sampling method for multivariate normal and log-normal distributions is presented. The method is exact for both multivariate normal and log-normal distribution, and can be extended to any combination of normally and log-normally distributed coupled parameters with arbitrary precision. Although the method is proven to be exact, an example of usage for random sampling of resonance parameters is briefly described. However, due to its generality, the applicability of the method reaches well beyond resonance parameters, nuclear data, and nuclear engineering. INTRODUCTION Along with the development of fast computer systems, accurate but numerically relatively inefficient Monte Carlo methods are frequently used in neutron transport calculations. One step further is the so-called Total Monte Carlo method, which couples random sampling of nuclear data (taing into account their uncertainties and correlations with Monte Carlo particle simulations. Nuclear data are usually given in the form of ected values and covariance matrices. The corresponding distributions are multivariate normal (for location parameters or log-normal (for scale or inherently positive parameters []. If relative uncertainties of the latter are small ( 0.3, normal distribution can be used with satisfactory accuracy for most applications []. However, this is not always the case [3]. Random sampling of single or independent arbitrarily distributed parameters is straightforward. When several parameters are correlated, the correlated sampling problem can be reduced to the problem of single parameter sampling by simply diagonalizing the covariance matrix. Note however, that non-gaussian distributions become distorted by the diagonalization, and furthermore, some negative values for inherently positive parameters are obtained after the transformation from the diagonal bac to the original set of parameters. For log-normal distribution, this can be overcome by a different approach, described below. Since the method can be extended to arbitrary combination of normally and log-normally distributed coupled parameters, it is the ultimate solution when dealing with parameter sets with given ected values and covariance matrices. 409.

2 409. THE CORRELATED SAMPLING METHOD The method is based on correlated sampling of parameters, which in principle allows use of arbitrary (yet carefully chosen to meet certain criteria probability distribution functions for parameters. The beauty of the method is that it exactly preserves the whole parameter distributions (including the ected values and uncertainties and correlations of the sampled parameters. The main limitation of the method is the need to solve a system of n equations, where n is the number of inter-correlated parameters, before applying the parameter sampling. In this wor, only normal and log-normal distributions are considered. Lucily, for these two distributions the system of equations is quadratic and may be written in a form of a matrix equation with one solution obtainable by performing the matrix square root operation. The basic idea of the method is very simple. We would lie to produce samples of n parameters distributed according to their distribution functions by sampling n independent (e.g. uniformly or normally distributed random variables ξ i. A sample would then be produced by calculating a nown function F of the random variables, which would depend on the parameter distributions and correlations. Formally, the m-th sample x (m would be x (m = F ( ξ (m. ( In practice, one has to derive F for every distribution or family of distributions, which is in general far from trivial. The function F is non-linear except for multivariate normal distribution of both x and ξ.. Normal distribution Due to its linearity, normal distribution is probably the most simple example. It is wellnown that any linear combination of normally distributed variables is again a normally distributed variable. If in ξ (m all components are normally distributed variables, the samples could be produced by x (m = A ξ (m + µ, ( where matrix A is defined such that mean values, standard deviations and correlations between the components of x are preserved. Without any loss of generality, all ξ can have zero means and unit standard deviations, i.e. can be distributed according to the so-called standard normal distribution. Consequently, µ represents the vector of ected values of x. Furthermore, matrix A has to satisfy V ij = lim M = A i A jl,l= m= lim ( ( x (m i µ i x (m j µ j = lim M M m= ξ (m ξ (m l = A i A jl δ l =,l= where V ij are the absolute covariances of the parameters x i and x j. Equality lim M m= m=,l= A i ξ (m A jl ξ (m l A i A j, (3 ξ (m ξ (m l = δ l (4 Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0

3 409.3 holds since the left hand side equals variance ( = l or covariance ( l of independent variables by definition assuming samples ξ (m are representative for the standard normal distribution. Eq. (3 defines n quadratic equations for n unnowns (the elements of the matrix A. When system Eq. (3 is solved, unlimited number of normally distributed parameter set x samples can be produced, taing into account all correlations between individual parameters. Probably the easiest way to find a solution A (any solution is sufficient for our purposes since though they all produce different random samples x (m, the latter all obey the same distribution is to first write Eq. (3 in matrix form: ( V ij = A i A j = A i A T = ( AA T = V = j ij AAT. (5 Since the covariance matrix V is real symmetric (because, of course, parameter i is correlated to parameter j as much as j is correlated to i, it is diagonalizable [4]: V = QDQ T, (6 where D is a diagonal matrix of the eigenvalues, and Q is an orthogonal matrix of the eigenvectors of V. Then, because Q T = Q and D T = D, A = QD / Q T (7 is a solution of Eq. (5. Here, we tae advantage of the fact that square root of a diagonal matrix is simply a matrix with square roots of diagonal elements. Furthermore, the diagonal elements of D are non-negative since the variance of arbitrary linear combination of initial parameters has to be non-negative. Hence, D / is real. To summarize, matrix Eq. (5 may have several solutions, one of them being the square root of matrix V : A = V /. (8 With licitly given A and µ, arbitrary random multivariate normally distributed samples can be produced by employing Eq. (.. Log-normal distribution When dealing with log-normally distributed variables, it is very useful to always bear in mind that their natural logarithms are normally distributed variables. In the following discussion, this property will be used several times. Let us start with independent random variables ξ (m i, i =,..., n, this time log-normally are normally distributed and so are all of their linear combi- distributed. Then variables ln ξ (m i nations: ln x (m i = j= A ij ln ξ (m j + µ i, (9 for practical reasons denoted by ln x (m i. Obviously, ( x (m i = A i ln ξ (m + µ i ( A i ln ξ (m ( ξ (m Ai (0 Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0

4 409.4 are again log-normal variables. Analogously to normally distributed case, we may assume all ln ξ i have zero means and unit standard deviations without limiting generality. Coefficients A ij have to be chosen in the way to correctly tae into account all covariances V ij : V ij = lim M = lim M m= ( M m= ( ( x (m i x i x (m j x j x (m i x (m j x j m= x (m i x i m= x (m j + x i x j. ( Let p(x = ( (ln x, x > 0 ( πx be the standard single variable log-normal probability distribution function. Note that ( x A x A p(xdx = (ln x e Az dx = ( z 0 0 πx π ( A ( (z A A = ( dz = π From that, it follows dz. (3 x i = lim M 0 m= x (m i = lim ξ A i p(ξ dξ M m= ( A i e µ i ( ξ (m Ai (4 and x i x j = lim e µ j M m= 0 x (m i x (m j = lim ξ A i+a j p(ξ dξ e µ j ( e µ i e µ j ξ (m Ai ( M m= ( [Ai + A j ] ξ (m Aj, (5 Inserting Eqs. (4 and (5 into Eq. ( we obtain ( V ij e µ [Ai + A j j ] x i x j ( ( e µ j A i A j (A i A j x i x j ( = x i x j A i A j x i x j. (6 Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0

5 Finally ( Vij A i A j = ln x i x j + = Ṽij, (7 where a new matrix Ṽ, which in fact is the absolute covariance matrix for the set of parameters ln x, is defined. Analogously to normal distribution case (see Eq. (5, we may again write a matrix equation Ṽ = AA T. (8 The coefficients of the matrix A can be calculated in exactly the same way as for the normal distribution, i.e. by calculating the square root of a matrix, this time Ṽ instead of the original covariance matrix V. Note that Ṽ is still symmetric and diagonalizable. Again, with licitly given A and µ, arbitrary random multivariate log-normally distributed samples can be produced by employing Eq. ( followed by the simple operation of..3 Combinations of normal and log-normal distribution In reality, cases may occur where some of the correlated parameters are normally while others are log-normally distributed (e.g. resonance energies are normally while the resonance widths are log-normally distributed. On any fixed confidence interval, log-normal converges to normal distribution in the limit of small relative uncertainties. Therefore, multivariate log-normal distribution can be assigned to the parameter vector x = x + X, (9 where X i + x i x i for normally distributed x i and X i = 0 for log-normally distributed x i, and samples of x produced as described in Section.. From these, final samples x can trivially be obtained. Arbitrary precision can obviously be achieved by increasing X i. Note that the covariance matrices of x and x are exactly the same. 3 REALISTIC EXAMPLE Since the correlated sampling method has been proven to be exact, it would in principle not be necessary to be supported by any concrete example whatsoever. However, the following instance will be given in order to illustrate the applicability of the method in reactor calculations. The random sampling method has been tested on two nuclear data evaluations for 55 Mn with different covariance data for resonance parameters, one with small relative uncertainties (less than 0% and full correlation matrix, the other with large relative uncertainties (up to 50% but no cross-resonance correlations. For the first evaluation, resonance parameters can very accurately be described by normal distribution, i.e. the diagonalization method can be employed with accuracy well within % and virtually no negative values. However, for the second evaluation, the correlated sampling method is to be used to avoid problems with negative parameter values and biased distributions. Table shows the comparison between the diagonalization method (diag. using different distributions for sampling of independent linear combinations of parameters in diagonal Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0

6 409.6 space and the correlated sampling method for all 55 Mn resonance gamma widths p in the resolved resonance range of the TENDL-00 evaluation [5] (accurate description of the input data can be found in [7]. The fraction of unphysical negative values is significant for normal distribution due to large relative uncertainties of some resonance widths. This fraction can be reduced by replacing the normal distribution with e.g. log-normal for sampling of individual independent linear combinations in diagonal space. However, when transformed bac into the original parameter space, the non-gaussian distributions become distrorted again introducing some negative values (a linear combination of non-negative log-normal variables can in principle be negative. Standardized mean values p p 0 σ p and relative standard deviations σ p p 0 σp the parameters are within the statistical errors around the ected values of 0 and, respectively, for both distributions. Negative values may be avoided by employing normal distribution in logarithmic parameter space ( log-space at an ense of producing large biases in mean values and standard deviations of the sampled parameters. As ected, the correlated sampling method (corr. sampl. with multivariate log-normal distribution produces samples with no negative parameters and no biases in the mean values and standard deviations. of Table : Negative values and biases for different sampling methods and distributions. Evaluation TENDL-00 [5], 55 Mn gamma widths, averaged over 000 random samples. Method/distribution p < 0 [%] p p 0 σ p σ p p 0 σp diag./normal ± ± log-normal ± ± log space ± ± corr. sampl./log-normal ± ± The generated random samples have further been used to estimate the uncertainty in the capture resonance integral for 55 Mn as a function of self-shielding level [6]. All relevant results are published in [7]. 4 CONCLUSIONS Nuclear data are usually given in the form of ected values and covariance matrices. The corresponding distributions are multivariate normal or log-normal. The simple diagonalization method for random sampling is exact for normal distribution, only. Alternative approach, described in this paper, is to calculate the parameters from independently sampled variables conserving the ected values, uncertainties and correlations. The dependence of parameters on the sampled variables is a function of the parameter distributions. Exact analytical ressions for multivariate normal and log-normal distributions have been derived. Also, employing a simple numerical tric, arbitrary combinations of coupled normally and log-normally distributed parameters can be sampled to any requested precision. Such combination of distribution appears for example when dealing with resonance parameters: the resonance energies are normally while the inherently positive resonance widths are log-normally distributed. The method has successfully been used to estimate the contribution of the uncertainty of the resonance parameters to the uncertainty in 55 Mn capture resonance integral for different levels of self-shielding. However, the method is general and can be used for any log-normally distributed set of correlated parameters. Therefore, its applicability reaches well beyond resonance parameters, nuclear data, and nuclear engineering. Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0

7 ACKNOWLEDGMENTS The research was partly supported by Slovenian Research Agency ARRS under contracts No and REFERENCES [] E. T. Jaynes, Prior Probabilities, IEEE Trans. Syst. Sci. Cyb., 4, 968, pp [] D. L. Smith, D. G. Naberejnev, Confidence Intervals for the Lognormal Probability Distribution, Nucl. Instr. Meth. A, 58, 004, pp [3] D. L. Smith, D. G. Naberejnev, L. A. Van Wormerb, Large Errors and Severe Conditions, Nucl. Instr. Meth. A, 488, 00, pp [4] S. Lipschutz, Linear Algebra, McGraw-Hill, New Yor, 974, pp. 88. [5] D. Rochman, A.J. Koning, TENDL-00: Reaching completeness and accuracy, OECD/NEA, JEFF document JEF/DOC-349, November 00; TENDL-00 nuclear data library, available at: [6] A. Trov, G. Žerovni, L. Snoj, M. Ravni, On the Self-Shielding Factors in Neutron Activation Analysis, Nucl. Instr. Meth. A, 60, 009, pp [7] G. Žerovni, A. Trov, R. Capote, D. Rochman, Influence of Resonance Parameters Correlations on the Resonance Integral Uncertainty; 55 Mn Case, Nucl. Instr. Meth. A, 63, 0, pp Proceedings of the International Conference Nuclear Energy for New Europe 0, Bovec, Slovenia, Sept.-5, 0