Fractal analysis of surface roughness by using spatial data

We develop fractal methodology for data taking the form of surfaces. An advantage of fractal analysis is that it partitions roughness characteristics of a surface into a scale-free component (fractal dimension) and properties that depend purely on scale. Particular emphasis is given to anisotropy where we show that, for many surfaces, the fractal dimension of line transects across a surface must either be constant in every direction or be constant in each direction except one. This virtual direction invariance of fractal dimension provides another canonical feature of fractal analysis, complementing its scale invariance properties and enhancing its attractiveness as a method for summarizing properties of roughness. The dependence of roughness on direction may be explained in terms of scale rather than dimension and can vary with orientation. Scale may be described by a smooth periodic function and may be estimated nonparametrically. Our results and techniques are applied to analyse data on the surfaces of soil and plastic food wrapping. For the soil data, interest centres on the effect of surface roughness on retention of rain-water, and data are recorded as a series of digital images over time. Our analysis captures the way in which both the fractal dimension and the scale change with rainfall, or equivalently with time. The food wrapping data are on a much finer scale than the soil data and are particularly anisotropic. The analysis allows us to determine the manufacturing process which produces the smoothest wrapping, with least tendency for micro-organisms to adhere.     

Fractal Function Estimation via Wavelet Shrinkage

In scientific studies objects are often very rough. Mathematically these rough objects are modelled by fractal functions, and the fractal dimension is usually used to measure their roughness. This paper investigates fractal function estimation by wavelet shrinkage. It is shown that wavelet shrinkage can estimate fractal functions with their fractal dimensions virtually preserved.     

基于分形理论的区域经济差异分析——以河南省为例

以河南省为例,首先,运用库茨涅茨不平衡系数并通过对其分解来分析河南省各地区经济发展相对差异的变化情况,得到河南省各区域的经济发展确实存在较大差异的结果;然后,又从分形理论中的位序-规模法则和分形维数的角度对河南省区域发展差异进一步分析,得出河南省城市体系等级结构具有分形特征;最后,结合分析结果,从分形理论的角度给出一些可行性的建议。     

An example of applying the asymptotic quasi-likelihood to dimension estimation for random spatial patterns

With reference to a specific example of a random spatial fractal and the modified box-counting method of dimension estimation, this paper aims to examine firstly the estimation of pointwise dimension via modification of the box-counting procedure, secondly the regression inspired estimation procedure, including generalised least squares and, finally, to develop a new estimation procedure – the asymptotic quasi-likelihood method – for the estimation of pointwise dimension. The main focus is on practicality – to arrive at an estimation method which is easy to use and robust.     

基于RS分析的中国债券市场分形特征研究

文章详细讨论了RS分析方法、概率空间分形维的测度,通过对上证国债指数和上证企债指数的RS分析,得到了两序列赫斯特指数和非周期循环长度,进一步得出中国债券市场为分形市场的结论.     

On the complex dynamics of functional-coefficients nonlinear autoregressive time series models

In this article, a new chaotic functional-coefficient nonlinear autoregressive time series model is formulated. Asymptotic stability of the equilibria of the skeleton is studied. Complex dynamics of the proposed model are investigated by means of the bifurcations, time series diagrams, and phase portraits. The effects of noise intensity on its dynamics and the intermittency phenomenon are also discussed via simulation. Two chaotic indicators, namely, the fractal dimension and the Lyapunov exponent methods are investigated for the model.     

Response and predictor folding to counter symmetric dependency in dimension reduction

In the regression setting, dimension reduction allows for complicated regression structures to be detected via visualisation in a low‐dimensional framework. However, some popular dimension reduction methodologies fail to achieve this aim when faced with a problem often referred to as symmetric dependency. In this paper we show how vastly superior results can be achieved when carrying out response and predictor transformations for methods such as least squares and sliced inverse regression. These transformations are simple to implement and utilise estimates from other dimension reduction methods that are not faced with the symmetric dependency problem. We highlight the effectiveness of our approach via simulation and an example. Furthermore, we show that ordinary least squares can effectively detect multiple dimension reduction directions. Methods robust to extreme response values are also considered.     

Estimating the structural dimension of regressions via parametric inverse regression

A new estimation method for the dimension of a regression at the outset of an analysis is proposed. A linear subspace spanned by projections of the regressor vector X , which contains part or all of the modelling information for the regression of a vector Y on X , and its dimension are estimated via the means of parametric inverse regression. Smooth parametric curves are fitted to the p inverse regressions via a multivariate linear model. No restrictions are placed on the distribution of the regressors. The estimate of the dimension of the regression is based on optimal estimation procedures. A simulation study shows the method to be more powerful than sliced inverse regression in some situations.     

Special Invited Paper: Dimension Reduction and Visualization in Discriminant Analysis (with discussion)

This paper discusses visualization methods for discriminant analysis. It does not address numerical methods for classification per se, but rather focuses on graphical methods that can be viewed as pre-processors, aiding the analyst's understanding of the data and the choice of a final classifier. The methods are adaptations of recent results in dimension reduction for regression, including sliced inverse regression and sliced average variance estimation. A permutation test is suggested as a means of determining dimension, and examples are given throughout the discussion.     

Linear censored quantile regression: A novel minimum-distance approach

In this article, we investigate a new procedure for the estimation of a linear quantile regression with possibly right-censored responses. Contrary to the main literature on the subject, we propose in this context to circumvent the formulation of conditional quantiles through the so-called “check” loss function that stems from the influential work of Koenker and Bassett (1978). Instead, our suggestion is here to estimate the quantile coefficients by minimizing an alternative measure of distance. In fact, our approach could be qualified as a generalization in a parametric regression framework of the technique consisting in inverting the conditional distribution of the response given the covariates. This is motivated by the knowledge that the main literature for censored data already relies on some nonparametric conditional distribution estimation as well. The ideas of effective dimension reduction are then exploited in order to accommodate for higher dimensional settings as well in this context. Extensive numerical results then suggest that such an approach provides a strongly competitive procedure to the classical approaches based on the check function, in fact both for complete and censored observations. From a theoretical prospect, both consistency and asymptotic normality of the proposed estimator for linear regression are obtained under classical regularity conditions. As a by-product, several asymptotic results on some “double-kernel” version of the conditional Kaplan–Meier distribution estimator based on effective dimension reduction, and its corresponding density estimator, are also obtained and may be of interest on their own. A brief application of our procedure to quasar data then serves to further highlight the relevance of the latter for quantile regression estimation with censored data.     

The efficacy of the correlation dimension technique in detecting determinism in small samples

This paper presents the results of a simulation study investigating the efficacy of the correlation dimension technique in detecting the presence of deterministic structure in small data sets. The results show that for some very simple structures a mere 100 data points suffice; but for some slightly more complicated structures, which arguably are still simple relative to real-world structures, even 2500 data points are inadequate. These results suggest that substantial caution be used in interpreting negative results on empirical data.     

3D dynamic forward modeling of polar plumes using Hidden Markov Trees

The polar plumes are very fine structures of the solar K-corona lying around the poles and visible during the period of minimum of activity. These poorly known structures are linked with the solar magnetic field and with numerous coronal phenomena such as the fast solar wind and the coronal holes. The SOHO space mission has provided some continuous observations to high cadence (each 10 min). From these observations the images of the K-corona have been derived and preprocessed with an adapted anisotropic filtering. Then, a peculiar type of sinogram called Time Intensity Diagram (TID) has been built. It is adapted to the evolution of polar plumes with the time. A multiresolution wavelet analysis of the TID has then revealed that the spatial distribution of the polar plumes as well as their temporal evolution were fractal. The present study consists in simulating polar plumes by forward modeling techniques in order to validate several assumptions concerning their nature and their temporal evolution. Our work involves two main steps. The first one concerns the simulation of polar plumes and the computation of their corresponding TID. The second one concerns the estimation of analysis criteria in order to compare the original TID and the simulated ones. Static and dynamic models were both used in order to confirm the fractal behavior of the temporal evolution of plumes. The most recent and promising model is based on a Hidden Markov Tree. It allows us to control the fractal parameters of the TID.     

A new process for modeling heartbeat signals during exhaustive run with an adaptive estimator of its fractal parameters

This paper is devoted to a new study of the fractal behavior of heartbeats during a marathon. Such a case is interesting since it allows the examination of heart behavior during a very long exercise in order to reach reliable conclusions on the long-term properties of heartbeats. Three points of this study can be highlighted. First, the whole race heartbeats of each runner are automatically divided into several stages where the signal is nearly stationary and these stages are detected with an adaptive change points detection method. Secondly, a new process called the locally fractional Gaussian noise (LFGN) is proposed to fit such data. Finally, a wavelet-based method using a specific mother wavelet provides an adaptive procedure for estimating low frequency and high frequency fractal parameters as well as the corresponding frequency bandwidths. Such an estimator is theoretically proved to converge in the case of LFGNs, and simulations confirm this consistency. Moreover, an adaptive chi-squared goodness-of-fit test is also built, using this wavelet-based estimator. The application of this method to marathon heartbeat series indicates that the LFGN fits well data at each stage and that the low frequency fractal parameter increases during the race. A detection of a too large low frequency fractal parameter during the race could help prevent the too frequent heart failures occurring during marathons.     

Spatio-temporal improvised explosive device monitoring: improving detection to minimise attacks

The improvised explosive device (IED) is a weapon of strategic influence on today's battlefield. IED detonations occur predominantly on roads, footpaths, or trails. Therefore, locations are best described when constrained to the road network, and some spaces on the network are more dangerous at specific times of the day. We propose a statistical model that reduces the spatial location to one dimension and uses a cyclic time as a second dimension. Based on the Poisson process methodology, we develop normalised, inhomogeneous, bivariate intensity functions measuring the threat of attack to support resourcing decisions. A simulation and an analysis of attacks on a main supply route in Baghdad are given to illustrate the proposed methods. Additionally, we provide an overview of the growing demand for the analysis efforts in support of operations in Afghanistan and Iraq, and provide an extensive literature review of developments in counter-IED analysis.     

Gaussian Regularized Sliced Inverse Regression

Sliced Inverse Regression (SIR) is an effective method for dimension reduction in high-dimensional regression problems. The original method, however, requires the inversion of the predictors covariance matrix. In case of collinearity between these predictors or small sample sizes compared to the dimension, the inversion is not possible and a regularization technique has to be used. Our approach is based on a Fisher Lecture given by R.D. Cook where it is shown that SIR axes can be interpreted as solutions of an inverse regression problem. We propose to introduce a Gaussian prior distribution on the unknown parameters of the inverse regression problem in order to regularize their estimation. We show that some existing SIR regularizations can enter our framework, which permits a global understanding of these methods. Three new priors are proposed leading to new regularizations of the SIR method. A comparison on simulated data as well as an application to the estimation of Mars surface physical properties from hyperspectral images are provided.     

Binary discrimination methods for high-dimensional data with a geometric representation

Four binary discrimination methods are studied in the context of high-dimension, low sample size data with an asymptotic geometric representation, when the dimension increases while the sample sizes of the classes are fixed. We show that the methods support vector machine, mean difference, distance-weighted discrimination, and maximal data piling have the same asymptotic behavior as the dimension increases. We study the consistent, inconsistent, and strongly inconsistent cases in terms of angles between the normal vectors of the separating hyperplanes of the methods and the optimal direction for classification. A simulation study is done to assess the theoretical results.     

A two-sample test for mean functions with increasing number of projections

We propose a test for equality of two means when data are functions and obtain the asymptotic properties of the test statistic as data dimension increases with the sample size. We also derive the asymptotic power of the test under some local alternatives and show that the test statistic is root-n consistent. A simulation study is conducted to evaluate the performance of the test numerically and to compare the proposed test with other existing four popular tests.     

上海股票市场分形特征的实证研究

以有效市场假说为基础的现代资本市场理论被越来越多的实践证明与现实情况不符,而分形理论则考虑到资本市场的复杂性和EMH的缺陷,以非线性范式为分析基础,解释了有效市场理论无法解释的许多市场现象,为更深入地分析资本市场提供了新的思路和方法。文章以上海股票市场为例,用分形理论对上海股市的分形特征进行了研究,其结果表明上海股票市场具有明显的分形特征。     

Inferences on a linear combination of K multivariate normal mean vectors

In this paper, the hypothesis testing and confidence region construction for a linear combination of mean vectors for K independent multivariate normal populations are considered. A new generalized pivotal quantity and a new generalized test variable are derived based on the concepts of generalized p-values and generalized confidence regions. When only two populations are considered, our results are equivalent to those proposed by Gamage et al. [Generalized p-values and confidence regions for the multivariate Behrens–Fisher problem and MANOVA, J. Multivariate Aanal. 88 (2004), pp. 117–189] in the bivariate case, which is also known as the bivariate Behrens–Fisher problem. However, in some higher dimension cases, these two results are quite different. The generalized confidence region is illustrated with two numerical examples and the merits of the proposed method are numerically compared with those of the existing methods with respect to their expected areas, coverage probabilities under different scenarios.     

Detecting the Dimensionality for Principal Components Model

A crucial issue for principal components analysis (PCA) is to determine the number of principal components to capture the variability of usually high-dimensional data. In this article the dimension detection for PCA is formulated as a variable selection problem for regressions. The adaptive LASSO is used for the variable selection in this application. Simulations demonstrate that this approach is more accurate than existing methods in some cases and competitive in some others. The performance of this model is also illustrated using a real example.