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.     

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.     

Accuracy of edge detection methods with local information in speckled imagery

We compare the accuracy of five approaches for contour detection in speckled imagery. Some of these methods take advantage of the statistical properties of speckled data, and all of them employ active contours using B-spline curves. Images obtained with coherent illumination are affected by a noise called speckle, which is inherent to the imaging process. These data have been statistically modeled by a multiplicative model using the G0 distribution, under which regions with different degrees of roughness can be characterized by the value of a parameter. We use this information to find boundaries between regions with different textures. We propose and compare five strategies for boundary detection: three based on the data (maximum discontinuity on raw data, fractal dimension and maximum likelihood) and two based on estimates of the roughness parameter (maximum discontinuity and anisotropic smoothed roughness estimates). In order to compare these strategies, a Monte Carlo experience was performed to assess the accuracy of fitting a curve to a region. The probability of finding the correct edge with less than a specified error is estimated and used to compare the techniques. The two best procedures are then compared in terms of their computational cost and, finally, we show that the maximum likelihood approach on the raw data using the G0 law is the best technique.     

Concordance-based estimation approaches for the optimal sufficient dimension reduction score

To characterize the dependence of a response on covariates of interest, a monotonic structure is linked to a multivariate polynomial transformation of the central subspace (CS) directions with unknown structural degree and dimension. Under a very general semiparametric model formulation, such a sufficient dimension reduction (SDR) score is shown to enjoy the existence, optimality, and uniqueness up to scale and location in the defined concordance probability function. In light of these properties and its single-index representation, two types of concordance-based generalized Bayesian information criteria are constructed to estimate the optimal SDR score and the maximum concordance index. The estimation criteria are further carried out by effective computational procedures. Generally speaking, the outer product of gradients estimation in the first approach has an advantage in computational efficiency and the parameterization system in the second approach greatly reduces the number of parameters in estimation. Different from most existing SDR approaches, only one CS direction is required to be continuous in the proposals. Moreover, the consistency of structural degree and dimension estimators and the asymptotic normality of the optimal SDR score and maximum concordance index estimators are established under some suitable conditions. The performance and practicality of our methodology are also investigated through simulations and empirical illustrations.     

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.     

Second-order inference for generalized least squares

Confidence regions for generalized least squares are commonly derived from a measure of departure calculated on the tangent plane at the MLE or on the tangent plane at the true value; the first gives approximate confidence regions, the second exact. For surfaces with curvature, indeed with varying curvature, the exact regions typically are not likelihood regions and can include parameter values of highest and of lowest likelihood. This paper develops an alternative approach to deriving exact confidence regions and uses both surface curvature and distance from the surface as supporting ingredients. For this, conditionality is invoked in two ways beyond that supported by the usual conditionality principle. For the case of normal error the ordinary chi-squared departure is replaced by a Von Mises-type angular (or cosine) departure which is assessed using curvature properties in the data direction and radial distance of the data from the regression surface. For the usual linear model (constant curvature equal to zero) the method coincides with the ordinary tests and confidence regions; for the case of constant nonzero curvature, the method generalizes to spheres and sphere-cylinders the Fisher (Statistical Methods and Scientific Inference, 1956) analysis of a rotationally symmetric normal on ?² with mean constrained to a circle. The effects of conditioning are indicated by a computer plot for obtaining 95% confidence.     

Canonical Correlation Analysis Through Linear Modeling

In this paper, we introduce linear modeling of canonical correlation analysis, which estimates canonical direction matrices by minimising a quadratic objective function. The linear modeling results in a class of estimators of canonical direction matrices, and an optimal class is derived in the sense described herein. The optimal class guarantees several of the following desirable advantages: first, its estimates of canonical direction matrices are asymptotically efficient; second, its test statistic for determining the number of canonical covariates always has a chi‐squared distribution asymptotically; third, it is straight forward to construct tests for variable selection. The standard canonical correlation analysis and other existing methods turn out to be suboptimal members of the class. Finally, we study the role of canonical variates as a means of dimension reduction for predictors and responses in multivariate regression. Numerical studies and data analysis are presented.     

Interpretable dimension reduction

The analysis of high-dimensional data often begins with the identification of lower dimensional subspaces. Principal component analysis is a dimension reduction technique that identifies linear combinations of variables along which most variation occurs or which best “reconstruct” the original variables. For example, many temperature readings may be taken in a production process when in fact there are just a few underlying variables driving the process. A problem with principal components is that the linear combinations can seem quite arbitrary. To make them more interpretable, we introduce two classes of constraints. In the first, coefficients are constrained to equal a small number of values (homogeneity constraint). The second constraint attempts to set as many coefficients to zero as possible (sparsity constraint). The resultant interpretable directions are either calculated to be close to the original principal component directions, or calculated in a stepwise manner that may make the components more orthogonal. A small dataset on characteristics of cars is used to introduce the techniques. A more substantial data mining application is also given, illustrating the ability of the procedure to scale to a very large number of variables.     

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。     

Testing Time Series for Nonlinearity

The technique of surrogate data analysis may be employed to test the hypothesis that an observed data set was generated by one of several specific classes of dynamical system. Current algorithms for surrogate data analysis enable one, in a generic way, to test for membership of the following three classes of dynamical system: (0) independent and identically distributed noise, (1) linearly filtered noise, and (2) a monotonic nonlinear transformation of linearly filtered noise.We show that one may apply statistics from nonlinear dynamical systems theory, in particular those derived from the correlation integral, as test statistics for the hypothesis that an observed time series is consistent with each of these three linear classes of dynamical system. Using statistics based on the correlation integral we show that it is also possible to test much broader (and not necessarily linear) hypotheses.We illustrate these methods with radial basis models and an algorithm to estimate the correlation dimension. By exploiting some special properties of this correlation dimension estimation algorithm we are able to test very specific hypotheses. Using these techniques we demonstrate the respiratory control of human infants exhibits a quasi-periodic orbit (the obvious inspiratory/expiratory cycle) together with cyclic amplitude modulation. This cyclic amplitude modulation manifests as a stable focus in the first return map (equivalently, the sequence of successive peaks).     

Use of estimated fractal dimension in model identification for time series

A fractal and its dimension has been a subject of great mathematical interest since the publication of Mandelbrot's manifestoes (1977, 1982). This paper discusses some empirical results indicating the potential usefulness of estimated fractal dimension in testing for white noise. These tests are applied for model identification in time series, and results for previously analyzed data are provided. A method for fractal interpolation of a continuous process from a finite number of observations is discussed, as well as some future research directions.     

Non-parametric estimation of data dimensionality prior to data compression: the case of the human development index

In many applications in applied statistics, researchers reduce the complexity of a data set by combining a group of variables into a single measure using a factor analysis or an index number. We argue that such compression loses information if the data actually have high dimensionality. We advocate the use of a non-parametric estimator, commonly used in physics (the Takens estimator), to estimate the correlation dimension of the data prior to compression. The advantage of this approach over traditional linear data compression approaches is that the data do not have to be linearised. Applying our ideas to the United Nations Human Development Index, we find that the four variables that are used in its construction have dimension 3 and the index loses information.     

Multinomial Logistic模型在区域高等教育资源结构优化中的应用

在区域确定的前提下,将各高等教育资源数据进行整理,建立Multinomial Logistic模型,分析各结构的相对发生比率以及各结构的最佳分布。分析该回归分析的多维发生比率,由此确定各种离散等级状态之间的调整方向及调整程度。对黑龙江省的各高等教育区域进行实证分析,结果表明,黑龙江省高等教育资源结构以教学型、教学研究型、研究教学型、研究型的比例来判断,高等教育效用有待于进一步挖掘。多维发生比率以及自变量对多维发生比率的变化影响这两个参数对区域高等教育资源结构优化起着关键作用。     

Hybrid Dirichlet mixture models for functional data

Summary.  In functional data analysis, curves or surfaces are observed, up to measurement error, at a finite set of locations, for, say, a sample of n individuals. Often, the curves are homogeneous, except perhaps for individual-specific regions that provide heterogeneous behaviour (e.g. 'damaged' areas of irregular shape on an otherwise smooth surface). Motivated by applications with functional data of this nature, we propose a Bayesian mixture model, with the aim of dimension reduction, by representing the sample of n curves through a smaller set of canonical curves. We propose a novel prior on the space of probability measures for a random curve which extends the popular Dirichlet priors by allowing local clustering: non-homogeneous portions of a curve can be allocated to different clusters and the n individual curves can be represented as recombinations (hybrids) of a few canonical curves. More precisely, the prior proposed envisions a conceptual hidden factor with k -levels that acts locally on each curve. We discuss several models incorporating this prior and illustrate its performance with simulated and real data sets. We examine theoretical properties of the proposed finite hybrid Dirichlet mixtures, specifically, their behaviour as the number of the mixture components goes to ∞ and their connection with Dirichlet process mixtures.     

Trace pursuit variable selection for multi-population data

Variable selection is a very important tool when dealing with high dimensional data. However, most popular variable selection methods are model based, which might provide misleading results when the model assumption is not satisfied. Sufficient dimension reduction provides a general framework for model-free variable selection methods. In this paper, we propose a model-free variable selection method via sufficient dimension reduction, which incorporates the grouping information into the selection procedure for multi-population data. Theoretical properties of our selection methods are also discussed. Simulation studies suggest that our method greatly outperforms those ignoring the grouping information.     

Dimension reduction in estimating equations with covariates missing at random

To estimate parameters defined by estimating equations with covariates missing at random, we consider three bias-corrected nonparametric approaches based on inverse probability weighting, regression and augmented inverse probability weighting. However, when the dimension of covariates is not low, the estimation efficiency will be affected due to the curse of dimensionality. To address this issue, we propose a two-stage estimation procedure by using the dimension-reduced kernel estimation in conjunction with bias-corrected estimating equations. We show that the resulting three estimators are asymptotically equivalent and achieve the desirable properties. The impact of dimension reduction in nonparametric estimation of parameters is also investigated. The finite-sample performance of the proposed estimators is studied through simulation, and an application to an automobile data set is also presented.     

Simultaneous estimation for semi-parametric multi-index models

Estimation of a general multi-index model comprises determining the number of linear combinations of predictors (structural dimension) that are related to the response, estimating the loadings of each index vector, selecting the active predictors and estimating the underlying link function. These objectives are often achieved sequentially at different stages of the estimation process. In this study, we propose a unified estimation approach under a semi-parametric model framework to attain these estimation goals simultaneously. The proposed estimation method is more efficient and stable than many existing methods where the estimation error in the structural dimension may propagate to the estimation of the index vectors and variable selection stages. A detailed algorithm is provided to implement the proposed method. Comprehensive simulations and a real data analysis illustrate the effectiveness of the proposed method.