Variance estimation with highly stratified sampling designs with unequal probabilities

It is common practice to design a survey with a large number of strata. However, in this case the usual techniques for variance estimation can be inaccurate. This paper proposes a variance estimator for estimators of totals. The method proposed can be implemented with standard statistical packages without any specific programming, as it involves simple techniques of estimation, such as regression fitting.

Variance estimation for systematic sampling from deliberately ordered populations

The systematic sampling (SYS) design (Madow and Madow, 1944) is widely used by statistical offices due to its simplicity and efficiency (e.g., Iachan, 1982). But it suffers from a serious defect, namely, that it is impossible to unbiasedly estimate the sampling variance (Iachan, 1982) and usual variance estimators (Yates and Grundy, 1953) are inadequate and can overestimate the variance significantly (Särndal et al., 1992). We propose a novel variance estimator which is less biased and that can be implemented with any given population order. We will justify this estimator theoretically and with a Monte Carlo simulation study.

Variance estimation with Chao's sampling scheme

We show that the Hájek (Ann. Math Statist. (1964) 1491) variance estimator can be used to estimate the variance of the Horvitz–Thompson estimator when the Chao sampling scheme (Chao, Biometrika 69 (1982) 653) is implemented. This estimator is simple and can be implemented with any statistical packages. We consider a numerical and an analytic method to show that this estimator can be used. A series of simulations supports our findings.

Adjusted jackknife for imputation under unequal probability sampling without replacement

Imputation is commonly used to compensate for item non-response in sample surveys. If we treat the imputed values as if they are true values, and then compute the variance estimates by using standard methods, such as the jackknife, we can seriously underestimate the true variances. We propose a modified jackknife variance estimator which is defined for any without-replacement unequal probability sampling design in the presence of imputation and non-negligible sampling fraction. Mean, ratio and random-imputation methods will be considered. The practical advantage of the method proposed is its breadth of applicability.

Using phylogeny to investigate the origins of the Cape flora: the importance of taxonomic, gene and genome sampling strategies

Phylogenetic methods hold great promise for the reconstruction of the transition from precursor to modern flora and the identification of underlying factors which drive the process. The phylogenetic methods presently used to address the question of the origin of the Cape flora of South Africa are considered here. The sampling requirements of each of these methods, which include dating of diversifications using calibrated molecular trees, sister pair comparisons, lineage through time plots and biogeographical optimizations are reviewed. Sampling of genes, genomes and species are considered. Although increased higher-level studies and increased sampling are required for robust interpretation, it is clear that much progress is already made. It is argued that despite the remarkable richness of the flora, the Cape flora is a valuable model system to demonstrate the utility of phylogenetic methods in determining the history of a modern flora.

Some new three-level orthogonal main effects plans robust to model uncertainty

In this paper, we list some new orthogonal main effects plans for three-level designs for 4, 5 and 6 factors in IS runs and compare them with designs obtained from the existing L-18 orthogonal array. We show that these new designs have better projection properties and can provide better parameter estimates for a range of possible models. Additionally, we study designs in other smaller run-sizes when there are insufficient resources to perform an 18-run experiment. Plans for three-level designs for 4, 5 and 6 factors in 13 to 17 runs axe given. We show that the best designs here are efficient and deserve strong consideration in many practical situations.

Probability density function estimation using orthogonal forward regression

Using the classical Parzen window estimate as the target function, the kernel density estimation is formulated as a regression problem and the orthogonal forward regression technique is adopted to construct sparse kernel density estimates. The proposed algorithm incrementally minimises a leave-one-out test error score to select a sparse kernel model, and a local regularisation method is incorporated into the density construction process to further enforce sparsity. The kernel weights are finally updated using the multiplicative nonnegative quadratic programming algorithm, which has the ability to reduce the model size further. Except for the kernel width, the proposed algorithm has no other parameters that need tuning, and the user is not required to specify any additional criterion to terminate the density construction procedure. Two examples are used to demonstrate the ability of this regression-based approach to effectively construct a sparse kernel density estimate with comparable accuracy to that of the full-sample optimised Parzen window density estimate.

Fully complex-valued radial basis function networks for orthogonal least squares regression

We consider a fully complex-valued radial basis function (RBF) network for regression application. The locally regularised orthogonal least squares (LROLS) algorithm with the D-optimality experimental design, originally derived for constructing parsimonious real-valued RBF network models, is extended to the fully complex-valued RBF network. Like its real-valued counterpart, the proposed algorithm aims to achieve maximised model robustness and sparsity by combining two effective and complementary approaches. The LROLS algorithm alone is capable of producing a very parsimonious model with excellent generalisation performance while the D-optimality design criterion further enhances the model efficiency and robustness. By specifying an appropriate weighting for the D-optimality cost in the combined model selecting criterion, the entire model construction procedure becomes automatic. An example of identifying a complex-valued nonlinear channel is used to illustrate the regression application of the proposed fully complex-valued RBF network.

Sparse kernel density estimator using orthogonal regression based on D-optimality experimental design

A novel sparse kernel density estimator is derived based on a regression approach, which selects a very small subset of significant kernels by means of the D-optimality experimental design criterion using an orthogonal forward selection procedure. The weights of the resulting sparse kernel model are calculated using the multiplicative nonnegative quadratic programming algorithm. The proposed method is computationally attractive, in comparison with many existing kernel density estimation algorithms. Our numerical results also show that the proposed method compares favourably with other existing methods, in terms of both test accuracy and model sparsity, for constructing kernel density estimates.

Parallel Monte Carlo approach for integration of the rendering equation

This paper is addressed to the numerical solving of the rendering equation in realistic image creation. The rendering equation is integral equation describing the light propagation in a scene accordingly to a given illumination model. The used illumination model determines the kernel of the equation under consideration. Nowadays, widely used are the Monte Carlo methods for solving the rendering equation in order to create photorealistic images. In this work we consider the Monte Carlo solving of the rendering equation in the context of the parallel sampling scheme for hemisphere. Our aim is to apply this sampling scheme to stratified Monte Carlo integration method for parallel solving of the rendering equation. The domain for integration of the rendering equation is a hemisphere. We divide the hemispherical domain into a number of equal sub-domains of orthogonal spherical triangles. This domain partitioning allows to solve the rendering equation in parallel. It is known that the Neumann series represent the solution of the integral equation as a infinity sum of integrals. We approximate this sum with a desired truncation error (systematic error) receiving the fixed number of iteration. Then the rendering equation is solved iteratively using Monte Carlo approach. At each iteration we solve multi-dimensional integrals using uniform hemisphere partitioning scheme. An estimate of the rate of convergence is obtained using the stratified Monte Carlo method. This domain partitioning allows easy parallel realization and leads to convergence improvement of the Monte Carlo method. The high performance and Grid computing of the corresponding Monte Carlo scheme are discussed.

A new solution of the rendering equation with stratified Monte Carlo approach

This paper is turned to the advanced Monte Carlo methods for realistic image creation. It offers a new stratified approach for solving the rendering equation. We consider the numerical solution of the rendering equation by separation of integration domain. The hemispherical integration domain is symmetrically separated into 16 parts. First 9 sub-domains are equal size of orthogonal spherical triangles. They are symmetric each to other and grouped with a common vertex around the normal vector to the surface. The hemispherical integration domain is completed with more 8 sub-domains of equal size spherical quadrangles, also symmetric each to other. All sub-domains have fixed vertices and computable parameters. The bijections of unit square into an orthogonal spherical triangle and into a spherical quadrangle are derived and used to generate sampling points. Then, the symmetric sampling scheme is applied to generate the sampling points distributed over the hemispherical integration domain. The necessary transformations are made and the stratified Monte Carlo estimator is presented. The rate of convergence is obtained and one can see that the algorithm is of super-convergent type.

Quasi Monte Carlo approach with uniform separation for solving of the rendering equation, ICNAAM 2007

This paper is directed to the advanced parallel Quasi Monte Carlo (QMC) methods for realistic image synthesis. We propose and consider a new QMC approach for solving the rendering equation with uniform separation. First, we apply the symmetry property for uniform separation of the hemispherical integration domain into 24 equal sub-domains of solid angles, subtended by orthogonal spherical triangles with fixed vertices and computable parameters. Uniform separation allows to apply parallel sampling scheme for numerical integration. Finally, we apply the stratified QMC integration method for solving the rendering equation. The superiority our QMC approach is proved.