Andrew T. A. Wood

Simulation of Stationary Gaussian Processes in [0, 1] d

Abstract A method for simulating a stationary Gaussian process on a fine rectangular grid in [0, 1]d ⊂ℝd is described. It is assumed that the process is stationary with respect to translations of ℝd, but the method does not require the process to be isotropic. As with some other approaches to this simulation problem, our procedure uses discrete Fourier methods and exploits the efficiency of the fast Fourier transform. However, the introduction of a novel feature leads to a procedure that is exact in principle when it can be applied. It is established that sufficient conditions for it to be possible to apply the procedure are (1) the covariance function is summable on ℝd, and (2) a certain spectral density on the d-dimensional torus, which is determined by the covariance function on ℝd, is strictly positive. The procedure can cope with more than 50,000 grid points in many cases, even on a relatively modest computer. An approximate procedure is also proposed to cover cases where it is not feasible to apply ...

Simulation of the von Mises Fisher distribution

Ulrichs (1984) proposal for simulating unit vectors from the von Mises-Fisher distribution is discussed and an alternative specification of the algorithm is given. Then we describe an application to the simulation of the von Mises-Fisher matrix distribution for rotations of R 3 or,equivalently, the Bingham distribution on the unit sphere in R 4The same idea also leadsto a procedure for simulating the Bingham distribution on the unit sphere in R q when q > 4.

Transcriptional Dynamics of Two Seed Compartments with Opposing Roles in Arabidopsis Seed Germination

Gene expression profiling in two seed compartments uncovers two transcriptional phases during seed germination that are separated by testa rupture. Seed germination is a critical stage in the plant life cycle and the first step toward successful plant establishment. Therefore, understanding germination is of important ecological and agronomical relevance. Previous research revealed that different seed compartments (testa, endosperm, and embryo) control germination, but little is known about the underlying spatial and temporal transcriptome changes that lead to seed germination. We analyzed genome-wide expression in germinating Arabidopsis (Arabidopsis thaliana) seeds with both temporal and spatial detail and provide Web-accessible visualizations of the data reported (vseed.nottingham.ac.uk). We show the potential of this high-resolution data set for the construction of meaningful coexpression networks, which provide insight into the genetic control of germination. The data set reveals two transcriptional phases during germination that are separated by testa rupture. The first phase is marked by large transcriptome changes as the seed switches from a dry, quiescent state to a hydrated and active state. At the end of this first transcriptional phase, the number of differentially expressed genes between consecutive time points drops. This increases again at testa rupture, the start of the second transcriptional phase. Transcriptome data indicate a role for mechano-induced signaling at this stage and subsequently highlight the fates of the endosperm and radicle: senescence and growth, respectively. Finally, using a phylotranscriptomic approach, we show that expression levels of evolutionarily young genes drop during the first transcriptional phase and increase during the second phase. Evolutionarily old genes show an opposite pattern, suggesting a more conserved transcriptome prior to the completion of germination.

Saddlepoint Approximations to the CDF of Some Statistics with Nonnormal Limit Distributions

Abstract In standard saddlepoint approximations to the cumulative distribution function of a random variable, the normal distribution has appeared to play a special role. In this article we consider what happens when the normal “base” distribution is replaced by an arbitrary base distribution. Generalized versions of several standard formulas, are presented. The choice of a chi-squared base or an inverse Gaussian base is then considered in detail. The generalized approximations are compared in two examples: a linear combination of chi-squared variables and the first passage time distribution for a random walk. The former example considers approximations using the chi-squared base that are slightly more accurate than their normal-based counterparts. In the latter example, approximations based on the inverse Gaussian are considerably more accurate than their normal-based counterparts.

Is my model too complex? Evaluating model formulation using model reduction

While mechanistic models tend to be detailed, they are less detailed than the real systems they seek to describe, so judgements are being made about the appropriate level of detail within the process of model development. These judgements are difficult to test, consequently it is easy for models to become over-parameterised, potentially increasing uncertainty in predictions. The work we describe is a step towards addressing these difficulties. We propose and implement a method which explores a family of simpler models obtained by replacing model variables with constants (model reduction by variable replacement). The procedure iteratively searches the simpler model formulations and compares models in terms of their ability to predict observed data, evaluated within a Bayesian framework. The results can be summarised as posterior model probabilities and replacement probabilities for individual variables which lend themselves to mechanistic interpretation. This provides powerful diagnostic information to support model development, and can identify areas of model over-parameterisation with implications for interpretation of model results. We present the application of the method to 3 example models. In each case reduced models are identified which outperform the original full model in terms of comparisons to observations, suggesting some over-parameterisation has occurred during model development. We argue that the proposed approach is relevant to anyone involved in the development or use of process based mathematical models, especially those where understanding is encoded via empirically based relationships.

A Guideline to Family-Wide Comparative State-of-the-Art Quantitative RT-PCR Analysis Exemplified with a Brassicaceae Cross-Species Seed Germination Case Study

Developmental processes like seed germination are characterized by massive transcriptome changes. This study compares seed transcriptome data sets of different Brassicaceae to identify stable expressed reference genes for cross-species quantitative RT-PCR normalization. A workflow is presented for improving RNA quality, quantitative RT-PCR performance, and normalization when analyzing expression changes across species. Comparative biology includes the comparison of transcriptome and quantitative real-time RT-PCR (qRT-PCR) data sets in a range of species to detect evolutionarily conserved and divergent processes. Transcript abundance analysis of target genes by qRT-PCR requires a highly accurate and robust workflow. This includes reference genes with high expression stability (i.e., low intersample transcript abundance variation) for correct target gene normalization. Cross-species qRT-PCR for proper comparative transcript quantification requires reference genes suitable for different species. We addressed this issue using tissue-specific transcriptome data sets of germinating Lepidium sativum seeds to identify new candidate reference genes. We investigated their expression stability in germinating seeds of L. sativum and Arabidopsis thaliana by qRT-PCR, combined with in silico analysis of Arabidopsis and Brassica napus microarray data sets. This revealed that reference gene expression stability is higher for a given developmental process between distinct species than for distinct developmental processes within a given single species. The identified superior cross-species reference genes may be used for family-wide comparative qRT-PCR analysis of Brassicaceae seed germination. Furthermore, using germinating seeds, we exemplify optimization of the qRT-PCR workflow for challenging tissues regarding RNA quality, transcript stability, and tissue abundance. Our work therefore can serve as a guideline for moving beyond Arabidopsis by establishing high-quality cross-species qRT-PCR.

Improved Pivotal Methods for Constructing Confidence Regions with Directional Data

Abstract The importance of pivoting is well established in the context of nonparametric confidence regions. It ensures enhanced coverage accuracy. However, pivoting for directional data cannot be achieved simply by rescaling. A somewhat cumbersome pivotal method, which involves passing first into a space of higher dimension, has been developed by Fisher and Hall for samples of unit vectors. Although that method has some advantages over nonpivotal techniques, it does suffer from certain drawbacks—in particular, the operation of passing to a higher dimension. Here we suggest alternative pivotal approaches, the implementation of which does not require us to increase the intrinsic dimension of the data and which in practice seem to achieve greater coverage accuracy. These methods are of two types: new pivotal bootstrap techniques and techniques that exploit the “implicit pivotalness” of the empirical likelihood algorithm. Unlike the method proposed by Fisher and Hall, these methods are also applicable to axia...

Simulation of stationary Gaussian vector fields

In earlier work we described a circulant embedding approach for simulating scalar-valued stationary Gaussian random fields on a finite rectangular grid, with the covariance function prescribed. Here, we explain how the circulant embedding approach can be used to simulate Gaussian vector fields. As in the scalar case, the simulation procedure is theoretically exact if a certain non-negativity condition is satisfied. In the vector setting, this exactness condition takes the form of a nonnegative definiteness condition on a certain set of Hermitian matrices. The main computational tool used is the Fast Fourier Transform. Consequently, when implemented appropriately, the procedure is highly efficient, in terms of both CPU time and storage.

Multivariate L-estimation

In one dimension, order statistics and ranks are widely used because they form a basis for distribution free tests and some robust estimation procedures. In more than one dimension, the concept of order statistics and ranks is not clear and several definitions have been proposed in the last years. The proposed definitions are based on different concepts of depth. In this paper, we define a new notion of order statistics and ranks for multivariate data based on density estimation. The resulting ranks are invariant under affinc transformations and asymptotically distribution free. We use the corresponding order statistics to define a class of multivariate estimators of location that can be regarded as multivariate L-estimators. Under mild assumptions on the underlying distribution, we show the asymptotic normality of the estimators. A modification of the proposed estimates results in a high breakdown point procedure that can deal with patches of outliers. The main idea is to order the observations according to their likelihoodf(X1),...,f(Xn). If the densityf happens to be cllipsoidal, the above ranking is similar to the rankings that are derived from the various notions of depth. We propose to define a ranking based on a kernel estimate of the densityf. One advantage of estimating the likelihoods is that the underlying distribution does not need to have a density. In addition, because the approximate likelihoods are only used to rank the observations, they can be derived from a density estimate using a fixed bandwidth. This fixed bandwidth overcomes the curse of dimensionality that typically plagues density estimation in high dimension.

Pivotal Bootstrap Methods for k-Sample Problems in Directional Statistics and Shape Analysis

We propose a novel bootstrap hypothesis testing approach for the problem of testing a null hypothesis of a common mean direction, mean polar axis, or mean shape across several populations of real unit vectors (the directional case) or complex unit vectors (the two-dimensional shape case). Multisample testing problems of this type arise frequently in directional statistics and shape analysis (as in other areas of statistics), but to date there has been relatively little discussion of nonparametric bootstrap approaches to this problem. The bootstrap approach described here is based on a statistic that can be expressed as the smallest eigenvalue of a certain positive definite matrix. We prove that this statistic has a limiting chi-squared distribution under the null hypothesis of equality of means across populations. Although we focus mainly on the version of the statistic in which neither isotropy within populations nor constant dispersion structure across populations is assumed, we explain how to modify the statistic so that either or both of these assumptions can be incorporated. Our numerical results indicate that the bootstrap approach proposed here may be expected to perform well in practice.