This paper introduces Latin supercube sampling (LSS) for very high dimensional simulations, such as arise in particle transport, finance and queuing. LSS is developed as a combination of two widely used methods: Latin hypercube sampling (LHS), and Quasi-Monte Carlo (QMC). In LSS, the input variables are grouped into subsets, and a lower dimensional QMC method is used within each subset. The QMC points are presented in random order within subsets. QMC methods have been observed to lose effectiveness in high dimensional problems. This paper shows that LSS can extend the benefits of QMC to much higher dimensions, when one can make a good grouping of input variables. Some suggestions for grouping variables are given for the motivating examples. Even a poor grouping can still be expected to do as well as LHS. The paper also extends LHS and LSS to infinite dimensional problems. The paper includes a survey of QMC methods, randomized versions of them (RQMC) and previous methods for extending Q...

The quasi-Monte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in high dimensional problems this benefit might not appear at feasible sample sizes. Substantial improvements from quasi-Monte Carlo integration have, however, been reported for problems such as the valuation of mortgage-backed securities, in dimensions as high as 360. We believe that this is due to a lower effective dimension of the integrand in those cases. This paper defines the effective dimension and shows in examples how the effective dimension may be reduced by using a Brownian bridge representation. 1 Introduction Simulation is often the only effective numerical method for the accurate valuation of securities whose value depends on the whole trajectory of interest Mathematics Departmen...

. A general theory on rates of convergence in multiple regression is developed, where the regression function is modeled as a member of an arbitrary linear function space (called a model space), which may be finite- or infinite-dimensional. A least squares estimate restricted to some approximating space, which is in fact a projection, is employed. The error in estimation is decomposed into three parts: variance component, estimation bias, and approximation error. The contributions to the integrated squared error from the first two parts are bounded in probability by Nn=n, where Nn is the dimension of the approximating space, while the contribution from the third part is governed by the approximation power of the approximating space. When the regression function is not in the model space, the projection estimate converges to its best approximation. The theory is applied to a functional ANOVA model, where the multivariate regression function is modeled as a specified sum of a constant te...

This paper introduces the dimension distribution for a square integrable function f on [0; 1]^s. The dimension distribution is used to relate several definitions of the effective dimension of a function. Functions of low effective dimension can be easy to integrate numerically.

Abstract — New families of Fisher information and entropy power inequalities for sums of independent random variables are presented. These inequalities relate the information in the sum of n independent random variables to the information contained in sums over subsets of the random variables, for an arbitrary collection of subsets. As a consequence, a simple proof of the monotonicity of information in central limit theorems is obtained, both in the setting of i.i.d. summands as well as in the more general setting of independent summands with variancestandardized sums. Index Terms — Central limit theorem; entropy power; information inequalities. I.

The mean dimension of a function is defined as a certain weighted combination of its ANOVA mean squares. This paper presents some new identities relating this mean dimension, and some similarly defined higher moments, to the variable importance measures of Sobol' (1993). The methods are applied to study some functions arising in the bootstrap, in computational finance, and in extreme value theory.

Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the author. Logarithmic CMOS image sensors capture high dynamic range scenes without saturation or loss of perceptible detail but problems exist with image quality. This thesis develops and applies methods of modelling and calibration to understand and improve the fixed pattern noise (FPN) and colour rendition of logarithmic imagers. Chapter 1 compares CCD and CMOS image sensors and, within the latter category, compares linear and logarithmic pixel designs. Chapter 2 reviews the literature on multilinear algebra, unifying and extending approaches for analytic and numeric manipulation of multi-index arrays, which are the generalisation of scalars, vectors and matrices. Chapter 3 defines and solves the problem of multilinear regression with linear constraints for the calibration of a sensor array, permitting models with linear relationships of parameters

. Consider repeated events of multiple kinds that occur according to a right-continuous semi-Markov process whose transition rates are influenced by one or more time-dependent covariates. The logarithms of the intensities of the transitions from one state to another are modeled as members of an arbitrary linear function space, which may be finite- or infinite-dimensional. Maximum likelihood estimates are used, where the maximizations are taken over suitably chosen finite-dimensional approximating spaces. It is shown that the L 2 rates of convergence of the maximum likelihood estimates are determined by the approximationpower and dimension of the approximating spaces. The theory is applied to a functional ANOVA model, where the logarithms of the intensities are approximated by functions having the form of a specified sum of a constant term, main effects (functions of one variable), and interaction terms (functions of two or more variables). It is shown that the curse of dimensionality c...

The analysis of variance is now often applied to functions defined on the unit cube, where it serves as a tool for the exploratory analysis of func-tions. The mean dimension of a function, defined as a natural weighted combination of its ANOVA mean squares, provides one measure of how hard or easy the function is to integrate by quasi-Monte Carlo sampling. This paper presents some new identities relating the mean dimension, and some analogously defined higher moments, to the variable importance measures of Sobol ’ (1993). As a result we are able to measure the mean dimension of certain functions arising in computational finance. We pro-duce an unbiased and non-negative estimate of the variance contribution of the highest order interaction, which avoids the cancellation problems of previous estimates. In an application to extreme value theory, we find 1 among other things, that the minimum of d independent U[0, 1] random variables has a mean dimension of 2(d +1)/(d +3).

The generalized wordlength pattern (GWLP) introduced by Xu and Wu (2001) for an arbitrary fractional factorial design allows one to extend the use of the minimum aberration criterion to such designs. Ai and Zhang (2004) defined the J-characteristics of a design and showed that they uniquely determine the design. While both the GWLP and the J-characteristics require indexing the levels of each factor by a cyclic group, we see that the definitions carry over with appropriate changes if instead one uses an arbitrary abelian group. This means that the original definitions rest on an arbitrary choice of group structure. We show that the GWLP of a design is independent of this choice, but that the J-characteristics are not. We briefly discuss some implications of these results. Key words. Fractional factorial design; group character; Hamming weight; multiset; orthogonal array