Multiscale methods for data on graphs and irregular multidimensional situations

Summary.  For regularly spaced one-dimensional data, wavelet shrinkage has proven to be a compelling method for non-parametric function estimation. We create three new multiscale methods that provide wavelet-like transforms both for data arising on graphs and for irregularly spaced spatial data in more than one dimension. The concept of scale still exists within these transforms, but as a continuous quantity rather than dyadic levels. Further, we adapt recent empirical Bayesian shrinkage techniques to enable us to perform multiscale shrinkage for function estimation both on graphs and for irregular spatial data. We demonstrate that our methods perform very well when compared with several other methods for spatial regression for both real and simulated data. Although we concentrate on multiscale shrinkage (regression) we present our new 'wavelet transforms' as generic tools intended to be the basis of methods that might benefit from a multiscale representation of data either on graphs or for irregular spatial data.     

Multiscale Adaptive Regression Models for Neuroimaging Data

Neuroimaging studies aim to analyze imaging data with complex spatial patterns in a large number of locations (called voxels) on a two-dimensional (2D) surface or in a 3D volume. Conventional analyses of imaging data include two sequential steps: spatially smoothing imaging data and then independently fitting a statistical model at each voxel. However, conventional analyses suffer from the same amount of smoothing throughout the whole image, the arbitrary choice of smoothing extent, and low statistical power in detecting spatial patterns. We propose a multiscale adaptive regression model (MARM) to integrate the propagation-separation (PS) approach (Polzehl and Spokoiny, 2000, 2006) with statistical modeling at each voxel for spatial and adaptive analysis of neuroimaging data from multiple subjects. MARM has three features: being spatial, being hierarchical, and being adaptive. We use a multiscale adaptive estimation and testing procedure (MAET) to utilize imaging observations from the neighboring voxels of the current voxel to adaptively calculate parameter estimates and test statistics. Theoretically, we establish consistency and asymptotic normality of the adaptive parameter estimates and the asymptotic distribution of the adaptive test statistics. Our simulation studies and real data analysis confirm that MARM significantly outperforms conventional analyses of imaging data.     

Approximate wavelet-based simulation of long memory processes

In this article, we investigate an algorithm for the fast O(N) and approximate simulation of long memory (LM) processes of length N using the discrete wavelet transform. The algorithm generates stationary processes and is based on the notion that we can improve standard wavelet-based simulation schemes by noting that the decorrelation property of wavelet transforms is not perfect for certain LM process. The method involves the simulation of circular autoregressive process of order one. We demonstrate some of the statistical properties of the processes generated, with some focus on four commonly used LM processes. We compare this simulation method with the white noise wavelet simulation scheme of Percival and Walden [Percival, D. and Walden, A., 2000, Wavelet Methods for Time Series Analysis (Cambridge: Cambridge University Press).].     

Estimation of global temperature fields from scattered observations by a spherical-wavelet-based spatially adaptive method

Summary.  The paper considers the problem of estimating the entire temperature field for every location on the globe from scattered surface air temperatures observed by a network of weather-stations. Classical methods such as spherical harmonics and spherical smoothing splines are not efficient in representing data that have inherent multiscale structures. The paper presents an estimation method that can adapt to the multiscale characteristics of the data. The method is based on a spherical wavelet approach that has recently been developed for a multiscale representation and analysis of scattered data. Spatially adaptive estimators are obtained by coupling the spherical wavelets with different thresholding (selective reconstruction) techniques. These estimators are compared for their spatial adaptability and extrapolation performance by using the surface air temperature data.     

Robust finite mixture modeling of multivariate unrestricted skew-normal generalized hyperbolic distributions

In this paper, we introduce an unrestricted skew-normal generalized hyperbolic (SUNGH) distribution for use in finite mixture modeling or clustering problems. The SUNGH is a broad class of flexible distributions that includes various other well-known asymmetric and symmetric families such as the scale mixtures of skew-normal, the skew-normal generalized hyperbolic and its corresponding symmetric versions. The class of distributions provides a much needed unified framework where the choice of the best fitting distribution can proceed quite naturally through either parameter estimation or by placing constraints on specific parameters and assessing through model choice criteria. The class has several desirable properties, including an analytically tractable density and ease of computation for simulation and estimation of parameters. We illustrate the flexibility of the proposed class of distributions in a mixture modeling context using a Bayesian framework and assess the performance using simulated and real data.

     

GOES-8 X-ray sensor variance stabilization using the multiscale data-driven Haar–Fisz transform

Summary.  We consider the stochastic mechanisms behind the data that were collected by the solar X-ray sensor (XRS) on board the GOES-8 satellite. We discover and justify a non-trivial mean–variance relationship within the XRS data. Transforming such data so that their variance is stable and its distribution is taken closer to the Gaussian distribution is the aim of many techniques (e.g. Anscombe and Box–Cox). Recently, new techniques based on the Haar–Fisz transform have been introduced that use a multiscale method to transform and stabilize data with a known mean–variance relationship. In many practical cases, such as the XRS data, the variance of the data can be assumed to increase with the mean, but other characteristics of the distribution are unknown. We introduce a method, the data-driven Haar–Fisz transform, which uses the Haar–Fisz transform but also estimates the mean–variance relationship. For known noise distributions, the data-driven Haar–Fisz transform is shown to be competitive with the fixed Haar–Fisz methods. We show how our data-driven Haar–Fisz transform method denoises the XRS series where other existing methods fail.     

Bayesian scale space analysis of temporal changes in satellite images

We consider the detection of land cover changes using pairs of Landsat ETM+ satellite images. The images consist of eight spectral bands and to simplify the multidimensional change detection task, the image pair is first transformed to a one-dimensional image. When the transformation is non-linear, the true change in the images may be masked by complex noise. For example, when changes in the Normalized Difference Vegetation Index is considered, the variance of noise may not be constant over the image and methods based on image thresholding can be ineffective. To facilitate detection of change in such situations, we propose an approach that uses Bayesian statistical modeling and simulation-based inference. In order to detect both large and small scale changes, our method uses a scale space approach that employs multi-level smoothing. We demonstrate the technique using artificial test images and two pairs of real Landsat ETM+satellite images.     

A semi-parametric method for estimating the beta coefficients of the hidden two-sided asset return jumps

ABSTRACT

We introduce a new methodology for estimating the parameters of a two-sided jump model, which aims at decomposing the daily stock return evolution into (unobservable) positive and negative jumps as well as Brownian noise. The parameters of interest are the jump beta coefficients which measure the influence of the market jumps on the stock returns, and are latent components. For this purpose, at first we use the Variance Gamma (VG) distribution which is frequently used in modeling financial time series and leads to the revelation of the hidden market jumps' distributions. Then, our method is based on the central moments of the stock returns for estimating the parameters of the model. It is proved that the proposed method provides always a solution in terms of the jump beta coefficients. We thus achieve a semi-parametric fit to the empirical data. The methodology itself serves as a criterion to test the fit of any sets of parameters to the empirical returns. The analysis is applied to NASDAQ and Google returns during the 2006–2008 period.     

Isotone additive latent variable models

For manifest variables with additive noise and for a given number of latent variables with an assumed distribution, we propose to nonparametrically estimate the association between latent and manifest variables. Our estimation is a two step procedure: first it employs standard factor analysis to estimate the latent variables as theoretical quantiles of the assumed distribution; second, it employs the additive models’ backfitting procedure to estimate the monotone nonlinear associations between latent and manifest variables. The estimated fit may suggest a different latent distribution or point to nonlinear associations. We show on simulated data how, based on mean squared errors, the nonparametric estimation improves on factor analysis. We then employ the new estimator on real data to illustrate its use for exploratory data analysis.     

The use of probabilistic models to produce optimal graphical displays of high-dimensional data sets

Summary Several techniques for exploring ann×p data set are considered in the light of the statistical framework: data-structure+noise. The first application is to Principal Component Analysis (PCA), in fact generalized PCA with any metric M on the unit space ℝ _p . A natural model for supporting this analysis is the fixed-effect model where the expectation of each unit is assumed to belong to some q-dimensional linear manyfold defining the structure, while the variance describes the noise. The best estimation of the structure is obtained for a proper choice of metric M and dimensionality q: guidelines are provided for both choices in section 2. The second application is to Projection Pursuit which aims to reveal structure in the original data by means of suitable low-dimensional projections of them. We suggest the use of generalized PCA with suitable metric M as a Projection Pursuit technique. According to the kind of structure which is looked for, two such metrics are proposed in section 3. Finally, the analysis ofn×p contingency tables is considered in section 4. Since the data are frequencies, we assume a multinomial or Poisson model for the noise. Several models may be considered for the structural part; we can say that Correspondence Analysis rests on one of them, spherical factor analysis on another one; Goodman association models also provide an alternative modelling. These different approaches are discussed and compared from several points of view.     

Sure independence screening for real medical Poisson data

The statistical modeling of big data bases constitutes one of the most challenging issues, especially nowadays. The issue is even more critical in case of a complicated correlation structure. Variable selection plays a vital role in statistical analysis of large data bases and many methods have been proposed so far to deal with the aforementioned problem. One of such methods is the Sure Independence Screening which has been introduced to reduce dimensionality to a relatively smaller scale. This method, though simple, produces remarkable results even under both ultra high dimensionality and big scale in terms of sample size problems. In this paper we dealt with the analysis of a big real medical data set assuming a Poisson regression model. We support the analysis by conducting simulated experiments taking into consideration the correlation structure of the design matrix.     

On the robustness properties for maximum likelihood estimators of parameters in exponential power and generalized T distributions*

Abstract

Examining the robustness properties of maximum likelihood (ML) estimators of parameters in exponential power and generalized t distributions has been considered together. The well-known asymptotic properties of ML estimators of location, scale and added skewness parameters in these distributions are studied. The ML estimators for location, scale and scale variant (skewness) parameters are represented as an iterative reweighting algorithm (IRA) to compute the estimates of these parameters simultaneously. The artificial data are generated to examine performance of IRA for ML estimators of parameters simultaneously. We make a comparison between these two distributions to test the fitting performance on real data sets. The goodness of fit test and information criteria approve that robustness and fitting performance should be considered together as a key for modeling issue to have the best information from real data sets.     

How to improve the fit of Archimedean copulas by means of transforms

The selection of copulas is an important aspect of dependence modeling issues. In many practical applications, only a limited number of copulas is tested and the copula with the best result for a goodness-of-fit test is chosen, which, however, does not always lead to the best possible fit. In this paper we develop a practical and logical method for improving the goodness-of-fit of a particular Archimedean copula by means of transforms. In order to do this, we introduce concordance invariant transforms which can also be tail dependence preserving, based on an analysis on the λ-function, l = \fracjj￠{\lambda=\frac{\varphi}{\varphi'}}, where j{\varphi} is the Archimedean generator. The methodology is applied to the data set studied in Cook and Johnson (J R Stat Soc B 43:210–218, 1981) and Genest and Rivest (J Am Stat Assoc 88:1043–1043, 1993), where we improve the fit of the Frank copula and obtain statistically significant results.     

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.     

Regression for censored survival data with lag effects

We consider regression modeling of survival data subject to right censoring when the full effect of some covariates (e.g. treatment) may be delayed. Several models are proposed, and methods for computing the maximum likelihood estimates of the parameters are described. Consistency and asymptotic normality properties of the estimators are derived. Some numerical examples are used to illustrate the implementation of the modeling and estimation procedures. Finally we apply the theory to interim data from a large scale randomized clinical trial for the prevention of skin cancer.     

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).     

运用柯布-道格拉斯生产函数模型分析研发活动的成效

本文采用柯布-道格拉斯生产函数模型,基于1995-2008年研发资金的投入和产出相关数据,对我国研发活动的成效进行了分析。总体而言,我国应该加大研发经费的支出。但如果我们更注重研发成果的质量,则研发资金的分配应该倾向于加大支持的强度;如果我们更注重研发成果的数量,则研发资金的分配应该倾向于支持更多的研发人员从事研发活动。同时,从长期来看,无论是以研发活动的质量还是数量为衡量标准,都应该增加对研发活动的投入,以实现研发活动的最佳规模。     

Multiscale detection and location of multiple variance changes in the presence of long memory

Procedures for detecting change points in sequences of correlated observations (e.g., time series) can help elucidate their complicated structure. Current literature on the detection of multiple change points emphasizes the analysis of sequences of independent random variables. We address the problem of an unknown number of variance changes in the presence of long-range dependence (e.g., long memory processes). Our results are also applicable to time series whose spectrum slowly varies across octave bands. An iterated cumulative sum of squares procedure is introduced in order to look at the multiscale stationarity of a time series; that is, the variance structure of the wavelet coefficients on a scale by scale basis. The discrete wavelet transform enables us to analyze a given time series on a series of physical scales. The result is a partitioning of the wavelet coefficients into locally stationary regions. Simulations are performed to validate the ability of this procedure to detect and locate multiple variance changes. A ‘time’ series of vertical ocean shear measurements is also analyzed, where a variety of nonstationary features are identified.     

Accelerated failure time models for the analysis of competing risks

Competing risks are common in clinical cancer research, as patients are subject to multiple potential failure outcomes, such as death from the cancer itself or from complications arising from the disease. In the analysis of competing risks, several regression methods are available for the evaluation of the relationship between covariates and cause-specific failures, many of which are based on Cox’s proportional hazards model. Although a great deal of research has been conducted on estimating competing risks, less attention has been devoted to linear regression modeling, which is often referred to as the accelerated failure time (AFT) model in survival literature. In this article, we address the use and interpretation of linear regression analysis with regard to the competing risks problem. We introduce two types of AFT modeling framework, where the influence of a covariate can be evaluated in relation to either a cause-specific hazard function, referred to as cause-specific AFT (CS-AFT) modeling in this study, or the cumulative incidence function of a particular failure type, referred to as crude-risk AFT (CR-AFT) modeling. Simulation studies illustrate that, as in hazard-based competing risks analysis, these two models can produce substantially different effects, depending on the relationship between the covariates and both the failure type of principal interest and competing failure types. We apply the AFT methods to data from non-Hodgkin lymphoma patients, where the dataset is characterized by two competing events, disease relapse and death without relapse, and non-proportionality. We demonstrate how the data can be analyzed and interpreted, using linear competing risks regression models.